NIST Technical Note 1337: Characterization of Clocks and Oscillators

Characterization ofClocks and OscillatorsEdited byD.B. SullivanD.W. AllanD.A. HoweF.l. WallsTime and Frequency DivisionCenter for Atomic, Molecular, and Optical PhysicsNational Measurement LaboratoryNational Institute of Standards and TechnologyBoulder, Colorado 80303-3328This publication covers portions of NBS Monograph 140 published in 1974.The purpose of this document is to replace this outdated monograph in theareas of methods for characterizing clocks and oscillators and definitionsand standards relating to such characterization.u.s. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, SecretaryNATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W Lyons, DirectorIssued March 1990

National Institute of Standards and Technology Technical Note 1337Natl. Inst. Stand. Techno!., Tech. Note 1337, 352 pages (Jan. 1990)CODEN:NTNOEFu.s. GOVERNMENT PRINTING OFFICEWASHINGTON: 1990For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402·9325

PREFACEFor many years following its publication in 1974, "TIME AND FREQUENCY: Theoryand Fundamentals," a volume edited by Byron E. Blair and published as NBS Monograph 140,served as a common reference for those engaged in the characterization of very stable clocks andoscillators. Monograph 140 has gradually become outdated, and with the recent issuance of anew military specification, MIL-0-55310B, which covers general specifications for crystal oscillators,it has become especially clear that Monograph 140 no longer meets the needs it so ablyserved in earlier years. During development of the new military specification, a process involvingdiscussion and input from many quarters, a key author of the specification, John Vig of the USArmy Electronics Technology and Devices, urged the National Bureau of Standards (now theNational Institute of Standards and Technology, NIST) to issue a revised publication to serve asreference for the characterization of clocks and oscillators. With NIST having agreed to thistask, the framers of the military specification used the nomenclature "NBS Monograph 140R" intheir document, anticipating a revised (R) volume which had not yet been prepared.Considering the availability of a number of newer books in the time and frequency field,the rewriting of a major volume like Monograph 140 seemed inappropriate. The real need hasnot been for rework of everything in Monograph 140, but only for those parts which providereference to definitions and methods for measurement and characterization of clocks and oscillators,subjects which are fully covered in a number of papers distributed through a variety ofconference proceedings, books, and journals. For the near term, we concluded that the mosteffective procedure would be to collect a representative set of these papers into one referencesource with introductory comments which permit the reader to quickly access material requiredto meet particular needs. Thus, we arrived at this particular collection. The editors' challengehas been to select representative papers, to organize them in a convenient manner, and to dealwith errata and notation inconsistencies in a reasonable manner. In the longer term, the materialin this volume needs to be more completely integrated. This task would profitably awaitfurther deVelopments in the area of phase noise measurements.Donald B. SullivanDavid W. AllanDavid A. HoweFred L. WallsBoulder, ColoradoFebruary 28, 1990iii

CONTENTSPagePREFACE "" "."" .. " IIITOPICAL INDEX TO ALL PAPERS , , .. " .. xABSTRACT "TN-IA. INTRODUCTIONA.l OVERVIEWTN-IA.2 COMMENTS ON INTRODUCTORY AND TUTORIAL PAPERS , TN-2A.3 COMMENTS ON PAPERS ON STANDARDS AND DEFINmONS TN-3A.4 COMMENTS ON SUPPORTING PAPERS " " 1N-3A.5 GUIDE TO SELECTION OF MEASUREMENT METHODS TN-5A.6 RELATIONSHIP OF THE MODIFIED ALLAN VARIANCE TO THE ALLAN VARIANCE. " TN-9A.7 SUPPLEMENTARY READING LIST " 1N-BB. INTRODUcrORY AND TUTORIAL PAPERS8.1 Properties of Signal SOlJlrces and Measurement MethodsDA. Howe, D.W. Allan, and JA. Barnes35th Annual Symposium on Frequency Control, 1981 Proceedings 1N-14I. The Sine Wave and Stability . TN-14II. Measurement Methods Comparison " . TN-20III. Characterization . TN-22IV. Analysis of Time Domain Data . TN-23V. Confidence of the Estimate and Overlapping Samples . TN-26VI. Maximal Use of the Data and Determination of the Degrees of Freedom . TN-2BVII. Example of Time Domain Signal Processing and Analysis . TN-3DVIII. Spectrum Analysis ....................".................................. TN-32IX. Power-Law Noise Processes . TN-38X. Pitfalls in Digitizing the Data " " " " . TN-39XI. Translation From Frequency Domain Stability Measurement to Time Domain StabilityMeasurement and Vice-Versa . . " " " . TN-48XII. Causes of Noise Properties in a Signal Source . TN-50XIII. Conclusion " . TN-55IU Frequency and Time - Their Measurement and CharacterizationSamuel R. SteinPrecision Frequency Control, Volume 2, Chapter 12, 1985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TN-6112.1 Concepts, Definitions, and Measures of Stability TN-6312.1.1 Relationship Between the Power Spectrum and the Phase Spectrum TN-6512.1.2 The IEEE Recommended Measures of Frequency Stability TN-6512.1.3 The Concepts of the Frequency Domain and the Time Domain TN-7312.1.4 Translation Between the Spectral Density of Frequency and the Allan Variance """"" TN-7312.1.5 The Modified Allan Variance " TN-7512.1.6 Detennination of the Mean Frequency and Frequency Drift of an Oscillator TN-7712.1.7 Confidence of the Estimate and Overlapping Samples TN-SO12.1.8 Efficient Use of the Data and Determination of the Degrees of Freedom. . . . . . . . . . . . . . . TN-8312.1.9 Separating the Variances of an Oscillator and the Reference" " " TN-86v

0 • • • • • • • • • • • • • • • • •0 • • • • • • • • • • • • • • • • • • • • • • • •0 • 0 •••••••••••• 0 • • • • • • • • • • • • • • • • • • • •0 0 •••••••••••••• _ • • • •12.2 Direct Digital Measurement IN-B712.2.1 Time-Interval Measurements IN-8712.2.2 Frequency Measurements IN-8912.2.3 Period Measurements IN-8912.3 Sensitivity-Enhancement Methods IN-9012.3.1 Heterodyne Techniques IN-9012.3.2 Homodyne Techniques IN-9112.3.3 Multiple Conversion Methods IN-9712.4 Conclusion - - . . . . . . _. . . . . _. . . . . . . . . . . _. . . . . . . . . . _. . . . . . _. . . . . . . . . . . . __ TN-WIB.3 Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of PrecisionClocks and OscillatorsDavid W. AllanIEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1987 _ o. TN-121Page0 ••••••• 0 _ ••• _ ••• 0 •• 0 •••••••••••••• _ •••••••• lli-1210L Introduction _ 0 _2. Systematic Models for Clocks and Oscillators .. _. 0 ••••••••••••• _ • • • • • • • • • • • • • • •• TN-1213. Random Models for Clocks and Oscillators _ 0 ••••••••••••••••••••••••••• _. TN-I234. Time-Domain Signal Characterization _ _ __ ••••••• lli-l235. Time and Frequency Estimation and Prediction _ TN-l266. Conclusions _ ••••••0 0 ••••••••••• 0 •••••• TN-127BA Extending the Range and Accuracy of Phase Noise MeasurementsF.L. Walls, AJ.D. Clements, C.M. Felton, MAo Lombardi and M.D. Vanek42nd Annual Symposium 011 Frequency Control, 1988 Proceedings _ _ lli-1290 •••••••••• 0 •••• _ • • • • • • • • • • • • • • • • • • • • • • • • ••L Introduction TN-129II. Model of a Noisy Signal TN-l290 •••••••••••••••••••••••• 0 •••••••••••HI. Two Oscillator Method TN-130IV. The New NBS Phase Noise Measurement Systems , TN-135V. Conclusion __ _ _. TN-l37C. PAPERS ON STANDARDS AND DEFINmONSCol Standard Terminology for Fundamental Frequency and Time MetrologyDavid Allan, Helmut Hellwig, Peter Kartaschoff, Jacques Vanier, John Vig, Gernot M.R. Winkler,and Nicholas Yannoni42nd Annual Symposium on Frequency Control, 1988 Proceedings _ 0• 0 TN-1390 ••••••••••••• _. lli-1390 ••••••••• 0 ••••••••••••1. Introduction _2. Measures of Frequency and Phase Instability TN-1393. Characterization of Frequency and Phase Instabilities _. TN-1394. Confidence Limits of Measurements 0 • 0 0 •• 0 •••••••••••• _ TN-l405. Recommendations for Characterizing or Reporting Measurements of Frequency andPhase Instabilities __. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IN-141vi

PageCo2 Characterixstaon of Frequency StabilityJames A. Barnes, Andrew R. Chi, Leonard S. Cutler, Daniel 1. Healey, David B. Leeson,Thomas E. McGunigal, James A. Mullen, Jr., Warren L. Smith, Richard L. Sydnor,Robert F.e. Vessot, and Gemot M.R. WinklerIEEE Transactions all Instrnmentation and Measurement, 1971TN-146I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IN-147II. Statement of the Problem TN-148m. Background and Definitions IN-149IV. Derlnition of Measures of Frequency Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IN-149V. Translations Among Frequency Stability Measures IN-1SlVI. Applications of Stability Measures . . . . . . . . . . . . . . . . . . . . . .. IN-1S3VII. Measurement Techniques for Frequency Stability TN-IS4VIII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. TN-IS7C.3 Characterization of Frequency and Phase NoiseComite Consultatif International Des Radiocomrnunications (CCIR)Report 580 of the CCIR, 1986TN-162L Introduction TN-1622. Fourier Frequency Domain TN-1633. Time Domain 1N-1634. Conversion Between Frequency and Time Domains TN-1M5. Measurement Techniques TN-l666. Confidence Limits of Time Domain Measurements TN-1667. Conclusion TN-167D. SUPPORTING PAPERSD.l Characteri:t8tion and Measurement of Time and Frequency StabilityP. Lesage and C. AudoinRadio Science, 1979;. TN-1711. Introduction TN-l?l2. Definitions: Model of Frequency Fluctuations TN-l713. Noise Processes in Frequency Generators TN-1734. Measurements in the Frequency Domain 1N-1745. Measurements in the Time Domain , TN-l776. Characterization of Frequency Stability in the Time Domain TN-1787. Characterization of Frequency Stability via Filtering or Frequency Noise TN-1828. Spectral Analysis Inferred From Time Domain Measurements TN-1839. Structure Functions of Oscillator Fractional Phase and Frequency fluctuations TN-l8410. Power Spectral Density of Stable Frequency Sources TN-18S11. Conclusion TN-IS7vii

0 •• 0 •• 0 •• 0 • • • • • • • • • • • ••0 0 •• 0 ••••••••••••• 0 • 0 .'Page0.2 Phase Noise and AM Noise Measurements in the Frequency DomainAlgie L. Lance, Wendell D. Seal, and Frederik LabaarInfrared and Millimeter Waves, Vol. 11, 1984. TN-1900 • 0 • 0 •••••••••••••••• 0 ••••I. Introduction TN-190II. Fundamental Concepts .. 00 •••••••• 0 ••• 0 •••• 0 •• 0 •••••••••• 0 •••••••••••••••TN-1910 •• 0 •••• 0 ••••• 0 ••••••••••••••••••••••••••••••0 •• 0 •••••••••••• 0 • 0 •••• 0 ••••• 0 ••••• 0 •• 0 0 ••••••• 0 •••0 •••• 0 0 •• 0 • 0 0 0 0 • 0 0 ••••0 •••••••••••••• 0 •••• 0 ••••• 0 ••••• 0 •••••••••0 ••••••••••••• 0 ••••A Noise Sidebands .. TN-194B. Spectral Density . TN-194C. Spectral Densities of Phase F1uctuations in the Frequency Domain 0 0 0 • 0 0 •••• 0 ••••• 0 TN-197D. Modulation Theory and Spectral Density Relationships . 0 0 •••••• 0 • 0 •• 0 • 0 •••••••• TN-199Eo Noise Processes 0 ••••••• 0 •••• 0 • 0 •• 0 ••••••••••••••• 0 • 0 •• 0 • TN-202F. Integrated Phase Noise o' 0 •••••••••••• 0 • 0 •••••• 0 ••••• 0 0 •••• 0 0 ••••••••• TN-203G. AM Noise in the Frequency Domain.. . 0 • 0 0 ••••• 0 0 • 0 • 0 •• 0 • 0 • 0 ••• 0 0 •• 0 • 0 0 • TN-20SIII. Phase-Noise Measurements Using the Two-Oscillator Technique TN-206A Two Noisy Oscillators TN-208B. Automated Phase-Noise Measurements Using the Two-Oscillator Technique 0" 0 •••••• TN-209C. Calihration and Measurements Using the Two-Oscillator System 0 •••• TN-210IV. Single-Osciliator Phase-Noise Measurement Systems and Techniques TN-218•• 0 0 • 0 0A The Delay Line as an PM Discriminator .... 0B. Calibration and Measurements Using the Delay Line as an FM Discriminator . TN-22SC. Dual Delay-Line Discriminator . 0 ••••• 0 ••• 0 ••••• 0 •••• 0 • 0 • 0 0 • 0 •• 0 •• TN-232• 0 •• 0 •••••••••••• 0 •• 0 •••• 0 •••• TN-2190 •• 0 • 0 •••••• 0 ••• 0 0 0 0 ••• 0 0 0 0 • 0 0 ••0 0 • 0 0 ••••• 0 ••• 0 ••••• 0 • 0 ••••• 0 0 •• 0 ••••••••••••• 0 ••••••••• 0 ••D. Millimeter-Wave Phase-Noise Measurements TN-234V. Conclusion TN-23803 Perlormance of an Automated High Accuracy Phase Measurement SystemS. Stein, Do Glaze, J. Levine, J. Gray, D. Hilliard, D. Howe, and L. Erb36th Annual Symposium on Fi-equency Control, 1982 ProceedingsTN-2411. Review of the Dual Mixer Time Difference Technique .. 0••••• 0 • 0 0 • 0 •••••••••••• 0 ., TN-2410 •••••••• ,0 • 0 •••• 0 0 ••• 0 ••••• 0 ••••••••••••••••• 0 •••••••• 0 .,0 •••••••••••••••••• 0 0 ••••• 0 • 0 • 0 • 0 ••••••••• 0 ••••••••• 0 0 • 0 • 02. Extended Dual Mixer Time Difference Measurement Technique TN-2423. Hardware Implementation . TN-2424. Conclusions . TN-243OA Biases and Variances of Several FliT Spectral Estimators as a Function of Noise Type andNumber of SamplesF.L. Walls, DoB. Percival and W.R. Irelan43,.d Annual Symposium on Frequency Control, 1989 ProceedingsTN-248••••••• 0 0 0 •••••••••••• 0 •••• 0 ••••••••••••••• 0 ••• 0 •••• 0 .,,,I. Introduction .. 0 TN-248n. Spectrum Analyzer Basics .... 0 ••••• 0 •••••••••••••••••• 0 •••••••••••••••••• 0 TN-248III. Expected Value and Bias of Spectral Estimates .. 0 ••••••• 0 ••• 0 • 0 •••• 0 • 0 •••• 0 •••• TN-249IV. Variances of Spectral Estimates 0 ••••• 0 ••••••• 0 •••••• 0 0 •••••• 0 • 0 ••• 0 •• 0 • 0 0 •• 0 TN-251V. Conclusions 0 ••• 0 •• 0 ••••••• 0 •••••• 0 •• 0 • 0 •••• 0 ••••••••••••• TN-252v···

PageD.S A Modified "Allan Variance" with Increased Oscillator Characterization AbilityDavid W. Allan and James A Barnes35th Annual Symposium on Frequency Control, 1981 Proceedings . TN-2541. Introduction . TN-2542. Defmition of "Allan Variance" and Related Concepts . TN-2543. Development of the Modified Allan Variance . TN-2554. Comparisons, Tests, and Examples of Usage of the Modified AHan Variance " . TN-2565. Conclusion . TN-257D.6 Charactel"ization of Frequency Stability: Analysis of the Modified Allan Variance andPl"operties of Us EstimatePaul Lesage and Theophane AyiIEEE Transactions on Instmmentation alld Measurement, 1984TN-259I. Introduction . TN-260II. Background and Definitions . TN-260III. The Modified AHan Variance . TN-260IV. Uncertainty on the Estimate of the Modified AHan Variance . TN-262V. Conclusion . TN-263D.7 The Measurement of Linear Frequency Drift in OscillatorsJames A. Barnes15th Annual PITI Meeting, 1983 ProceedingsTN-264I. Introduction . TN-264II. Least Squares Regression of Phase on a Quadratic . TN-265III. Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . TN-266IV. Drift and Random Walk FM . TN-270V. Summary of Tests . TN·270VI. Discussion . TN-276VII. Conclusions . TN-276D.8 Variances Based on Dahl With Dead Time Between the MeasurementsJames A. Barnes and David W. AllanNIST Technical Note 1318, 1990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. TN-2961. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1N-2972. The Allan Variance TN-2983. The Bias Function B 1(N,r,JL) 1N-2994. The Bias Function E 2(r,JL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. TN-3005. The Bias Function E 3(N,M,r,JL) 1N-3006. The Bias Functions TN-3027. Examples of the Use of the Bias Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. TN-3038. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1N-304AppendixTN-312E. APPENDIX· Notes and El"rata 1N-336IX

TOPICAL INDEX TO ALL PAPERSThe following index is designed for quick location of substantial discussion of a particular topic. The page numbers arethose where the topic is first cited in the reference.Aliasing. .Allan VarianceAmplitude Modulation NoiseBias FunctionsClock-Time Prediction . .Common Measurement ProblemsComponents Other than Oscillators (Methods)Confidence of the Estimate ....Conversion Between Time and Frequency DomainCorrelation Method .Dead Time ..Degrees of Freedom .Digitization of Data .Direct Measurementso Measurements at the Fundamental Frequencyo Measurements after Multiplication/DivisionDriftEfficient Use of DataFilter Method. . .Frequency Synthesis .HeterOdyne or Beat-Frequency Methodso Single-Conversion Methodso Multiple-Conversion Methods. . .o Time-Difference (Time-Interval-Counter)- Dual-Mixer, Time-Difference MethodHomodyne Measurementso Phase-Lock-Loop Methods.{]I Discriminator Methods. .- Cavity-Discriminator Method- Delay-Line MethodLeast Squares (Regression)Modified Allan VarianceOverlapping SamplesPickett-Fence Effect. .Power Law Spectra (Random Noise)Quantization Uncertainty . . . .Reference-Phase-Modulation TechniqueSeparating Oscillator and Reference VariancesSpectral Density . . . . . . .Systematic Fluctuations (non-Random-Noise) .Window Function (Leakage) . . .TN-40TN-24, TN-71, TN-l23, TN-140, TN-150, TN-163, TN-179, TN-254,TN-260, TN-298TN-194, TN-205, TN-235TN-162, TN-l64, TN-299TN-l26, TN-140TN-156TN-37, TN-131, TN-176TN-26, TN-80, TN-140, TN-156, TN-l66, TN-lBO, TN-262TN-48, TN-73, TN-142, TN-1S1, TN-l64TN-132, TN-176, TN-216TN-140, TN-l64, TN-lBO, TN-296TN-28, TN-83TN-39TN-15, TN-B7TN-I01, TN-135TN-3D, TN-77, TN-l22, TN-151, TN-l84, TN-2MTN-28, TN-83TN-182TN-97TN-17, TN-90, TN-154, TN-l77TN-17, TN-90, TN-154, TN-177TN-97TN-20, TN-87, TN-177TN-17, TN-99, TN-l77, IN-241TN-18 TN-34, TN-93, TN-BO, TN-176, TN-209TN-n, TN-B3, TN-174, IN-218TN-133, TN-174, TN-219TN-92, TN-134, TN-176, TN-219TN-31, TN-77, TN-265TN-75, TN-143, TN-l64, TN-255, TN-260TN-26, TN-80, TN-l40TN-46TN-38, TN-73, TN-l23, TN-141, TN-167, TN-173, TN-200TN-40, TN-89TN-133TN-86TN-32, TN-43, TN-65, TN-BO, TN-139, TN-150, TN-163, TN-l72,TN-194TN-22, TN-121, TN-150TN-44x

CHARACTERIZATION OF CLOCKS AND OSCILLATORSD.B. Sullivan, D.W. Allan, D.A. Howe and F.L. Walls, EditorsTime and Frequency DivisionNational Institute of Standards and TechnologyBoulder, Colorado 80303This is a collection of published papers assembled as a reference for those involvedin characterizing and specifying high-performance clocks and oscillators. Itis an interim replacement for NBS Monograph 140, Time and Frequency: Theoryand Fundamentals, an older volume of papers edited by Byron E. Blair. Thiscurrent volume includes tutorial papers, papers on standards and definitions, anda collection of papers detailing specific measurement and analysis techniques.The discussion in the introduction to the volume provides a guide to the contentof the papers, and tables and graphs provide further help in organizing methodsdescribed in the papers.Key words: Allan variance, clocks, frequency, oscillators, phase noise, spectraldensity, time, two-sample variance.A. INTRODUCTIONA.I OVERVIEWThe papers in this volume are organized into three groups: Introductory and TutorialPapers, Papers on Definitions and Stand~rds, and Supporting Papers. The three sections (A.2,A.3, and A.4) immediately following this introduction provide overviews of each of the threegroups of papers with comments on each paper. The arrangement of the papers in this particularorder is somewhat arbitrary, since, for example, the first two papers under Supporting Paperscould be included with the Introductory and Tutorial Papers, while paper B.4 could easily beplaced with the Supporting Papers. Our rationale for the first group of papers (discussed inmore detail in section A.2) is that, taken as a group, they provide reasonably complete coverageof the concepts used in characterizing clocks and oscillators. Several of the papers, taken individually,are good introductory papers, but, for this publication, need to be complemented withadditional material to provide coverage of an appropriate range of topics.The second group (section C) of three papers discussed in section A.3 were specificallywritten to address definitions and standards. This is a particularly important section, sinceconsistency in specification of performance can only be achieved if manufacturers and users referto the same measurement and characterization parameters.The Supporting Papers in section D provide additional discussion of topics introduced inthe first group. The first papers in this group (D.l and D.2) also provide good introductorymaterial which might be used with section B to gain a better understanding of the concepts.Section A.S provides a table and graph designed to help the reader select a measurementmethod to meet a particular need. To make this useful, it was kept simple and must thereforebe used with care. Such tabular information can never be arranged well enough to anticipate allTN-I

of the wide range of measurement situations which might be encountered. However, it can serveas a starting point for the decision-making process.Section A6 contains some new material which should be helpful in understanding therelationship between the Allan variance and the modified Allan variance. Since these ideas areunpublished, we include them here rather than with the papers on those topics. This section isfollowed by a reading list with references to major articles and books which can be used assupplementary resources. Because some of the papers include extensive reference lists, we havelimited our list to works which are either very comprehensive or only recently published. Particularlyextensive reference lists are included with papers 8.1, B.2, C.l, C.3, D.l, and D.2.Since notation and definitions have changed over the period bridged by these papers, wehave highlighted problem areas on the papers with an asterisk (*). A note directs the reader tothe Appendix where the particular problem is discussed. We have also used this device tohighlight inconsistencies and the usual typographic and other errors which creep into the literature.The page numbers of the original publications are retained, but we have also used acontinuous page numbering to simplify location of items in the volume.The topical index on page xi organizes much of the material in the papers under a fewkey subject headings. This index provides a shortcut to locating material on a particular topic.A.2 COMMENTS ON INTRODUCTORY AND TUTORIAL PAPERSPaper B.l in this section, by Howe, Allan, and Barnes, was originally prepared andpresented as a tutorial paper and has been used with success as an introductory paper in ourannual Time and Frequency Seminar. This paper is now 9 years old, so there are a substantialnumber of notes which relate to updates in notation. The paper is nevertheless highly readableand introduces many of the key measurement methods, providing circuit diagrams with enoughspecific detail to be useful in real laboratory situations. Furthermore, it includes discussion andexamples on handling of data which are useful for practical application of the concepts. Thepaper presents a particularly useful discussion of the pitfalls encountered in digitizing data, aproblem which is often overlooked.The second paper (B.2) by Stein is more advanced and those familiar with the generalconcepts may find it a better starting point. This and other papers in this collection cite earlierIEEE recommendations on measures of frequency stability and, while much of this has notchanged, there is a new IEEE standard (paper C.l). In general, the reader should consult theoverview and papers of section C if there is any question concerning definitions or terminology.Paper B.2 is quite comprehensive, introducing topics (not covered in paper B.l) such as themodified Allan variance, the delay-line-phase-noise-measurement system, and the use of frequencysynthesis to reach frequencies far from normally available reference frequencies.The materials in papers B.l and B.2, aside from differences in level of presentation, areorganized in quite different ways. The Howe-Allan-Barnes paper goes directly to the measurementconcepts and then describes the means for analyzing the output data and understanding theconfidence of the measurements. On the other hand, the Stein paper carefully lays out thetheoretical background needed to analyze the data before introducing the measurement concepts.Both papers cover time-domain and frequency-domain measurements.Paper B.3 by Allan reviews the concepts of the two-sample or Allan variance and themodified Allan variance showing how classical statistical methods fail to usefully describe thetime-domain performance of good oscillators. The Allan variance concept is also introduced inTN-2

papers B,l and B-2, and the modified Allan variance is described further in papers D.4 and 0.5.The presentation in paper B,3 is particularly useful in that it discusses general aspects of performanceof different types of oscillators (quartz, rubidium, hydrogen, and cesium) providing thebasis for prediction of time errors, a topic which may prove useful to those who must developsystem specifications.Paper BA by Walls, Clements, Felton, Lombardi, and Vanek adds to the discussion offrequency-domain measurements providing information on methods which can be used to increasethe dynamic range for both carrier frequency and for Fourier frequencies up to 10 percentfrom the carrier. With some aerospace hardware now carrying phase-noise specifications, this isan important addition to the literature.A.3 COMMENTS ON PAPERS ON STANDARDS AND DEFINITIONSThe first paper in this group (paper C.l) outlines the standard terminology now used forfundamental frequency and time metrology. This document was widely circulated for commentduring the draft stage and, with its acceptance by IEEE as a standard, supersedes the earlierreference (paper C.2) which had served as the foundation for characterization of frequencystability. This latter paper is included because it is so widely cited, and the reader will probablybe confronted with specifications based on its recommendations. Paper C2 contains additionalmaterial on applications of stability measures and measurement techniques including a usefuldiscussion of some of the common hazards in measurements. Paper Cl restricts itself to veryconcise statements of the definitions.The reader will note that the updated terminology in the first paper (Cl) varies in anumber of minor ways from the earlier paper (C2). A notable addition to definitions is theintroduction of script "ell", J?(f), which has become an important measure of phase noise. Thisquantity was previously defined as the ratio of the power in one sideband, due to phase modulationto the total signal power. For Fourier frequencies far from the carrier, this quantity canbe simply related to the usual spectral densities which are the quantities that are generally measured,but the relation breaks down in the important region near the carrier. To resolve thisproblem, the new standard defines the approximate relation between 2(f) and spectral density asbeing exact and applicable for any Fourier frequency.The third paper in this group (paper C.3), from the 1986 report of the InternationalRadio Consultative Committee (CCIR), presents the definitions and terminology which havebeen accepted for international use by this body. The material in this particularly readabledocument is fairly consistent with the IEEE standard and would be useful to those involved inspecification of performance for international trade. A number of minor changes to this documenthave been recommended by different delegations to the CCIR and these will likely bemade in their next publication.A.4 COMMENTS ON SUPPORTING PAPERSPapers D.l and D.2 are included in this collection for a number of reasons. First, theyprovide alternative introductions to the general topic of oscillator characterization. And second,they include material not fully covered by introductory papers, B.l and B.2.TN-3

The first of these (D.l) by Lesage and Audoin covers most of the same ground as papersB.l and B.2, but, in addition, includes a nice discussion of characterization of frequency stabilityvia filtering of phase or frequency noise, a method which may be especially useful for rapid,automated measurements where high accuracy is not required. Furthermore, this paper discussescharacterization of stable laser sources, a topic not covered in any of the other reference papers.Paper D.2 by Lance, Seal, and Labaar limits itself to discussion of the measurement ofphase noise and amplitude-modulation (AM) noise. The paper presents a detailed discussion ofdelay-line measurement methods which can be used if a second reference oscillator is notavailable. The delay-line concept is also introduced in B.2 and BA, but in much less detail.While the delay-line method is less sensitive (for lower Fourier frequency) than two-oscillatormethods, it is easier to implement The paper contains many good examples which the readerwill find useful.A complete discussion of AM noise is beyond the scope of this volume. AM noise isusually ignored in the measurement and specification of phase noise in sources under the assumptionthat the AM noise is always less than the phase noise. This assumption is generallytrue only for Fourier frequencies dose to the carrier. At larger Fourier frequencies the normalizedAM noise can be the same order of magnitude as the phase noise. In systems with activeamplitude leveling, the normalized AM noise can be higher than the phase noise. Under thiscondition the AM to PM conversion in the rest of the system may degrade the overall phasenoise performance. For these reasons, we cannot ignore amplitude noise altogether. Paper D.2provides a useful discussion of amplitude noise. Note 1 in appendix E provides further informationon defmitions, notation and, in particular, the specification of added phase noise andamplitude noise for signal-handling components.The next contribution (D.3) provides substantially more detail on the extension of thetime-domain, dual-mixer concept for higWy accurate time and time-interval measurements. Thebasic dual-mixer ideas are included in papers B.l and B.2.Paper DA, published recently, provides the first quantitative treatment of confidenceestimates for phase-noise measurements. To the best of our knowledge, this is the only availabletreatment of this important subject. We expect to see additional papers on this topic in thefuture.Paper D.5 discusses the modified Allan variance in more detail than the introductorypapers B.l, B.2, and R3. It is followed by the Lesage-Ayi paper (D.6) which provides analyticalexpressions for the standard set of power-law noise types and also includes discussion of theuncertainty of the estimate of the modified Allan variance.Linear frequency drift in oscillators is treated by Barnes in paper D.7. As noted in thispaper, even with correction for drift, the magnitude of drift error eventually dominates all timeuncertainties in clock models. Drift is particularly important in certain oscillators (e.g., quartzoscillators) and a proper measure and treatment of drift is essential. As with other topicstreated by this group of papers, introductory papers Rl, B.2, and R3 present some discussion offrequency drift, but D.7 is included because it contains a much more comprehensive discussion ofthe subject.The final paper (D.8) by Barnes and Allan contains the most recent treatment of measurementsmade with dead times between them. Paper C.2 introduced the use of bias functions,B 1 and B 2 , which can be used to predict the Allan variance for one set of parameters based onanother set (for the power~law noise models). This last paper extends those ideas, introducing athird bias function, B 3 , which can be used to translate the Allan variance between cases whereTN-4

dead time is accumulated at the end and where dead time is distributed between measurements,a useful process for many data-acquisition situations.AS GUIDE TO SELECTION OF MEASUREMENT METIIODSThe table and graph in this section are quick guides to performance limits of the differentmethods as well as indications of advantages and disadvantages of each. The ideas presentedhere are drawn from papers B.l and BA, but we have modified and expanded the material tomake it more comprehensive. The table has been kept simple, so the suggested methods shouldbe viewed as starting points only. They cannot possibly cover all measurement situations.For a given measurement task, it is often best to start with a quick, simple measurementwhich will then help to define the problem. For example, faced with the need to characterize anoscillator, a good starting point might be to feed the output of that oscillator along with theoutput of a similar, but more stable, oscillator into a good mixer and then look at the output. Ifthe two can be brought into quadrature by tuning one of the oscillators or by using a phaselockedloop, then the output can be fed to a spectrum analyzer to get an immediate, at leastqualitative idea, of the performance of the oscillator. This mixing process, which brings thefluctuations to baseband where measurement is much more straightforward, is basic to many ofthe measurement methods. A large number of measurement problems can probably be resolvedwith this simple, single-conversion, heterodyne arrangement. If the simplest approach is insufficient,then some of the more advanced methods outlined below can be used.There are many ways to go about categorizing the various measurement methods. Sincethis volume is aimed at practical measurements, we choose to use the characteristics of themeasurement circuit as the basis for sorting. In this arrangement we have (1) direct measurementswhere no signal mixers are used, (2) heterodyne measurements where two unequal frequenciesare involved, and (3) homodyne.measurements where two equal frequencies are involved.These methods are listed below.I. Direct MeasurementsL Measurements at the Fundamental Frequency2. Measurements after Multiplication/Division11 Heterodyne MeasurementsL Single-Conversion Methods2. Multiple-Conversion Methods3. Time-Difference Methoda. Dual-Mixer, Time-Difference MethodIII. Homodyne Measurements1. Phase-Lock-Loop Methods (two oscillators)a. Loose-Phase-Lock-Loop Methodb. Tight-Phase-Lock-Loop Method2. Discriminator Methods (single oscillator)a. Cavity-Discriminator Methodb. Delay-Line MethodTN-5

Table 1. Guide to Selection of Measurement Methods.MeasurementMethodTimeAccuraclTimeStability(1 day)Frequen

Table 1. Guide to Selection of Measurement Methods. (continued)Measurement Time Time Frequen'1 Frequencg'Method Accuracy a Stability Accuracy Stability Advantages Disadvantages(1 day) 0'/7)m. Homodyne Methods Particularly useful for measurement Generally no used for measurementof phase noise.of time.1. Phase-Lock-Loop Methods Assure continuous quadrature of Care needed to assure that noise ofsignal and reference.interest is outside loop bandwidth.a. LODse-Phase-Lock LoopUseful for short-term time stabilityDepends on -10,7/(11 07) analysis as well as spectrum analysiscalibration which at 10 MHz and the detection of periodicity inof varicap is _10,14/ 7 noise as spectral lines; excellentsensitivity.Long-term phase measurements(beyond several seconds are notpractical.Measurement noise typically less Need voltage controlled referenceDepends on -10,7/(11 7) 0than oscillator instabilities for oscillator; frequency sensitivity isb. Tight-Phase-Lock Loop calibration which at 10 MHz 7 = 1 s and longer; good measure­ a function of varicap tuning curve,:t , of vancap · . 10. 14 IS - / 7 ment system bandwidth control; hence not conducive to measuring-...l dead time can be made small or absolute frequency differences.negligible.Requires no reference oscillator. Substantially less bandwidth than2. Discriminator Methods two-oscillator, homodyne methods;sensitivity low at low Fourier frequency.Depends on Depends on Requires no reference oscillator; Requires more difficult calibrationcharacteris­ characteris­ very easy to set up and high in to obtain any accuracy over evena. Cavity Discriminator tics of dis­ tics of dis­ sensitivity; practical at microwave modest range of Fourier frequencies;criminator criminator frequencies. accurate only for Fourier frequenciesless than 0.1 x bandwidth.Requires no reference oscillator;Depends on dynamic range set by propertiesb. Delay Line characteris­ of delay line; practical at microticsof delay wave frequencies.lineSubstantially less accurate than twooscillator,homodyne methods; cumbersomesets of delay lines neededto cover much dynamic range; considerabledelay needed for measurementsbelow 100 kHz from carrier.aAccuracy of the measurement cannot be better than the stability of the measurement. Accuracy is limited by the accuracy of the reference oscillator.t>rms is for a measurement bandwidth of 10 4 Hz; 11 0= frequency; 7 = measurement time.The dash (--) means that the method is not generally appropriate for this quantity.

10- 1410- 15c-een10- 1610- 1710- 1810- 20 oFigure LCurve ACurve B.Curve CCurve D.Curve E.Curve F.10 1 10 2 10 3 10 4Fourier Frequency (Hz)Comparison of nominal lower noise limits for differentfrequency-domain measurement methods.The noise limit (resolution), Sct>(f), of typical double-balancedmixer systems at carrier frequencies from 0.1 MHz to 26 GHz.The noise limit, Sct>(f), for a high-level mixer.The correlated component of Sct>(f) between two channelsusing high-level mixers.The equivalent noise limit, Sct>(f), of a 5 to 25 MHz frequencymultiplier.Approximate phase noise limit for a typical delay-line systemwhich uses a 500 ns delay line.Approximate phase noise limit for a delay-line system whichachieves alms delay through encoding the signal on anoptical carrier and transmitting it across a long optical fiber toa detector.TN-8

Table 1 organizes the measurement methods in the above manner giving performancelimits, advantages and disadvantages for each. Figure 1 following the table provides limitationsfor phase noise measurements as a function of Fourier frequency. The reader is again remindedthat the table is highly simplified giving nominal levels that can be achieved. Exceptions can befound to almost every entry.A.6 RELATIONSHIP OF THE MODIFIED ALLAN VARIANCE TO THE ALLAN VARIANCEIn sorting through this set of papers and other published literature on the Allan Variance,we were stimulated to further consider the relationship between the modified Allan variance andthe Allan variance. The ideas which were developed in this process have not been published, sowe include them here.Paper D.6, "Characterization of Frequency Stability: Analysis of theModified Allan Variance and Properties of Its Estimate," by Lesage and Ayi adds new insightsand augments the Allan and Barnes paper (D.5), "A Modified Allan Variance with IncreasedOscillator Characterization Ability." In this section we extend the ideas presented in these twopapers and provide further clarification of the relationship between the two variances, both ofwhich are sometimes referred to as two-sample variances.Figure 2 shows the ratio [mod 0ir)/oy(r)]2 as a function of n, the number of time orphase samples averaged together to calculate mod Oy( r). This ratio is shown for power-law noisespectra (indexed by the value of ex) running from f to f2. These corrected results have a somewhatdifferent shape for ex = -1 than those presented in either paper D.5 or paper D.6. Furthermore,this figure also shows the dependence on bandwidth for the case where ex = L For allother values of ex shown, there is no dependence on bandwidth. Table 2 gives explicit values forthe ratio as a function of n for low n as well as the asymptotic limit for large n. For ex = 1, theasymptotic limit of the ratio is considerably simplified from that given in papers D.5 and D.6.With these results it is possible to easily convert between mod air) and oir) for any of thecommon power-law spectra.The information in figure 2 and table 2 was obtained directly from the basic definitions ofaye r) and mod 0/ r) using numerical techniques. The results of the numerical calculations werechecked against those obtained analytically by Allan and Barnes (D.5) and Lesage and Ayi (D.6).For ex = 2, 0 and -2 the results agree exactly. For ex = 1 and -1 the analytical expressions arereally obtained as approximations. The numerical calculations are obviously more reliable.Details of our calculations can be found in a NIST report [1]. A useful integral expression (notcommonly found in the literature) for modo~( f) is2 2 !lJA sylf) sin 6 ( 1t 1;/)mooa,('t) - -- ~----d/.n41t2't~ 0 f2 sin 2 (1t't o f)Figure 2 and table 2 show that, for the fractional frequency fluctuations, mod Oy( r)always yields a lower value than Oy(.,.). In the presence of frequency modulation (FM) noisewith ex ~ 0, the improvement is very significant for large n. This condition has been examined indetail by Bernier [2]. For white FM noise, a = 0, the optimum estimator for time interval .,. isto use the value of the times or phases separated by r to determine frequency. This is analogousto the algorithm for calculating Oy(r) which yields the one-sigma uncertainty in the estimateTN-9

Table 2. Ratio of mod a~('r) to a~( T) versus n for common power-law noise types Sy(t) = haf. n is thenumber of time or phase samples averaged to obtain mod a~( T = n T 0 where To) is the minimum sample time.W his 21r times the measurement bandwidth f h•= mod a~(f)R(n) vs. n r = nT oa~(r)n a = -2 a = -1 cr = 0 cr = +1 Q = +2~ TO = ) wt, TO = 10 ~ TO = 100 I."hr o = 10 41 1.000 1.000 1.000 1 . 000 1.000 1.000 1.000 1.0002 0.859 0.738 0.616 0.568 0.543 0.525 0.504 0.5003 0.840 0.701 0.551 0.481 0.418 0.384 0.355 0.3304 0.831 0.681 0.530 0.405 0.)59 0.317 0.284 0.250- i 5 0.830 0.684 0.517 0.386 0.324 0.279 0.241 0.2006 0.828 0.681 0.514 0.349 0.301 0.251 0.214 0.16707 0.827 0.679 0.507 0.343 0.283 0.235 0.195 0.143I8 0.827 0.678 0.506 0.319 0.271 0.219 0.180 0.12510 0.826 0.677 0.504 0.299 0.253 0.203 0.160 0.10014 0.826 0.675 0.502 0.274 0.230 0.179 0.137 0.071420 0.825 0.675 0.501 0.253 0.210 0.163 0.119 0.050030 0.825 0.675 0.500 0.233 0.194 0.148 0.106 0.033350 O. '825 0.675 0.500 0.210 0.176 0.134 0.0938 0.0200100 0.825 0.675 0.500 0.186 0.159 0.121 0.0837 0.0100Limit 0.825 0.675 0.500 ( ).37 ) linor-- 1.04 +3 In~T -+

jo0N:; IR(n) = mod a~(f)a~(r)VS. n r = nroI I I1'I I ,,'.2 I 11111' 1 1 I ""~ 0825(i -I 0.6755~ IX 0 0.5022I>-'>-'----c(( 0.15Ir2" S (f) =h f exy ex'"0.01! I I ! I I ! I I I ! I ! I ! ! ! N ! I ! ! I I ! ! ,1 2 5 1a 2 5 100 2 5 1000n, Number of Samples AveragedFigure 2. Ratio of mod a~( r) to a~( r) as a function of the number of time samples averagedtogether to calculate either variance.

of the frequency measured in this manner over the interval r. The estimate of frequency,obtained by averaging the phase or time data, is degr~ded by about 10 percent from that obtainedby using just the end points [3]. This is analog~us to the algorithm for calculating modair) which, in this case, underestimates the uncerta~ty in measuring frequency by J2. Forwhite phase modulation (PM) noise, a = 2, the optimu~ estimator for frequency is obtained byaveraging the time or phase data over the interval r. i This is analogous to the algorithm forcalculating mod ay( r) which yields the one-sigma unbertainty in the estimate for frequencymeasured in this manner. This estimate for frequency ~s In better than that provided byay(r).Based on these considerations, it is our opinion that mod a y(r) can be profitably usedImuch more often than it is now. The presence of significant high-frequency FM or PM noise inthe measurement system, in an oscillator, or in an oscillator slaved to a frequency reference, isvery common. The use of mod CTy(r) in such circums~ances allows one to more quickly assesssystematic errors and long-term frequency stability. In lother words, a much more precise valuefor the frequency or the time of a signal (for a given m~surement interval) can be derived usingmod air) when n is large.iThe primary reasons for using ay( r) are that it ~s well known, it is simple to calculate, itis the most efficient estimator for FM noise (a ~ 0), :tnd it has a unique value for all r. Theadvantages of mod aye r) are cited in the above paragrkph. There are some situations where astudy of both air) and mod air) can be even more Gevealing than either one. The disadvantagesof using mod av(r) are that it is more complex tp calculate and thus requires more computertime and it has 'not been commonly used in the lit~rature, so interpretation of the results ismore difficult to reconcile with published information. Another concern sometimes raised is thatmod a/ r) does not have a unique value in regions ~ominated by FM noise (a ~ 0). Withrapidly increasing computer speeds, the computational d~sadvantage is disappearing. The correctedand expanded information presented in figure 2 ahd table 2 addresses the concern aboutuniqueness. :The primary disadvantage of using a/ r) is that ~he results can be too conservative. Thatis, if the level of high-frequency FM noise is high, the~ the results are biased high, and it cantake much longer (often orders of magnitude longer) to! characterize the underlying low-frequencyperformance of the signal under test. :References to Section A.6I[1] Walls, F. L, John Gary, Abbie O'Gallagher, [Roland Sweet and Linda Sweet, "TimeDomain Frquency Stability Calculated from the! Frequency Domain Description: Use ofthe SIGINT Software Package to Calculate T~e Domain Frequency Stability from theFrequency Domain," NISTIR 89-3916, 1989.I[2] Bernier, LG., Theoretical Analysis of the Modified Allan Variance, Proceedings of the41st Frequency Control Symposium, IEEE Cataldgue No. 87CH2427-3, 116, 1987.I[3] Allan, D.W. and J.E. Gray, Comments on the dctober 1970 Metrologia Paper "The U.S.Naval Observatory Time Reference and Perfoimance of a Sample of Atomic Clocks,"International Journal of ScientifiC Metrology, 7, 7~, 1971.!IN-12

A.7 SUPPLEMENTARY READING LISTGerber, Eduard A. and Arthur Ballato (editors),IPreciSion Frequency Control, Volumes 1and 2 (Academic Press, New York, 1985).iJespersen, James and Jane Fitz-Randolph, Froim Sundials to Atomic Clocks, (DoverPublications, New York, 1982).,Kartaschoff, P., Frequency and Time, (Academic Press, New York, 1978).iIKruppa, Venceslav F. (editor), Frequency Stability:[Fundamentals and Measurement, (IEEEPress, New York, 1983).IRandall, R.B., Frequency Analysis, (Brliel and Kj~r, Denmark, 1987).IVanier, J. and C. Audoin, The Quantum Physics df Atomic Frequency Standards, Volumes1 and 2 (Adam Hilger, Bristol, England, 1989). IISpecial Issues of Journals devoted to Time and Frequen? TopicsIProceedings of the IEEE, Vol. 54, February, 1966. iiProceedings of the IEEE, VoL 55, June, 1967.Proceedings of the IEEE, Vol. 74, January, 1986.IEEE Transactions on Ultrasonics, Ferroelectrics, ~nd Frequency Control, Vol. UFFC-34,November, 1987. !IRegular Publications dealing with Time and Frequency TopicsIEEE Transactions on Ultrasonics, Ferrolectrics ami, Frequency Control.IProceedings of the Annual Symposium on FrequenGJ( Contro~IEEE.Proceedings of the Annual Precise Time and Time [Interval (PITI) Applications and PlanningMeeting, available from the U.S. Naval Obsertatory.Proceedings of the European Frequency and Time ~01um.IN-13

From: Proceedings of the 35th Annual Symposium on Frequency Control, 1981.PROPERTIES OF SIGNAL SDURCE~MEASUREMENT METHODSiANDD. A. Howe, D. W. Allan, and J.IA. BarnesSummaryThis paper is a review of frequency stabilitymeasurement techniques and of noise properties offrequency sources.First, a historical development of the usefulnessof spectrum analysis and time domain measurementswi 11 be presented. Then the rati ana1e wi 11be stated for the use of the two-sample (Allan)variance rather than the classical variance.Next, a range of measurement procedures wi 11 beoutlinea with the trade-offs given for the various,-echni ques emp 1oyed. Methods of i nte,,:lreti ng themeasurement results will be given. In particular,the five commonly used noise models (white PM,flicker PM, white FM, flicker FM, and random walkFM) and thei r causes wi 11 be di scussea. Methodsof characterizing systematics will also be given.Confidence intervals on the various measures willbe discussed. In addition, we will point outmethods of improving this confidence interval fora fixed number of data points.Topics will be treated in conceptual detail.Only light (fundamental) mathematical treatmentwi 11 be gi ven.Although traditional concepts. will be detailed, two new topi cs will be introduced in thi spaper: (1) accuracy limitations of digital andcomputer-based analysis and (2) optimizing theresults from a fixed set of input data.The final section will be devoted to fundamental(physical) causes of noise in commonly usedfrequency standards. Also transforms from time tofrequency domain and vice-versa will be given.Key Words. Frequency stabil ity; Oscill ator noi semodeling; Power law spectrum; Time-domain stabi1ity; Frequency-domain stabil ity; White noise;Flicker noise.IntroductionPrecision oscillators play an important rolein high speed communications, navigation, spacetracking, deep space probes and in numerous otherimportant applications. In this paper, we willrevi ew some preci sian methods or measuri ng thefrequency and frequency stability of precisionoscillators. Development of topics does not relyheavily on mathemati cs. The equipment and set-upfor stabi 1i ty measurements are out1i ned. Ex amp 1esand typical results are presented. PhysicalTime and Frequency Divis~onNational Bureau of Standa~dsBoulder, Colorado 8030B,interpre~ations of common noise processes arediscusseb. A table is provided by which typicalrrequencw domain stability characteristics may beitranslated to time domain stability characteristicsand vicetversa.L THEI SINE WAVE AND STABILITYA , Is i ne wave signal generator produces avoltage [that· changes in time in a sinusoidal wayas show~ in figure 1.1. The signal is an oscillatingsi;gna1 because the sine wave repeats itself.A cycle! of the oscillation is produced in oneperi ad 'rr". The phase is the ang1 e ""," wi thi n acycle c~rresonding to a particular time "t"..,,: FIGURE 1.1Itiis convenient for us to express angles inradians! rather than in units of degrees, andpositi\(~ zero-crossings will occur at even multip1es!orinumber iorrecipro~a1express{on!sine wa~en-radi ans. The frequency: "u" is thecycles in one second, which is theof period (seconds per cycle). Thedescribing the voltage "V" out of asignal generator is given by Vet) ::: V pis the peak voltage amplitUde.sin ["'(t)] where VipEquival+nt expressions are1TN-14

andConsider figure 1.2.V(t) = V sin (Lnvt).pLet's assume tnat the maximumvalue of "V" equals 1, hence "V " = 1 We sayp .that the voltage "\I(t)" is normalized to unity.If weknow the freguency of a signal and if thesignal is a sine ~, then we can determine theincremental change in the period "Til (denoted byAt) at a particular angle of phase."Frequency stabil ity is the degree to whichan oscillating signal produces the same valueof frequency for any interval. At, throughouta speci fi ed peri od of time".Let's examine the t ....o waveforms shown infigure 1.3. Frequency stability depends on theamount of time involved in a measurement. Of thetwo oscillating signals, it is evident that "2" ismore stable than "1" from time t 1 to t:) assumingthe horizontal scales are linear in time.+1, I ,v_l~i '~--1II ill "2. '

where V0 ;: nomi na I peak vo Itage amp I i tude,e(t) ;: deviation of amplitude from nominal,"0 ;: nominal fundamental frequency,~(t) ;: deviation of phase from nominal.Ideally "e" and "dl" should equal zero for alltime. However, in the real world there are noperfect oscillators. To determine the extent ofthe noise components "e" and "ill", we shall turnour attention to measurement techniques.The typical precision oscillator, of course,has a very stable sinusoidal voltage output with afrl!quency v and a period of oscillation T, whichis the reciprocal of the frequency (v =1/T). Onegoa1 is to measure the frequency and/or the frequencystability of the sinusoid. Instability isactually measured, but with little confusion it isoften called stability in the literature. Naturally,fluctuations in frequency correspond tofluctuations in the period. Almost all frequencymeasurements, with very few exceptions, are measurementsof phase or of the period fluctuationsin an osci 11 ator, not of frequency, even thoughthe frequency may be the readout. As an example,most frequency counters sense the zero (or nearzero) crossing of the sinusoidal voltage, which isthe point at which the voltage is the most sensitiveto phase fluctuations.One must also realize that any frequencymeasurement involves two oscillators. In someinstances, one oscillator is in the counter. Itis impossible to purely measure only one osc11­, ator. In some instances one osci 11 ator may beenough better than the other that the fluctuationsmeasured may be considered essentially those ofthe latter. However, in general because frequencymeasurements are always dual, it is useful todefine:v - vY (t)·C" 1 a (1.2)vaas the fractional frequency difference or deviationof 05cillator one, v l ' with respect to a referenceoscillator V o divided by the nominal frequency v o 'Mow, Yet) is a dimensionless quantity and uSl!fu1in describing oscillator and clock performance;e.g., the time deviation, x(t), of an oscillatorover a period of time t, is simply given by:tx(t) = f y(t')dt' (1. 3)oSince it is impossible to measure instantaneousfrequency, any frequency or fracti ona1 frequl!ncymeasurement always involves sOflle saJllPle time, Ator "t"--some time window through which the oscillatorsare observed; whether it's a picosecond, asecond, or a day, there is always some sampletime. So when determining a fractional frequency,y(t). in fact what is happening is that the timedeviation is being measured say starting at sometime t and again at a later time, t + t. Thedifference in these two time deviations, dividedby t gi ves the average fracti ana1 frequency overthat period t:yet) = x(t + t) - x(t) (1.4)tTau, t, may be called thl! sample time or averagingtime; e.g., it may be determined by the gate timeof a counter.What happens in many cases is that one samplesa number of eyc1 es of an osci 11 ation duri n9 thepreset gate time of a counter; after the gate timehas elapsed, the counter latches the value of thenumber of cyc 1es so that it can be read out,printed, or stored in some othl!r way. Then thereis a delay time for such processing of the databefore the counter anns and starts again on thenext cycle of the oscillation. During the delaytime (or process time), infonnation is lost. Wehave chosen to call it dead time and in someinstances it becomes a problem. Unfortunately fordata processing in typical oscillators the effectsof dead time often hurt most when it is the hardestto avoi d. In other words, for times that areshort comparl!d to a second when it;s very difficultto avoid dead time, that is usually wOeredead time can make a significant difference in thedata analysis. Typically for many oscillators, if3TN-16

the sample time is long compared to a second, thedead time makes little difference in the dataanalysis, unless it is excessive. 1 New equipmentor techniques are now avai lab1e which contributezero or negligible dead time. 2In reality, of course, the sinusoidal outputof an oscillator is not pure, but it containsnoise fluctuations as well.withThis section dealsthe measurement of these fluctuations todetermine the quality of a precision signal source.We wi 11 descri be fi ve di fferent methodS ofmeasuring the frequency fluctuations in precisionosci 11 ators.B. Dual mixer time difference (DTMD) svstemThis system shows some significant promise andhas just begun to be exploited. A block diagram isshown is figure 1.5.1.1 Common Methods of Measuring Frequency Sta­bilityA. Beat frequency methodThe first system is called a heterodynefrequency measuring method or beat frequencymethod.The signal from two independent oscil­lators are fed into the two ports of a doublebalanced mixer as illustrated in figure 1.4.aSCIU~TUuoO[1 TUTHETERODYNEFREQUENCYMEASUREMENT METHODFIGURE 1. 41"1 • Yo I • ", • t ~aU""PU "a • 1 Nz'Til • 1 I• Yo • 1 tThe di fference frequency or the beat frequency,Vb' is obtained as the output of a low pass filterwhich follows the mixer.This beat frequency isthen amp 1i fi ed and fed to a frequency counter andprinter or to some recording device.The frac·tiona1 frequency is obtained by dividing Vb by thenominal carrier frequency va' This system hasexcellent precision; one can measure essentiallyall state-of-the-art oscillators.,.". Ofntlut!: 1l~1) tJ "t ~ c, .. ~ • ~'ItACTIOtiAl ,.tDuuu: ,(tl • A:tL!J. .. 1'11·11 ~... ,,2.,FIGURE 1. 5To preface the remarks on the DMTD,'UI)it should bementi oned that if the time or the time f1 uctuations can be measured di rect1y, an advantage isobtai ned over just measuri ng the frequency.reason is that on~Thecan calculate the frequencyfrom the time without dead time as well as knowthe time behavior.The reason, in the past, thatfrequency has not been inferred from the time (forsample times of the order of several seconds andshorter) is that the time difference between apair of oscillators operating as clocks could notbe measured with sufficient precision (commerciallythe best that is available is 10- 11 seconds).Thesystem described in this section demonstrates apreci 5i on of 10- 13 seconds.Such preci s i on opensthe door to making time measurements as well asfrequency and frequency stability measuements forsample times as short as a few milliseconds andlonger, all without dead time.In fi gure 1. 5, osci 11 ator 1 caul d be consideredunder test and oscillator 2 could beconsidered the reference oscillator. These signalsgo to the ports of a pa i r of daub1e ba 1ancedmixers.Another oscillator with separate symmetricbuffered outputs is fed to the remaining other two4TN-17

ports of the pair of double balanced mixers. This the cycle ambiguity. It is only important to knowcommon oscillator's frequency is offset by a k if the absolute time difference is desired; fordes i red amount from the other two osci 11 ators. frequency and frequency stability measurements andThen two di fferent beat frequenci es come out of for time fluctuation measurements, k may be assumedthe two mi xers as shown. These two beat frequen­ zero unless one goes through a cycle during a setcies will be out of phase by an amount proportional of measurements. The fractional frequency can beto the time difference between osci 11 ator 1 and derived in the normal way from the time fluctua­2--exc1uding the differential phase shift that may tions.be inserted. Further, the beat frequencies differinfrequency by an amount equa 1 to the. frequency"l(i, t) - "Z(i, t)difference between oscillators 1 and Z. " 0This measurement technique is very usefulxCi + 1) - xCi)Y1,Z(i, t) = (1. 6)where one has oscillator 1 and oscillator 2 on thetsame frequency. This is typical for atomic stan­ ~t(1 + 1) - at(i)t 2dards (cesium, rubidium, and hydrogen frequencyb "0standards).Illustrated at the bottom of figure 1.5 is In eqs (1.5) and (1.6), assumptions are madewhat mi ght represent the beat frequenci es out of that the transfer (or common) oscillator is set atthe two mi xers. A phase shifter may be inserted a lower frequency than osci 11 ators 1 and Z, andas illustrated to adjust the phase so that the two that the voltage zero crossing of the beat "I - "cbeat rates are nominally in phase; this adjustment starts and that "2 - "c stops the time intervalsets up the nice condition that the noise of the counter. The fractional frequency difference maycommon osci 11 ator tends to cance 1 (for certai n be averaged over any integer multiple of to:types of noise) when the time difference is determined. After amp 1ifyi ng these beat signals, thestart port of a time interval counter is triggered= xCi + m) - xCi)mt b(1.7)with the positive zero crossing of one beat andthe stop port with the positive zero crossing of where m is any positive integer. If needed, tbthe other beat. Taki ng the time di fference be­ can be made to be very small by having very hightween the zero crossings of these beat frequencies, beat frequencies. The transfer (or common) oscilonemeasures the time difference between oscillator lator may be replaced with a low phase-noise1 and oscillator 2, but with a precision which has frequency synthesizer, which derives its basicbeen amp 1ified by the ratio of the carri er fre­ reference frequency from osci 11 ator Z. In thi squency to the beat frequency (over that normally set-up the nominal beat frequencies are simplyachievable with this same time interval counter). gi ven by the amount that the output frequency ofThe time difference x( i) for the i th measurement the synthesi zer 15 offset from "2' Sample timesbetween oscillators 1 and Z is given by eq (1.5). as short as a fE'll milliseconds are easiliy obtained.Logging the data at such a rate can be a(1. 5)problem without special equipment. The latest NBStime scale measurement system is based on the DMTDand is yielding an excellent cost benefit ratio.where ~t(i) is the i th time difference as read onthe counter, tbis the beat peri od, "0 is the C. Loose ohase lOCK loop methodnominal ~arrier frequency, ~ is the phase delay in This first type of phase lock loop method isradians added to the signal of oscillator I, and k illustrated in figure 1.6. The signal from anis an integer to be determi ned in order to remove oscillator under test is fed into one port of a5TN-18

mixer. The signal from a reference oscillator isfed into the other port of this mixer. The signalsare in quadrature, that is, they are 90 degrees out+--­OUTPUT OFPlL FilTERFIGURE 1. 6of phase so that the average 1101 tage out of themixer is nominally zero, and the instantaneousvoltage fluctuations correspond to phase fluctuationsrather than to amplitUde fluctuationsbetween the two signals. The mixer is a keyelement in the system. The advent of the Schottkybarrier diode was a significant break.through inmaking low noise precision stability measurements.The output of this mixer is fed through a low passfi lter and then amp1 ified in a feedback loop,causing the voltage controlled oscillator (reference)to be phase locked to the test oscillator.The attack time of the loop is adjusted such thata very loose phase lock (long time constant)condition exists. This is discussed later insection VIII.The attack time is the time it takes theservo system to make 70% of its ultimate correctionafter being sl ightly disturbed. The attack timeis equal to l/nw h, where w his the servo bandwidth.If the attack time of the loop is about a secondthen the voltage fluctuations will be proportionalto the phase fluctuations for sample times shorterthan the attack time. Depending on the coefficient of the tuni ng capacitor and the qual i ty ofthe oscillators involved, the amplification usedmay vary significantly but may typically rangefrom 40 to 80 dB via a good low noise amplifier.In turn this signal can be fe.d to a specturmanalyzer to measure the Fourier components of thephase fluctuations. This system of frequencydomainanalysis is discussed in sections VII! to X.It is of particular use for sample times shorterthan one second (for Fourier frequencies greaterthan 1 Hz) in analyzing the characteristics of anoscillator. It is specifically very useful if onehas di screte side bands such as 60Hz or detai 1edstructure in the spectrum. How to characterizepreclslon oscillators using this technique will betreated in detail later in section IX and XI.One may also take the output lIo1tage from theabove amp 1ifier and feed it to an AID converter.This digital output becomes an extremely sensitivemeasure of the short term time or phase fl uctuationsbetween the two oscillators. Precisions ofthe order of a picosecond are easily achievable.D. Tiaht phase lock 1000 methodThe second type of phase lock loop method(shown in figure 1. 7) is essentially the same asthe fi rst in fi gure 1. 6 except that in thi s casethe loop is in a tight phase lock condition; i.e.,the attack time of the loop should be of the orderof a few mi 11 i seconds. In such a case, the phasefluctuations are being integrated so that thelIo1tage output is proportional to the frequencyflCIlLATORYIDU nITTIGHT ~HAS£.LOCl( LOO~MfTHOO OF MEASURINGFREOUENCY STABILITYFIGURE 1. 7fluctuations between the two oscillators and is nolonger proportional to the phase fluctuations forsample times longer than the attack time of theloop. A bias box is used to adjust the voltage onthe lIaricap to a tuning point that is fairlylinear and of a reasonable value. The voltage6TN-19

fluctuations prior to the bias box (biased slightlyaway from zero) may be fed to a voltage to frequencyconverter whi ch in turn ;s fed to a frequencycounter where one may read out the frequencyfluctuations with great amp1ific3tion of theinstabi 1ities bet..een this pair of oscillators.The frequency counter data are logged with a datalogging device. The coefficient of the varicapand the coefficient of the voltage to frequencyconverter are used to determine the fractionalfrequency fluctuations, Yi' between the oscillators,where i denotes the i th measurement asshown in figure 1.7. It is not difficult toachieve a sensitivity of a part in 1014 per Hzresa1uti on of the frequency counter. so one hasexcellent precision capabilities with this system.E. Time difference methodThe last measurement method we will illustrateis very commonly used, but typically does not havethe measurement precision more readily availablein the first four methods illustrated above. Thismethod is called the time difference method, andis shown in figure 1.8. Because of the wideI,all. 312.'" UIIIf·I...·-----'-".ocS1IlII '.tt81 ! +N ~ I IiG~FIGURE 1.8bandwidth needed to measure fast rise-time pulses,this method is limited in signa1-to-noise ratio.However, some counters are commercially availableallowing one to do signal averaging or to doprecision rise-time comparison (precision of timedi fference measurements in the range of 10 ns to10 piS are now available). Such a method yields adirect measurement of x(t) without any translation,conversion, or multiplication factors. Cautionshould be exercised in using this technique evenif adequate measurement precision is availablebecause it is not uncommon to have significantinstabi 1ities in the frequency dividers shown infigure 1. a--of the order of severa1 nanoseconds.The techno 1ogy exi sts to build better frequencydividers than are commonly available, but manufacturershave not yet availed themselves of state-ofthe-arttechniques in a cost beneficial manner. Atrick to by-pass divider problems is to feed theoscillator signals directly into the time intervalcounter and' observe the zero voltage crossing intoa well matched impedance. (In fact, in all of theabove methods one needs to pay attention to impedancematchi ng, cab 1e 1engths and types, and connectors).The divided signal can be used toresolve cycle ambiguity of the carrier, otherwisethe carrier phase at zero volts may be used as thetime reference. The slope of the signal at zerovolts is 2TcV h: 1 • where t 1 = l/Ulp(the period ofoscillation). For Vp= 1 volt and a 5 MHz signal,this slope is 3m vo1ts/ns, whiCh is a very goodsensitivity.II. MEASUREMENT METHODS COMPARISONWhen maki n9 measurements between a pair offrequency standards or clocks, it is desirable tohave less noise in the measurement system than thecomposite noise in the pair of standards beingmeasured. This places stringent requirements onmeasurement systems as the state-of-the-art ofprecision frequency and time standards has advancedto its current 1eve1. As wi 11 be shown, perhapsone of the greatest areas of di spari ty betweenmeasurement system noise and the noise in currentstandards is in the area of time difference measurements.Commercial equipment can measure timedifferences to at best 10_ 11 S, but the time fluctuationssecond to second of state-of-the-artstandards is as good as 10_ 13 S.The disparity is unfortunate because if timedifferences between two standards could be measuredwith adequate precision then one may also know thetime fluctuations, the frequency differences, andthe frequency fluctuations. In fact, one can set7TN-20

EJtHllle ofDeu deducibl. fl'Qll the ..uur...ntlIi.nchy Meuur_ntSUtus Metl\Q(l Freq.Duel IIiur Ti.. Offf.,,(t) ,,~ 6 ,,(t,I) y(t.I) .. o$y(t,I)"or tv " (t + t)-,,(t) ~ y(t+t,I)-y(t.t)Ti.. Inu,."el Countert2Laase Phu.-lack.d 6 ,,(t,I) "...f ....nc:. alei lletar an't ....u...~Het.rodyne ar Mat an't ....u... een'r ....u...frequ.ncy!y(t.t)1 .."beet"a4Ti gilt plllls.-lo

up an interesting hierarchy of kinds of measurementsystems: 1) those that can measure time, x(t); Z)those that can measure changes in time or timefluctuations ox(t); 3) those that can measurefrequency, v(y = (v-vo)/v ); and 4)othose that canmeasure changes in frequency or frequency fluctuations,ov (oy =ov/v o)' As depicted in table 2.1,if a measurement system is of status 1 in thishierarchy, i.e., it can measure time, then timefluctuations, frequency and frequency fluctuationscan be deduced. However, if a measurement systemis only capable of measuring time fluctuations(status 2 - table 2.1), then time cannot be deduced,but frequency and frequency fluctuationscan. If frequency is being measured (status 3 ­table 2.1), then neither time nor time fluctuationsmay be deduced with fidel ity because essentiallyall commercial frequency measuring devices have"dead time" (technology is at a point where thatis changing with fast data processing speeds thatare now available). Dead time in a frequencymeasurement destroys the opportunity of integratingthe fractional frequency to get to "true" timefluctuations. Of course, if frequency can bemeasured, then trivially one may deduce the frequencyfluctuations. Finally, if a'system canonly measure frequency fluctuations ~status 4 ­table 2.1), then neither time, nor time fluctuations,nor frequency can be deduced from the data.If the frequency stabil ity is the primary concernthen one may be perfectly happy to employ such ameasurement system, and similarly for the otherstatuses in this measurement hierarchy. Obviously,if a measurement method of Status 1 could beemployed with state-of-the-art precision, thiswould provide the greatest flexibility in dataprocessing. From section 1, the dual mixer timedifference system is purported to be such a method.Table 2.2 is a comparison of these differentmeasurement methods. The values entered arenominal; there may be unique situations wheresignificant departures are observed. The time andfrequency stabil iti es 1i sted are the nomi na 1second to second rms values. The accuracieslisted are taken in an absolute sense. The costslisted are nominal estimates in 1981 dollars.Figure 2.1 is a diagram indicating thesample time regions over which the various methodsNII

If, for example, one has the values of thefrequency over a period of time and a frequencyoffset from nominal is observed, one may calculatedirectly that the phase error will accumulate as aramp. If the frequency values show some lineardri ft then the time f1 uctuations wi 11 depart as aquadratic. In almost all oscillators, the abovesystematics, as they are sometimes called, are theprimary cause of time and/or frequency departure.A useful approach to determine the. value of thefrequency offset is to calculate the simple meanof the set, or for determining the value of thefrequency drift by calculating a linear leastsquares fit to the frequency. A least squaresquadratic fit to the phase or to the time derivativeis typically not as efficient an estimator ofthe frequency drift for most oscillators.3.2 Random FluctuationsAfter calculating or estimating the systematicor non-random effects of a data set, these may besubtracted from the data leaving the residualranctom fluctuations.These can usually be bestcharacterized statistically. It is often the casefor precision oscillators that these random fluctuationsmay be well modeled with power law spectraldensities.sections VIII to X.This topic is discussed later inWe haveS(f)=hfl,y(3.1)awhere S (f) is the one-sided spectral density ofythe fracti ona 1 frequency f1 uctuations, f is theFouri er frequency at whi ch the densi ty is taken,h 1s the intensity coefficient, and a is a numberamode 1i ng the most appropri ate power 1aw for thedata. It has been shown 1 ,3 that in the timedomain one can nicely represent a power law spectraldensity process using a well defined timedomainstability measure, a (-t), to be explainedyin the next section. For example, if one observesfrom a log C1 let) versus t diagram a particularyslope (call it IJ) over certain regions of sampletime, 1:, this slope has a correspondence to apower 1awspectral dens i ty or a set of the samewith some amplitude coefficient h .aIn particular,IJ = -cr -1 for -3 < a

is determi ned by the measurement system.If thetime difference or the time fluctuations areavailable then the frequency or the fractionalfrequency fl uctuations may be ca 1cu latt!tl frOli oneperiod of sampl ing to the next over ttle datalength as indiciated in figure 4.1. Supposefurther there are M values of the fractionalfrequency Yi'these data.Now there are many ways to analyzeHistorically, people have typicallyused the standard deviation equation shown infigure 4.1, a td d (t),s . ev.where y is the averagefractional frequency over the data set and issubtracted from each value of Yj before squaring,summing and dividing by the number of values minusone, (M-1), and taki ng the square root to get thestandard deviation.At NBS, we have studied whathappens to the standard deviation when the dataset may be characteri zed by power 1aw spectrawhich are more dispersive than classical whitenoise frequency fluctuations.In otne'T' words, ifthe fluctuations are characterized by flickernoise or any other non-white-noise frequencydeviations, what ha~pensto the standard deviationfor that data set? One can show that the standarddeviation is a function of the number of datapoints in the set; it is also a function of thedead time and of the measurement system bandwidth.For example, using flicker noise frequency modulationas a model, as the number of data pointsincreases, the standard deviation monotonicallyincreases without limit. Some statistical measureshave been deve loped whi ch do not depend upon thedata length and which are readily usable forcharacterizing the random fluctuations in precisionoscillators. An IEEE subcommittee on frequencystability hasrecommended what has come to beknown as the "A11 an vari ance" taken from the setof useful variances developed, and an experimentalestimation of the square root of the Allan varianceis shown as the bo~tom right equation infi gure 4.1. Thi 5 equation is very easy to imp 1ementexperimentally as one simply need add up thesquares of the differences between adjacent valuesof Yi' divide by the number of them and by two, and'take the square root. One then has the quantitywhich the IEEE subcommittee has recommended for

e able to generate values for a y(m.) as a functionof III. from which one may be able to infer a modelfor the process that is Characteristic of thispa i r of oscillators. If one has dead time in themeasurements adjacent pairs cannot be averaged inan unambiguous way to simply increase the sampletime. One has to retake the data for each newsample time--often a very time consuming task.This is another instance where dead time can be aproblem.How the classical variance (standard deviationsquared) depends on the number of samples is shownin figure 4.2. Plotted is the ratio of the standarddeviation squared for N samples to the standarddeviation squared for 2 samples;

Since each average of the fractional frequencyfl uctuati on va 1ues is for one second, then thefirst variance calculation will be at t = 1s. Weare given /1 = 8 (eight values); therefore, thenumber of pairs in sequence is /1-1 = 7. We have:aata v_lues Ffnt d1f1"tr1"eftC1l first 41ffor-onc:••_\"OilY k(. 10-» (Yk-I - YkJ (n 10-') (Yk+l - YkJ' (. W-'D)t. J6t.61 0.25 0.06J.l~ .1. 42 2.02t.21 1.02 1. att.47 0.26 0.D7J.96 -0.51 0.264.10 0.1" 0.02J.08 -1.02 LOt-nrMoo 1~ (y - y )Z = 4 51 X 10- 10&1 k+1 k .V. CONFIDENCE OF THE ESTIMATE AND OVERLAPPINGSAMPLES 4One can imagi ne taki ng three phase or timemeasurements of one oscillator relative to anotherat equally spaced intervals of time. From thisphase data one can obtain two, adjacent values ofaverage frequency. From these two frequency measurements,one can calculate a single sample Allan(or two-sample) variance (see fig. 5.1). Ofcourse this variance does not have high precisionor confidence since it is based on only one frequencydifference.t-+ * *+-l----r----I--­~J1 ...I+- 1 -+1+- 2......f'lTherefore,and4 51 X 10- 10a~(ls) = . 2(7) = 3.2 X 10- 11cr (t) = [a2(ls)]~ = [3.2 x 10_11]~ = 5.6 X 10- 6y YUsing the same data, one can calculate thevariance for t = 2s by averaging pairs of adjacentvalues and using these new averages as data valuesfor the same procedure as above. For three secondaverages (t = 3s) take adjacent threesomes andfind their averages and proceed in a similiarmanner. More data must be acqui red for longeraveraging times.One sees that wi th 1arge numbers of datavalues, it is helpful to use a computer or progr!llimablecalculator. The confidence of the estimateon a (t) improves nominally as the square root ofythe number of data values used. In this example,M=8 and the confi dence can be expressed as bei ngno better the l/$' x 100% = 35%. This then is theall owab 1e error in our estimate for the T = Isaverage. The next section shows methods of computingand improving the confidence interval.FIGURE 5.1Statisticians have considered this problem ofquantifying the variability of quantities like theAllan Variance. Conceptually, one could imaginerepeating the above experiment (of taking thethree phase points and calculating the AllanVariance), many times and even calculating thedistribution of the values.For the above ci ted experiment we know thatthe results are distributed like the statistician'schi-square distribution with one degree of freedom.That is, we know that for most common asci 11 atorsthe first difference of the frequency is a normallydistributed variable with the typical bell-shaoedcurve and zero mean. However, the square of anormally distributed variable is NOT normallydistributed. That is easy to see since the squareis always positive and the normal curve is completelysymmetric and negative values are aslikely as positive. The resulting distribution iscalled a chi-square distribution, and it has ONE"degree of freedom" since the distribution wasobtained by considering the squares of individual(i. e., one independent sample). normally distributedvariables.In contrast, if we took five phase values,then we could calculate four consecutive frequencyvalues, as in figure 5.2. We could then take the13TN-26

t ....chi -square. d. f. is the number of degrees ofAfL +.. "* .... freedom (possibly not an integer), and a 2 is the/f 4---~---4----~---~---"'t 1"""true" Allan Variance we're all interested ink- 1-I+- 2_1+- ~~ Jof ~ knowing--but can only estimate imperfectly.I+- ~ BUR: -+I+-~~-+'Chi-square is a random variable and its distri­r----------, bution has been studied extensively. For some:14-~~~ : reason, chi-square is defined so that d. f., the... _--------_ ....number of degrees of freedom. appears explicitlyFIGURE 5.2 in eq (5.1). Still, X 2 is a (implicit)first pair and calculate a sample Allan Variance.function of d. f., also.and we could calculate a second sample Allan The probability density for the chi-squareVariance from the second pair (i.e.• the third anddistribution is given by the relationfourth frequency measurements). The average ofthese two sample Allan Variances provides an d. f.-1 X1 2improved estimate of the "true" Allan Variance, p(x 2 ) = (x 2 T e (5.2)and ~ewould expect it to have a tighter confidenceinterval than in the previous example.This couldbution with TWO degrees of freedom.However, the re is another option. We cou1dalso consider the sample Allan Variance obtainedfrom the second and third frequency measurements.That is the middle sample variance.Now. however,Chi-square distributions are useful in deterwe'rein trouble because clearly this last sampleA11 an Vari ance is NOTtwo.independent of the otherIndeed, it is made up of parts of each ofthe other two.Thi s does NOT mean that we can'tuse it for improving our estimate of the "true"Allan Variance, but it does mean that wecan'tjust assume that the new average of three sampleAllan Variances is distributed as chi-square withcounter chi-square distributions with fractionaldegrees of freedom.And as one might expect. thenumber of degrees of freedom will depend upon theunderlying noise type, that is, white FM. flick.erFM, or whatever.Before going on with this, it is of value toreview somebution. Sample variances (like sample Allantion:2d. f. r(d/.)where r(d;/ ") is the gamma function, defined bybe expressed with the aid of the chi-square distritheintegralOIlt-l ·xr (t) = J0 X e dx (5.3)mining specified confidence intervals for variancesand standard deviations. Here is an example.Suppose we hav~a sample variance S2 = 3.0 and weknow that this variance has 10 degrees of fre~dom.(Just how we can k.now the degrees of freedom wi 11be discussed shortly.) Suppose a1so that we wantto know a range around our sample value of S2 = 3.0which "probably" contains the true value, a 2 .three degrees of freedom. Indeed, we will endesiredconfidence is, say, 90%.TheThat is, 10% ofthe time the true value will actually fall outsideof the stated bounds.to a 11 ocate S%The usual way to proceed isto the low end and 5% to the hi ghend for errors, leaving our 90% in the middle.This is arbitrary and a specific problem mightdictate a different allocation. We now resort toconcepts of the chi-square distritablesof the chi-square distribution and findthat for 10 degrees of freedom the 5% and 95%Variances) are distributed according to the equapointscorrespond to:x2(.05) :: 3.94x2. = (d. f. ) . 52(5.1)at for d.f. = 10 (5.4)where 52 is the sample Allan Variance, x2. is X 2 (.95) =18.314TN-27

Thus. with 9~ probability the calculated samplevariance, S2, satisfies the inequality:3.94 ~ (d.f~~.s2 < 18.3 (5.5)and this inequality can be rearranged in the form1. 64 ~ 0 2 < 7. 61 (5.6)or, taking square roots:1. 28 ~ " .: 2.76 (5.7)Now someone might object to the form ofeq (5.7) since it seems to be saying thatthe true sigma falls within two limits with 90%probabil ity.Of course, this ;s either true ornot and is not subject to a probabilistic interpretation.Actually eq (5.7) is based onthe idea that the true sigma is not known and weestimate it with the square root of a samplevariance, S2.This sample variance is a randomvariable and is properly the subject of probability,and its value (Which happened to be 3.0 inthe example) will conform to eq (5.7) ninetimes out of ten.Typically, the sample variance is calculatedfrom a data sample using the relation:NS2 = N=l 1: (x - x)2 (5.8)n=1 nwhere it is implicitly assumed that the xn's arerandom and uncorre1ated (i.e., white) and where xis the sample mean calculated from the same dataset. If all of this is true, then S2 is chi-squaredistributed and has N-l degrees of freedom.Thus, for the case of white x ' and a convenntional sample variance (i.e., eq (5.8», thenumber of degrees of freedom are given by theequation:d. f. N-l (5.9)The prob 1em of interest here is to obtai n thecorresponding equations for Allan Variances usingoverlapping estimates on various types of noise(i. e. , white FM, fl iclc.er FM, etc.).Yoshimura)Other authors (Lesage and AUdoin, andhave considered the question of thevariance of the Allan Variances without regard tothe distributions.re1ated prob1emresults.This is, of course, a closelyand use will be made of thei rThese authors considered a more restrictiveset of overlapping estimates than will beconsidered here, however.VI.MAXIMAL USE OF THE DATA AND DETERMINATION OFTHE DEGREES OF FREEDOM.6.1 Use of DatacomparedCons i der the case of two asci 11 ators bei ngin phase and exactly N values of thephase di fference are obtai ned.Assume that thedata are taken at equally spaced intervals, to'Fromthese N phase values, one can obtain N-lconsecuti ve values of average frequency and fromthese one can compute N-2 individual, sample AllanVariances (not all independent) for t = to' TheseN-2 values can be averaged to obtain an estimateof the Allan Variance at t = to' The variance ofthis variance has been calculated by the aboveci ted authors.Using the same set of data, it is also possibleto estimate the Allan Variances for integermultiples of the base sampling interval, t =nt 'oNow the possibilities for overlapping sample AllanVariances are even greater.For a phase data setof N points one can obtain exactly N-2n sampleAllan Variances for t =nt ' Of course only aofracti on of these are genera11 y independent.Still the use of ALL of the data ;s well justified(see fig. 6.1).Consider the case af' an experiment extendingfor several weeks in duration with the aim ofgetting estimates of the Allan Variance for tauva1ues equal to a week or more. As always thepurpose is to estimate rel iably the "true" AllanVariance as well as possib1e--that is, with astight an uncertainty as possible. Thus one wants15TN-28

The approach used is based on three equationsrelating to the chi-square distribution:dfJ 2 52=:;'1" (6.1)(d. f. 0E[x 2 ) = d. f. (6.2)Var[x 2 ) = 2(d.f.) (6.3)FIGURE 6.1to use the data as efficiently as possible sinceobtai ni ng more data can be very expens i ve. Themost efficient use is to average all possiblesample Allan VaTiances of a given tau value ttlatone can compute from the data.The problem comes in estimating how tight theconfidence intervals really are--that is, in estimatingthe number of degrees of freedom. Clearly,if one estimates the conf;dence intervals pessimistically,then more data is needed to reach aspecified tolerance. and that can be expensive.The other error of over-confidence in a ques~ionablevalue can be even more expensive. Ideally,one has realistic confidence estimates for themost efficient use of the data, which is theintent of this writing.6.2 Determining the Degrees of FreedomIn principle, it should be possible to determine ana 1ytically the equations corresponding toeq (5.9) for all cases of interest. Unfortunate1ythe ana1ysi s becomes qui te comp Heated.Exact computer algorithms were devi sed for thecases of white phase noise, white frequency modulationand random walk FM. For the two f1 ickercases (i.e., flicker FM and PM) a completelyempirical approach was used. Due to the complexityof the computer programs, empirical fits weredevised for all five noise types.where the expression E[x 2 ) means the "expectation,"or average value of x. Var[x 2 ) is the variance ofx 2 , and d.f. is the number of degrees of freedom.A computer was ,used to s imu1ate phase datasets of some 1erigth, N, and then Allan Varianceswith t = ntowere calculated for all possiblesamples. This "experiment" was repeated at least1000 times using new simulated data sets of thesame spectral type, and always of the same length,N. Since the data were simulated on a computer,the "true" All an Yari ance, 02.. was known for manyof the noise models and could be substituted intoeq (6.1). From the 1000 values of s2./a2.. distributionsand sample variances were obtained. The"experimental" distributions were compared withtheoretical distributions to verify that theobserved distributions true1y conformed to thechi-square distribution.The actual calculation of the degrees offreedom were made using the relation:=~~~ (6.4)d.f. ~which can be deduced from eqs (6.1), (6.2),and (6.3). The Var(s2) was estimated by thesample variance of the 1000 values of the averageA11 an Vari ances, each obtai ned from a phase dataset of length N.Of course this had to be repeated for variousvalues of Nand n. as well as for each of the fivecOlllllon noise types: white PM, flicker PM, whiteFM, flicker FM, and random walk FM. Fortunately,certain limiting values are known and these can beused as checks on the method. For examp 1e, when(N-l)/2=n. only one Allan Variance is obtainedfrom each data set and one shou 1d get about onedegree of freedom for eq (6.4), which was observed16TN-29

in fact. Also for n=l the "experimental" conditionscorrespond to those used by Lesage andAudoin, and by Yoshimura. Indeed, the method alsowas tested by verifying that it gave resultsconsistent with eq (5.9) when applied to theconventi ona1 samp 1e vari ance. Thus, combi ni ng eq(6.4) with the equations for the variance ot theAllan Vari ances from Lesage and Audoi nandYoshimura, one obtains:18(N-Z)2White PM d. f. :: , for H > 435N-88Flicker PM d. f. :: ?2(N-Z)2White FM d. f. = (6.5)3N-7Fl i cker FMRandom Walk FM2(N-Z)2d. f. = 2.3N-4.9d. f. = N-Zfor n=l. Unfort.unate1y. their results are nottotally consistent with each other. Where inconsistency arose the value in best agreement withthe "experimental" results was chosen.The empirical equations which were fit to the"experimental" data and the known values aresummarized below:White PMd. f. _ (N+1)(N-2n)Z(N-n).)f Fl i cker PM d.f. _ exp (In ~~1It is appropriate to give some estimate ofjust how well these empirical equations approachthe "true" values. The equations have approximateIy (a few percent) the correct as symptoti cbehavior at n=1 and n=(N-1)/2. In between, thevalues were tested (using the simulation results)over the range of fo#::S to N=1025 for n=l ton=(N-1)/2 changing by octaves. In general, thefit was good to within a few percent. We mustacknowledge that distributional problems with therandom number generators can cause problems,although there were several known values whichshould have revealed these problems if they arepresent. Also for three of the noi se types theexact number of degrees of freedom were calculatedfor many values of Nand n and compared with the"Monte Carlo" calculations. The results were allvery good.Appendix I presents the data in graphicalform. All va 1ues are thought to be accurate towithi n one percent or better for the cases ofwhite PM, white FM, and random walk FM. A largerto1erence should be allowed for the flicker cases.VII. EXAMPLE OF TIME-DOMAIN SIGNAL PROCESSING ANOANALYSISWe will analyze in some detail a commercialportable clock, Serial No. 102. This cesium wasmeasured against another commercial cesium whosestabil ity was well documented and verified to bebetter than the one under test. Platted in figure7.1 are the residual time deviations after removing(6.6)White FM4nd.f. - 2n N 24n2 + 5= [lQ!:.ll -2(N-Z) JFTSPC 'IS CSl-01* Fl icker FMRandom Walk FM d.t. S N~2(N-l)2-3n(N-1)+4n 2(N-3)2The figures in Appendix I demonstrate the fit tothe "experimental" data.0-4- ...l.-__..)""L_1...- -..L__o10~'Y$FIGURE 7.1* See Appendix Note # 3817TN-3D

a mean frequency of 4.01 parts in 10 13 . Applyingthe m~thods described in section IV and section V,....e generalted the a (t)ydiagram shown in figure).2.• //ID'10'"!ifCS 'PC \02 ~~ CS (,()1Dl!I'T~FIGURE 7.3FIGURE 7.2One observes that the last two points are proportionalto t"+1 and on~ is suspicious 01 a significantfr@qulI!ncy drift.'~ "J T1 "1.";..: ~1.4f.J:l5f~"t'-Hr,10." I,If one ca1clJlatll!s the drift k.nowing that 10o)r} is equal to theefrift times 12 a drift ofl~ 22 )( 10-11. per day ;s obtained. A 1inear leastsquares to the frequency was removed and sections FIGURE 7.4IV and V were applied again. The linear leastsquares fit sho\oted a drift of 1.23 x 10-14 perday, which is in excellent agreement with thepreviou$ calculated valu@obtained 1rom 0y(r).Typically, the linear least squalres will give amuch bll!tter estimate of the 1inear frequency drHtthan will the estimate from a (r) being propor""·1 ytional to rFigure 7.3 gives the plot 01 the time residualsafter removing thE! linear llast squares andfigure 7.4 is the corresponding 0y(t) YS. t diagram.From the 33 days 01 data, we have used thE!90% conti dence i nterva1 to bracket the stabi 1i tyest.imates and Dne sees a reasonable f1t cerrll!spendingto white noise frequency mDdulation at a't-,,~"9­'t,~Slevel of 4.4 x 10- 11 t-~. This seemed excl!!Ssive FIGURE 7.5')t1O-I2't~~18TN-3l

1n tenns of the typical performance of this pltrticl.Ilarc~silnl\' and 1n as much as we were doingsome other testing within the environment, such as'Working on power supplies and cnarg1ng and dischargingbatteries. we did some later tests.Figure 7.S i. a plot of 0y(t) after the .tandarA'E" tmOO. ---'"FIGURE 8.2One can plot a graph showing MIlSfrequency for a g11/en signal.is called the power spectrum.f1 gure 8.1 the power spectrwn wi 11power vs.This kind of plotFor the wavefontl ofhave a hi gnvalue at Voand will have lower values for thes1gnals produced by the glitches. Claser analysisreveals that there i5 a recognizable, someWhatCDnstant repetition rate associated with theglitches.In fact, \tI'e can deduce. that there is asignificant amount of power fn another slgna1whose period is the p!riod of the glitches asshown in figure 8.2. Letls call the frequency ofthe glitches "s. Since this is the case, we willobserve a noticeable amount of power in the sp@c"trum at "s with an amp 1itude which is related tothe Characteristics of the gl itChes,The powerspectrum shown in fi gure 8.3 has this feature.predomi nant "5 component has been depicted, butother harmonics also exist.A-1FIGURE B.lHere \Il'e have a sine wave which's perturnedfor short fnstances by noise. Some 100cely referto these types of noises as uglitches ll • The~aveformhas a nominal f~~uency ov~r one cyc:le"",hich we'll call !lOUOII (\'1'0 =~). At times, noisecaUSes the 1nstantaneous frequency to differQFIGURE S.319TN-32

Some noi se wi 11 cause thE!! instantaneous fre"quency to "jitter" aro ....nd u o • with probabl1ity ofbeing higher or lower than ""0' We thus usuallyfind a I/lJedestal " associated with "0 as shown infigurE!' 8.4..,--·0FIGURE a.4ihe orocess of break.ing 110wna signal intoall of its various components of frequency iscalled Fourier exoansion (see 'Sec. X). Inother wordS,the. addition of all the frequencycomponent.s, called Fourier frequency components,produr:es the original signal. ihe value of aFouri er frequency i.sthe di fference between thefrequency component and the fundamental fr@quency.The power spectrum can be normalized to unity suchthat the tota1 area under the curve equa 1sane.The power spectrum normalized in this way is thepower spectral density,The power spectrum. often call ed the RFspectrum, of Vet) is ",ery useful in many appl i­cations~Unfortunately, if one is given the RFspectrum, it ;5 impossible to determine whetherthe power at different Fourier frequencies is aresult of amplitude fluctuations lIe(t)1I or phasefluctlJations '1¢J(t).The RF spectrum can be separatedinto two independent spectra, one being the'Spectral densi..!l of ~ often called the AMpower spectra) density and the other being thespectral densi~ of ~.For the purposes here~the phase-f1uctuationcomponents are the ones of interest. The spectraldensity of phase- fluctuations is denoted by S~(f)wherE!! IIf" is Fourier frequency. for the fre"quently encountered case where the AM power spec~tral d@nsity is negligibly small and the t.otalmOdulation of the phase fluctuations is small(mean-square value is much less than one rad2),the RFspectrum has approximately the same shapeas the phase spectral density_However, a maindifference in the representation is that the RFspectrum includes the fundamental $ignal (carrier),and the phase spectral density does not.Another major di fference is that the RFspectrumis a power spectral density and is measured inunits of watts/hertz.The phase spectral densityinvolves no II powerl' measurement of the electricalsignal. The units are radians·/hertz. It iste'"Pting to think. of S~(f) as a IIpowe~' spectraldens i ty because in practice it is measured bypassing Vet) through a phase detector and measuringthe detector's output power spectrum.The measure~ment technique makes use of the relation that forsmall deviations (6~ « 1 radian),5 (f) = (-,Vrm~s_(f_) )' (8.3)$ V,Where Vrms(f) is the root-mean"square noise.voltageper ./HZ at a Four1 er frequency II fll. and V ; s thessensitivity (volts per radian) at the phase quadratureoutput of a phase detector ~hlch15 comparingthe two oscillators. In the next section, we willlook at a scheme for directly measuring 5$(f).One question 'Wemight ask, is. IIMow do fre"quency changes relate to phase fluctuations?lIAfter all it's the frequency stability of anoscillator that ;s a major consideration in manyapplications.The frequency is equal to a rate ofchange. in the phase of a 'Sine wave.11'11'5 tells usthat fluctuations in an oscillator's output frequencyare related to phase fluctuations since wemust change the rate of ".(t)" to accomplish ashift in "V(t)ll, the frequency at time t. A rateof change of total u4'J~e have thenr(t)U is denoted by "*T(t)".lm>(t) ~ ~T(t) (8.4)The dot denotes the mathematical operation ofdifferentiation on the function 4'J with respect torits independent variable t.;II From eq (8.4)• As an analogy, the same operation relates theposition of an object with its velocity.20TN-33

and eq (1.1) we get2m>(t) = ~T(t) = 2m>0Rearranging, we have2m>(t) - 2m>0 = ~(t)or"It) - "0 =~+ ~(t)(8.5Jspectral density of frequency fluctuations isdenoted by S (f) and is obtained by passing theysignal from an oscillator through an ideal FMdetector and performing spectral analysis on theresultant output voltage. S (f) has dimensions ofy(fractional frequency)llHz or Hzal. Differentiationof ~(t) corresponds to multiplication by -!"0in terms of spectral densities. With further calculation,one can derive thatThe quantity u(t) -va can be more convenientlydenoted as 6v(t). a change in frequency at time t.Equation (8.5) tells us that if.we differentiateth.e phase f1uctuatlons ~(t) and divide by 2n, wewill have calculated the frequencyOU (t) .fluctuationRather than specify; ng a frequency f1 uctuationin terms of shift in frequency, it ;suseful to denote O\1'(t) ....ith respect to the nominalfrequency ~O- The quantity ~ is called theVofractional freguency fluctuation"'· at time t and;s signified by the variable y(t), We have(8.6)The fractional frequency fluctuation yet) isa dimensionless quantity.Wtu!O ta1!(;ng about frequencystabil ity, its approoriateness becomesclearer if weconsider the following example.Suppose in two oscillators 6,,(t) is consistentlyequal to + 1 Hz and we have sampled this value formany times t. Are the two oscillators equal intheir ability to producefrequencies?their desired outputNot if one oscillator is .operatingat 10 Hz and the other at 10 MHz.In one case Ith@ average value of t~e fractional frequencyfluctuation is 1110, and in the second case is1/10,000,000 or 1 x 10- 7 . The 10 MHz oscillatoris then lIore precise.If frequencies are multipliedor divided IJsingfractional stability is not changed.ideal electronics, theIn ttle frequency domain. we can measure the.spectrum of frequency fluctuations yet). The~~ Some international recommendations replaceUfractional ll by " norma 11zed u •(8.7)We ~ill address ourselves primarily to S~(f). thatis, the spectral density of phase fluctuations.For no; se-measurement purposes, Slj)( f) can bemeasured with a straightforward, easily duplicatedequipment set-up.Whether one measures phase orfrequency spectral densities is of minor importancesince they bear a direct relationship. It isimportant, however, to make the distinction and touse eq (8.7) if ~ecessary.8ft 1 The Loose Phase-locked loopSection 1, 1.1, C described a method afme-asuring phase fluctuations between two phaselock.edoscillators. How we win detail the procedurefor measuring S~(f),Suppose: we have a noisy oscillator. We wishto measure the oscjllator's phase fluctuationsrelative to nominal phase. One can do this byphaselocldng another oscillator (called the referenceoscillator) to themixing the two oscillator signals 90~test oscillat.or andout of phase(phase quadrature). Th; s ; $ shown schematicallyin figure 8.9.The two oscillators are at thesame frequency in long term as guaranteed by theph.s.-loc~ loop (PLL). A low-pass filter (tofi1ter ttle R.F. sum component) is used after themixer since the difference (baseband) signal isthe one of interest.By holding the two signalsat a relative phase difference of 90~.short-termphase fluctuations between the test and referenceoscillators will appear as V"oltage fluctuationsout of ~he mi~er.21TN-34

FIGURE 8.9'With a PLL, ; f we can maKe the ser'llo timeconstant very lang, then the Pll bandWidth as afiller will be small.This may be done by loweringthe gain A of the loop amplifier. We want tovtranslate the phase modulation spectrum to basebandspectruJI 50 that it is easily measured on alow frequency spectrum analyler. With a PLLfilter, wemust keen in mind that the referenceoscillator sl10uld be as good or better than thetest oscil1ator.1h'iS is because the output ofthe PLL represents the noise from both oscillators.and if nat properly chasen, the reference can ha'llenoise maSKing the noise from the test osc'i 11 atar.Often 1the reference and test asci 11 aters are ofthe same type and have, therefore, apprOXimatelythe same noise. We can acquire a meaningfulmeaSurement by noting t~at the noise we measure isfrom two oscillators.Many times a good approximationis to assume that the noise power is twicethat which is associated with one oscillator.S$(f) is general notation depicting spectral d@nsityon a p@r hertz basis.A PLL filter outputnece$sar;ly yields noise from two oscillators.The output of thE! PLL filter at Fourierfr@quencies abov@ the loop bandwi dth is a vo 1tag@represent ing phase f1 uctuat ions oetwe@n referenceand test osc; 11 ator.1t ; s necessary to mak.e thetime"'constant of the loop long compar@d with th@inverse of th@ lowest Fourier fr@quency woe wiSh tom@asure. That is, te > 2n fclowest)· This meansthat if we want to measure S~(f) down to 1 Hz, theloop time-constant must be gr~~ter tha~ ~ Seconds.One can measur@ the time-constant byperturbing the loop (momentarily disconnecting thebattery is convE!n;@nt) and noting the time ittakes for tt1@contro1 \/01 tage to reach 70% of itsfinal valu@. Th@ signal from the mixer can thenb@ inserted into a spectrum analyzer. A preampmay be necessary before the 50ectrum analyzer.• See Appendix Note , 3*22The analyzer determines the mean square volts thatpass throug~the analyzerls bandwidth centeredaround a pre-chasen Fourier frequency f.desireable to normalize results to a 1 HzIt isbandwidth.Assuming white phase no; se ( ....hite PM),this can be done by dividing the mean squarevoltage by the analyzer bandWidth in Hz.One mayhave to approximate for ather noi!;e processes.(The phase noise sideband levels will usually beindiCAted in rms volts-per-root-Hertz on mostanalyzers. )B.2 Equipment for Frequency !Jomai n Stabi 1i tyMeasurements(1) low-noise mixerThis should be a high quality, douolebalancedtype.,but single-ended typesmay be used. The oscillators shouldhave ~ell-buffered outputs to b@able toisolate the coupling between the twoinput RF ports of the mixer. Results **that are too good may be obtained if th@two oscillators couple tightly viasignal injection through the inputports. We want the PLL to contrallocking. One should read t~e specificationsin order to prevent exceedingth@ maximum allowable input power to themixer.It is best to operate near th@maximum for b@st signal-to-nois@ out ofth@cases~IF port of the mixer and, in someit is possible to drive th@ mixerinto saturation without burning out thedevice.FIGURE 8.10(2) Law-noi,. DC amplifi.rThe amount of gain A needed in the toopvamplifie:r 'Will depend on the amplitudeof th! mixer output and the d@gree of•• See Appendix Note" 4TN-35

varactor control in th@ reference oscil­ be ~uff;cient to maintain phase-loc~ oflator. we may need only a small amount the reference. In general, low qual i tyof gain to ac:quire lock. On the other test osci 11 ators may hav@ lIaractQrhand, it may oe neces58ry to add as ~uchcontrol of as much as 1 x 10- 6 fractionalas 80 dB af gain. Good lo...-no;se DC frequency Change per volt. Same prov;­ampl ifiers are available from a number ~ion 5hould be available on the ref@r@nceof saurces~ and with cascading ttages of os~illatQr for tuning the mean frequencya.,."l;fication, each contr1buting noise~over a frequency range that will enableit will b. til. noise of tile first stag. phase-lOCK. Many factors enter into thewhich will add most significantly to thechoice of the refer@nce oscillator, andnohe being measured. If a suitable often it is convenient to simply use twolow-noise first"'"sUge amplifier is not*test oscillators phese"'locked t.ogether.readily available, a schematic of an In thi sway t one can assume that theamplifier with 40 dB 01 gain ;! ~hown in noise out of the PLL filter ;s no worsef1 gure B. 11 whi en wi 11 serve ni eely for than 3 dB greater than the noise fromthe first stage. Amplifiers ....ith very each oscil1ator. If it is uncertainlow equivalent input noise performancethat both asci 11 ators are contri but; ngare also available from many IMnufac:­ approximate ly equal no; se, then oneturers. The response of the amp 1; fi er should perform measurements on threeshould be flat from DC to the highest oscillators taking two at a till'll!. TheFourier frequency one wishes to measure. noisier-than-average oscillator willThe loop time-constant is inversely reveal ltself.related to the gain A vand the determinationof A. hi best made by experimen­ (4) Sp.otrum analyz,rtat;on knowing • tllat to < 2nf (low.st)" 1Til. signal analy,.r typ;0.11y should b.capa!)le of measuring the noise in nns•'.'Dvolts in a narrow bandYiidth from near1 Hz to the highest Fourier frequencyof interest. This may be 50 kHz forcarrier frequencies of 10 MHzFor voltage measuring analyzers~or lower.it istypical to use units of " vo lts per JHZu .The spectrum analyzer and any aS50c:iatedinput amplifier will exhibit high-frequencyroll off.The Fouri er frequencyat WhiCh the voltage has ~roppedby 3 dB1s tile m••sur....nt syst.m bandwidtll f ll,LPI .:USE M'IfIlIFIEior w~ =lxf . hThis can be measuredFrGURE 8" 11directly with a variable signal gener­(3) VOltaq.-oontroll.d r.fer.nce quart, ator.asci l1ator"."This oscillator should be a good on4 Section X de5cribes how analysis can bewith specifications available on its pt!rformed usi~ a discrete fourier transformfrl!ql.lency domain ~bhi lity. The r"'4!tfer"'­ analyser. Expanding digital teChnology has madeence shou14 be no worse tnan the test the use of fast~fourier transform analysis afforoscillator"'.The v«ractor control should dable and compact~• See Appendix Note i s­'"• t. HUTfUIU1~ lUlU".CWI!IITTUttil,",..uTrIlT23TN-36

Ratner than measure the spectral density Gfphase fluctuations between two oscillators, it ispossible to measure the phase fluctuations introducedby a device such as an acti ....e filter oramplifier. Only a slight modification of theexisting PLL ffltlr equipment set up is needl!!!d.The scheme is shown in fi~ure B.12.FIGURE 8.12Figure 8.12 is a differential phuE! .!22.i!!measurement set-up. The output of the referenceo!;cillator is split so that part of the signalpasses through the device IJnder test. We want thetwo signals going to the mixer to be 90° out of~nase,thus. phase f1 uctuations between the t\rlOinput ports cause voltage fluctuations at theoutput. The voltage fluctuations then can bemeasured at various Fourier frequencies on aspectrum analyzer.To estirnate the noise inherent in the tes.tset-up. one can in principle bypass the de.... iceunder test and compensate for any change in amplitudeand pnase at the nixer. The PLL filtertechnique must be con....erted to a differentialpnase noise tecnnique in order to measure inherenttest equipment noise. It is a good practice tomeasure the system noise before proceeding to measurementof device noise.A frequency domain measurement set-up isshown schematically in figure 8.13. The componentvalues for the low-pass filter out of the mixerare suitable for 05cillatnrs operating at around5 MHz.The active gain element (a ) of the loop is avDC ~lifi@r with flat frequency response. Onemay replace this element by an integrator toachieve high gain near DCand hence, maintainbetter lock of the reference oscillator in longterm. Otherwise long-term drift between thereference and test oscillators might reqUiremanua 1 re- adj ustJllent of the frequency of One orthe other oscillator. S8.3 Procedure and EKample *At the input to the spectrum ana 1yzer, thevoltage varies as the phase fluctuations in short­te".5 (f) = (~Vr.i!!mo-s_(f_ l \'~ Vs JV sis the phase senSitivity of the mixer in voltsper radian.Using the previously described eQuip­ment s.et-up, V can be measured by disconnectingsthe feedback loop to the varactor of the referenceoscillator. The peak voltage swing is equal to V sin units of volts/rad if the resultant beat noteis a sine wa ....e. This may not be the case forstate-of-tne-art 5~(f)measurements where one mustdrive the mixer very hard to achieve low mixernoise levels. Hence, the output will not be asine wave. and the volts/rad sensitivity must beestimated by the slew-rate (through zero ....olts) ofthe resultant square-wa....e out of the mixer/ampli­fier.Tne valLe for the measured S~(f)Is g;ven by:in dec1belsVrmsVoltage at fV fu11-scale ~detector ....oltagesIF1GURE 8.13EXAMPLE,Given a PLL with two oscillators suchthat, at the mixer output:V = 1 volt/rads• See Appendix Note # 624IN-37

V nn5(45 Hz) = 100 nV por root hortz Power 1aw noi se processes are characterized by501vo for 5~(45 Hz). the i r functi ana J dependence on Fouri er frequency.Equat;on 8.7 rolates S~(f) to Sy(f), the ,pectraldensity af frequency fluctuations. Translation ofS (45 HZ) = 100 nV (Hz) = 10­( -'I)' ( ')' rad1 HZ_l Sy(f) to time-domain data 0y(t) far the five model~ 1 Vireo 1In decibels,=: 10-14 rad 2HzS~(45 Hz) = 20 log 10~ ~v = 20 Jog 1~~;= 20 (-7) = -140 dB at 45 HzIn the @xample. nate that the mean frequencyof the oscillators in the PLL was nat essential tocomputing S4l(f). However, in th@ application ofS~(f)ttion. Along with an S41(f). one should alwaysattach \,10.naveIn the example above "0 = 5 MHZ, so weS~ (45 Hz) = 10-" r~~z, "0 = 5 MHz.From oq (8.7). 5 (f) can bo computod asyIX.Sy (45 Hz) = (5 45 )' 10- 1 'x 10~6Hz5 y (45 Hz) = 8.1 x 10-" Hz-'. "0 =5 MHz.POWER-LAW NOISE P!OCESSESPower-law noise processes are models ofprecision oscillator noise ttlat produce a parti~noise processes is co .... ered later in section :

such as in d.1gital servo loops where correctionsar~ needed for fatt changing errQrs.A portion of the conversio"... time @rror is afunction of the rate of change ~ of the processif tne sample-and-hold portion of the ADCrelieson the charging of a capacitor during an aperturetime. This is true because the charge cycle wi 11have a finite time-constant and because of aperturetime uncertainty.For example, if the time'"'con'"stant is 0.1 ns (given by say a 0.1 n sourceres; stance charg1 ng .. 0.001 I"dd capacitorL thena 0.1% nominal error will exfst for slope ~ =IV/~s due to charging. With good design, thiserror can be reduced.The samp 1i ng circuit (be'"fore cl'1arge) ;s usual1y the dominant source oferror and logic gate-delay jitter creates anaperture time uncertainty. The jitter typical1yis between 2-5 ns which means an app 1i ed signa1slewing at, !i.ay. 1 V/IJSof 2.-5 mV.producl!ls an uncertaintySince i;o* is directly proportionalto signal slewing rate. it can ce anticipated thathigh-levRl. nigh-frequency components of Yet) .il1have the greatest error in conversion.For typicalAOe IS, 1ess than 0.1% error ca.n be achi eved byholding ~ to less than 0.2 V/~s.The continuous process y(t) is partitionedinto 2" discrete ranges for n""bit conversion.Allanalog values within a given range are representedby the same digital code, usually assigned to thenominal micirange value.There is. therefore. aninnerent quantization uncertainty of t: ~1eastsignificantbit (LSB), in addition to ather conversionerrors.For example, a lO-bit AOe has a totalof 1024 discrete ranges with a lowest order b1tthen representing about 0.1% of full scah andquantizatio" uncertainty of t: O.OS%.We def1 na the dynamic range of a d1 gi to 1system as the ratio between the lIlaximucn allowablevalue of the process (prior to any overflow condttion)and the minimum d1scernable value. Thedynamic range when digitizing the data is. set bythe quantizing uncertainty. or resolut1on o and1s the ratio of 2" to 50s: LSI!L (If additive noisemakes coding ambiguous to the J,: LSB level, tnenthe dynamic range is the ratio at Zn to the noiseuncertainty, but this is ucually not the case.)For uample. the dynami c range of a 10-bit systami. 2 n = 1024 to ~. or 2048 to 1. E)('pressed indBls, this isZOlog 2048 =66.2 dBif referring to a vo]tage-to·coae converter"The converter linearity specifies the degreeto which the voltage-to~code transfer approximatesa straight l1ne. T~e nonlinearity is the deviationfrom a straight 1fne drawn between the end points(a11 zeros to: all ones code). It is usually notacceptable to nave nonlinearity greater than ~ LSBwhich means that the sum of the positive errors orthe sum of the negative errors of the individualbits must not exceed ~ LSB (or " ~ LSB). Thelinearity specification used in ttl1s contex.tincludes all effects such as tel1lperll.ture errorsunder expected operating temperature extremes andpowersupply sensitivity errors under expectedoperating supply variations.10.3 AliasingFigure 10.1 illustrates equispaced samplingof continuous process y(t).It is important tohave a sufficient number of samples/second to prop~erly describe information in the high frequencies.On the other Mand, sampling at too high a rate nayunnecessarily increase the processing labor.we reduce the rate. we see that samp 1e valuescould represent low or high frequencies in y(t).This property ;s called aliasing and constitutes asource Of error sillil;a,. to "imaging" which occursin analog frequency mixing schemes (1. e., in ttlemultiplication of two different signals).If the time between samples (k) is T seconds,then the sanrp 11 ng ratei:5 f 5Allq) h:s per second.Than usaful data in y(t) will ba from 0 to iT HIand r""guancies Mghar than h HI wi 11 be fa1dadinto the l~~r range from 0 to }r Hz and confusedwith data in this lower range.The cutoff frequencyis then given byAs1 (l0.4)nand is sometimes called the !lMyquist frequency. II27TN-40

We can use the convolution theorem to simplyi11ustrate the existence of atiases. Thh theoremstates that multiplication in the time domaincorresponds to convolution in the frequency domain,and the time domain end frequency domain representationsare Fourier transform pairs) TheFourier tran"form of yet) In figure lO.l(a} isdanotod by Y(f); thus:Y(fl convolvad with Af ;s danotad by Y(f)'a(f) and is shown in figura 10.2(c). Wo sao thatthe transform ....(f) is repeated with origins atf =y. Conversely, high frequency data with informationaround f = f will fold into thQ data aroundthe origin bil!tween -f and "'f ' In the computations sof pow~r spectra, we encounter error5 as shown infiguro (10.3).andTheY(f) : j-y(t).j2rrftdt (10.5)=y(t) : -l f Y(f)oj2rrftdf (10.6)Zn-=function Y(f) is depicted tn f1gure lO. Z(a).The Fourier transform of ~(t) ;s shown in figure10.2(b) and is gillen by .6.(f) wh@re applying thediscrete transform yields:(10.7)recall1ng that---'". '.,=Mt): E 6(t -nT). (10.8)n=fromeq (10.2).L ~..I...-.-+" f+.-.J\.--,----+-r'"'. '. +--1C4) {I.}FIGURE 10.2Is'-~FIGURE 10.3Al lased power spectra dtle to folding.Spactra, (b) Aliasod Spoctr._(a) TrueTwo pionil!ers in infonnation theory, HaroldNyquist and Claude Shannon, developed designcriteria for discrete-continuous processing sys~tems. Given a specified accuracy, 'We can conveytime-domain process Yet) through a finite bandwidthwhose up~er limit f is the highest ~ignificantspectral component of y(t). ForNdiscretecontinuousprocess Yk(t) Iideally the input signalspectrum should not extend beyond f ' ors(10.9)where f is givE'!:n by eq 00.4). Equatio" (10.9)sis refered to as the lIShannon limit. /IIn practice, there is never a case in whichthere is absolutely no signal or noise compon@ntabove f,.. Filtars are used befor!' the ADC inorder to suppress components above f which foldNinto tho lower bandwidth of intarost. rhis 00­called anti-aliasing fi1ter usually must be quiteSOphisticated in order to have low ripple in thepassband, constan't phase delay in the passband?and steep rolloff characteristics. In examiningthe rollotf requirements of the anti-aliasingfilter, we Can apply a fundamental filter propertythat the output spectrum is equal to the input28TN-41

spectrum multiplied by the sqllare of the frequencyresponse function; that is,5(f) [H(f)]' • 5(f) (10.10)outThe filter response must be flat to f and at~Nt@n~ate aliased nois~ compone~ts at f % f =2nf %sf. In digitizing the data. the observedspectra will be the sum of the baseband sp@ctrum(to f N) and all spectra '!oIhich are folded into thebaseband spectrum5(f) • 5 0(f) + 5. 1 (2f, - f) + 5., (2f, + f)Observed.5.> (4f, - f).". + S; (2; U, • t f))(f, + f))where M ;s an appropriate finite limit.(10.11)For a ghen rejettion at an upper freQuency.clearly the cutoff freqllency f for the anti~c~liasing filter Should be as low as possible torelax the rolloff requirements.Recall that annth order lClw"'pass filter has frequency responsefunctionand output spectrumS fS(f) = --Tff~c"f)"2n:-out 1 +(10.12)(10.13)and ~f~er sampling, we haye (applying eQ (10.11))5(t)observed(10.14)If fcis cnosen ta be higher than f then theN ,first tenn (baseband spectrum) is negl igiblyaffected by the filter, which is our hope.It;sthe second term (the sum of the folded in spectra)which causes an error.As an example of the rol10ff requirement.consider the measurement of noise process net) atf ~ 400 Hz in a 1 Hz. bandwidth on a digital spectrumanalyzer. Suppose net) 15 white; that is,k constantO =Suppose further that '!ole(10.15 )w; sh to on ly measure thenoi 51!" from 10 Hz to 1 kHl.; thus f N = 1 kHz.. Letus assume a sampling frequency of f s = 2f Nor2 k.Hz. If we impose a 1 dB error limit inSobserved and have 60 dB of dynaralc range, then wecan tolerate an error limit of 10~6 du~ to aliasingeffects in this measurement. and the second term; neQ (10.14) must be reduced to this lelJel.choose f =1.5kHz and obtainc5( f)observedWe can2 (f + ~f) 2n1· --...-;';---'-'-­f c(10.15)The term in the series which contributes mas':. isat i :: -1. the nearest fold-in.The denominatormust be 10 6 or more to real;!e the allowable error11mit and at n > a this condition is met.Thenext most contributing term is i = +: at whlch theerror is < 10- 1 for n = 8, a negfigible contribution.The error increases as f increases for afixed n because the Dearest fold-in (i = ~1)coming dO>tnin frequency (note fig. 10.2(c)) andpower there is filtered less by the anti-aliasingfilter. Let us look. at the worst case (f ::: 1kHz)to determine a design criteria for this example.At f =1 kHz, we must haye n ~ 10.10~poleThus the requirement in this example is for alo",-pass filter (60 dB/octave rol1off).10.4 Some History of Spectrum Analysis Leading tothe Fast Fourier TransformNewton in his Principia (1687) documented thefirst mathematical treatment of wave motion al­is29TN-42

though the concept of harmonics in nature waspointed out by Pythagoras., Kepler, and Galileo.However, it wa~ the work of Josepn Fourier in 1807"'hicn showed that almost any function of a realvariable could be represented as t!'le sum af $inesand cosines. The theory was rigorously treated ina document in 1822.In using Fourier1s hchnique, thlli!! periodicnature of a process or signal ;s analyzed. Fourieranalysis assumes we can apply fixed amp11tudes?frequencies, and phases to the signal.In the early 1900 l s two r.latively independentdevelopments took place: (1) radio electronicsand electric power hardware were fast growingtechnologies; and (2) statistical analysis ofevents or processes which were not periodic becamei ncreas ; ngl y understood. The radi 0 engineerexolored signal and noise properties of a voltageor current into a load by means of the spectrumanalyzer and measurement of the power spectrum.On the other hand, statisticians explored deter"ministic and stochastic properties of a process bym~ansof the variance and sel f-correlation propertiesof the process at different times. Wiener(1930) showed that the variance spectrum (Le••the breakdown of the variance w1th Fourier frequency)was the Fourier transform of the autocorrelationfunction of the process. He also theor"i zed that the vari ance spectrum was the same asthe power spectrum normaliz@d to unit area. Tukey(1949) advocated the use of the variance spectrumin the stat1stical treatment of all processesbE-cause (1) it is mare easily interpreted thancorrelation-type functionsj and (2) it fortuitouslyis readily mea:sureable by the radio engine~r.The 1950 l s saw rigorous application of statisticsto communication theory. Parallel to this~asthe rapid advancement of d1gital computerhardware. Blackman and TUk~y (1959) and W@lch(1961) elaborated on other ~seful methoas ofderiving an estimate for the variance spectrum bytaking the ensemble t1me-average sampled, discrete1i ne spectra. The approach a.ssumes the randomprocess is ergodic. Some digital approachesestimate th@ variance spectrum using Wienerlstheorem if corr@!ation.. type functions are usefulin the analysis, but in general the time-averaged,sample sp@ctrum is the approach taken since its1mplementation is dir@ct and straigtltfo~ard.Most always, @rgodicity can be assumed. 8,9Th@variance of process yet) is related tothe total power spectrum bySincecr2[y(t)] = f 5 (f) df. (10.17).~ YTcr2[y(t)] _ lim 1 f y2(t) dt (10.18)- T- rr -Twe see that if Yet) is a voltage or c~rr~nt into 3I-ohm load, then the mean power of Yet) is theintegral of Sy(f) with respect to frequency overthe entire range of frequencies (-0l.00).Sy(f) is,therefore. the power spectrum of process Yet).The power spectrum curve shows how the variance ;sdistributed with frequency and should be expressedin units of watts per unit of frequency. or voltssquared per unit of frequency when the load is notconsidered.Direct estimation of power spectra has beencarried out for many years through the use ofanalog instruments. These have variously beenreferred to as sweep spectrum analyzers, harmonicanalyzers, filter banks, and wave ana1yzers.These devices make use of the fact that the spectrum of the output of a 1; near system (analogfilter) ;s the spectrum of the input multiplied ~ythe square of the system1s frequency responsefunction (real part of the transfer characteristic).Note eq (10.10). If yet) has spectrumS (f) feeding a filter with frequency responseyfunction H(f), then its output ;sS(f) = [H(f)]2 S (f) (10.19)filteredyIf H(f) is rectangular in shape with width ~f,then we ~an measure the contribution to the totalpo~er spectrum due to Sy(f ± ~f).The development of the fast Fourier transform(FFT) in 1965 made digital methods of spectrum30TN·43

estimation increasingly attracti~e. Today thechoice between digital or analog methods dependsmore on the objectives of the analysis rather thanon technical 1imitation!ii. However, many aspectsof digital spectrum analysis are not well known bythe casual user io the laboratory while the analoganalysi5 methods and their limitations are understoodto a greater l!xtent.Digital spectrum analysis is realized usingthe di5cre-te Fourier transform (OFT). a IIOdifiedversion of the cont1nuous transform depicted ineq. (10.5) and (10.6). By sallplinq the inputwaveform yet) at discrete intervals of t1~e t •1~t representing the sampled waveform by eq (10.2)and integrating eo (10.5) yields~Y(f) = l:t=~.y(lt)e'J20flT (10.20)Equation (10.20) is a Fourier series expansion.Becayse t(t) ;s specified as being bandlimited,the Fourier transform as calculated by eq (10.20)is as accurate as eq (lO~ 5); however, it cannotextend beyond the Nyquist fr!quency, eq (10.4).In practice ..,e cannot compute the Fouriertransform to an infinite extent, and we are restrictedto some observation time T consisting ofnAt intervals.This produces a spectrum which isnat eont1 nuolJs in f but rather is computed ",ithresolution Af ",hereM=.1..= 1 (10.21)nAt 'fWith this change. we g.t the discrlte finitetransforllN-1 .Y{mAf) = l: y (t)e·JZroo.l.fnt (10.22)","0 1The OFTcomputes a sampled Fourier series,end eq (10.22) assumes that the function y(t)repeats ibelt with period T. Y(mAf) is calledthe "1 fne spectrum." A cmnparison of the OFT withthe continuous Fourier transform 15 shown later inpart 10.7.The fast Fourier tr.nsform (FFT) is ,n algorithm""hi ch efficientiy computes the 1i ne spectrumby reducing the number of adds and 1I1l1itipl iesinvolved in eq (10.22). If we choose T/At toequal a rational power of 2, ttlen a symmetricmAtrix can be derived through which YR,(t) passesand quickly yieldS YClMf). An M-point tronsformationby the direct method requires a processingtime proportional to N2 whereas the FFT requires atime proportional to N 10g2 N.The approximateratio of FFT to direct comput1ng time is given bywhere H = 2 Y ,Mlog. N log, HHZ =--N- = if (10.23)For e.ample, if H = 2'0, the FFTrequires less than 1/100 of the norlllal processingtime.Wemust calculate both the magnitude andphase of a frequency in the line spectrum, L e. ,the real and imaginary part at the given frequency,N poi nts in the time damaina11 ow ~/2 comp Texquantities in the frequency domain.The power spectrWl of y(t) is computed bysquaring the real and imaginary components, addingthe two together and dividing by the total time T.We haveS (mAf) = R[Y(mAf)]2 .. I[Y(mAfll' (10,24)yfThis qua.ntity is the sampled power spectrumand again assumes periodicity in proces5 yet) withtot.l period T.1010. 5 LeakagoSampledinvolvesdigitaltransformingContlnucus precess )I(t)through a data windowdescribed byspectrum analysis alwaysa finite bloc~of data.15 "looked at" for T timewhich can functiona11y bey' (t) = w(t)'y{t) (10,25)The time­where wet) is thlf time domain window.discrete counterpart to eq (10.25) is31TN-44

and wz(t) is now the si!npled version of w(t)Oeri.eo ,imil.rly to eq (10.2). Equ.tiDn (10.26)is equivalent to convolution in the frequencydam.!!;;n, orV'(mof) =W(mAf)'V(mAf) (10.27)Vl(mAf) is called thl'!' I'modified" line spectrum dueto convolution of the or;g;nal linl'!' spectrum withthe F'ourie.r transform of the time-domain w;ndowfuncti on.andSuppose the window function f$ rectangular.The transform process (eq 10.22) treats thesample signal as if it wer~ periodically extended.Oiscontinuit;l'!'s usually DCClJr at the ends of thewindow function in thl'!' extended version of thesampled waveform as in figure 10.5(1:). Samplespectra thus represent a p!riodical1y extendedsalllP1ed waveform, complete with £liscontinuftes atits ends, rather than the original wavefo~.Cal(10.28)This windOW is shown in f~oJure 10.4(a). TheFOlJrier transfonn of this. Iojindow is(b)---WJ\j--­(c) r- T -1W(mof) =T sinn mAf NT (10.29)nmofHT.nd is shown in figure 10.4(b).If y(t) is a sine~ave. we convol~e the spectrum of the sinusoid, adl'!'lta fu"ction, with W(mAf).f' ~T/2 Tj2I(0)••F!GURE 10.5Spurious components appear near the sinusoidspectrum and thls is referred to as 1I1@ak2lge.'1Leakage rl'!'sutts from discontinuites in the periOdicallyextended sample w~~eform.Leak-age cannot be eliminatl'!'d entirely, butone can choose an appropriate window function wet)in order to minimize its effect. This is usuallydone at the expense of resolutiQn in the fr9quencydomain. An Q"timum window for most cases 1s theHanning window gi~en by:11 2lttaIi(t) =[2 - 2 cos (,)) (10.30)F1GURE 10.4for 0 .5. t ~ T and "all designates tile number oftimes the window is implE!f1'1ented.Figure lO.6(a)shows the window function and lO.6(b) shows theHanning line shape in the frequenc.y domain for'iarious numbers of "Hanlls. nNote that this window32TN·45

FIGURE 10.6eliminates discontinuities due to the ends ofsample length T.Each time the Hanni ng wi ndow is app 1i ed, theside lobes in the transform are attenuated by 32aB/octave, and the main lobe is widened by 2tl.f.The amp 1itude uncertainty of an arbitrary sinewave input is reduced as we increase the number ofHanns; however. we trade off reso1ution in frequency.The effective noise bandwidth indicates thedeparture of the fi 1ter response away from a truerectangularly shaped filtered response (frequencydomain). Table 10-1 lists equivalent noise bandwidth corrections for up to three app 1i cati on5 ofthe Hanning window. 11Number of HannsEquivalent NoiseBandwidth1 1.5 af2 1.92 Af3 2.31 AfTABLE 10. I10.6 Picket-Fence EffectThe effect of leakage discussed in the previoussection gives rise to a sidelobe type responsethat can be tailored according to thetime-window function through which the analyzedsignal passes as a block to be transformed to thefrequency domain. Using the Hanning window diminishesthe amplitudes of the sidelobes. however, itincreases the effective bandwidth of the passbandaround the center frequency. This is because theeffective time-domain window length is shorterthan a perfect rectangular window. Directlyrelated to the leakage (or sidelobe) effect is onecalled the "piCket-fence" effect. This is becausethe sidelobes themselves resemble a frequencyresponse which has geometry much 1ike a picketfence.The existence of both sidelobe leakage andthe resultant picket-fence effect are an artifactof the way in which the FFT analysis is performed.Frequency-domain analysis using analog filtersinvolves a continuous signal in and a continuoussignal out. On the other hand, FFT analysisinvolves a continuous signal in, but the transformto the frequency domain is performed on blocks ofdata. In order to get di screte frequency i nformationfrom a block, the assumption is made that theblock represents one peri od of a periodi c signa1.The picket-fence effect is a direct consequence ofthis assumption. For example, consider a sinewavesi gna1 whi ch is transformed from a t ime-varyi ngva 1tage to a frequency-domai n representationthrough an FFT. The block of data to be transformedwill be length, T, in time. Let's say that theblock, T. represents only ~ cycles of the inputsinewave as in figure 10.5. Artificial sidebandswill be created in the transform to the frequencydomain, whose frequency spacing equals f, or thereciprocal of the block length. This representsa worst-case condition for sidelobe generationand creates a large number of spuri ous discretefrequency components as shown in figure 10.7(b).If, on the other hand, one changes the blocktime, T. so that the representation is an integralnumber of cycles of the input sinewave, then thetransform will not contain sidelobe leakage compo­33TN-46

10.7 Time Domain-Frequency Domain TransformsA. Integral transformFigure 10.8 shows the well-Known integraltransform, which transforms a continuous timedomainsignal extending over all time into a.. rlf ~:ca 1 .". :" f" :i :J~ ;4; ,.. 1\ 1'. f'. r~:' ;: H:: ::, I , , , " .: ,t " .1 .. ,f I' U• :: i: H \: :: :1 :: :: = :' 14il ._ " .. ,. I ., ,.. '"FIGURE 10.7nents and the artificial sideband frequency componentsdisappear. In practice, when looking atcomplex input signals, the blOCk time, T, is notsynchronous wi til any component of the transformedpart of the signal. As a resul t, di screte frequencycomponents in the frequency domain haveassociated with them sidebands which come and godependi ng on the phase of the time wi ndow, T,re 1ati ve to the sine components of the i ncomi ngsignal. The effect is much like looking through apicket fence at the sidebands. 12An analogy to the sidelooe leakage and picketfenceeffect is to record the incoming time-varyingsignal on a tape loop, which has a length of time,T. The loop of tape then repeats itself with aperiod of T. This repeating signal is then coupledto a scanni ng or fil ter-type spectrum ana lyzer.The phase d,i scanti nui ty between the end of onepassage of the loop and the beginning of the loopon itself again represents a phase-modulationcomponent, which gives rise to artificial sidebandsin the spectrum analysis. A word of caution ­this is not what actually happens in a FFT analyzer(i.e., there is no recirculating memory). However,the Fourier transform treats the incoming block asif this were happening.FIGURE 10.8continuous frequency-domain signal extending overall frequency. This is the ideal transform. Inpractice, however, one deals with finite times andbandwidths. The integral transform then, at best,is an estimate of the transform and is so for onlyshort, well-behaved signals. That is, the signalgoes to zero at infinite time and at infinitelyhigh frequency.B. Fouri er seri esThe Foud er-seri es transform assumes peri 0­dicity in the time-domain signal for all time.Only one period of the signal (for time T) isrequired for this kind of transform. The Fourierseries treats the incoming signal as periodic withperiod, T, and continuous. The transformed spec~trum is then discrete with infinite harmoniccomponents with frequency spacing Of~. This isshown in figure 10.9.~~Y{\). fl.:.;.i)o.~ JI:~~ .r ""~J)c i~\~.ll~.-._.~.. , ,'.FIGURE 10.9-"/I ,, .34TN--47

XI.TRANSLATION FROM FREQUENCY DOMAIN STABILITYMEASUREMENT TO TIME DOMAIN STABILITY MEASURE­MENT ANO VICE-VERSA.ll.l ProcedureKnowing how t.o measure SolI(f) or Sy{f) for apair of asci 11 at01's, 1et us see how to trans latethe ~ower-law noise process to a plot of a 2 (t).yFirst, convert the spectrum data to Sy(f) , thespectral density Df frequency fluctuations (seesections III and VIII).There are two quantities..midl t:DlllPlrtely Sl)ecify S y(f) for a particularpower-law noise precess: (1) the s)ope on alog-log plot for a given range of f and (2) theFIGURE 10.10 amplitUde. Tile sl~ we shall denote by "or";Figure 10.10 shows the transfoTlll from a Si!lll"" therefore ~ is the straight line (on log-logpled t i me- domainsi gna1 to the f1'til1M!ocy domain. scale) which relat.es S (f) to f. The amplitudeyNote that in the frequency domain, t1le result is will be denoted "h "; it is simply the coefficientarepetitive in frequency. This effect, CtmIIIonly of f fDr a range at f. When we examine a plot ofcalled aliasing, is discussed ..arlil~T in sect.iDn spectral density of frequency fluctuations, we are10.3. Figure 10.9 and figure 10.10 show the sym­ looking at .a representation of the addition of allmetry between the time- and frequency-dmIIain trans­ t.ile power-law pT'l)a5ses (see sec. IX). We haveforms of discrete lines.CItC. Discrete fourier transform S (f) =YI:(U.l)Fi na11 y , we have the saJlll'l ed, period; c a=­(assumed) time-domain signal which ;s t.ransformeddomain representation.10.nThis;s shown in figureIn section IX, five power-law noise processesto a discrete and repetitive (aliased) frequencyweTeoutlined with respect to SolI(f).ar!! the cDllllllonoscillators.Thes!! fiv!!ones encountered wi th preci s i onEquation (8.1) relates these noiseprocesses to Sy(f). One obtainsFIGURE 10. Uwith respect toSif)1. Random walk FM (f-2. ) II =-2(f_l )2. Flicker fM a =-I3. White fH (1) a = 04. Flicker ~ (f) a = 1S. White~ (fZ) a = 2slop!! onlog-logpaperTable ll.l is a list of coefficients fortranslation from a 2(t) to S (f) and from S~(f) toy y '"3STN-48

o/(t). In the table, the left column is the EXAMPLE:designator for the pOloler-law process. Using the In the phase spectral density plot of figuremiddle column, we can solve for the value of Sy(f) 11.1, there are two power-lalol noise processes forby computing the coefficient "a" and using the asci 11 ators bei ng compared at 1 MHz. For regi onmeasured time domain data a 2(t). yThe rightmost 1, w@ see that when f increases by one decadecolumn yields a solution for ° 2(t) given frequency (that is, from 10 Hz to 100 Hz), S~ (f) goes downydomain data S~(f) and a calculation of the appro- by thN!e decades (that is, from 10-11 to 10- 14 ).priate "b" coefficient. Thus, S~(f) goes as I/f 3 '" f- 3 • For region 1, we5 (f) '" H f' S (f) '" a aZ(t) aZ(t) '" b S...(f)y a '" a y a = Y y b '" 'I'2(white phase)1 (271:)% t Z f(flicker noise)a(white frequency)2 t-1 1 2 In (2) f3(flicker frequency) 2 In(2) . f v2o-2(random walk frequency)(27t)2 t f46 vZaTABLE 11.1Conversion table from time domain to frequency domain and from frequencydomain to time domain for common kinds of interger power law spectral densities;f b(= whlln) is the measurement system bandwidth. MeasuN!ment reponse should bewrthin 3 dB from D.C. to f h(3 dB down high-frequency cutoff is at f h).identify this noise process as f1 icker FM. Thev 2 rightmost column of table 11.1 relates a~(t) toS~(f) '" ~ Sy(f)S~(f). The row designating flicker frequencynoise yields:a2(t) '" 2£n(21 . f3 S... (f)Y v 0 'l'S+l~-10I()One can piclc (aroitrarily) a convenient Fourierfrequency f and determine the corresponding values10" of S.(f) given by the plot of figure 11.1. Say,.,'"tj'"f '" la, thus 5.(10) '" 10- 11 • Solving for a~(t),given Vo '" 1 MHz, we obtain:OZ(t) '" 1 39 X 10- 20il' y •dtherefore, a (t) '" 1.18 yx 10- 10 . For region 2, weFIGURE 11.1 have white PM. The relationship between a~(t) and36TN-49

Again, we choose a Fourier frequency. say f = 100,land see that S~ (100) = 10- 14 • Assumi ng f h = 10 4Hz, we thus obtain:a~(L) = 7.59 x 10- 2t . ~2therefore,The resultant time domain characterization isshown in figure 11.2.16 'IO~ei'Of'10""rjI'10-0................r..........rDl .e..l 0·........... .:t:lr.. - o-atrd~ rfJ. 0'" q 10'....~"l'I1II:,'tl~FIGURE 11.2The translation of S~(f) of figure 11.1 yields thisCJ (t) yplot.Dr. ....._.J-_--I__..L-_......._-J'--_..l-_--'10'" J:)-5 10"" r:j~ Ji~ 10i (Ht>FIGURE 11.4Figures 11.3 and 11.4 show plots of timedomainstability and a translation to frequencydomain. Since table 11.1 has the coefficientswhich connect both the frequency and time domains,it may be used for translation to and from eitherdomain.XII. CAUSES OF NOISE PROPERTIES IN A SIGNAL SOURCE12.1 Power-law Noise ProcessesSection IX pointed out the five commonly usedpower-law models of noise. With respect to S~(f).one can estimate a staight line slope (on a log-logscale) which corresponds to a particular noisetype. This;s shown in figure 12.1 (also fig.9.1).·]0...... ...... -'OS""..... ~ -.c:o:"""'".........., ~..~"" ........... ....'a.Gues. '" ..........It/!> '----...... -'-__......J. ..l-__......l, r:r os~."...,t1S)·\]0.1SO~-------~--....:::::.;:::~~·170 "-:'---i.:---~i.:--_-.l~-_-..I100 10 1 10 2 10 J 104M1ruulIU ruouuCT (IIfLlCIU P.70FIGURE 11.3FIGURE 12.137TN-50

Weean make the following general remarks aboutpower-law noise processes:1. Random walk FM (1/f4) noise is difficultto measure since it is usua 11 y veryclose to the carrier. Random walk FMusually relates to the OSCILLATOR'SPHYSICAL ENVI RONMENT. If random lola 1k FMis a predominant feature of the spectraldensity plot then MECHANICAL SHOCK,VIBRATION, TEMPERATURE, or other environmentaleffects lIlay be causing "random"shifts in the carrier frequency.2. Flicker FM (1/f 3 ) is a noise whosephysical cause is usually not fullyunderstood but may typically be relatedto the PHYSICAL RESONANCE MECHANISM OFAN ACTIVE OSCI LLATOR or the DESIGN ORCHOICE OF PARTS USED FOR THE ELECTRONICS,or ENVIRONMENTAL PROPERTIES. Flicker FMis common in high-quality oscillators,but may be masked by white FM (1/f 2 ) orfliCker PM (1/f) in lower-quality oscillators.3. White FM (1/f2) noise is a common typefound in PASSIVE-RESONATOR FREQUENCYSTANDARDS. These contain a slave oscillator,often quartz, which is locked toa resonance feature of another devi cewhich behaves much like a high-Q filter.Cesium and rubidium standards have whiteFM noise characteristics.4. Fl iclter PM (1/t) noi se may relate to aphysical resonance mechanism in anoscillator, but it usually is added byNOISY ELECTRONICS. Thi s type of noi seis common, even in the highest qualityoscillators, because in order to bringthe signal amplitUde up to a usablelevel, amplifiers are used after thesignal source. Flicker PM noise may beintroduced in these stages. It may alsobe introduced in a frequency multiplier.• See Appendix Note /I 7*38Flicker PM can be reduced with gDodlow-noise amplifier design (e.g., usingrf negative feedback) and hand-selectingtransistors and other electronic components.5. Whi te PM (f0) noi se is broadband phasenoi se and has 1itt1e tD do ....ith theresonance mechanism. It is probablyproduced by similar phenomena as flickerPM (1/f) noise. STAGES OF AMPLIFICATIONare usually responsible for ....hite PMnoise. This noise can be kept at a very10.... value ....ith good amp 1ifier design,hand-selected components, the additionof narrowband fi lteri ng at the output.or increasing, if feasible, the pDwer ofthe primary frequency sDurce. 1312.2 Other types of nDiseA common1y encDuntered type of no i se from asignal source or measurement apparatus is thepresence of 60 Hz A.C. line noise. Sho....n infigure 12.2 is a CDnstant ....hite PM noise source....i th 60 Hz, 120 Hz and 180 Hz components added.This kind of nDise is usually caused by AC powergetti ng into the measurement system or the. sourceunder test. In the plot of S(j)(f) , one observesdiscrete line spectra. Although S$(f) is a measureof spectral density, one can interpret the linespectra ....ith no loss of general i ty. although oneusually does not refer to spectral densities whenCharacterizing discrete lines. Figure 12.3 is thetime domain representation of the same white phasemodulation level ....ith 60 Hz noise. Note that theamplitUde of ay(t) varies up and down depending onsampling time. This is because in the time domainthe sensitivity to a periodic ....ave varies directlyas the sampling interval. This effect (which isan alias effect) is a very powerful tool forfiltering out a periOdic wave imposed on a signalsource. By sampling in the time domain at integerperiods, one is virtually insensitive to theperiodic (discrete line) term.IN-51

ttl·010·'110 .t....,.. ,;,'"',,,,,, t' ltl.~.Je 1O~-f'.IO,0'".1504S10-w ,i1'~ ~ fila. 10,,& ~ U»w"~~,tFIGURE 12.2FM behavior mask.s the ",hite PM (with the superimposedvibration characteristic) for long averagingtimes.J8-iID 16'-0'·10'0-t1O.ott·12-120 S-(t) ~I!l~ '0-fifO 10"45.~ 10·I(,Q fa"")fa:)Hr, ~ 7141. ~ ~ tmt~~~.~FIGURE 12.4, ,"%,., 13K ~_ m- COOllll: t:nX>ESAI"F'I.£ i1I"'C,'r(,)FIGURE 12.3For example, diurnal variations in data due to dayto day temperature, pressure, and other envi ronmentaleffects can be eliminated by sampling tnedata once per day.data with only one periodic term.Thi s approach is usefu1 forFigure 12.4 shows the kind of plot one mightsee of S,(f) with vibration and acoustic sensitivityin the signal source with the deviceunder vibration.Figure 12.5 shows the translationto the time domain of this effect. Alsonoted in figure 12.4 is a (typical) flicker FMbehavior in the low frequency region. In thetranslation to time domain (fig. 12.5), the flickerFIGURE 12.5Figure 12.6 shows eXallples of plots of twopower law processes (S,«(» with a change in theflicker FM level. (Example 1 is identical to theexample given in sec. XI.) Figure 12.1 indicatesthe effect of a lower flicker FMtranslated to the time domain.level asHote again theexistence of both power law noise processes.However for a given averaging time (or Fourierfrequency) one noise process may dominate over theother.39

-flO• r.1OSP1 t(f2-,~ 1Ci~-/'10 f6"I~t~1---__-00-w~II ,FIGURE 12.6a FIGURE 12.7a~'2,~l\~l'!'1.~?iJ".10'\oI1-flO l(/l-f2O ~ J1-00 tP.f'tO rd'-eo d>-1100 .",.~-. 1-. 10-. 1(lOo,. _ ,~.~ 1I'C,t'l~}FIGURE 12.6b FIGURE 12.7b~-10 td-100 ,,fJ-110 ttJf-m ~(j f(/J-00 t6~-f1IO ,eI'.~ t1':>-w Id'... .. .. ,FIGURE 12.6c FIGURE 12.7c40TN-53

.""-1= 10-flO ,(;042-1.:20 S () 10IS f -~-,~ 10-/"10 rbl'1-150 rbI'S....-1(,0 10A- 'lO-100.11)10to'..,fO~o:a.ss t>J;:VIC£ '10S£ (_1ST""-.tt!>'~ CA'P.6C.1OlHl~'~'f NOSE ~ UP__.:l'SIS·to10~,6'~ (t) ~ .~~ Iif'161il fS10'"FIGURE 12.10Y1:lOH:. Xol-lz 1Hz. ~ IDJ< jlXXl"j,:~,~~,+FIGURE 12.1141TN-54

frequency extent of the analysis band and improvingthe anti-aliasing filter's stopband rejection.Section X has an example of the filter requirementsfor a particular case.XIII.CONCLUSIONThis writing highlights major aspects oftime-domain and frequency-domain oscillator signalmeasurements. The contents are patterned after1ectures presented by the authors. The authorshave tried to be general in the treatment oftopics, and bibliography is attached for readerswho would like details about specific items.References1. J. A. Barnes, Andrew R. Chi, Leonard S.Cutler, Daniel J. Healey, David B. Leeson,Thomas E. McGuniga1, James A. Mullen, Jr.,Warren L. Smith, Richard L. Sydnor, Robert F.C. Vessot, and Gernot M. R. Winkler, "Characterizationof Frequency Stability," NBSTech. Note 394 (1970); Proc. IEEE Trans. onInstrum. and Meas. IM-2o, 105 (1971).2. D. W. Allan, "The Measurement of Frequencyand Frequency Stability of Precision Oscillators,"NBS Tech. Note 669 (1975); Proc. 6thPTTI Planning Meeting, 109 (1975).3. D. W. Allan, "Statistics of Atomic FrequencyStandards," Proc. IEEE 54,221 (1966).4. Sections V and VI are from notes to be publishedby J. A. Barnes.5. S. R. Stein, "An NBS Phase Noise MeasurementSystem Built for Frequency Domain MeasurementsAssociated with the Global Positioning System,NBSIR 76-B46 (1976). See also F. L. Wallsand S. R. Stein, "Servo Technique in Oscillatorsand Measurement Systems," NBS Tech. Note692 (1976).6.7. E. O. Brigham, The Fast Fourier Transform,Prentice-Hall, Inc., New Jersey (1974).B. Private communication with Norris Nahman,June 1981.9. G. M. Jenkins and D. G. Watts, SpectralAnalysis and Its Application, Holden-Day, SanFrancisco, 1968.10. P. F. Panter, Modulation, Noise, and SpectralAnalysis, McGraw-Hill, New York, 1965, pp.137-141.11. D. Babitch and J. Oliverio, "Phase Noise ofVarious Oscillators at Very Low Frequencies,Proc. 28th Annual Symp. on Frequency Control,May 1974. U. S. Army Electronics Command,Ft. Monmouth, NJ.12. N. Thrane, "The Discrete Fourier Transformand FFT Analyzers," Brue1 and Kjaer TechnicalReview, No. 1 (1979).13. Much of the material presented here is containedin NBS Tech. Note 679, "FrequencyDomain Stability Measurements: A TutorialIntroduction," D. A. Howe (1976).General ReferencesBibliographY1. November or December of even-numbered years,IEEE Transactions on Instrumentation andMeasurement (Conference on Precision ElectromagneticMeasurements, held every two years).2. Proceedings of the IEEE. Special issue onfrequency stability, vol. 54, February 1966.3. Proceedings of the IEEE. Special issue ontime and frequency, vol. 60, no. 5, May 1972.4. Proceedings of the IEEt-NASA Symposium on theDefinition and Measurement of Short-TermFrequency Stability at Goddard Space FlightCenter, Greenbelt, MD, November 23-24, 1964.Prepared by Goddard Space Flight Center(Scientific and TeChnical InformationDi vi s ion, Nat i ona1 Aeronautics and SpaceAdministration, Washington, DC, 1965).Copies available for $1.75 from the US GovernmentPrinting Office, Washington, DC 20402.5. Proceedings of the Annual Symposium onFrequency Control. Prepared by the US ArmyElectronics Command, Fort Monmouth, NJ.Copies from Electronics Industry Association,2001 Eye Street, N. W., Washington, DC20006, starting with 32nd Annual or call (201)544-4878. Price per copy, $20.00.6. J. A. Barnes, A. R. Chi, L. S. Cutler, eta1., "Characterization of Frequency Stabi 1­ity," IEEE Trans. on Intsrum. and Meas.1M-2o, no. 2, pp. 105-120 (May 1971). Preparedby the Subcommi ttee on Frequency Stabilityof the Institute of Electrical andElectronics Engineers.7. Time and Frequency: Theory and Fundamentals,Byron E. Blair, Editor, NBS Monograph 140,May 1974.8. P. Kartaschoff, Frequency and Time, London,England: Academic Press, 1978.9. G. M. R. Winkler, Timekeeping and Its Application,Advances in Electonics and ElectronPhysics, Vol. 44, New York: Academic Press,1977.42TN-55

10. F. M. Gardner, Phaselock Techniques, NewYork: Wiley &Sons, 1966.11. E. A. Gerber. "Precision Frequency Controland Selection, A Bibliography," Proc. 33rdAnnual Symposium on Frequency Control, pp.569-728, 1979. Available from ElectronicsIndustries AS$ociation, 2001 Eye St. NW,Washington, DC 20006.12. J. Rutman, "Characterization of Phase andFrequency Instabilities in Precision FrequencySources; Fifteen Years of Progress,"P'I'oc. IEEE, YOlo 66, no. 9, September 1978,pp. 1048-1075.Additional Specific References (selected)10 O. w. A1lan, "Statistics of Atomfc FrequencyStandards," Proc. IEEE, vol. 54, pp. 221­230, February 1966.2. D. W. Allan, "The Measurement of Frequencyand Frequency Stability of Precision Oscillators,"NBS Tech. Note 669, July 1975.3. J. R. Ashley. C. B. Searles, and F. M. Palka,"The Measurement of Oscillator Noise atMicrowave Frequencies," IEEE Trans. on MicrowaveTheory and Techniques MIT-16 , pp. 753­760, September 1968. -----­4. E. J. Baghdady, R. D. Lincoln, and B. D.Ne1 in, "Short-Term Frequency Stabi 1ity:Theory, Measurement, and Status," Proc. IEEE,vol. 53, pp. 704-722.2110-2111,1965.5. J. A. Barnes. "Models for the Interpretationof Frequency Stabil i ty Measurements," NBSTechnical Note 683, August 1976.6. J. A. Barnes, " A Review of Methods of AnalyzingFrequency Stability," Proc. 9th AnnualPrecise Time and Time Interval (PTTI) Applicationsand Planning Meeting, Nov. 1977, pp.61-84.7. J. A. Barnes, "Atomic Timekeeping and theStatistics of Precision Signal Generators,"Proc. IEEE, vol. 54, pp. 207-220, February1966.8. J. A. Barnes, "Tables of Bias Functions. B 1and B 2 , for Variances Based on Finite Samplesof Processes with Power Law Spectral Oensities."NBS Tech. Note 375, January 1969.9. J. A. Barnes and O. W. Allan, "A StatistfcalModel of Flicker Noise," Proc. IEEE, vol. 43,pp. 176-178, February 1966.10. J. A. Barnes and R. C. Mock1 er, "The PowerSpectrum and its Importance in Precise FrequencyMeasurements," IRE Trans. on Instrumentation1:2. pp. 149-155, September 1960.11. W. R. Bennett, "Methods of Solving NoiseProblems." Proc. IRE, May 1956, pp. 609-637.12. W. R. Bennett, Electrical Noise, New York:McGraw-Hill, 1960.13. C. Bingham. M. O. Godfrey, and J. W. Tukey,"Modern Techni ques of Power Spectrum Estimation,"IEEE Trans. AU, vol. 15, pp. 56-66,June 1967.14. R. B. Blackman and J. W. Tukey, The Measurementof Power Spectra, New York: Dover,195B.15. E. O. Brigham and R. E. Morrow, "The FastF,'urier Transform." IEEE Spectrum. vol. 4,rl.). 12, pp. 63-70, December 1967.16. J. C. Burgess, "On Digital Spectrum Analysis0';: Periodic Signals," J. Acoust. Soc. Am.,".1. 58, no. 3, September 1975, pp. 556-557.17. ri. A. Campbell, "Stability Measurement Techniquesin the Frequency Domain," Proc. IEEE­NASA Symposium on the Definition and Measurementof Short-Term Frequency Stability, NASASP-BO, pp. 231-235, 1965.18. L. S. Cutler and C. L. Searle, "Some Aspectsof the Theory and Measurement of FrequencyFiuctuations in Frequency Standards," Proc.I£EE, vol. 54, PP. 136-154. February 1966.19. W. A. Edson, "Noise in Oscillators," Proc.IRE, vol. 48, pp. 1454-1466. August 1960.20. C. H. Grauling, Jr. and D. J. Healey, III,"Instrumentati on for Measurement of theShort-Term Frequency Stabi 1ity of Mi crowaveSources," Proc. IEEE. vol. 54, pp. 249-257.February 1956.21. D. Halford, "A General Mechanical Modelfor If ,CI Spectral Dens i ty Random Noi se wi thSpecial Reference to Flicker Noise 1/lfl."Proc. of the IEEE, vol. 56, no. 3, March196B, pp. 251-258.22. H. Hellwig, "Atomic Frequency Standards: ASurvey," Proc. IEEE, vol. 63, pp. 212-229,February 1975.23. H. Hellwig, "A Review of Precision Oscillators."NBS TeCh. Hate 662, February 1975.24. H. Hellwig, "Frequency Standards and Clocks:A Tutoria1 Introduction," NBS Tech. Note616-R. March 1974.25. Hewlett-Packard, "Precision Frequency Measurements,"Application Note 116, July 1969.26. G. ,M. Jenkins and D. G. Watts, SpectralAnalysis and its Applications, San Francisco:Holden-Day, 1968.27. S. L. Johnson, 8. H. Smith, and D. A. Calder,"Noi se Spectrum Charact.eristi cs of Low-No; seMicrowave Tubes and Solid-State Devices,"Proc. IEEE, vol. 54, pp. 258-265, February1966.43TN-56

28. P. Kartaschoff and J. Barnes, "Standard Timeand Frequency Generation," Proc. IEEE, vol.60, pp. 493-501, May 1972.29. A. L. Lance, W. D. Seal, F. G. Mendoza, andN. W. Hudson, "Automating Phase Noise Measurementsin the Frequency Qomai~," Proc. 31stAnnual Symposium on Frequency Control, 1977,pp. 347-358.30. D. B. Leeson, "A Simple Model of FeedbackOscillator Noise Spectrum, 'I Proc. IEEE L.,vol. 54, pp. 329-330, February 1966.31. P. Lesage and C. Audoin, "A Time DomainMethod for Measurement of the Spectral Densityof Frequency Fluctuations at Low FourierFrequencies," Proc. 29th Annual Symposium onFrequency Control, 1975, pp. 394-403.42. F. L. Walls, S. R. Stein, J. E. Gray, and D. J.Glaze, "Design Considerations in State-ofthe-ArtSignal Processing and Phase NoiseMeas urement Sys tems •" Proc. 30th Annua 1Symposium on Frequency Control, 1976, pp.269-274.43. F. Walls and A. Wainwright, "Measurement ofthe Short-Term Stability of Quartz CrystalResonators and the Imp 1i cations for QuartzOscillator Design and Applications," IEEETrans. on Instrum. and Meas., pp. 15-20,March 1975.44. G. M. R. WinKler, "A Brief Review of FrequencyStability Measures," Proc. 8th Annual PreciseTime and Time Interval (PITI) Appl icationsand Planning Meeting, Nov. 1976, pp. 489-527.32. P. Lesage and C. Audoin, "Estimation of theTwo-Sample Variance with Limited Number ofData," Proc. 31st Annual Symposium on FrequencyControl,1977, PP. 311-318.33. P. Lesage and C. Audotn, "Characterization ofFrequency Stabil tty: Uncertai nty due toFinite Number of Measurements," IEEE Trans onInstrum. and Meas. IM-22, pp. 157-161, June1973. -----­III... ,I!AppendixDEGREES OF FREEDOM FOR ALLAN VARIANCE_34. W. C. Lindsey and C. M. Chie, "Specificationand Measurement of Oscillator Phase NoiseInstability," Proc. 31st Annual Symposium onFrequency Control, 1977, pp. 302-310.35. D. G. Meyer, "A Test Set for Measurement ofPhase Noi se on Hi gh Qual i ty Si gna 1 Sources,"Proc. IEEE Trans on Instrum. and Meas. IM-19,No.4, pp. 215-227, November 1970. -­36. C. D. Motchenbacher and F. C. Fitchen,Low-Noise Electronic Desion, New York.: JohnWiley &Sons, 1973.37. D. Percival. "Prediction Error Analysis ofAtomi c Frequency Standards ," Proc. 31stAnnual Symposium on Frequency Control, 1977.pp. 319-326.38. D. Percival, "A Heuristic Model of Long-TermClock Behavi or," Proc. 30th Annua 1 Sympos i umon Frequency Control, 1976, pp. 414-419.39. J. Shoaf, "Specification and Measurement ofFrequency Stabi lity," NBS Internal Report74-396, November 1974.E..g= IIII.u......o 110--­IIIIII= e,~....oI.III !"i I::l IZ_L._~lJTau (in unit. or data .pacing)---lIII40. J. L. Stewart, "Frequency MOdulation Noise inOscillators," Proc. IRE, vol. 44, pp. 372­376, March 1956.41. R. F. C. Vessot, L. Mueller, and J. Vanier,"The Specification of Oscillator Characteristicsfrom Measurements in the FrequencyDomain," Proc. IEEE, pp. 199-206, February1966.44TN-57

"r---"-­IDEGREES OF FREEDO~ FOR ALLAN VARIANCE~i_ '" ....1..1....] 1IftI! ........ fer !lhIU FN----r-------.-----,IIjDEGREES OF FREEDOM FOR ALLAN VARIANCESWhite Phe•• Noia."0-0OJOJI.­........0l!)IIII'llI-enIIICl.....0IIl­ll>..0EO::l:z iIL.__. _ __---ḷ .E0-cIIIIIII­.........0..IIIIIII-mIIIc.....0I­III.0EO::lZ"SJ• - :l. 11-513..II-2S'I...12llIIoe...33.•,~,~,-- L­IIl1liJ. I..J_Tou (In unlta of data apecing)DEGREES OF FREEDOM FOR ALLM~VARIANCEDEGREES Of FREEDOM FOR ALLAN VARIANCES-..g"(IIIIII­,.........0..._ .._'----­IIIIIII-tilIIIC.«4- I0 IIl-III..cJI!J------j••Tau (in unit. of data .paeing)Eo-c • IIL.... l1li ----

_--_._~--~II..gIIIIII1.......oIIIIIIIIIIn~....o1.izDEGREES OF FREEDOM FOR ALLAN VARJANCESFlicker Pho•• Modulation- ~-'--'-.T"TTT1r-r-r I n..,.,.---, ...,--,-TTT'T:f.. , ! :::)r ! I· ~-IIr I I ..;ị..~oee&,eC...-.... .ojJDEGREES Of FREEDOM FOR ALLAN VARIAI:CESRQnC/OIIl Walk FHII ..Teu (in units of dete spec ins)-•Tau (in units of dote epccins)-- ::;DEGREES OF FRWJOM FOR ALWj VAR1M~::ESFlicker Frequency HedvlationI!Q"'0IIIIII1.....-..__... I.... •::;0III4l4l&,IIIC....0i.J:l~:zI - I T r~-.··· I ,. 1 rTT't!IIJ--!.,ul•...;CONFIDENCE FACTORS FROM CH1-SO~ARE-r-I .T,-ntnr .... r-r--- -90% IntervelsI._~---I.. II! It J: i"I~~ 1.-1-""!----l'--'-,L..IIc..l.J1IU..L"'_-'--'-''-~• • ­...;::;;- :::l46TN-59

_~_O_"O_LOWER CDNFIOEN:E FACTORSUPPER CONFIDENCE FACTORSrTrnrr-'I II r 1 II it f i I IiI,onT'lIIJiJ ;• 1i t I1""!... & :II: Ii---_._- .~.._-! ...8 -----.:k!.l-I- ..:::l...­0•~ ,L.. ..tilo.I." , • I. .., ,"' rTl" -..,.--,--rTT3...- 6 ~ ruC;..oo__-j--- i, 5 I&*~2U.. I L"..o... i- IS. I- ltc ., ~ III!i L Ii!,..u.uJ._-l._ I , , " .I0~ I~ I~J ,J ,~,~u] LAJ'" • - • -IINumb.~ of D.~... of Fr..domNumb.r of o.9r-.. ot FreedomUillll r ------,-------::::::::::==-_---...,.th.~.~lool R.~to of.., Chl-Squ.~. ~o O.gr...., rr"04doMt01"0 ..... AftlOF_ do f. -11. elD(,II..lDc..56'J/--------t:n------------------lI~_'""""':..-__---I:-------------------I, 113Chi SqvQre/Oe9r.e of Freedom471N-60

From: Precision FrequenC)' Control, Vol. 2, edited by E.A. Gerber and A. Ballato(Academic Press, New York, 1985), pp. 191-416.12Frequency and Time-Their Measurementand CharacterizationSamuel R. SteinTime and Frequency DirisionNaTional Bureau ofSrandardsBOlllder, ColoradoPRECISIO.\; FREQLE."CY CO:-.TROLV..Jlu:r.c =O....... i1..1[.,)rs .1nd St;md:uJlIoList of Svmbols19212.1 Concepts. Definitions. and Measures of Stability 19312.1.1 Relationship between the Power Spectrum and thePhase Spectrum19512.1.2 The IEEE Recommended Measures of FrequencyStability19512.1.3 The Concepts of the Frequency Domain and theTime Domain20312.1.4 Translation between the Spectral Density ofFrequency and the Allan Variance20312.1.5 The Modified Allan Variance 20512.1.6 Determination of the Mean Frequency andFrequency Drift of an Oscillator20712.1.7 Confidence of the Estimate and OverlappingSamples21012.1.8 Efficient Use of the Data and Determination of theDegrees of Freedom21312.1.9 Separating the Variances of an Oscillator from theReference21612.2 Direct Digital Measurement 21712.21 Time-Interval Measurements 21712.2.2 Frequency Measurements 21912.2.3 Period Measurements 21912.3 Sensitivity-Enhancement Me!hods 2201231 Heterodyne Techniques 22012.32 Homodyne Techniques 221123.21 Discriminator and Delay line 22212.32.2 Phase-lOCked loop 22312.3.3 Multiple Conversion Methods 22712.331 Frequency Svnthesis 22712.33:2 The Dual-Mixer Time-DifferenceTechnique229123.3.3 Frequency Multiplication 23112.4 Conclusion 231191TN-61

192 SAMUEL R STEINLIST OF SYMBOLSdf Number of degrees of freedomfFourier (cycle) frequencyG•.uw) Open-loop transfer function of a phase-locked loopf\sl Transfer function of the loop filter of a phase-locked loopH(J - fol Transfer function of a filterK. Sensitivity of a linear phase detector in volu per radianpNumber of counts accumulated by a time-interval counterQ Quality factor of a resonance. equal to the ratio of the stored energy to theenergy lost in one cycle5 Independent variable of the Laplace transform527Estimate of the two-sample or Allan varianceS,IJ) One-sided power spectral density of the fractional-frequency deviationsS.l!l One-sided power spectral density of the phase deviationsS~SU) Two-sided power spectral density of the phase deviationsS~~,U) Two-sided power spectral density of the phase deviations of the carrierS~pU) Two-sided power spectral density of the phase deviations of the pedestalSrU) Two-sided power spectral density of the instantaneous voltageV o Peak voltage of a signal generatorv.. Output voltage of a phase detectorVI Voltage applied to the tuning element of the voltage-controlled oscillator of aphase-locked loopV(rl Instantaneous voltage of a signal generator or other deviceX(I) Time deviation required by a signal generator operating at nominal frequency Voto accumulate phase equal to 4>(f).Xt!l Fourier transform of x(f).11 I) Instantaneous fractional-frequency offset from nominal)\ Mean fractional-frequency offset over the kth interval7 Exponent off for a power-law spectral densityal'Q Frequency uncertainty due to the quantization of measurements~ Damping constant of a phase-locked loopP. Exponent of r for a power-law Allan variance'it) Instantaneous frequencyVaNominal frequency of a signal generatorV(r ,: ( 2) Mean frequency over the interval f\ ~ f ~ f 1V H Heterodyne frequency or dill'ercnce frequency between two oscillatorsPI/' Probability density of the chi-squared distributiona;(N, T. r) Sample variance of N fractional-frequency deviations. each averaged over aperiod r spaced at intervals Ta;(r) The two-sample or Allan variance of the fractional-frequency deviationsmod a;frJ Modified Allan variancerAveraging timePeriod of the time base of a counter.plr) Instantaneous phase deviationlChi-squared distributionwFourier (angular) frequencyNatural frequency of a phase-locked loopTN-62

12 FREQUENCY AND TIME MEASUREMENT 19312.1 CONCEPTS, DEFINITIONS, AND MEASURESOF STABILITYThis chapter deals with the measurement of the frequency or time stabilityof precision oscillators. It is assumed that the average output frequency isdetermined by a narrow-band circuit so that the signal is very nearly a sinewave. To be specific, it is also assumed that the output is a voltage, which isconventionally (Barnes et at., 1971) represented by the expressionV(c) = [Vo + e(t)] sin[21tv o t + q'>(t)], (12-1)where Vo is the nominal peak voltage amplitude, £(t) the deviation ofamplitude from nominal, Vo the nominal fundamental frequency, and q'>(t) thedeviation of phase from nominal.When either specifying or measuring the noise in an oscillator, one mustconsider the nature of the reference. This may be either a passive circuit suchas a narrow-band filter, another similar oscillator, or a set of oscillators,synthesizers, and other signal-generating equipment. A reference with lowernoise than the device under test may be available, and in this case theexpressions developed in this chapter describe the noise in the oscillatoralone. However, a state-of-the-art device will have lower noise than anyavailable reference. In this case all the expressions below refer to the sum ofdevice and reference noise. The most common approach to solving thisproblem is to compare two or more nearly identical devices. Under mostcircumstances it is then reasonable to assume that each oscillator contributeshalf of the measured noise.The most direct and intuitive method of characterizing the properties ofa signal is to determine the two-sided spectrum of V(c), which is denotedSV(f) (Rutman, 1978). The variable f is called a Fourier frequency and isvery closely related to the concept of a modulation frequency. A positivef indicates a frequency above the carrier frequency vo, while a negative findicates a frequency lower than the carrier. Since the noise can in theorymodulate the carrier at all possible frequencies, a continuous function isrequired to describe the modulation of V(c). S is called a spectral density andS~s(f) is the mean-square voltage

194SAMUEL R. STEIN)----1BANDPASS FILTER v'm MEAN - SQUAREMet - '0)METERFIG. 12-1 An rf speclrum analyzer. The device produces an output proponional 10 themean-square value of the signal passing through a tunable narrow-band filter centered atfrequency10'in Fig. 12-1. The spectrum of the filtered voltage V'(t) is equal to the squareof the magnitude of the filter transfer function H(f - fo) multiplied by thespectrum of the input signal (Cutler and Searle, 1966). The variance of thefiltered voltage is obtained from Parseval's theorem:Cl~·(fO) = f~ IH(f - fo)f S~(f) df. (12-2)If the bandpass filter is sufficiently narrow, so that S!i'(j) changes negligiblyover its bandwidth, then Eq. (12-2) may be inverted. With this assumption,the power spectrum is estimated from the measurement using Eq. (12·3):S!i'(fo) = (1~,(fo)/B, (12-3)where B = J~ '" IH(f' - 10)1 2 df' is the noise bandwidth of the filter andfo itscenter frequency. Figure 12-2 shows a typical two-sided rf spectrum. Formany oscillators the spectrum has a Lorentzian shape, that is,STS = 2< y2>/7r AfJdBIi (f) 1 + U/(A!3dB'''l))2 . (12-4)The Lorentzian lineshape is completely described by the mean square voltage and the full width at half maximum A!JdB'-f e 0 f eFIG. 12-2 The rf spectrum of a signal. It is orten useful 10 divide Ihe speclrum inlo thecarrier and the noise pedeslal. The speclral densily of the carrier exceeds that of Ihe noisepedestal for Fourier rrequencies smaller in magnitude IhanJ;.IN-64

12 FREQUENCY AND TIME MEASUREMENT19512.1.1 Relationship between the Power Spectrum and thePhase SpectrumThe power spectrum differs from a delta function b(j) due to the presenceof the amplitude- and phase-noise terms, c(c) and ¢(t), respectively, includedin Eq. (12-1). Usually the noise modulation separates into two distinctcomponents, so that one observes a very narrow feature called the carrierabove the level of a relatively broad pedestal. The frequency that separatesthe carrier and pedestal is denoted fe' Below this frequency the spectraldensity of the carrier exceeds that of the pedestal. Assuming that the amplitudenoise is negligible compared to the phase noise and that the phasemodulation is small, the relationship between the power and phase spectrais given by (Walls and DeMarchi, 1975)STS(j) ~ VJ e-1(fcl x {J(f), if -fc

196 SAMUEL R STEINperforming measurements or the description of measurement results. ThisIsituation was complicated by the difficulty of comparing the various:~eSCriPtions. A measure of stability is often used to summarize someImportant feature of the performance of the standard. It may therefore not befpossible to translate from one measure to another even though the respective!measurement processes are fully described and all relevant parameters aregiven. This situation resulted in a strong pressure to achieve a higher degreeof uniformity.In order to reduce the difficulty of comparing devices measured in separatelaboratories, the IEEE convened a committee to recommend uniformmeasures of frequency stability. The recommendations made by the committeeare based on the rigorous statistical treatment of ideal oscillators thatobey a certain model (Barnes ee ai., 1971). Most importantly, these oscillatorsare assumed to be elements of a stationary ensemble. A random process isstationary if no translation of the time coordinate changes the probabilitydistribution of the process. That is, ifone looks at the ensemble at one instantof time, then the distribution in values for a process within the ensemble isexactly the same as the distribution at any other instant of time. The elementsof the ensemble are not constant in time, but as one element changes valueother elements of the ensemble assume previous values. Thus, it is notpossible to determine the particular time when the measurement was made.The stationary noise model has been adopted because many theoreticalresults, particularly those related to spectral densities, are valid only for thiscase. It is important for the statistician to exercise considerable care sinceexperimentally one may measure quantities approximately equal to either theinstantaneous frequency of the oscillator or the instantaneous phase. But theideal quantities approximated by these measurements may not both bestationary. The instantaneous angular frequency is conventionally defined asthe time derivative of the total oscillator phase. Thus,and the instantaneous frequency is writtendw(r) = dr [2rrvoe + et>(e)], (12-7)1 det>v(e) = Vo +--. (12-8)2n dtFor precision oscillators. the second term on the right-hand side is quitesmall, and it is useful to define the fractional frequency\'le) - Vo 1 det> dxy(t)= =--=- (12-9)• Vo 2n\'0 de de 'TN-66

12 FREQUENCY AND TIME MEASUREMENT197wherex(r) = ¢(r)/2rrvo (12-10)is the phase expressed in units of time. Alternatively, the phase could bewritten as the integral of the frequency of the oscillator:¢(r) = ¢o + {2ir[V(e) - vo] de. (12-11)However, the integral of a stationary process is generally not stationary.Thus, indiscriminate use of Eqs. (12-7) and (12-11) may violate the assumptionsof the statistical model. This contradiction is avoided when oneaccounts for the finite bandwidth of the measurement process. Although amore detailed consideration of the statistics goes beyond the scope of thistreatment, it is very important to keep in mind the assumption that lie behindthe statistical analysis of oscillators. In order to analyze the behavior of realoscillators, it is necessary to adopt a model of their performance. The modelmust be consistent with observations of the device being simulated. To makeit easier to estimate the device parameters, the models usually include certainpredictable features of the oscillator performance, such as a linear frequencydrift. A statistical analysis is useful in estimating such parameters to removetheir effect from the data. It is just these procedures for estimating thedeterministic model parameters that have proved to be the most intractable.A substantial fraction of the total noise power often occurs at Fourierfrequencies whose periods are of the same order as the data length or longer.Thus, the process of estimating parameters may bias the noise residuals byreducing the noise power at low Fourier frequencies. A general technique forminimizing this problem in the case of oscillators actually observed in thelaboratory is discussed below.It has been suggested that measurement techniques for frequency and timeconstitute a hierarchy (Allan and Daams, 1975), with the measurement of thetotal phase of the oscillator at the peak. Although more difficult to measurewith high precision than other quantities, the total phase has this statusowing to the fact that all other quantities can be derived from it.Furthermore, missing measurements produce the least deleterious effect on atime series consisting of samples of the total phase. Gaps in the data affect thecomputation of various time-dependent quantities for times equal to orshorter than the gap length, but have a negligible effect for times much longerthan the gap length. The lower levels of the hierarchy consist of the timeinterval, frequency. and frequency fluctuation. When one measures a quantitysomewhere in this hierarchy and wishes to obtain a higher quantity, it isnecessary to integrate one or more times. In this case the problem of missingdata is quite serious. For example, if frequency is measured and one wants to1N-67

198 SAMUEL R. STEINknow the time of a clock, one needs to perform the integration in Eq. (12-1l).The missing frequency measurements must be bridged by estimating theaverage frequency over the gap, resulting in a time error that is propagatedforever. Thus, it is preferable to always make measurements at a level of themeasurement hierarchy equal to or above the level corresponding to thequantity of principle interest. In the past this was rather difficult to do.Measurement systems constructed from simple commercial equipment sufferedfrom dead time, that is, they were inactive for a period after performinga measurement. To make matters worse, methods for measuring time orphase had considerably worse noise performance than methods for measuringfrequency. As a result, many powerful statistical techniques weredeveloped to cope with these problems (Barnes, 1969: Allan, 1966). The effectof dead time on the statistical analysis has been determined (Lesage andAudoin, 1979b). Other techniques have been developed to combine shortdata sets so that the parameters of clocks over long periods of time could beestimated despite missing data (Lesage, 1983). The rationale for theseapproaches is considerably diminished today. Low-noise techniques for themeasurement of oscillator phase have been developed. Now, commercialequipment is capable of measuring the time or the total phase of an oscillatorwith very high precision. Other equipment exists for measuring the timeinterval. These devices use the same techniques that were previouslyemployed for the measurement of frequency and are very competitive inperformance.The proliferation of microcomputers and microprocessors has had anequally profound effect on the field of time and frequency measurement.There has been a dramatic increase in the ability of the metrologist to acquireand process digital data. Many instruments are available with suitablestandard interfaces such as IEEE-583 or CAMAC (IEEE, 1975) andIEEE-488 (IEEE, 1978). As a result, there has been a dramatic change indirection away from analog signal processing toward the digital, and thisprocess is accelerating daily. Techniques once used only by nationalstandards laboratories and other major centers of clock development andanalysis are now widespread. Consequently, this chapter will focus first onthe peak of the measurement hierarchy and the use of digital signalprocessing. But the analysis is directed toward estimating the traditionalmeasures of frequency stability. Considerable attention will be paid toproblems associated with estimating the confidence of these stability measuresand obtaining the maximum information from available data.The IEEE has recommended as its first measure of frequency stability theone-sided spectral density Sy(f) of the instantaneous fractional-frequencyfluctuations J(t). It is simply related to the spectral density of phase fluctuationssince differentiation of the time-dependent functions is equivalent toTN-68

12 FREQUENCY AND TIME MEASUREMENT199multiplication of their Fourier transforms by jw:SyU) = (fjv O )2S¢U) = (2rr.f) 2 S",U)· (12-12)Section l2.1.1 on the relationship bl.'tween the power spectrum and thephase spectrum described the analog method for the measurement of aspectral density. If a voltage Vis the output of the oscillator, then the result ofthe measurement is proportional to the rf power spectral density. But if thevoltage were proportional to the frequency or phase of the oscillator, then theresult of the measurement would be proportional to the spectral density ofthe frequency or phase. The most common units of S¢U) are radians squaredper hertz.Alternatively, the spectral density can be obtained by digital analysis of thesignal. For example, the quantity SAf) can be calculated from the Fouriertransform of x(t). The relevant continuous Fourier-transform pair is definedas follows:(12-13)and1 roc " fX(t) = - XU)eJ-~ [df (12-14)2rr. ~-xHowever, one does not generally have continuous knowledge of the phase ofthe oscillator. Since it is relatively easy to measure x(t) at equally spaced timeintervals, we assume the existence of the series Xl, where Xl = x{lr) for integervalues of I. The discrete Fourier transform is defined by analogy to the-continuous transform (Cochran et aI., 1967):XU) = I 00 x(/)e- j2~fl. (12-15)I =- XlIn practice the time series has finite length T consisting of N intervals oflength r, and it is not possible to compute the infinite sum, Nevertheless, itremains possible to compute a spectrum that is not continuous inf but ratherhas resolution AI, whereAf = l/T = l/NT:. (12-16)The need to sum over all values of the index I is removed by assuming that thefunction x(c) repeats itself with period T. The resulting spectrum contains noinformation on the spectrum at Fourier frequencies less than 1 T. Truncationof the time series also introduces spurious effects due to the turn-on and turnofftransients. These problems can be minimized through the use of a windowfunction. The computed spectrum is actually the square of the magnitude ofIN-69

200 SAMUEL R. STEINthe window function multiplied by the desired spectrum. The use ofa windowfunction reduces the variance of the spectrum estimate at the expense ofsmearing out the spectrum to a small degree. With these changes but nowindow function, we arrive at the discrete finite transform1 N- tX(n 6.1) = - L x(kr)e-j2·ft~f't. (12-17)N .=0The spectral density of x(t) is computed from Eq. (12-17) by squaring the realand imaginary components, adding the two together, and dividing by thetotal time T:2S,,(m 6.f) = {R[X(m 6.f)J} ; {I[X(m 6.1)J}2(12-18)The digital method of estimating spectral densities has many advantagesover analog signal processing. Most important is the fact that it may becomputed from any set of equally spaced samples of a time series. As a result,the technique is compatible with other methods of characterizing the signal,that is, the sampled data can be stored and processed using a variety ofalgorithms. In addition, each record of length Tproduces a single estimate ofthe spectrum for each of the N frequencies 6.1, 2 6.1, _.. , N !1f It is thereforepossible to estimate the entire spectrum much more quickly using the digitaltechnique than it would be using analog methods. The fast Fouriertransform, a very efficient algorithm for the computation of the discrete finitetransform, has opened the way to versatile self-contained, commercialspectrum analysis. It is also very straightforward to compute the spectrumfrom data acquired by computerized digital data acquisition systems.A result of the finite sampling rate is that the upper frequency limit of thedigital spectrum analysis is 1/2T, called the Nyquist frequency (Jenkins andWatts, 1968). Power in the signal being analyzed that is at frequencies higherthan the Nyquist frequency affects the spectrum estimate for lower frequencies.This problem is called aliasing. The out-of-band signal is rejected byonly approximately 6 dB per octave above the Nyquist frequency. Thus,when significant out-of-band signals exist, they must be reduced by analogfiltering. One or more low-pass filters are usually sufficient for this purpose.As its second measure of frequency stability, the IEEE recommended thesample variance c;;(-r) of the fractional-frequency fluctuations. It is a measureof the variability of the average frequency of an oscillator between twoadjacent measurement intervals. The average fractional-frequency deviation5'. over the time interval from t. to r. + r is defined as1 J'.+t Y. = - y(r) dr, (12-19)r I.TN-70

12 FREQUENCY AND TIME MEASUREMENT 2Q1x(t)I II IIt-r-I'I II,T-ITIMEFIG.12-3 lIiklsuremenr process for the computation of the sample ,·ariance. The phasedifference between two oscillators is plotted on the ordinate. The measurement yields a set offrequencies averaged over equal intervals t separated by dead time T - t.from which it follows that(12-20)The quality r is often referred to as the sampling time or the averaging time.Eq uations (12-19) and (12-20) are not the only way to define mean freq uency.but they are the simplest. Other definitions lead to alternative measures ofstability that may have desirable properties.Suppose that one has measured the time or frequency fluctuations betweena pair of precision oscillators and a stability analysis is desired. The process isillustrated in Fig. 12-3. These are N values of the fractional frequency y;.Each one is measured over a time t, and measurements are repeated afterintervals of time T If the measurement repetition time exceeds the averagingtime, then there is a dead time equal to T - t between each frequencymeasurement, during which there is no information available.There are many ways to analyze these data. A fairly general approach is theN-sample variance defined by the relation(12-21)where the angle brackets denote the infinite time average. Frequently, Eq.(12-21) does not converge as N ...... x, since some noise processes in oscillatorsdiverge rapidly at low Fourier frequencies. This implies that the precisionwith which one estimates the variance does not improve simply as the samplesize is increased. For this reason, the two-sample variance with no dead timeis preferred. Also called the Allan variance, it converges for all the majornoise types observed in precision oscillators. It may be written as(12-22)TN-71

202 SAMUEL R. STEIN~1­!10 2 k.­t10- 1 ' I I I ","1 I 1 I II !III !! 11I!!11 1~ 1~ 1~NUMBER OF SAMPLES NFIG. 12-4 lli-sample varIance versus Allan vanance. The two-sample variance converges forthe important types of noise observed in frequency standards but the ratio of the traditionalvariance to the two-sample variance is an increasing function of sample size for flicker frequencynoise and random-walk frequenq noise.The dependence of the classical variance on the number of samples is shownin Fig. 12-4 for the case of no dead time. The quantity plotted is the ratio ofthe N-sample variance to the Allan variance. Note that O";(r) has the samevalue as the classical variance for the white-noise frequency modulation.However, the classical variance grows without bound for flicker-frequencyand random-walk-frequency noises.One may combine Eqs. (12-20) and (12-22) to obtain an equation for O"ir)in terms of the time-difference or time-deviation measurements:u;(r) =

12 FREOU::NCY AND TIME MEASUREMENT20312.1.3 The Concepts of the Frequency Domain and the Time DomainSpectral densities are measures of frequency stability in what is called thefrequency domain since they are functions of Fourier frequency. The Allanvariance, on the other hand, is an example of a time-domain measure. In astrict mathematical sense, these two descriptions are connected by Fouriertransform relationships (Cutler and Searle, 1966). However, for many yearsthe inadequacy of measurement equipment created artificial barriers betweenthese two characterizations of the same noise process. As a result, manyspecialized techniques have been developed to translate between the variousmeasures ofstability (Allan, 1966: Burgoon and Fischer, 1978). The precedingsections have demonstrated how easily both types of stability measures canbe computed from the same data provided that the measurement processprovides complete information. For example, both u;(mro) and S.x(m 6./) canbe computed from evenly spaced samples of x(t). However, incompleteinfonnation can result from either measurement dead time or interruptions inthe data acquisition process. In these cases translation techniques remainvaluable.Both the spectral density and the Allan Variance are second-momentmeasures of the time series x(r). However, it is only possible to translateunambiguously from the spectral density to the Allan variance, not thereverse. To calculate the spectral density it is necessary to use the autocorrelationfunction of the phase. The following discussion on power-law noiseprocesses further demonstrates this dichotomy. As we shall see, the Allanvariance for a fixed measurement bandwidth does not distinguish between allof the noise processes that are commonly observed in precision oscillators.12.1.4 Translation between the Spectral Density of Frequencyand the Allan VarianceThe power-law model is most frequently used for describing oscillatorphase noise. It assumes that the spectral density of frequency fluctuations isequal to the sum of terms, each of which varies as an integer power offrequency. Thus, there are two quantities that completely specify Sy(f) for aparticular power-law noise process: the slope on a log-log plot for a givenrange of/and the amplitude. The slope is denoted by:x and thereforej' is thestraight line on a log-log plot that relates Sif) tof. The amplitude is denotedh,. When we examine a plot of the spectral density of frequency fluctuations,we represent it by the addition of all the power-law processes (Allan, 1966:Vessot et aI., 1966) with the appropriate coefficients:Sylf) = I x; It,!'. :12-26)2:::.-:r:TN-73

204 SAMUEL R. STEINTABLE 12-1Correspondence between Common Power-Law Spectral Densities and theAllan Variance"Noise type SJIl a:(rj\\'bite phase hd 2 3f.h 2 /(211)2 r2flicker phasehI![1.038 + 31n1211:1. r)] 1h,­(211)2 .2White frequency ho !holl/.)flicker frequency h.d-' 2In(2/h_ 1Random-walk frequency h. 2 F 2 !121t)2h_ 2r• Where necessary for convergence the spectral density has been assumed tobe zero for frequencies greater than the cutotrfrequency f •.This technique is most valuable when only a few terms in Eq. (12-26) arerequired to describe the observed noise and each term dominates over severaldecades of frequency. This situation often prevails. Five power-law noiseprocesses (Allan, 1966: Vessot et aI., 1966) are common with precisionoscillators:(1) random-walk frequency modulation :x = - 2(2) flicker frequency modulation ex = -1(3) white frequency modulation 0:=0(4) flicker phase modulation 11: = 1(5) white phase modulation 0:=2The spectral density of frequency is an unambiguous description of theoscillator noise. Thus, the spectrum can be used to compute the Allanvariance (Barnes er ai., 1971):O";('r) = ~ r:r: S~(j) sin4.(rrfr) df (12-27)(rrvort J o *However, Eq. (12-27) shows that the Allan variance is very sensitive to thehigh frequency dependence of the spectral density of phase, thereby necessitatinga detailed knowledge of the bandwidth-limiting elements in themeasurement setup. The integral has been computed for each of the powerlawnoise processes, and the results are summarized in Table 11-1 (Barnes etaI., 1971). For:x in the range -2 ::::; ex ::::; 0, the Allan variance is proportionalto Til, where f.i = - ex - 1. When the log of the Allan variance is plotted as afunction of the log of the averaging time, the graph also consists of straightlinesegments with integer slopes. However, Table 12-1 also shows that even if.. See Appendix Note 1/ 8TN-74

12 FREQUENCY AND TIME MEASUREMENT205the oscillator is reasonably modeled by power-law spectra, it is not practicalto distinguish between white phase noise and flicker phase noise from thedependence of the Allan variance on t. In both cases 0-; == l/t 2 •12.1.5 The Modified Allan VarianceTable 12-1 also shows that the Allan variance has very different bandwidthdependence for white phase noise and flicker phase noise. Therefore, thesenoise types have been distinguished by varying the bandwidth of themeasurement system. Ifx(r) were measured, the noise type could be identifiedby computing the spectrum. However, both the approach of makingmeasurements as a function of bandwidth and the computation of thespectrum can be avoided by calculating a modified version of the Allanvariance. The algorithm for this variance has the effect of changing thebandwidth inversely in proportion to the averaging time (Snyder, 1981; Allanand Barnes, 1981).Each reading of the time deviation Xi has associated with it a measurementsystembandwidthfb. Similarly, we can define a software bandwidth!. = Aln,which is l/n times narrower than the hardware bandwidth. It can be realizedby averaging n adjacent x;'s. Based on this idea it is possible to define amodified Allan variance that allows the reciprocal software bandwidth tochange linearly with the sample time t:(12-28)where", = mo. Equation (12-28) reduces to Eq. (12-23) for n = 1. One can seethat mod o-;(t) is the second difference of three time values, each of which is anonoverlapping average of n of the Xi'S. As n increases the softwarebandwidth decreases asft,/n.For a finite data set of N readings of Xi (i = 1 to N), mod o-;(t) can beestimated from the expression(12-29)which is easy to program but takes more time to compute than thecorresponding equation (12-24) for o-;It).Table 12-2 gives the relationship between the time-domain measuremod O";(r) and its power-law spectral counterpart. In the right-hand columnare the asymptotic values of the ratio of the modified Allan variance to theAllan variance. It is clear from the table that mod O";(t) is very useful for white• See Appendix Note # 9TN-75

206 SAMUEL R. STEINTABLE 12-2Correspondencc between Common Power-law Spectral Dcnsitics and thc Modified AllanVariancc'Noise typeWhite phasenFlicker phase[1.038 + 3 In(2,q.rl] Ih, ::-----:-'--=-=-.:.::.(21t)' r 1White frequency h o .4r 0.5Flicker frequency h_ 1 (O.936) 0.674Random-walk frequency L 1(5.42)r 0.824• Where necessary the spectral density has been assumed to be zero for frequencies greater thanthe cutoff frequency f. Thc constant n i~ the number of adjacent phase values that are averagedto produce the bandwidth reduction. The values in the last two columns arc for thc asymptoticlimit n - x.. In practice. n onIy needs to be 10 or larger before the asymptotic limit is approachedwithin a few percent. When n = I the ratio in the last column is I in all cases.phase modulation and flicker phase modulation, but for :I ~ I the conventionalAllan variance gives both an easier-to-interpret and an easier-tocalculatemeasure of stability.It is interesting to make a graph of:t versus J.1 for both the ordinary Allanvariance and the modified Allan variance, such as the one shown in Fig. 12-5.al'2'- "I "i ,1~JI-+-2-3-4 -3 -2 -, 0f-LFIG. 12-5 Relationship between a power-law spectral density whose slope on a log-log plotis " and the corresponding sample variance whose slope on a log-log plot is jJ.. The solid linedescribes the behavior of the Allan variance, while the dashed line shows the advantag.e of themodified Allan variance for white phase noise and nicker phase noise.TN-76

12 FREQUENCY AND TIME MEASUREMENT207This graph allows one to determine power-law spectra for noninteger as wellas integer values of a. In the asymtotic limit the equation relating p and a forthe modified Allan variance isa = -p - 1 for -3 < 7. < 3. (12-30)12.1.6 Determination of the Mean Frequency and Frequency Driftof an OscillatorBefore the techniques of the previous four sections can be meaningfullyapplied to practical measurements, it is necessary to separate the deterministicand random components of the time deviation x(t). Suppose, forexample, that an oscillator has significant drift, such as might be the case for aquartz crystal oscillator. With no additional signal processing, the Allanvariance would be proportional to ,2. The variance of the Allan variancewould be very small, further demonstrating that deterministic behavior hasbeen improperly described in statistical terms and the oscillator's predictabilityis much better than the Allan variance indicated. Unfortunately, itis difficult to estimate the oscillator's deterministic behavior without introducinga bias in the noise at Fourier frequencies comparable to the inverseof the record length. In practice, it has been sufficient to consider twodeterministic terms in x(c):(12-31)The first term on the right-hand side is the synchronization error. The secondterm is due to imperfect knowledge of the mean frequency and is sometimescalled syntonization error. The quadratic term, which results from frequencydrift, is the most difficult problem for the statistical analysis because the Allanvariance is insensitive to both synchronization and syntonization errors.For white noise, the optimum estimate of the process is the mean.Therefore, a general statistical procedure that can be followed is to filter thedata until the residuals are white (Allan et aI., 1974: Barnes and Allan, 1966).For example. at shoft times the frequency fluctuations of atomic clocks areusually white. Taking the first difference of Eq. (12-31), we find that(12-32)and a linear least square fit to the frequency data yields the optimum estimateof .1\'. However, the drift in atomic clocks is generally so small that the valueobtained for D will not be statistically significant when r is small enough toTN-77

208 SAMUEL R. STEINsatisfy the assumption of white frequency noise. Thus, we are led to considerthe first difference of the frequency,Ji(t + r) - Ji(t) = D + x1(t + 2r) - 2x1(t + r) + X1(t).r r 2 (12-33)Many atomic clocks are dominated by random walk of frequency noise forlong averaging times. Thus, the first difference of the frequency data (thesecond difference of the phase data) is white, and the optimum estimate of thedrift is just the simple mean. Ifinstead, a linear least square fit were removedfrom the frequency data in this region of r, then the random-walk residualswould be biased, and it is likely that an optimistic estimate of O"y(r) would beobtained.The optimum procedure would be different if the dominant noise typewere flicker of frequency, rather than random walk. But there is no simpleprescription that can be followed to estimate the drift in that case.Fortunately, a maximum likelihood estimate of the prameters for sometypical cases has shown that the mean second difference of phase is still agood estimator of frequency drift in the sense that it introduces negligible biasin the Allan variance. Thus, in practice there is a simple prescription forcomputing the Allan variance in the presence of significant drift. Startingwith the phase data, one forms the second difference and uses the simpleaverage to estimate the mean. The value of r chosen for creating these seconddifferences must be long enough so that the predominant noise process israndom-walk frequency modulation. After subtracting this estimate, thesecond-difference data is integrated twice to recover phase data with driftremoved, and further analysis, including the computation of the Allanvariance, may proceed. Figures 12-6 through 12-10 illustrate the estimationof drift. The quadratic dependence of the phase data in Fig. 12-6 nearlyobscures the noise. The first difference of this data produces the nearly linearfrequency dependence shown in Fig. 12-7, and the second difference producesthe residuals shown in Fig. 12-8, which appear to be nearly white. Rigorousstatistical analysis of this data indicates that the first difference of thefrequency is indeed white with 90~/~ confidence. Next, the mean frequencydifference is subtracted. Then the residuals of Fig. 12-8 are integrated twice,and the result is the estimate of the phase deviation with drift removed shownin Fig. 12-9. Fig. 12-10 illustrates the Allan variance of this data calculated bythree techniques. The squares were computed from the data of Fig. 12-6,while the open circles were computed following the recommended procedurefor estimating the drift. The validity of the approach is illustrated by the blackdots, which are the result of a statistically optimum parameter estimationprocedure.TN-78

12 FREQUENCY AND TIME MEASUREMENT20920-30:c~------------,-J.TIME(DAYS)FIG. 12-6 Measured phase difference between a frequency standard and a reference duringa 140-day experiment. The nearly quadratic form of the data effectively obscures the noise.~3'0~>­

210 SAMUEL R. STEIN2-2~O---_--------""""'U'OTIME (DAYS)FIG. 12-9 Phase variations of the frequency standard due 10 the residuals. oblained byperforming two integrations OD Ihe data of Fig. 12-8. The ordinate scale is expandedappcollimatc:ly 10 times compared 10 Fig. 12-6.-"[f-"I• """T! •• II I.." "• " " .• . "a ~ ~-'4 :::'_-:_---,!:-I--ll,---+---:!I;----!:I---!.Io 23 .. 5 IS 7log T(sec)FIG.12-10 logarithm of the square root of the Allan \'ariance as a function of thelogarithm of the averaging time for three different computation methods. The squares werecomputed from the data of Fig. 12-6 and show the effect of the drift. The open circles werecomputed from the data of Fig. 1~-9. The closed circles were computed using an optimumparameterestimation procedure.12.1.7 Confidence of the Estimate and Overlapping SamplesConsider three phase or time measurements of one oscillator relative toanother at equally spaced intervals of time. From this phase data one canobtain two adjacent values of average frequency and one can calculate asingle sample Allan variance (see Fig. 11-11). Of course, this estimate does nothave high precision or confidence, since it is based on only one frequencydifference.For most commonly encountered oscillators, the first difference of thefrequency is a normally distributed variable with zero mean. However, thesquare of a normally distributed variable is not normally distributed. This isso because the square is always positive and the normal distribution is"TN-SO

12 FREQUENCY AND TIME MEASUREMENTI~I----.------;-----+­211TIMEFIG.12-11 Calculation of two average frequencies.vl and Yz by measuring the phase of anoscillator x(/) at times I,. t 2 • and r l .completely symmetric, with negative values being as likely as positive ones.The resulting distribution is called a chi-squared distribution, and it has one"degree of freedom" since the distribution was obtained by considering thesquares of individual (i.e., one independent sample), normally distributedvariables (Jenkins and Watts, 1968).In contrast, from five phase values four consecutive frequency values canbe calculated, as shown in Fig. 12-12. It is possible to take the first pair andcalculate a sample Allan variance. A second sample Allan variance can becalculated from the second pair (i.e., the third and fourth frequencymeasurements). The average of these two sample Allan variances provides animproved estimate of the true Allan variance, and one would expect it to havea tighter confidence interval than in the previous example. This could beexpressed with the aid of the chi-squared distribution with two degrees offreedom.However, there is another option. One could also consider the sampleAllan variance obtained from the second and third frequency measurements,that is, the middle sample variance. This last sample Allan variance is notindependent ofthe other two, since it is made up ofparts ofeach ofthe others.But this does not mean that it cannot be used to improve the estimate of thetrue Allan variance. It does mean that the new average of three sample Allanvariances is not distributed as chi squared with three degrees offreedom. The~i~IIIIII II, I. I, I. I.TIMEYz. Yl' and Y. from five phaseFIG.12·12 Calculation of four frequency values ji"Feasurements at limes C, • C,. Cl. I•• and Is. The sample variance formed from YI and Y2 and theone formed from Y3 and j. are independent. The sample variance fonned from Y2 and YJ is nolindependent of the other two but does contain some additional information useful in estimating~he true sample variance.IN-81

212 SAMUEL R. STEINnumber of degrees of freedom depends on the underlying noise type, that is,white frequency. flicker frequency. etc.• and may have a fractional value.Sample Allan variances are distributed as chi square according to theequation(12-34)where s; is the sample Allan variance, df the number of degrees of freedom(possibly not an integer~ and

12 FREQUENCY AND TIME MEASUREMENT213allocation. By rderring to tables of the chi-squared distribution, one findsthat for 10 degrees of freedom (df = 10) the 5% and 95~~ points correspond to(12-37)Thus, with 90~~ probability the calculated sample variance s; = 3 satisfies theinequalityand this inequality can be rearranged in the form3.94 < (df)s;/CT; < 18.3, (12-38)1.64 < a; < 7.61. (12-39)The estimate s; = 3 is a point estimate. The estimate 1.64 < a; < 7.61 isan interval estimate and should be interpreted to mean that 90% of the timethe interval calculated in this manner will contain the true 11;.12.1.8 Efficient Use of the Data and Determination of theDegrees of FreedomTypically, the sample variance is calculated from a data set using therelation1 NS2 ::::::-- L (~ - Z)2 (12-40)- N - 1 11=1 -II ,where it is implicitly assumed that the z..'s are random and uncorrelated (i.e.,white) and where zis the sample mean calculated from the same data set. Ifallof this is true, then S2 is chi-squared distributed and has N - 1 degrees offreedom.Consider the case of two oscillators being compared in phase with N valuesof the phase difference obtained at equally spaced intervals To' From these Nphase values one obtains N - 1 consecutive values of average frequency, andfrom these one can compute N - 2 individual sample Allan variances (not allindependent) for r = To. These N - 2 values can be averaged to obtain anestimate of the Allan variance at T = To'The variance of this Allan variance has been calculated (Lesage andAudoin, 1973; Yoshimura, 1978). This approach is less versatile than themethod of the previous section since it yields only symmetric error limits.However, it is simple and easy to use. let 6(N) be the relative differencebetween the sample Allan variance and the true value. Thus,(12-41)IN-83

214SAMUEL R. STEINTABLE 12·3Variance of the Relative Difference between the SampleAllan Variance and the True Value (C./N)·Noise type CI C.White phase 2 3.88Flicker phase 1 3.88White frequency 0 2.99Flicker freq uency -I 2.31Random-walk frequency -2 2.25• N is the number of phase measurements. The result isaccurate to beller than 10 ~~ for N larger than 10.The quantity A(N) has mean zero. For N larger than 10, the variance of A isapproximately0- 2 (.1.) = CJN. (11-41)Table 11-3 gives the constant C a for the five major noise types.Using the same set of data it is also possible to estimate the Allan variancesfor integer multiples of the base sampling interval t = mto. Now thepossibilities for overlapping sample Allan variances are even greater. For adata set of N phase points, one can obtain a maximum of exactly N - 2msample Allan variances for r = mro. Of course only (N - 1)2m of these aregenerally independent. Still, the use of all of the data is well justified since theconfidence of the estimate is always improved by so doing. Consider the caseof an experiment extending for several weeks in duration with the aim ofgetting estimates of the Allan variance for r values equal to a week or more.As always, the purpose is to estimate the "'true" Allan variance as well aspossible, that is, with as tight an uncertainty as possible. Thus, one wants touse the data as efficiently as possible. The most efficient use is to average allpossible sample Allan variances ofa given r value that one can compute fromthe data. This procedure is illustrated in Fig. 12-14.In order to calculate confidence intervals for a sample variance, it isnecessary to know the number of degrees of freedom. This has been done byboth analytical and Monte Carlo techniques, and empirical equations havebeen found that are accurate to 1% for white phase, white frequency, andrandom-walk frequency modulation. The tolerance is somewhat larger forflicker frequency and phase modulation (Howe et al., 1981). The empiricalequations for the degrees of freedom are given in Table 12-4. Table 12-5 givesthe degrees of freedom for selected values of N, the tot31 number of phasevalues, and m, the number of intervals averaged. Figure 12-15 illustrates thenumber of degrees of freedom for all noise processes as a function of t for thecase of 101 total phase measurements.IN-84

12 FREOUENCV AND TIME MEASUREMENT215x(t),.!.., ,~~ I"IIIII ,! l' I I I !---..lI I ' :t ~:a .. 5 • 7'.. • 10 11 12 13 14'"Tl I NONOYERLAPPING y~ :~....."I--;"i FULLY OYERLAPPING i~..."aIt'TIMEFIG. 12-14 Illustration of the case of ': = 4'0_ for which the ralio of the number of fullyoverlapping to nonoverlapping estimates of the variance is more than 8 for the Si phase pointsshown. When the averaging time for the computation of mean frequencies 1 exceeds thesampling time 10. the number of fully overlapping mean frequencies is far larger than the numberof nono'·erlapping frequencies. In general. for large N approximately 2m times as manyestimates of the sample variances.can be computed using the fully overlapping technique.* TABLE12-4Number of Degrees of Freedom for Calculation of the Confidence of theEstimate of a Sample Allan Variance'!'ioise typeWhite phaseFlicker phaseWhite frequencydfIN + lX.'I/ - 2m)2lN - m)eX{In('\: I) In«1m + I~X,\j -I))J3(N - 1) 2(N - 21J 4m'[ ~ - --N-- 4m 2 + 52(N - 21Flicker freq uency for m = 12.3N - 4.9Random-walk frequency5N'4rrnN ~ 3mlfor m :2: 2N - 2 (N - 1)2 - 3m(.'V - I) + 4m 2m (N - 31=• For': = mro from .'1 phase points spaced '0 apart.* See Appendix Note # 39IN-85

216 SAMUEL R. STEINTABLE 12-5Number of Degrees of Freedom for Calculation of the Confidence of the Estimate of a SampleAllan Variance for the Major Noise Types·White Flicker White Flicker Random-walkN m phase phase frequency frequency frequency9 1 3.665 4.835 4.900 6.202 7.0002 3.237 3.537 3.448 3.375 2.8664 1.000 1.000 1.000 1.000 0.999129 1 65.579 79.015 84.889 110.548 127.0002 64.819 66.284 71.642 77.041 62.5244 63.304 52.586 42.695 36.881 29.8228 60.310 37.306 21.608 16.994 13.56716 54.509 22.347 9.982 7.345 5.63132 44.761 9.986 4.026 2.889 2.04764 1.000 1.000 1.000 1.000 1.0001025 1 526.373 625.071 682.222 889.675 1023.0002 525.615 543.863 583.621 636.896 510.5024 524.088 459.041 354.322 316.605 253.7558 521.038 366.113 186.363 156.492 125.39~16 514.952 269.849 93.547 76.495 61.24132 502.839 179.680 45.947 36.610 29.21064 478.886 104.743 21.997 16.861 13.288128 432.509 50.487 10.003 7.281 5.516256 354.914 17.419 4.003 2.861 2.005512 1.000 1.000 1.000 1.000 1.000• N is the number of equally spaced phase points that are taken m at a time to form the averagingtime.12.1.9 Separating the Variances of the Oscillator and the ReferenceA measured variance contains noise contributions from both the oscillatorunder test and the reference. The individual contributions are easily separatedif it is known a priori that the reference is much less noisy than thedevice under test or equal to it in performance. Otherwise, the individualcontributions can be estimated by comparing three devices (Barnes, 1966).The three possible joint variances are denoted by 0"&. UJb and ufk' while theindividual device variances are uf, uf, and u~.The joint variances arecomposed of the sum of the individual contributions under the assumptionthat the oscillators are independen t:2 2 2Uij = O"j + Uj.UJk = uJ + u~. (12-43)O";k = 15; + 15;.TN-86

12 FREQUENCY AND TIME MEASUREMENT217l/)WWO::;:Ec:lW 0Ow oU-w00::0::U­WU­COO5z100~10­...~ ...'\~"'-""~""" "'~"" ,\""'.'-\".'\ .~' ....\' :'" \'.\::..1;-1------'10-:-----=>"'--,-!100T (IN UNITS OF DATA SPACING)FIG.12-15 Number ofdegrees offreedom as a function ofaveraging time for the case of 101phase measurements: The heavy broken line is for random-walk frequency noise, the lightbroken line is for flicker frequency noise, the dotted line is for white frequency noise. the heavysolid line is for flicker phase noise, and the light solid line is for white phase noise,An expression for each individual variance is obtained by adding two jointvariances and subtracting the third:Uf = !(ut + ui - a]k)'uf = !(O}k + u~ - a!k), (12-44)uf = t(U7k + Urk - 0'5)·This method works best if the three devices are comparable in performance.Caution must be exercised since Eqs. (12-44) may give a negative sampleAllan variance despite the fact that the true Allan variance is positive definite,This is possible because the confidence interval of the estimate is sufficientlylarge to include negative variances. Such a result is an indication that theconfidence intervals of the sample Allan variances are too large and thatmore data is required.12,2 DIRECT DIGITAL MEASUREMEJ\T12.2.1 Time-Interval MeasurementsA common technique for measuring the phase difference between oscillatorshaving nearly equal nominal frequencies is the use of direct timeintervalmeasurements. In this section and those that follow, the symbols Vtaand Vl0 are used to indicate the nominal values of \'1 and v 2 , respectively. Inthe simplest form of this technique, a time-interval counter is started on someIN-87

12 FREQUENCY AND TIME MEASUREMENT219occurs to N/v lO , where N is the divisor. Such measurement systems are usedat many standards laboratories for the long-term measurement of atomicclocks, whose output is usually divided down to I pulse/sec. Since timeintervalcounters with resolution better than 0.1 nsec are available, thismeasurement scheme is suitable for long-term performance monitoring,yielding frequency-measurement precision of 10- 14 for 1-day averages.12.2.2 Frequency 1\;leasurementsA'/erage frequency is measured most directly using a frequency counter.Used this way, the counter determines the number of whole cycles Moccurring during a time interval T given by the counter's time base. Thusv(O; T) = (M + 6..\-1)/T ~ M/T, (12-46)where V(tl: t 1 ) denotes the average frequency over the interval from t 1 to t 2and 6.""f, the fractional cycle, is not measured by the counter. The startingtime is arbitrarily called t = O. Thus, the quantization error is given by~VQ!

12 FREQUENCY AND TIME MEASUREMENT221point on the waveform in order to count cycles. The positive-going zerocrossings are normally chosen in order to provide immunity from changes inboth the amplitude and symmetry of the waveform.Consider two signals whose frequency difference is much less than thefrequency of either oscillator:andVI(t) = VIO sin[~rrvl0t+

222SAMUEL A. STEINThe analysis of phase noise is accomplished by arranging that¢IO - ¢20 = n/2, which can be achieved with a phase shifter. Then,(12-59)There are various methods by which one can control the signal V 2(c) so that"10 = \120 without producing significant correlation between ¢2(t) and ¢I(t).When anyone of these methods is used, it is possible to use Vet) as a measureof ¢(t). Two methods, delay lines and phase-locked loops (Gardner, 1966), aredescribed below.12.3.2.1 DISCRIMINATOR AND DELAY LINEThe circuit of a discriminator or delay-line system for measuring phasenoise is illustrated in Fig. 12-19. The delayed signal is given byV 2 (t) = VIet - rd) = V 20 sin[21t\lIo(t - rd) + ¢,(r - Cd) + ¢10 + ¢.J.(12-60)When the phase shifter is set for quadrature, ¢. -2n\'10td = n.'2 andV2(r) = V 20 sin[l1t\l'or + ¢,(r - rd) + ¢'O + 1!,'2J. (12-61)The output of the phase detector is given bySubstituting Eq. (12-62) into Eq. (12-20), we obtain(12-62).nr - t d : r) = - V(r)/21t\·0 VOrd (12-63)and we see that the delay-line method can be used to produce samples of.~imro) by varying the delay time. However, the technique is used morefrequently with a fixed delay by restricting its application to the region of rmuch greater than the delay time, so that .Wt - td: r) is a good approximationfor the instantaneous frequency. Under this assumption spectrum analysis ofPHASE SHIFTERfaVIltl DISCRIMINATOR ORDELAY LINEOSCILLATORFIG. 12-19 A delay-line phase-noise-measurement system. When the phase shifter isadjusted so that 1'2(r) is in phase quadrature with Volt), the output of the phase detector isapproximately equal to the instantaneous frequency deviation of the oscillator. The spectraldensity of the source may be estimated for Fourier frequencies smaIl compared to the inverse ofthe delay' lime.TN-92

12 FREQUENCY AND TIME MEASUREMENT223the signal from the mixer can be used to estimate the spectral density of thefrequency fluctuations:1S7(!) ~ C' t )2 Sv,vo(f) for f 4, I/1tC d • (12-64)... 7tVo dFrequency discriminators are applied in an analogous fashion. A resonantcircuit is often used to provide discrimination since it produces a phase shiftproportional to the frequency deviation from the resonant frequency. Forexample, the phase shift on reflection from a resonance with loaded qualityfactor Qis¢ = arctan(2Qy) ~ 2Qy, (12-65)provided that the frequency deviation is small compared to the bandwidth ofthe resonance and the applied signal is nearly at the center frequency of thediscriminator. This can be accomplished either manually or with a frequencylockedloop. The design of such a loop is similar to the phase-locked loop ofthe next section. Once again, one can spectrum analyze the signal from themixer to obtainfor f 4, vo/Q· (12-66)The noise floor for measurements made with either a delay line or discriminatornormally results from white voltage noise in the analysis circuitry andis independent of the Fourier frequency. We denote the noise floor Sv,vo(minimum) and find the noise floor for frequency or phase measurements by2 2vvSq,(noise limit) = f~ S7(noise limit) = P(;Q)2 Sv/Vo(minimum).(12-67)Consequently, the discriminator or delay-line technique is limited in sensitivitysince the output voltage is proportional to the frequency deviations.Greater sensitivity is possible using two oscillators in a phase-locked loop.The noise in the reference is an important consideration, even though thereference is passive in the case of a discriminator or a delay line. If theoscillator has sufficiently low noise, then the circuits described measure thevariations of the discriminator center frequency or the delay variations in thedelay line.12.3.2.2 PHASE-LOCKED LOOPThe block diagram for the most general phase-locked loop that will beconsidered here is shown in Fig. 12-20. The noise voltage summed into theloop is a schematic way of representing ¢nU), the open-loop phase noise of theTN-93

224 SAMUEL R. STEINPHASEDETECTO~OSCILLATO~ /v o -< ~ V,UNDE~ TEST 8...LI NOISEVOLTAGEFIG. 12-20 Block diagram of a phase-locked loop. The order of Ihe loop is delermined bythe filter transfer function. For convenience, noise in the oscillator under test is introduced at thesumming junction.oscillator under test. Phase noise in the reference oscillator is denoted bytPr.rrjw) + S",Jw)].(12-72)(12-73)TN-94

12 FREQUENCY AND TIME MEASUREMENT225cINPUT ~-.,t;--4-j,>--"---0 OUTPUTFIG. 12-21 Circuit diagram of the most common loop filter for a second-order phaselockedloop. Resistor R 1 is required for stable operation. Capacitor C provides the lowfrequencygain needed to reduce the phase errors of the first-order loop.FIG. 12-22 Bode plot for the loop filter of Fig. 12-21.Thus, if we know the behavior of Geq(jw), then we can relate the measuredspectrum of the voltage at the output of the phase detector or at the varaC10rtuner to the sum of the spectral densities of the phase noise of the twooscillators.The loop filter is often chosen to be a pure gain. The resulting first-orderloop has a significant drawback: the two oscillators are offset from quadratureby a phase shift proportional to their open-loop frequency difference. Inorder to maintain system calibration, the operator must remove the frequencyoffset from time to time. This problem can be eliminated by using asecond-order loop. Figure 12-21 illustrates one loop filter that can be used toachieve the desired frequency response. The transfer function of this filter isF(s) == (1 + S!2)!S!I' (12-74)where !z == RzC and !I = RIC. Figure 12-22 shows the Bode plot of thefrequency-response function of this filter. Substitution of Eq. (12-74) into Eq.(12-69) yields the open-loop frequency-response function. w~ + 2j(w n wGeq{jw) == - l' (12-75)w-whereand(12-76)(12-77)TN-95

226 SAMUEL R. STEIN10010FIG. 12-23 Bode plOI ofthe open-loop frequency-response function ror a phase-locked loophaving the loop filter of Fig. 12-21. Parameters were chosen to illustrate a stable condition.The first requirement to be satisfied by the loop parameters is that theclosed loop be stable. Since the transfer function G.q(s) has no poles or zerosfor s > 0, a sufficient requirement for the phase-locked loop to be stable isthat the slope of the Bode plot of IGcqUw)! be less steep than -12 dB/octave atthe point where IGcqUw)1 = 1. The Bode plot of /G.qUw)1 is shown in Fig.12-23 for a case where the loop operation is stable.I t is desirable for the loop to be nearly critically damped, that is, s= 1. Atcritical damping the natural frequency of the loop is related to '2 bywwn.~=t = 2/'2' (12-78)Under the same conditions the unity gain frequency is(12-79)The second requirement to be satisfied by the phase-locked loop is relatedto the accuracy with which spectral-density measurements can be made.Substitution of Eq. (12-75) into Eq. (12-72) yieldsK~W4S~)CJ) = (2 2)2 + 4- 2 1 2 [S4>.e'(W) + S",Jw)]. (12-80)w - W n ~ W W nSince the proportionality factor has a high pass response, it is possible to usean essentially constant calibration to relate Sva(w) and S.p(w). For example, ifwe require that(12-81)with no more than 1O~~ error for all Fourier frequencies greater than2n rad/sec, then for the critically damped loop the requirement on '2 is'2 > 1.4 sec.TN-96

228 SAMUEL R. STEIN9.3 GHzOSCILLATORUNDERTESTx 1838COUNTERFIG. 12-24 Use of frequency synlhesis to measure oscillators whose frequency differssignificantly from the available low-noise reference. h may be necessary to use a frequencymultiplier to bring the signal into the range of the available synthesizer or to overcome thesynthesizer's phase noise.appropriate range and to enhance the oscillator noise compared to the shorttermphase noise of the synthesizer. Figure 12-24 demonstrates both aspectsof the technique_The initial mixing stage from the microwave frequency to the rf results in asubstantial heterodyne factor, 77.5 for the example chosen. The output of thefirst conversion stage lies within the range of low-noise commercial frequencysynthesizers, which makes it possible to obtain a fixed, low beat frequencyover a wide range of input frequencies. The initial mixing stage also reducesthe frequency synthesizer's contribution to the measurement-system noise.Figure 12-25 shows the typical phase excursions of a high-quality commercialsynthesizer operated near 5 MHz.Under some circumstances a frequency divider may be used to providethe signal for the second mixing stage, as shown in Fig. 12-26. This techniquehas the disadvantage of requiring a custom divider but results in muchlower measurement noise than the direct use of a synthesizer with a singleheterodyne stage.• (t)O~ \-10-.1 !o 80,000TIME (SEC)FIG. 12-25 Typical phase excursions of a commercial frequency synthesizer.IN-98

12 FREOUENCY AND TIME MEASUREMENT2295.00688 MHz8-----~OSCILLATORUNDERTESTL.---'SMHzREFERENCEFIG.12-26 Use ofa simple divider as a substitute for a commercial frequency synthesizer ina heterodyne measurement system. Better noise performance can result from the initial mixingstage.12.3.3.2 THE DUAL-MIXER TIME-DIFFERENCETECHNIQUEThere is no best answer to the question of how to make freq uency-stabilitymeasurements. However, by combining versatility with low-noise perfonnance,the dual-mixer time-difference technique (Cutler and Searle, 1966: Allanand Daams, 1975) shown in Fig. 12-27 comes close to the ideal. The originalmotivation for this method was to use a transfer oscillator and two mixers inparallel to permit short-term frequency-stability measurements betweenoscillators that have an inconveniently small frequency difference. Thetransfer oscillator is most easily realized with a frequency synthesizer lockedto one of the oscillators, designated oscillator I in Fig_ 12-27. By conventionthe frequency of the synthesizer is set low compared to the oscillator undertest, so we write the frequency of the synthesizer asV = VI (1 - IIR). (12-83)sThe constant R is equal to the heterodyne factor, which can be seen bycalculating the beat frequency between oscillator 1 and the synthesizer:V BI = VI - V. = vt/R. (12-84)OSCILLATOR 1fREQUENCYSYNTHESIZERZERO­CROSSINGDETECTORZERO­CROSSINGDETECTORpOSCILLATOR 2FIG. 12-27 A dual-mixer measurement syslem. The scalars measure the number of wholecydes of elapsed phase. while the time-interval counter measures the fractional cycle.

230 SAMUEL R. STEINThe combination of oscillator, frequency synthesizer, and mixer functions asa divider and scaler I functions as the system clock, recording elapsed time inunits of cycles of oscillator 1.The signals from oscillators 1 and 2 are represented according to Eq.(12-52) with 4>J 0 = 4>20 = 0, and the signal from the synthesizer is writtenv,,(t) = v"o cos[2XVsof + 4>.(r)]. (12-85)The phase of the synthesizer retards nearly linearly in time compared to thephase of oscillators 1 and 2. At time t..., the synthesizer reaches phasequadrature with oscillator I and the beat signal crosses zero (in the positivedirection), producing a pulse from the zero-crossing detector and startingthe time-interval counter. At a later time t N the continued sweep of thesynthesizer has brought it into quadrature with oscillator 2, and a pulse isproduced that stops the time-interval counter. The phase difference betweenthe oscillators can be written in terms of the three counter readings:(12-86)where N is the reading of scaler 2, M the reading of scaler 1, P the reading ofthe time-interval counter, and '0 the period of its time base (Stein er aI., 1983).Comparison with Eq. (12-45) for direct time-interval measurements revealsthat the role of the scalers is to accumulate the coarse phase differencebetween the oscillators, while the time-interval counter provides fine-grainresolution of the fractional cycle. This process is illustrated in Fig. 12-28. Theadvantage of the technique over direct time-interval measurements is that thenoise performance is improved by the large heterodyne factor, allowing timeresolution of 0.1 psec to be obtained. The synthesizer degrades the noiseperformance very little since it contributes to the noise only over the interval'eP'_______ .1:--- -o:~""bJI 000III0000000TIMEFIG.12-28 Total elapsed phase measured b} the dual-mixer system of Fig. 11-17 (solidlinel. This phase mea~urement consists of two components: the number of full cycles that haveelapsed is [he step function ploned as a dashed line: the fractional cycle is the saw-tooth functionploned as open circles.TN-lOO

12 FREQUENCY AND TIME MEASUREMENT 231The average beat frequency VB2(t.v; IN) cannot be known exactly, but it maybe estimated with sufficient precision if it changes slowly compared to theinterval between measurements. If the primed and unprimed variablesrepresent two independent measurements, then12.3.3.3 FREQUENCY MULTIPLICATION(12-87)A frequency multiplier produces n fulI cycles of the output signal for eachcycle of the input signal, where n is an integer determined by the design of thedevice. Such a device is also a phase multiplier, that is, the total phaseaccumulation of the output signal is n times as great as the phase accumulationof the input signal:(12·88)It follows that the spectral density of the output signal is enhanced by a factorof n 2 compared to the input signal,2SiPouJf) = n S¢'lft(f),making it easier to perform the necessary noise measurements. Similarly, it isalso easier to make Allan-variance measurements. If the oscillator under testand the reference are both multiplied by the same factor, the beat frequencywill be n times larger than with no multiplication but the heterodyne factorwill be the same. The zero crossings that must be detected by the counterhave n times higher slope and more easily overcome the voltage noise in thecounter trigger circuits. The ability to measure frequency stability is onlyenhanced if the multipliers have extremely low phase noise themselves. This isthe case for many modern multipliers that are triggered by the zero crossingsof the input signal. As a result, the use of multipliers can reduce theperformance requirements on the phase detector and the following low-noiseamplifiers.12.4 CO~CLUSIO;.'>iThe IEEE recommendations have achieved the goal of introducingsubstantial uniformity in the specification of oscillator performance. TheAllan variance and the one-sided power spectral density of phase have provedsufficient to evaluate oscillators for all common applications. In a few casesmore specialized measures are helpful in relating performance to the specificapplication. For example, the rms time-prediction error is helpful in judging aclock·s ability to keep time over long intervals (Allan and Hellwig, 1978).TN-lOl

232 SAMUEl R STEINHowever, the specialized performance measures are generally calculable interms of the IEEE recommended measures.Significant progress has been made during the last 15 years in measurementtechniques and data processing. These advances have obscured thedividing line between the frequency domain and the time domain. Today thespectral density and the variance are most often computed from the identicalinput data set, the equally spaced time series of the phase deviations. Thechoice of a specific measurement setup can be made mostly on a cost versusperformance basis. Perhaps the biggest advance in commercially availableequipment is the introduction of heterodyne measurement techniques fortime-domain (counter-based) measurements. As a result, the noise performanceof these systems has improved dramatically.One recommendation that should be made is to perform measurements ashigh up in the measurement hierarchy as possible. Direct measurement of thephase deviation is most desirable. This approach places the largest share ofthe burden on the measurement equipment, minimizes long-term errors, andmaximizes data processing flexibility.TN·102

Chapter 12399Ablowich. D .. Jr. (1966) Some observations conccrnin~ frcqucncy and tlmc standardization.Proc. lis/Ins/rum. Soc. Am. Con! 21. 20Al:imochkin. I. K .. Dcnisov. V. P.• and Chcrcnl:ova. E. M (1973). Lon~·tcrm frcquencycomparator. M~as. Tech (Eng/. Trans/.) 16. 83-85Aldridl!c. D. (1975).5 MHz dll!ital frcqucncy mctcr. Eltc/ron (8~1. 61. 63-64. 66, 68Aldridl!c, D. (1978).5 MHz dil!ital frequcncy mcasurinJ! instrumcnt EI~k/ron. An: 10. 15-18. InGcrmanAllan. D W. (1966). Statistics of atomic frcqucncy standards. Proc.IEEE S4. 2.21-~30Allan. D (1975). "The Mcasurcmcnt of Frcquency and Frequency Slabillty of PreCIsionOscillators." NBS Technical Note 669. National Bureau of Standards. Boulder. Colorad0Allan. D. W .• and Barnes. J. A. (19811. A modificd .. Allan vanancc" .....ith Increased OSCillatorcharactcriution ability Proc. jJ/h Ann" Fr~q Control Symp.. pp 4~O-4'3TN-I03

400 CHAPTER BIBLIOGRAPHIESAllan. D. W.. and Daams. H. I 1975). Picosecond time difference measurement system. Proc.191JrAnnu. Freq. Concrol Symp .• pp. 404-411. .Allan. D. W .. and Hellwig. H. (\978). Time deviation and the prediction error for clockspecification, characterization and application. Proc. Position Localion and NQt:igalionSymp.. pp. 29-36.Allan. D. W .. Gray. 1. E.. and Machlan. H. E. (1972). The National Bureau of Standards atomictime scales~ generation. dissemination. stability, and accuracy. IEEE Trans. Instrum . .\feas.IM-21.388-391.Allan. D. W.. Gray. J. E.. and Machlan, H. E. (\974). The National Bureau of Standards atomictime scales: generation. stability. accuracy and accessibility. In "Time and Frequency:Theory and Fundamentals" (B. E. Blair. ed.), NBS Monogr. NO. pp. 205-231.Allan. D. W .. Hellwig, H.. and Glaze. D. J. (1975). An accuracy algorithm for an atomic timescale. Merrologia 11. 133-138.Andres. I. (1977). Straight talk on frequency counters. Communications 504. 56-60.Andriyanov. A. V., Glebovich. G. V., Krylov. V. V.. Korsakov. S. Ya.. and Ponomarev, D. M.(1980). Automated measurements in the time domain and a method of increasing theiraccuracy I:mer. TekJr. 23,42-44. In Russian.Anonymous (1967). Definition. realisation and use of time and frequency. Colloq. Dig. lEE.Paper no. I J·8 7.Anon~mous (1975a l. Electromagnetic radiation in frequency' measurements. Electron. AlLSr. 37,31. nAnonymous (\975b). Analogue frequency meter (rev. counter). Elektor (GB) I. 737.Anon~mous 11975c). l'ni"ersal frequency reference. Elektor (GB) 1. 715.Anonymous (l975d). A programmable counter type frequency meter. Elecrron. Microelectron.Ind. (2131. 45-49. In French.Anonymous (l976a). High accuracy timing devices. Elektron. Elekrroll!ch. 31. 9-11. In Dutch.Anon~mous (\976b). Time and frequency; two important quantilies 10 modem measurements.Tl!cnol. Eleltr. (\2), 66-69. In Italian.Anon~mous (\977a). 50 MHz counter 'timer. CSIO Commun. I/ndial4. 3~.Anonymous (1977b). A un"'ersal counting and time measuring device. Elekron Inc. IS). 167-169.In German.Anonymous (1977c). Precision timebase for frequency counter. Elekror 1GB) 3.22-25..-\nonymous (1977d). Which counter' Auromation 12. 21. 23. 25Anonymous (1978a). Time measurement where are we? M1!5. Regul. Autom. 43. 19-23. InFrench.Anonymous (1978b). A universal counter on one chip. Elekrronikschau 504. ~8. In German.Anonymous (1978c). Yaesu ~Iusen YC·500S frequency counter. Electron. AlLSt. 40. 103.Anon~mous (1979a). A system of measurement and display of frequency for a AM FM receiver.Electron. Appl. Ind. 271. 55-56. In French...t.nonymous (l979b). A frequency counter for radio receivers. Elekrronikschau 55. 56-57. InGerman.Anonymous (19~9c '. A speciallC revolutionizes the design of universal counters. Elekron. Pra.t.14. 14-18. In German.Anon~mous \ 1979d). Microprocessor and lSI: two features of a universal 100 MHz count~r.Electron Appl. Ind. 270. :23-26. In French.Anonymous (197ge). Time frequency measurements in radJr equipment-which technique?Onde Elecrr. 59. 47-5·t In FrenchAntyufee.... V. I.. Breslavets. V. P.. Fedulov. V. M.. and Fertik. N. S. (19801. Application ofdigitalliltratlon for measuring instability R

CHAPTER 17. 401Arnoldl. M. (1974). A frequency counter with time-multiple indication. L Funk· Tech. 20­721-724. In German.Arnoldt, M, (1977), Frequency counter for ultra short wave radio receivers. Funk· Tech. 32,237-242, 244-245, In German,Arnoldt, M. (1979), A frequency counter for YHF receivers with frequency control loops.Fundschau 51,1371-1374, In German.Artyukh, Yu. N., Amit, M"A" Gotlib. G. Land Zagurskii, Y, Ya, (1978), Instrument formeasuring time intervals to an accuracy of 2 nsee, Instrum. Exp. Tech, (Engl. Transl,) 20,1039-1041.Ashley. 1. R.. Searles, C. B.. and Palka, F, M. (1968), The measurement of oscillator noise atmicrowave frequencies. IEEE Trans. Microv,are Theory Tech, MIT-16. 753-760.Auchterlonie. L J., and Bryant. D, L (1978), A direct method ofmeasuring small frequency shiftsusing two open resonators at millimetre wavelengths. Int, J, Electron. 45, 113-122.Ausejo, R. (1976), Frequency and time meters, Mundo Electron, 50, 59-66, In Spanish,Azoubib. L Granveaud, M., and Guinot. B. (1977), Estimation of the scale unit duration of timescales, Metrologia 13. 87-93,Raev, A. V"~ Rozkina, M, 5" and Torbenkov, G, R (1973), Capacitive method for measuringfrequency. Meas. Tech, (Engl, Transl,) 16. 1034-1036,Bahadur. H.. and Parshad. R. (1977), Measurement of time. J, Inst, Electron Telecommun, Eng.(;Vel>' Delhi) 17. 103-108,Baidakov, I. G .. Pushkin, S, B.• and Titov, Y, P. (1973), Securing unanimity oftime and frequencymeasurement in the USSR at the highest section of the reference system, ,'>leas, Tech, (Engl,Trans/,) 16. 60-63,Baird, K. M .. Hanes, G, R,. Evenson, K, M .. Jennings. D. A" and Petersen, F, R. (1979),Extension of absolute·frequency measurements to the visible: frequencies of ten hyperfinecomponents of iodine, Opt. Lerr, 4, 263-264,Balph. T (1976). A 200 MHz autoranging frequency counter. Electron, Equip, News, April. p. 17,Barber, R. E. (l9711. Short-term frequency stability of precision oscillators and frequencygenerators. Bell Syst, Tech, J, 50, 881-915.Barnaba, J. F. (1972). USAF time and frequency calibration program. Proc. 26th Annu, Freq.Control Symp.. pp. 260-263,Barnes. J, A. (1966), Atomic timekeeping and the statistics of precision signal generators, Proc,IEEE 54. 207-219,Barnes, J, A. (1969). Tables of bias functions B I and B 1 for variances based on finite samples ofprocesses with power law spectral densities. ,'IiBS Tech, Note (C.S.) 375.Barnes. J. A. (1972), Problems in the definition and measurement of frequency stability. Proc.26th Annu. Freq. Conrrol Symp .. p. 20.Barnes, J. A. 11976a). A simulation of the fluctuations of International Atomic Time. ,IiBS Tech.Sote ( l'.S,) 609, ::0 pp.Barnes, J. A. (1976b), Models for the interpretation of frequency stability measurements. NBSTech. ,Vote (L'S) 689,39 pp.Barnes. J. A., and Allan. D. W. (1966), A statistical model of flicker noise. Proc. IEEE 54,I ~6-178,Barnes, J. A.. and Winkler. G. M. R, 119~2). The standard of time and frequency in the IJSA,Prof. :!6th Annu. Freq. Control Symp .. pp 269-::78.Barnes, J A.. Chi, A, R .. Cutler. L S, Healey, D. 1. Leeson, D. B. McGunigal, T. E.. Mullen,J. A, Jr, Smith, W. L.. Sydnor. R. L.. Vessot. R. C F.. and Winkler, G, M. R. (\9711.Characterization of frequency stability. IEEE Trans. Instrum, J/eus. D·I·20. 105-120.B:lrrier. B.. and Humbert. J .-P, (197 3). ~umericaI frequency meters from A to Z, Int. Elecrron. 28.33-39. In French,Bastelberger. J. 11%8). Normal frequencies, their investigation and measurement. Fernme/de·1",

402 CHAPTER BIBLIOGRAPHIESBaugh. R. A. (1971). Frequency modulation analysis with the Hadamard variance. Proc. 25/hAnnu. Freq. Control Symp.. pp. 222-225.Baugh. R. A. (1972). Low noise frequency multiplication. Proc. 26/h Annu. Freq. Control Symp..pp.50-54.Bava. E.. De Marchi. A.. and Godone. A. (1977). A narrow outputlinewidth multiplier chain forprecision frequency measurements in the I THz region. Proc. 3!s/ Annu. Freq. ControlS.I'mp.. pp. 578-582.Bay. Z.. and Luther. G. G. (1968). Locking a laser frequency to the time standard. App!. Phys.Lett. 13. 303-304.Becker. G. (1976). The second as standard unit of time. L'msch. Wiss. Tech. 76. 706-707. InGerman.Becker. G .. and Hubner. C. (1978). The generation of time scales. Radio Sci. 14.593-603.Beckman. D. (1974). How accurate is your timer-counter? E1ec/ron. Equip. News IS. 32-33.Begley. W. W.. and Shapiro. A. H. (1970). Precision timing systems. I. Standard scales. Ins/rum.Control Sys/. 43. 87-89.Behnke. H. (1973). Standard frequency comparison by error multiplication. Radio FernsehenElektron. 22. 642-643. In Gennan.Beisner. H. M. (19S0). Clock error With a Wiener predictor and by numerical calculation. IEEETrans. Ins/rum. !>leas. IM-29. 105-113.Belenov. E. M .. and Uskov. A. V. (1979). Measurement oflaser frequencies using superconductingpoint contacts. SOL'. J. Quantum Electron. (Engl. Transl.) 9. 1519-1522.Belham. N. D. N. (1980). How accurate is a digital frequency meter? Radio Commun. 56. 368-371.Belomestnov. G. L. Dmitrienko. A. D .• Emakov. A. S.• and Korzh. V. F. (1976). Automaticsystem fvr acquiring. recording. and processing information by computer for secondarytime-and-frequency standards. !>feas. Tech. (Engl. Transl.) 19. 1572-1574.Belotserkovskii. D. Yu.. lI·in. V. G .. and Stepanova.l. V. r1976). International cooperation in thefield ofe.~act time and frequency measurements. Meas. Tech. (Engl. Transl.) 18. 517-519.Berger. F. (1975). Time metrology and quartz application.!. Time metrvlogy. Rer. Gen. Eleetr.8-1.907-918. In French.Bezdrukov. S. M.. and Preobrazhenskii. N. I. (1979). Increased accuracy in audio-frequencymeasurements with the 54- I 2 and S4-53 spectrum analyzers. Ins/rum. Exp. Tech. (Engl.Transl.) 21. 714~716.Bichler. H.. and Fenk. J. (1981 a). Simple universal 0 to 30 M Hz frequency counter with SDA5680A. Siemens Components (Engl. Transl.) 16. 21-24.Bichler. H.. and Fenk. J. (1981 b). The IC unit. SDA 5680A. universal frequency counter covering100 kHz to 30 MHz. E1ektron Int. 3--4. 89-92. In Gennan.Blachman. I\. M. (1976). The elfect of noise upon polar-display instantaneous frequencymeasuremenl.lEEE Trans. Instrum. .Heas.I"I-25. 214-21.Blair. B. E. (1974). Time and frequency: theory and fundamentals. .vBS .Honogr. ([:.S.) 1-10.460 pp.Blaney. T. G. Cross. N. R.. Knight. D. J. E.. Edwards. G. 1.. and Pearce. P R (1980). Frequencymeasurement at 4.25 THz (70.5 Ilm) using Josephson harmonic mi.'er and phase-locktechniques. J. Phys. D 13. 1365-1370.Bodea. M.. and Danila. T. (19'79). Autoranging. digital time-interval meter. Bull. Inst. Politelt.Gheorghe Gheorglriu-Dej Bucuresti Ser. E1ec/roteh. -10. 57-63. In Rumanian.Body. l.. and Szemok. L (1974). General problems of frequency measurement .\feres AUlOm. 22.422-428. In Hungarian.Bologlu..-\. ( 1975). Automatic 4.5-G Hzcounter provides I-Hz resolution. He\\'lett-PackllrJ1. 27.14-18.TN-106

CHAPTER 12 403Bologlu. A. (1980). A new method of measuring the frequency of microwave signals. usingmicroprocessors. Nachr. E/eklron. 34. 149-152. In German.Bologlu. A.. and Barber. V. A. (1978). Microprocessor-controlled harmonic heterodynemicrowave counter also measures amplitudes. Hew/ell-Packard J. 29. 2-4. 6-7. 9-12. 14-16.Bordt. A. (1973a). Universal digital counter for frequencies up to 12 MHz. I. Radio FernsehenE/eklron. 22. 365-368. In German.Bordt. A. (l973b). Universal digital counter for frequencies up to 12 MHz. II. Radio FernsehenE/eklron. 22. 402-406. In German.Borisochkin. V. V. (1972). Checking the frequency of highly stable generators. Meos. Tech. (Eng/.Trans/.) 15.460-463.Bosnjakovic. P. (1976). A new solution of the precision line frequency measurement. AUlOma/ika17. 194-196. In Serbo-Croatian.Bowman. 1. A. (1973). Short and long term stability measurements using automatic data recordingsystem. Proc. 27/h Annu. Freq. Control Symp.. pp. 440-445.Bowman. M. J .. and Whitehead. D. G. (1977). A picosecond timing system. IEEE Trans. Ins/rum.Meas. IM-26. 153-157.Brandenberger. H.. Hadom. F.. Halford. D .• and Shoaf, J. H. (1971). High quality quartz crystaloscillators: frequency domain and time domain stability. Proc. 25/h Annu. Freq. Con/rolSymp.. pp. 226-130.Brown. D. E. (1957). Frequency measurement of quartz crystal oscillators. Marconi Ins/rum. 11.12-15.Budreau. A. J.. Siobodnik. A. 1.. Jr.. and Carr. P. H. (1982). Fast frequency hopping achievedwith SAW synthesizers. Micro .....au J. 25. 71-73. 76-80.Burgoon. R.. and Fischer. M. C. (1978). Conversion between time and frequency domain ofintersection points of slopes of various noise processes. Proc. 32nd Annu. Freq. ControlSymp.. pp. 514-519.Buchuev. F. I.. and Ivakin. V. M. (1979). Electronic converter of the unit time dimension. .~{eas.Tech. (Eng/. Trans/.) 21. 788-789.Buxton. A. J. (1976), Digital frequency meter. Prac/. E/ecrron. 12.376-382.Buzek. 0 .. Cermak. J.. Smid. 1.. and Medellin. H. (1980a). A time base controlled by the OMA50 kHz transmitter. Sde/ovaci Tech. 28. 362-365. In Czech.Buzek. 0 .. Cermak. 1.. and Tolman. J. (I 980b). Accurate time and frequency. S/ahoproudy Ob=.41. 263-266. In Czech.Cantero. J. A.. Pollan. T.. and Barquillas. 1. (1978). Digital frequency meter uses variablemeasuring and automatic scale conversion. Mundo E/ecrron. 71. 34-35. In Spanish.CardwelL W. T.. Jr. (1978). Frequency counter in a probe. Radio E/ec1ron Eng. 49. 67. 72-73.100-10I. 103.Charriere. J.-J. (1977). A digital chronometer. I. Ret'. Poly/echo (6). 585. 587. In French.Chebotayev. V. P.. Goldort. V. G .. Klementyev. V. M.. Nikitin. M. V .• Timchenko. B.A.. andZakharyash. V. F. (1982). Development of an optical time scale. App/. Phys. B 29.63-65.Coates. P. B. (1968). Fast measurement of short time intervals. 1. Sci. [nsrrum. l. 1123-1127.Cochran. W. T .. Cooley. J. W.. Fallin. D. L.. Helms. H. D.. Kaenel. R. A .• Lang. W. W., Maling.G. c., Jr.. i\ielson, D. E., Rader. C. M.. and Welch. P. D. (1967). What is the fast Fouriertransform? I££E Trans. Audio E/ec/roacousr. .-\V-15. 45-55.Colburn, J.. and Owen. B. (1979).600 MHz portable frequency counter. Radio E/ecrron. Eng. 50.39-43.90.Cole. H. ;v!. (1975). Time domain measurements. E/ecrron. E/ecrro-Opric In! Counrermeas. l.38-42. 55.Con.... ay. A .. and L'rdaneta. N. (l978a). Considerations in selecting a frequency counter.CommumCQfions 15. 22-:3.TN-to?

404 CHAPTER BIBLIOGRAPHIESConway. A.. and Urdaneta. N. (1978b). Selecting a frequency counter. T"I"ph. Eng. Jlalltlg"m"nt82.77-78.Cordwell. C. R. (1968)... Frequency Measurements." Vacation school on electrical measurementpractice. Institution of Electrical Engineers. London. 13 pp.Cordwell. C. R. (1969). Frequency measurements. Electron. Radio Tech. 3. 95-103.Cosmelli. C. and Frasca. S. (1979). Fast new technique for measuring the resonance frequenciesof high Q systems. Ret·. Sci.lnstrum. SO. 1650-/651.Cutler, L. S.• and Searle. C. L. (1966). Some aspects of the theory and measurement of frequencyfluctuations in frequency standards. Proc. IEEE 54. 136-154.Czarnecki. A. (19801. Test set for the comparison of standard frequencies. type FZ-530. Pro Inst.Tele-Radiotech. (83). 27-32. In Polish.Danilevich. V. V.. Chernyavskii. A. F.. and Yakushev. A. K. (1977a). Seleclro-type time intervalanalyzer with a measurement range of 10- 10 to 10- 2 sec. Instrum. Exp. Tech. (Engl. Trans!.)20.326-327.Danilevich. V. V.. Kvachenok. V. G .. Priimak. M. A.. Chernyavskii. A. F.. and Yakushev. A. K.(1977b). Signal generator for adjusting and calibrating time measurement systems. Instrum.Exp. Tech. (Engl. Transl.) 20. 600.Danzeisen. K. (1977). Remote frequency measurement using VHF-UHF receivers. Funk-Tech.32.196-197. In German.Davis. D. D. (1975). Calibrating crystal oscillators with TV color-reference signals. Electronics48.107-112.De Costanzo. L. (1979). Radio receiver frequency counter clock. Electron. Ind. 5. 16-17.De Jong. G .. and Kaarls, R. (1980). An automated time-keeping system. IEEE Trans. Instrum.!tieas. Il\I·29. 230-233.Denbnovetskii, S. V.. and Kovtun. A. K. (1970). Time measurements. Digital time-intervalmeters. I:mer. Tekh. 10. 29-31. In Russian.Denbnovetskii. S. V.. and Shkuro. A. N. (1975). Principles of eliminating ambiguity in digitaltime·interval meters (review). Instrum. Exp. Tech. (Engl. Transl.) 18.1659-1670.De Prins. J. (1974). Timing systems. In .. ELF-VLF Radio Wave Propagation Spatind. Norway,"pp. 385-398. Reidel. Dordrecht. The Netherlands.De Prins. J. (1977). Aspects of time and frequency measurement. Ret·. HFIO. 123-134. In French.De Prins. 1.. and Cornelissen. G. (1974). Critique and discussion about frequency slabilitydefinitions with respect to various kinds of noises. CPEJf 74 Dig., pp. 54-56.Dietzel. R. (1975). Passive resonators with noise feeding in frequency-analogue measuringsystems. Jfess.. Steuern. Regeln 18. 389-393. In German.Dombrovskii. A. S.. Zaitsev. V. N .. and Pashev. G. P. (1974). Apparatus for time and frequencymeasurement. .\1eas. Tech. (Engl. Trans!.) 17.919-9:1.Domnin. Y. S.. Kosheljaevsky. N. B.. Tatarenkov. V. M .. and Shumjatsky. P. S. (1980a). Precisefrequency measurements in submillimeter and infrared region. IEEE Trans. Instrum. Meas.1:\-1-29.264-26'Damnin. Yu. S.. Kosheljaevsky. N. B.. Tatarenkov. V. M .. and Shumjatsky. P. S. (1980bl Precisefrequency measurements in submililmeter and infrared region [lasers]. CPE.H Dig .. p. 85.Doring. F. W. and Hollenberg. K. (1981). Eight.channel time measurement system with 2 nsresolution. J. Phys. E 14. 638-640.Doronina. O. M .. and Petukh, A. M (1980). Interpolation method of increasing measurementaccuracy for low frequencies. A.L"(ometriya IS). 89-9:. In RUSSIan.Drake. J. M.. and Rodriguez-Izquierdo. G. (1978). Analogue setup for fast-response frequencymeasurement. Electron. Eng. SO. 41. 43.Drake. 1. M .. ROdriguez-Izquierdo. G. (1980). Fast analogue measurement oflow frequencies.ReI'. Tdegr. Electron. 69. 1359-1360. In Spanish.TN-lOB

CHAPTER 12 405Egidi. C (1969). Frequency:lime metrology al the I.E.1'

406 CHAPTER BIBLIOGRAPHIESGardner. F. M. (\966). "Phaselock Techniques." Wiley. New York.Godone. A.. De Marchi. A.. and Bava. E. (\979). An improved multiplier chain for precisefrequency measurements up to 20 THz. Proc. JJrd Annu. Freq. Control Symp.. pp. 498-503.Gonzalez.. R. L.. and Olivero. M. A. (1980). A digital frequency meter. Ret'. Telegr. Electron. 68.67-72. In Spanish.Granveaud. M. (\ 980). Contribution of the computation algorithm to the instability of atomictime scales. CPEM Dig.• pp. 7-8.Granveaud. M.. and Azoubib. J. (1974). The problem of ageing in the derivation of TAl[International Atomic Time]. CPEM Dig.• pp. 279-280.Granveaud. M.. and Guinot. B. (1978). The role of primary standards in establishing theInternational Atomic Time TAL CPEM Dig.. pp. 39-41.Grauling. C. H.. Jr.• and Healey. D. J.• III (\ 966). Instrumentation for measurement of the shortt.::rmfrequency stability of microwave sources. Proc.IEEE54. 249-257.Gray. J. E. (1976). Clock synchronization and comparison: problems. techniques and hardware.NBS Tech. Note (U.S.) 691. 9 pp.Greivulis. Ya. P.. and Blumbergs. E. A. (1976). Frequency meter. I=v. Vyssh. Vcheb". Zat·ed.Priborosrr. 19. 13-15. In Russian.Grenchik. T. J.• and Fang. B. T. (1977). Time determination for spacecraft users of the NavstarGlobal Positioning System (GPS). Proc. 31st Annu. Freq. Control Symp.. pp. 1-3.Groslambert. J. M. (1971). Automatic measurement system of oscillator frequency stability.Eurocon Dig. (IEEE). p. I.Groslambert. J., Olivier. M .• and Uebersfeld. J. (1972). Automatic plotting systems for frequencystability curves and discrete spectral density of frequency standards. IEEE Trans. Instrum.Meas. IM-ll. 405-409.Groslambert, J .• Olivier. M .. and Uebersfeld. J. (l974a). Spectral and short term stabilitymeasurements. IEEE Trans.lnstrum. Meas.IM-23. 518-521.Groslambert. J., Olivier. M .• and Uebersfeld. J. (1974b). Spectral and short term stabilitymeasurements. CPEM Dig.. pp. 60-61.Guignard. A .• and Berney. J. C. (1967). A new series of electronic instruments for timemeasurements and standard-time distribution. Mes .. Regul.. Autom. 32. 100. In French.Guinot. B. (1967). Mean atomic time-scales. BIIII. Astron. 1. 449-4&4. In French.Guinot. B. (197 I l. Characteristics of a scale of time. ,.lira Fre,]. 40. 684-686.Guinot. B. (\974). Long term stability and accuracy of time scales. CPEM Dig .• pp. 333-335.Guinot, B. (\ 980). Problems of the generation. quality. and availability of the InternationalAtomic Time Scale. CPEM Dig.. pp. 5-6.Guinot. B.. and Granveaud. M. (\ 972). Atomic time scales. IEEE Trans. Ins/rum. Meas. IM-li.396-400.Guinor. B.. Granveaud. M .. and Azoubib. J. (1981). Stability and accuracy of the internationalatomic time. 1. Ins1. Electron. Telecommun. Eng. LVe'" Delhi) 27. 514-517.Gutnikov. V. S.. Nedashkovskii. A. I.. and Palkin. V. S. (1977). A specialised digital frequencymeter used for operation with measuring frequency convertors. Prib. Sisto Cpr. 5. 21-14. InRussian.Hafner. E. (\969). Frequency and time. Proc. Conf Ne ..... Hori=ons in Measuremems-Trends inTheary and Application (IEEE). pp. 58-68.Halford. D. Wainwright. A. E.. and Barnes. J. A. (1968\. Flicker noise of phase in RF amplifiersand frequency multipliers: characterization. cause and cure. Proc. 22nd Annu. Freq. CantrolSymp.. pp. 340-341.Halford. D .. Shoaf. J. H.. and Risley. A. S.1I973). Spectral density analysis: frequency domainmeasurements of frequency stability. Proc. :lith Annu. Freq. Control Symp.. pp. 411-43/TN-ll0

CHAPTER 12 407Hamanaka. A. (1978). New frequency counter series features high reliability. JEE. J. Electron.Eng. (142).50-54.Harry. E. (1978). Counter. scope team up for precise timing measurements. Electronics 51.126-127.Healey. D. 1.. III (1972). Flicker of frequency and phase. and white frequency and phasefluctuations in frequency sources. Proc. 26th Annu. Freq. Control Symp., pp. 29-42.Healey. D.l.. III (1974). L(j) measurements on UHF sources comprising VHF crystal controlledoscillator followed by a frequency multiplier. Proc. 28th Annu. Freq. COn/rol Symp., pp.190-202.Hellwig. H. (1981). Reliability. serviceability and performance of frequency standards. J. Inst.Electron. Telecommun. Eng. (Nel

408 CHAPTER BIBLIOGRAPHIESJackisch. C. Schubert. H. J., and Hartmann. H. L. (1980). Measurement of time fluctuations insynchronous communication networks. CPEM Dig .• pp. 34-39.Jayapal, R. (1977). Measure power frequency with a conventional counter. Electron. Eng. 49.17.Jenkins. G. M .. and Watts. D. G. (1968). ,. Spectral Analysis and Its Applications." Holden-Day,San Francisco.Jennings, D. A.• Petersen, F. R.. and Evenson. K. M. (1975). Extension of absolute frequencymeasurements to 148 THz: frequencies of the 2.0 and 3.5 J.lm Xe laser. Appl. Phys. Lell. 26,510-511.Jennings. D.A., Petersen. F. R., and Evenson. K. M. (1979). Frequency measurement of the 260­THz (1.15 J.lm) He-Ne laser. Opt. Lell. 4.129-130.Jimenez, J. J., and Petersen, F. R. (1977). Recent progress in laser frequency synthesis. InfraredPhys. 17.541-546.Job. A. (1969). Short term stability ofsinusoidal generators. Onde Electr. 49, 528-538. In French.Jones. C H. (1979). A test equipment for the measurement of phase and frequency instability inVHF and SHF sources. Radio Elec/ron. Eng. 49. 187-196.Jutzi. K. H. (1973). Frequency measurement by signal sampling. Frequenz 21, 309-311. InGerman.Kahnt. D. (1977). ASMW time and frequency signals and their applications. Radio FernsehenElek/ron.26. 112-113. In German.Kalau, M. (1980). The GDR atomic time scale. CPEM Dig., pp. 11-14.Kamas, G., and Howe, S. L. (1979). Time and frequency user's manual. NBS Spec. Publ. (U.S.)559,248 pp.Kamyshan. V. V., and Valitov, R. A. (1969). Heterodyne method of frequency measurement inthe millimeter range. Instrum. Exp. Tech. (Engl. Transl.) (4), 925-928.Karpov, N. R. (1980). Vernier method of measuring time intervals. Meas. Tech. (Engl. Transl.)23.817-822.KartaschofT. P. (1973). Terms and methods to describe frequency stability. Tech. Mill. PTT .51,520-529. In German.KartaschofT, P. (1977). Frequency control and timing requirements for communications systems.Proc. 31s/ Annu. Freq. Control Symp.• pp. 478-483.Kazi. A. M., and Perini. G. J. (1971). Program execution time measurement. IBM Tech.Disclosure Bull. 14. 428.KendalL W. 8., and Parsons. P. L. (1968). A precision frequency measurement technique for amoving signal with a low signal to noise ratio. Conf Precision Elec/romagn. Meas (IEEE),pp.54-55.Kesner, D. (1973). Take the guesswork out of phase-locked loop design. EDN 18, 54.Kiehne. F. (1967). FEGR-A mUlti-purpose 500 kHz counter. News Rhode and Schwar: 1.15-16.Kimel'blat, V. I. (1978). Evaluation of measuring accuracy of middle frequency and bandwidthfor radio-receiving apparatus. .l4eas. Tech. (Engl. Transl.) 21,541-543.Kirchner, D. (1983). Precision timekeeping at the Observatory lustbueheL Graz, Austria. Proc.37/h Annu. Freq. Control Symp.. pp. 67-77.Kirianaki. N. V.. Kazantsev. E. M.. and Mosievich. P. M. (1977). Multirange fast automaticdigital frequency meter. I:t·. Vyssh. C:chebn. Zar:ed. Priboros/r. 20. 9-14. In Russian.Klimov, A. Land Meleshko. E. A. (1977). Monitoring unit for time-interval analyzers ofnanosecond range. Ins/rum. Exp. Tech. (Engl. Transl.) 20. 1058-1060.Knight. D. J. E.. Edwards. G. L and Blaney. T. G. (1977). Developments in laser frequencymeasurement towards realising the metre from a unified frequency and length standard.European Conf Precise E1ectr. Meas. (lEE). p. 40.Kobayashi, M.. Osawa. H., Morinaga.'N.. and Namekawa, T. (1973). Output signal·to-noiseratios in mean frequency measurement system. Tech. Rep. Osaka Unit·. 23, 515-526.TN-112

CHAPTER 12 409Kobayashi. M.. Osawa. H.. Morinaga. N.. and Namekawa. T. (1976). Mean frequency and meanpower measurement using correlation detection. Trans. Soc. Instrum. Control Eng. 12.613-618. In Japanese.Kocaurek. R. (1968). Frequency comparison receiver EF 151 k. Ne"..s Rohde and Sch •.-ar: 8.23-24.Koch. E. (1979). An all-waveband receiver with a digital frequency display and a quartz clock.Funkschau 51.236-239. In German.Koga. I. (1970). Precision measurement of crystal Frequency by means of" center line method."Proc. 24th Annu. Freq. Control Symp.. pp. 168-171.Kogan. L. R. (1970). Oscillator stability requirements in radio interFerometers. Radiotekh.Eleklron. 15. 1292-1294. In Russian.Kolbasin. A. I.. and Pavlenko. Yu. F. (1978). TransFer of the dimension of a unit deviation ofFrequency from the special state standard to prototype means of measurement. Meas. Tech.(Engl. Trans'.) 21. 806-809.Koppe. H. (1979). Frequency measurement using a microcomputer-a program and aninstruction program. Elekrron. Prax. 14. 30. 35-36. 39. In German.Kotel'nikov. A. V.• Rubailo. G. T.. and Uvarov. N. E. (1975). An instrument For measuring theratio of time intervals. Te!ecommun. Radio Eng. (Engl. Transl.), ParI 129.60-61.Kourigin. E. I.. Chemyshev. V. S.. and Shumkin. Yu. V. (1979). High-speed device for timeinterval measurement. Inslrum. Exp. Tech. (Engl. Transl.) 22. 484-486.Kramer. G. (1970). An electronic time intervalometer with high resolution. Nachrichlentech. Z.23.433-436. In German.Krause. W. H. U. (1968). Accurate time and frequency measurement. Frequen: 22. 178-182. InGerman.Kuchynka. K.. and Preuss. A. (1981). Electronic counters facilitate time measurement. SiemensPower Eng. 3.179-182.Kul'gavchuk. V. M. (1975). Frequency meters using devices with S-shaped volt-amperecharacteristics. Inslrum. Exp. Tech. (Engl. Transl.) 18. 514-516.Kuznetsov. V. P.. and Reznikov. A. L (1974). Adjacent time intervals measured by two Frequencymeters. Meas. Tech. (Engl. Transl.J 17.474-475.Laker. K. R.• and Slobodnik. A. 1.. Jr. (1975). "Impact of SAW Filters on RF. Pulse FrequencyMeasurement by Double Detection." Report AFCRL-PSRP-621. Air Force CambridgeResearch Laboratories. Hanscom Air Force Base. Massachusetts. 194pp. Available fromthe National Technical InFormation Service. Springfield. Virginia.Lance. A. L.. Seal. W. D .. Mendoza. F. G .. and Hudson. N. H. (1977). Automating phase noisemeasurements in the frequency domain. Proc. 31s1 Annu. Freq. Control Symp.. pp. 347-358.Leah. P. (1977). Frequency counter timer. Pracl. Elalron. 14.22-28.Leikin. A. Ya.. and Orlov. E. Z. (1973). Equipment for comparing frequencies of highly-stablegenerators. JIeas. Tech. (Engl. Transl.) 16. 1469-1470.Leikin. A. Ya .. Tomashenko. I. V.. and Fertik. N. S. (\974). Application of a rubidiumdiscriminator for the measurement of frequency flucluations. 1:1'. Vl·ssh. Cchehn. ZaredRadlO(i:. 17.851-854. In Russian.Lemesev. S. G.. Rachev. B. T .. Strugach. V. G .. and Shub. S. M. (1975). Statistical method formeasuring a recurrent time interval using a random pulse signal. 1:1'. Vyssh. Uchebn. Zared.Priboroslr. 18. 16-21. In Russian.Lesage. P. (1982). Characterization oFfrequency stability: bias due to the addition of time intervalmeasureme:nts. CPEJf Dig.. pp. M.8-9.Lesage. P. (1983). Characterization of frequency stability: bias due to the: juxtaposition of tlmeintervalmeasurements. IEEE Trans. Inslrum. l.leas. 1:\-1-32. 204-207.TN-113

410 CHAPTER BIBLIOGRAPHIESlesage. P.. and Audoin. C. (1973). Characterization of frequency stability: uncertainty due to thefinite number of measurements. IEEE Trans. Instrum. Meas. IM-22. 157-161. withcorrections from IEEE Trans. Instrum. Ideas. IM-25. 270.lesage. P.. and Audoin. C. (1975a). A time domain method for measurement of the spectraldensity of frequency fluctuations at low Fourier frequencies. Proc. 29th Annu. Freq. ControlSymp.. pp. 394-403.Lesage. P.. and Audoin. C. (1975b). Comments on "Characterization of frequency stability:uncertainty due to the finite number of measurements," IEEE Trans. Instrum. Meas. 1:\1-24.86.lesage. P.. and Audoin. C. (1977). Estimation of the two-sample variance with a limited numberof data [signal generator frequency stability]. Proc. 31st Annu. Freq. Control Symp.. pp.311-318.Lesage. P., and Audoin. C. (1979a). Characterization and measurement of time and frequencystability. Radio Sci. 14, 521-539.lesage. P.• and Audoin C. (1979b). Effect of dead time on the estimation of the two-samplevanance. IEEE Trans. Instrum. Meas. 1:\1·28.6-10.Lo Faso, G. (1968a). Digital frequency meter and chronometer in RTL micrologic. Int. £lekrron.Rundsch. 22. 63-66. In German.Lo Faso. G. (l968b). RTL Micrologic in a digital frequency and time counter. Int. Elektron.Rundsch. 22.47-50. In German.Lomas, G. (1974). Signal-frequency meter. Wireless World 80, 429-434.Loose. P. (1979). Modern electronic-based timing systems. Electron. Eng. 51. 81. 83-84.Lubentsov. V. F. (1973). The errors of the absolute method for measuring difference-signalfrequency. TeIecommun. Radio Eng. (Eng/. Transl.), Part 2 28. 96-99.Lubentsov. V. F., and Ashitkov. I. N. (1968). Device for measuring the frequencies and errors ofcrystal oscillators. Meas. Tech. (Engl. Transl.) (7), 906-911.Lukas. M. (1968). An intermediate frequency measuring equipment with an unusual frequencyindication. Siaboproudy Ob;. 29.323-326. In Czech.MacLeod. K. J. (1973). A portable high-resolution counter for low-frequency measurements.Hewlell-Packard J. 25. 10-15.Malell. W. (1974).100 MHz universal counter FET 100. News Rohde & Schwar; 14. 5-7.Markowitz. W. (1968). Time and frequency measurement. atomic and astronomical. Proc. 5thAnnu.ISA Test Meas. Symp.. lnstrument Society ofAmerica. Pittsburgh. Pennsylvania. p. 6.~larlin. D. (1971). Frequency stability measurements by computing counter system. He ....lell­Packard 1. 23. 9-14.Martin. D. (1974). Measure time interval precisely. Electron. Des. 22. 162-167.Martin. J. D. (1972). Digital methods of frequency measurements: a comparison. Radio Electron.Eng. 42. 285-294.Maton. G. W. (1978). Thoughts on frequency meters and counter-timers. Electron (134),24,27.Maxwell. B. J. (1980). Perception of errors in instruments and applications. L Time, our mostaccurate measurement. Autom. Control (Ne... Zealand) 10, 27-29.McCarthy, E. P. (1979). A digital instantaneous frequency meter. IEEE Trans. Instrum. ;\feas.1\-1-28.224-226.McCaskill. T. B.. and Buisson. J. A. (1975). NTS,/ (TIMATION IIIl [satellite] quartz andrubidium oscillator frequency stability results. Prot'. 29th Annu. Freq. Controi Symp., pp.425-428.McCaskill. T.. White. 1.. Stebbins. S. and Buisson, J. (1978). NTS-2 cesium frequency stabilityresults. Proc. 32nd Annu. Freq. Control Symp.. pp. 560-566.',;fcCellan, G. (1978\. Prescaler and LSI chip form 135-MHz counter. Electronics SI. 97. 99.~leer. P. (1977). Four fune'tlon calculator can measure time. Electron. Eng. 49. 35.TN-114

CHAPTER 12 411Melsa.J. L.. and Schultz. D. G. (1969).•. Linear Control Systems:' McGraw-HilI. New York.Mende. F. F.. and Charkin. V. A. (1973). Superhigh frequency measurement with electroniccounterfrequency meters. Meas. Tech. (Engl. Transl.) 16. 1360-1362.Millham. E. H. (1974). High-accuracy time meter. IBM Tech. DisdoSilre Bull. 17.1143-1144.Mints. M. Ya.. and Chinkov. V. N. (\980). Accuracy of time interval measurement by pulsecounting. Meas. Tech. (Engl. Transl.) 23. 603-606.Mitsenko. I. D., and Borisov. S. K. (\975). Phase-pulse discriminator for measurement ofnanosecond time intervals. Instrum. Exp. Tech. (Engl. Transl.) 18. 144-145.Mohos. Z. (1973). Time interval measurement by averaging. Meres Autom. 21. 268-272. InHungarian.Mroz. T. J. (1977). Time-interval meter reads digitally to 99.9 ms on a DVM Ie. Electron. Des. 25.108. 110.Muchiauri, A. A. (1973). A digital method offrequency measurement for hannonic signals. Meas.Tech. (Engl. Transl.) 16.1710-1712.Muhlenbroich. H. P. (1980). The mode of operation of time interval measurement. Elektron. J.15. 32-34. In Gennan.Mungall. A. G. (1971). Atomic time scales. Metfologia 7, 146-153.Murashov. B. P., and Tsvetkov. Yu. P. (1976). Instrument for checking quartz-crystal oscillatorswith standard frequency signals in the long-wave and very-long-wave ranges. Meas. Tech.(Engl. Transl.) 19,1012-1015.Muzychuk. O. V .• and Shepelevich. L. G. (1974). On the detennination of short time signalfrequency instability. b·. Vyssh. Uchebn. Zaved. Radiofi:. 17. 855-863. In Russian.Nak.adan. Y.• and Koga. Y. (1977). Measurements of frequency stability of precision oscillators.Dyo Bursuri 46.572-579. In Japanese.Navarro-Fernandez. A. (1974). Adapting a frequency meter into a digital clock. Mundo Electron.(34). 92. In Spanish.Niki, S. (1975). Automatic frequency counter uses TRAHET technique to count to 32 GHz with- 20 dBm sensitivity. JEE. J. Electron. Eng. (107). 20-23.Offennann, R. W.• Schultz. S. E.• and Trimble. e. R. (1975). Active probes improve precision oftime interval measurements. Hewlell·Packard J. 27, 11-16.Ohman. B. (1978). l'ew task for the telecommunication authority-national measurementstandards for time and frequency. Tele 84. 14-24. In Swedish.Orte. A. (l976). Comparison of various fundamental methods of defining time. with specialreference to astronomical application. NaL"igation 14. 72-84.Osakabe. M.. Sasao. K.. and Kato, E. (l973). Field strength and frequency measuringinstruments. Anritsu Tech. Bull. (29). 52-56. In Japanese.Pacheu. Y. (1979). Analogue frequency meter. Electron. Appl. Ind. (262S). 15-16. In French.Page. R. (1979). Time and frequency counting techniques. Electron. Eng. 51. 47-48. 51. 53, 57-58.Palamaryuk. G. 0 .. and Kholkin, I. I. (1967). Digital smoothing frequency meter for variablesignpulse signals. Meas. Tech. (Engl. Transl.) (7). 845-847.Palii. G. N.. and Pushkin. S. B. (1980). Improvement of time and frequency stabilization. Meas.Tech. (Engl. Transl.) 23.115-118.Parkmson. B. A. R. (1970). Precise time and frequency measurement. H. Electron. Equip. Sews12,60-65.Pashev. G. P.. and Parfeno\', G. A. (1977). Analysis of the errors of statistical devices measuringrepeated time intervals. ,"'Ieas. Tech. (Engl. Transl.) 20. 347-349.Patankar. A. R.. Nambiar. K. P. P .. and Sltharaman. M. R. (1974). An automatic electronicfrequency comparator. J. Inst. Electron. Telecommun. Eng. (N..... Delhi) 20. 405-407.Patykov. V. G .. Podlesnyi. S. A.. and Chmykh. M. K. (1976). Digital frequency meter with highnoise immunity.lnstrum. Exp. Tech. (Engl. Transl.J 19. 1581.TN-llS

412 CHAPTER BIBLIOGRAPHIESPaul. F. (1973). Reciprocal frequency meters. Electron. Microelectron. Ind. (\68). 55-59. InFrench.Peltz. l. G. (\ 971). Highly stable audio generator for frequency measuremenlS on clock crystals.Elektronik (Munich) 20. 345-348. In German.Percival. D. B. (1978a). The U.S. Naval Observatory clock time scales. IEEE Trans. Instrum.Meas. IM-27, 376-385.Percival. D. B. (1978b). The U.S. Naval Observatory clock time scale. CPEM Dig.. Con!Precision Electromagn. Meas., pp. 42-44.Percival. D. B. (1982). Characterization of time and frequency instabilities: prediction errorvariance. CPE,I,[ Dig.. pp. N;7-9.Percival. D. B., and Winkler. G. M. R. (1975). Timekeeping and the reliability problem. Proc.29th Annu. Freq. Control Symp., pp. 412-416.Peres. A. (1980). Measurement of time by quantum clocks. Am. J. Phys. 48, 552-557.Perkinson. R. E. (1970). Time.frequency technology in system development. Proc. 24th Annu.Freq. Control Symp., pp. 315-321.Petersen, A. P. (1980). A new apparatus for the precise detennination of time and frequency onthe IF band using transmitter DCF 77. CPEM Dig.. pp. /7-19.Petrov. A. F.. Zhukov, E. 1., Kosharnovskii, G. V., Breslavets. V. P., and Fedulov, V. M. (1977).Frequency comparator based on a delay line. Meas. Tech. (Engl. Transl.) 20.1456-1458.Pokorny. L (1976). Modern frequency counters. Radio Elekrron. Schau (Austria) 52, 20-24. InGennan.Pokorny. l. (1977). Hewlwtt-Packard 5328A universal counter with DVM options.Elektroniskschau (Austria) 53. 56.Potschke. H. (1976). Time measurement. Nachr. Elekrron. 30, 107-108. In Gennan.Powers. R. S., and Snyder, W. F. (1969). Radio-frequency measurements in the NBS Institute forBasic Standards NBS Tech. Note (V.S.) 373. pp. 37-42.Pritchard. N. B. (1979). A frequency counter for a 144 MHz transmitter. Radio Commun. 55,410-415.Prusakov. V. K.. and Kilna. A. A. (1974). About digital hannonic signal frequency measuringaccuracy. Liet. TSR Mokslu Akad. Darb., Ser. B (80), 141-148. In Russian.Putkovich. K. (1972). Automated timekeeping. IEEE Trans. Instrum. Meas. IM-21. 401-405.Ramamurthy. 1. V.• and Kumar. M. K. (1979). A measurement technique forlow frequencies. J.Phys. E. 12. 187-188.Reeder. R. (1977). The not so simple counter. Electron. Equip. News (November), 9-10.Reider. G. A. (1980). Standard deviation of averaged digital time interval measurement [errorvalues]. Ret·. Sci. Instrum 51. 1423-1424.Reid. J. H. (1972). The measurement of time. 1. R. Astron. Soc. Can. 66. 135-148.Reinhardt. V.. and lavanceau. J. (974). A comparison of the cesium and hydrogen hyperfinefrequencies by means of Loran C and portable clocks. Proc. 28th Annu. Freq. Control Symp.,pp. 379-383.Reussner. 1.. and Weiss. H.-J. (1973). Standard frequency receiver for calibrating frequencycounters. Radio Fernsehen Elektron. 22. 638-640. In Gennan.Ribenyi. A. (1967). Digital frequency measuring instruments. Meres Automat. 15.436-439. InHungarian.Rikhter. V.. and Fogler. G. (1977). Influence of stresses due to discontinuous jamming on theaccuracy of digital frequency measuring devices. Prib. Sisto Vpr. ( I). 21-23. In Russian.Risley. A. S. (\976). The national measurement system for time and frequency. NBS Spec. Publ.(CS) 445-1. 64 pp.Rivamonte. J. M. (1972). U.S. Army calibration program. Proc. 26th .4nnu. Freq. Control Symp..pp. 258-259.Roth. R. (1977). Alignment generator AS5 with quartz controlled frequency indicator. GrundigTech. In! 24. 209-216. In Gennan.TN-116

CHAPTER 12 413Rozwadowski. M. (1971). Modem frequency stabilisation methods. Pr:egl. Telekomun 44.335-338. In Polish.Ruderfer. M. (1979). One-way doppler effects in atomic timekeeping. Speculations Sci. Technol.2.387-404.Runyon. S. (1977). Focus on electronic.counters. Electron. Des. 25. 54-63.Rusconi. L.. and Sedmak. G. (1977). Estimates of the limiting standard deviation in the absolutetiming of periodic sources. Astrophys. Space Sci. 46. 301-308.Rutman. 1. (\972a). Comment on "Characterization of frequency stability," IEEE Trans.Instrum. Mea.>. 1~1-21. 85.Rutman. 1. (1972b). Short tenn frequency instability of signal generators. Onde Elecrr. 52.480-487. In French.Rutman. 1. (1974a). Characterization of frequency stability: a transfer function approach and itsapplication to measurements via filtering ofphase noise. I EEE Trans.lnsrrum. Meas.I:\

414 CHAPTER BIBLIOGRAPHIESSorden. J. L. (1974). A new generation in frequency and time measurements. Hew/ell-Packard J.25.2-8.Spapens. W. (1972). Accurate time measurements with a counter and an oscilloscope. Philips TeslMeas. NOI. (2l. 4-8.Speake. B. (1979). The accurate setting of microwave frequencies. Commun.Inl. 6.19.21.Stecher. R. (1973). Electronic time interval measurement with nanosecond resolution.Nachrichlenrech. Eleklron. 23.413-416. In German.Steele. 1. McA. (1967). Frequency measurement using standard frequency emissions. Conf. Freq.Generalion and Control Radio SySI.. lEE Conference Publication No. 31.Steer, G. (1973). A new UHF frequency measuring device. Fernme/de-Praxis SO. 571-600. InGerman.Stein. S.. Glaze, D .• Levine, J., Gray. L Hilliard. D., Howe. D .• and Erb, L. A. (1983). Automatedhigh-accuracy phase measurement system. IEEE Trans. Inslrum. Meas. IM-32. 227-231.Steiner. C. I. ( 1969). Electronic methods oftime measurement. E/ek Ironiker 8. 3-16. In German.Stoyko. A. (1967). Ephemeris time according to the revised lunar theory ofBrown. Bull. ASlron. 2,411-416. In French.Szata. R. (1968). New method of oscilloscopic frequency measurement. Pomiary. Au/om.,Kontro/a 14. 200--30 I. In Polish.Szepan. R. (1978). Determining accuracy ofreceiver-controlled frequency standards. Ne ....s Rohde& Sch ....ar: 18. 23-26.Taylor, D. (1968). Time measurement. Inslrum. Rn:. 15. 173-176.Taylor. G. W. (1977). Time-a precious commodity. E/eclron. Power 23, 61-64.Tolman. J. (1970). Accurate frequency and time measurement at the Institute of RadioEngineering and Electronics. S/aboproudy Db•. 31, 286-290. In Czech.Tooley. M. (1979). In-line crystal calibrator. Pracl. Wireless 55.49-50,55.Triplet. J. M. (1967). Ephemeris time determination by lunar photography. Bull. Astron. 2.465-527. In French.Trkal, V. (1968). A new definition of the second on a quantum basis. Cesk. Cas. Fy:. A. 18,641-669. In Czechoslavakian.Turrini, A. (1974). Heterodyne frequency meter No. 351C E.I.P. Antenna 46, 62-68. In Italian.Tyrsa, V. E., and Dyunyashev, V. V. (1981). Accuracy of frequency measurement based on thepulsed coincidence principle. Meas. Tech. (Eng/. Transl.) 24,308-312.Underhill. M. L Sarhadi. M .. and Aitchinson. C S. (1978). Fast-sampling frequency meter.Eleclron. Lell. 14, 366-367.Upton, R. (1977). An objective comparison of analog and digital methods of realtime frequencyanalysis. Brue/ Kjaer Tech. Rev. (Denmark) l. 18-26.Vakman, D. Yeo (1979). The evaluation of frequency stability. Te/ecommun. Radio Eng. (Eng/.Trans/.), ParI 2 34.49-53.Vakman. D. Yeo (1979). Measuring the frequency of an analytical signal. Radio Eng. Eleclron.Phys. (Engl. Trans/.) 24.63-69.Valaskova. E. (1971). Measuring the frequency of damped oscillations. Sdelol'aci Tech. 19.174-175. In Czech.Valverde. R. M. (1968). A digitallOstantaneous frequency-meter. AUloma:. Slrum. 16.216-221.In Italian.Van Blerkom. R.. Freeman, D. G .. and Crutchfield, R. C. Jr. (1968). Frequency measurement·techniques. IEEE Trans. InSlrum. Meas. IM-17. 133-145.Van Den Brock. H. G. 1. (1977). Two frequency counters for routine measurements. PolYlech.Tijdschr. E/eklrolech. Elektron. 32. 517-518. In Dutch.Van den Steen. L. (1979). A low frequency meter with instantaneous response. IEEE Trans.Biomed. Eng. B:\1E-26. 137-140.TN-118

CHAPTER 12 415Van Straalen. J. (1979). A one-chip LSI frequency meier in Ihe I'L lechnology for FM and AMreceivers. IEEE Trails. COIUJIm. Elulroll. CE-2S. 134-143.Vessol. R.• Mueller. l.. and Vanier, J. (1966). The specificalion of oscillalor characlerislics frommeasuremenlS made in Ihe frequency domain. Proc. IEEE Sot. 199-206.Viles. R. S. (1972). SlIIepl frequency lechniques for aCCUrllle r.f. measuremenls. Moreolli IIIslT1Uf1.13.88-92.Voss, R. F. (\979). lifnoise. Proc. JJrd AMII. Frtq. COII/rol Symp.• pp. 40-46.Wamwrighl. A. E.• Walls. F. l.. and McCaa. W. D. (1974). Direcl measuremenl of the inherenlfrequency stabilily of quanz crystal resonalors. Proc. 281h AllnLl. Frtq. Conlrol Symp.• pp.177-180.Walls. f. l.. and DeMarchi. A. (1975). Rf spectrum of a signal after frequency multiplication;measurement and comparison wilh a simple calculalion.IEEE Trans. Inslrum. MtQS.IM-24.310-317. *W.alls. f. L.. and Stein. S. R. (1976). Servo lechniques in oscillators and measurementsystems. NBS Tech. No/e (CS.) 69:.Walls. f. L.. Stein. S. R.. Gray. J. A.. and Glaze. D. J. (1976). Design considerations in state-ofthe-artsignal processinl! and phase noise measurement systems. Proe. 30,h .-fMII. Freq.COnlrol Symp.. pp. 269-274.Ward. R. B. (1982). VH f /UHF frequency measurement using the surface acoustic wave delaydifference device. Appl. Phys. Ull. 40.1010-1012.Weiss. C 0 .. and Godone. A. (l98~). Extension offrequency measurements with Schonley diodesto the 4 THz ranl!e Appl. Phys. B 27, 167-168.Weiss. R. (1976). Time-signal and standard·frequency receiver for the DCF 77 wilh reserve. l.FLinksehoLi 48. 964-968. In Gennan.Wessel. R. (1973). A digital frequency meter. Eleklromeisler DIsch. Elektrohand..'erk 48.1066-1068. In Gennan.Westlund, R. C. (1978). Very short time interval measurements. Ne..· £lee/rOil. It 13-14. 18.Wheeler. K. G. (1976). An optimum solution for couming and timing. He .. Electron. 9. 24. 28. 3~.Whinaker. A. E.. and Westbury. P. R. (\976). Measuring low frequencies. Ne... Eleclron. 9. 36.Wiley. R. G. (977). A direct time-domain measure of frequency stability: the modified Allanvariance. IEEE Tro/'lS. 1/'ISlrum. Mea!. IM-26, 38-41.Woschni. E. G. (975). Errors in high-precision frequency measurements due to interactiveeffects. Wiss. Z. Tech. Hochseh. Karl-Marx-Suuft 17. 633-639. In GermanWoschni. E. G. (1976). On the optimum difference frequency in hi~ accuracy frequencymeasurements of industrial parameters. Mess.. St~ern. Rtgtln Alllomalisitrllllgsprax. 19.334-335. In Gennan.Woschni. E. G. (978). Application of microprocessors to the calculation of error correction forthe synchronisation effect in hil!hly accurate frequency measurements. Win. Z. Tuh.Hochsch. Karl-Marx-Slodt 20. 665-668. In Gennan.Wrzesien. M. (\976). Instantaneous frequency measuring circuit. Pomiory. Autom. Kontrolo 22.90-91. In Polish.Ya~uda. Y. (1974). Accuracy of estimating c10ckface and mean rate of the frequency standard bynormal regression theory. J. Radio Res. UJb. (Japan) 21. 39-60.Yoshimura. K. (1972). The generation of an accurate and uniform time scale with calibrationsand prediction. ""BS Tech. "'-ole lC.S.) 626.59 pp.Yoshimura. K. (1978). Characterization of frequency stability: uncertainty due to the autocor·relation of the frequency fluctuations. IEEE Trans. Inslrum. Meas. IM-27. 1-7.Yoshimura. K. (1980). Calculation of unbIased clock-variances in uncalibrated atomic time scaleal~orithms. Mnrologio 16. 133-139.• See Appendix Note # 10TN-119

416 CHAPTER BIBLIOGRAPHIESYoshimura. K. Harada. K.. and Kobayashi. M. (1971). Measuremenl of short-term frequencystability of frequency standards. Rrt·. Radio Rl!s. LiJbs. 17. 437-4·B. In Japanese.Yuva!. I. (1982). Frequency comparalor uses synchronous detel:tion. Elufronics ~~. 160.Zander. H. (19771. Measurement of low frequencies with short duration. Elrklronik (Munich) 26.6J-64. In German.Zelenlosov·. V. L Sulakshin. A. 5.. and Fedorov. K P. (1976). Meter for measurinl! the carrierfrequency of smgle ultrahigh-frequency pulses. Insfrum. E.tp. Tl!ch. (Ellgi. Trans!.) 19.476-4"77.Zhilin. No S. and Subbotin. l. S. (1977). Devices for reproducinl! reference time inter-als. .4Jl!as.Tl!ch. (Engl. Trans!.) 20. 995-999.Zhuk. A V.. KJZ3nlseli. E. M.. Klrianakl. ~. K.. and Mosieliich. P. M. 119771. Computerinvestil!ation 01 the metroIOl!1Cal charac!enstlcs of frequency meters ....·I\h digital positivef~dback and measures to improve them. albor Prrrdacha lrrf. 50. 110-115. In RussianTN-120

tEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRIC:;. AND FREQUENCY CONTROL. VOL. VFFC-J4. NO.6. NOVEMBER 1987647Time and Frequency (Time-Domain) Characterization,Estimation, and Prediction of PrecisionClocks and OscillatorsInvited PaperDAVID W. ALLANAbstract-A tutorial review of some time-domain methods of characterizingthe performance of precision clocks and oscillators is presented.Characterizing botb the s)'stematic and random deviations isconsidered. The Allan variance and the modified Allan variance aredefined. and methods of ulilizing them are presenled along "'ilh rangesand areas of applicabilily. The standard de"iation is contrasted andsho.. n not 10 be. in general. a good measure for precision clocks andoscillators. Once a proper characterization model has been developed,t!len optimum estimation and prediction techniques can be employ'ed.Some important cases are illustrated. As precision clocks and oscillatorsbecome increasingly' important in society. communication of theircharacteristic, and specifications among the vendors. manufacturers,design engineers. managers. and metrologists of this equipment becomesincreasingly important.hTRODUCTIO:-l, 'WHAT THEN." asked St. Augustine. "is time?If no one asks me. I know what it is. If I wishto explain it to him who asks me. I do not know." ThoughEinstein and others have taught us a lot since St. Augustine.there are still many unanswered questions. In particular.can time be measured? It seems that it cannot; whatis measured is the time differellce between two clocks.The time of an event with reference to a particular clockcan be measured. If time cannot be measured. is it physical,an abstraction. or is it an artifac(lWe conceptualize some of the laws of physics with timeas the independent variable. We attempt to approximateour conceptualized ideal time by inverting these laws sothat time is the dependent variable. The fact is that timeas we now generate it is dependent upon defined origins,a defined resonance in the cesium atom. interrogatingelectronics. induced biases. timescale algorithms. andrandom perturbations from the ideal. Hence. at a significantlevel. time-as man generates it by the best meansavailable to him-is an artifact. Corollaries to this are thatevery dock disagrees with every other clock essentiallyalways. and no clock keeps ideal or "true" time in anabstract sense except as we may choose to define it. Frequencyor time interval. on the other hand. is fundamental10 nature; hence the definition of the second can approach1>bnu,cnpt r~~~:\~J ~(ay 11. 1987: r~\'iseJ June 15. 1987.The authur IS \\;th Ihe Time and Frequency Division. Sattonal Bureauof SlanJards. 3~5 Broadll.·ay. Boulder. CO 80303.IEEE l'lg :"umber 8716461.the ideal-down to some accuracy limit. Noise in natureis also fundamenul. Characterizing the random variationsof a clock opens the door to optimum estimation of environmentalinfluences and to the design of optimum combiningalgorithms for the generation of uniform time andfor providing a stable and accurate frequency reference.Let us define V( t) as the sine-wave voltage output of aprecision oscillator:V(t) == V o sin ep(t) ( I )where ep (t) is the abstract but actual total time-dependentaccumulated phase starting from some arbitrary origin ep (1== 0) == O. We assume that the amplitude fluctuations arenegligible around Vo- Cases exist in which this assumptionis not valid, bUI we will not treat those in the contextof this paper. This lack of treatment has no impact on thedevelopment or the conclusions in this paper. Since infinitebandwidth measurement equipment is not availableto us, we cannot measure instantaneous frequency; therefore11(l) == (1/2'll") dep/dt is not measurable We canrewrite this equation with 110 being a constant nominal frequencyand place all of the deviations in a residual phasecP(r):V(r) = Vo sin (2"Yol + ¢(r»). (2) *We then define a quantity y (r) == (II (r) - vo)/ 110. whichis dimensionless and which is the fractional or normalizedfrequency deviation of II (t) from its nominal value. Integratingy( t) yields the time deviation x( r), which hasthe dimensions of timex(r) == 1~ y(r') dr'. (3)From this. the time deviation of a clock can be written asa function of the phase deviation:SYSTE\IAT1C MODELS FOR CLOCKS ASD OSCILLATORS(4 )The next question one may ask is why does a c1o.ckdeviate from the ideal? We conceptualize two categones* See Appendix Note :(I 11TN-121

648IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CONTROL. Val. UFFC-'4. NO.> 6. NOVEMBER 1987y( t)+ FREQUENCY OFFS£1OSCILLATOR, au RB H H< o. ) CS""1(a)J•TE,*"ERATU~ESE"fStrIvITY / d.g KFig. 2. Nominal values for temperature coefficient for frequency standards:QU = quanz crystal. RB = rubidium gas cell. H = active hy.drogen maser. H(pas) = passive hydrogen maser. and CS = cesiumbeam.OSCILLATOR, au RB H H< ce)I.~I •CSINEGATIV£ FREQuENCY DRIFT(b)Fig. l. Frequency y(1) and time x(t) deviations due to frequency offsetand to frequency drift in clock. (a) Fractional frequency error versustime. (b) Time error versus time.of reasons. the first being systematics such as frequencydrift (D). frequency offset ( Yo), and time offset (xo). Inaddition. there are systematic deviations that are often environmentallyinduced. The second category is the randomdeviations E( n. which are usually not thought to bedeterministic. In general. we may writex(r) = Xo + -,"or + 1/2 Dr;' + €(r). (5) *Though generally useful, the model in (5) does not applyin all cases~ e.g.. some oscillators have significant frequency-modulationsidebands, and in others the frequencydrift D is not constant. In some clocks and oscillators.e.g.. cesium-beam standards, setting D = 0 isusually a better model.Note that the quadratic D term occurs because x (r) isthe integral of y (t), the fractional frequency. and is oftenthe predominant cause of time deviation. In Fig. 1 wehave simulated two systematic-error cases: a clock withfrequency offset. and a clock with negative frequencydrift. Figs. 2-6 summarize some of the important systematicinfluences on precision clocks and oscillators. In additionto Figs. 1-6, important systematic deviations mayinclude modulation sidebands, e.g., 60 Hz, 110 Hz, daily.and annual dependences, which can be manifestations ofenvironmental effects such as deviations induced by vibrations.shock. radiation, humidity, and temperature... See Appendix Note # 12,.'1,-11,.­1.­1·Fig. 3. Nominal values for magnetic field sensitivity for frequency standards:QU = quartz crystal. RB = rubidium gas cell, H = active hydrogenmaser. H (pas) = passive hydrogen maser. and CS = cesiumbeam.OS:: I ~.L A TOR, au RB H H < os ) CS•••.8I,Fig. 4. Nominal capability of frequency standard to reproduce same frequencyafter period of time for standards: QU = quartz crystal. RB =rubidium gas cell. H = active hydrogen maser. H (pas) = passive hydrogenmaser. and CS = cesium beam...­ '"I1OSCilLATORau RB H H ( o.) CSABSOLUTE AC:uRACYFig. 5. Nominal capability for frequency standard to produce frequencydetermined by fundamental constants of nature for standards: QU =quartz crystal. RB = rubidium gas cell. H = active hydrogen maser.H (pas) = passive hydrogen maser. and CS = cesium beam.TN-122

ALLAN: PRECISION CLOCKS AND OSCILLATORS649TABLE IApPLICABLE OSCILLATORS AND RANGE OF ApPLICABILITYTypical Noise TypesName Cs H-Active'"H-Passive Qu Rb210-I-2white-noise PMflicker-noise PMwhile-noise FMflicker-noise FMrandom-walk FMO!: 10 SO!:days~weekss 100 s100 SST O!:~ II)" sO!:weeks10' s O!: 1 5~daysoeweekssl mssIs~ 1 soehO!: 1 S~ 10'~daysli"Ii"I'~"OSCILLATOR--=r----"RcrB-;- auH H ( "e) CSthen the average fractional frequency for the ith measurementinterval is-10 Xi+ I - XiYi= (6)TOle'·,i·rfig. 6. Nominal values (ignonng sign) for frequency drift for frequencystandards: QU = quanz crystal, RB = rubidium gas cell, H .., aClivehydrogen maser, H (pas) = passive hydrogen maser, and CS = cesiumbeam.POWER LAW SPECTRAf-II"V·/\~'w.itr ..,\f{'-'~.-'V'"vf-'~f-'~Fig. 7. Simulated random processes commonly occurring in output signalof alomic clocks. Power law spectra S (f) are proponional to w 10 some *nponenr. where f is Fourier frequency «w .., 2rfl and Sy( /)..,";s..( /).RANDOM MODELS FOR CLOCKS AND OSCILLATORSThe random-frequency deviations of precision clocksand oscillators can often be characterized by power-lawspectra S,. (1) - 10., where1is the Fourier frequency andOi typicaliy takes on integer values, i.e., - 2, - I, 0, I, 2[lJ-[4]. Fig. 7 shows noise samples corresponding tothese different power law spectra, and Table I shows thenominal range of applicability of these power-law models.TIME-DOMAIN SIGNAL CHARACTERIZATIONGiven a discrete set of time deviations X j taken in sequencefor the measurable time difference between a pairof clocks or between a clock and some primary reference,and given that the nominal spacing between adjacent timedifference measurements is TO (see Fig. 8 for an example),'" See Appendix Note # 13where -TO over Yi denotes the average over an interval TO.We can thus construct a set of discrete frequency valuesfrom such a time-difference data set. If the standard deviationis calculated for this set of values, one can showthat for some kinds of power-law spectra encountered inprecision oscillators the standard deviation is divergent[1], [2], [5], i.e., it does not converge to a well-definedvalue and is a function of data length. Hence the standarddeviation is seldom useful and can be misleading in characterizingclocks. An IEEE subcommittee has recommendedSy (1) in the frequency domain and a measure(J~ ( 1) in -the time domain [I]. Sy (1) is the one-sidedspectral density ofY as a function of Fourier frequency f.The latter is often called the Allan variance or two-samplevariance. The convergence of 0',( 1) has been verified [1]­[4] for the power law spectra o(interest in precision clocksand oscillators. The measure (J;( 1) is defined as [1]-T " ••(7)where Ay IS the difference between adjacent fractIOnalfrequency measurements, each sampled over an interval1, and the brackets ( >indicate an infinite time averageor expectation value. A pictorial description is shown inFig. 9 for a finite data set. A data set of the order of 100points is more than adequate for convergence of (J\" (1),though of course the confidence of the estimate wili typicallyimprove as the data length increases [6].Given a discrete set of stored evenly spaced data, thevalue of 1 can be varied in the software [7]. If 10 is theminimum data spacing for the original stored data set y[O,then one can change the sampling time to 1 "" nTo by averagingn adjacenl values OfYi- m to obtain a new fractional **frequency estimate yr. with sample time T as input to (7).Note this is different from averaging adjacenl values of x.Hence in a very convenient way one can calculate (J,. (1)as a function of T, which will be shown to be very useful.For a finite data set of M values of yj', (7) for general Tbecomes (see Fig. 8 for an example computation of (J.\"( 1•• See Appendix Note # 14TN-123

650 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS. AND FREQUENCY CONTROL. VOL. UFFC-34, NO.6. NOVEMBER 1987• ( t)Yet)Y2 Y;0M(). 1 h( -2 i:as TD DE V Y T M- 1 i • 1 Yi - y)Fig. 8. Simulated lime deviation plot .r(r) with indicated sample time 1 over which each adjacent fnctional frequency y, ismeasured. Equations are for standard deviation and for estimate of U,e 1) for finite data set of M frequency measurements.Often standard deviation diverges as data length increases when measuring long-tenn frequency stability of precision oscillators.whereas uy( 1) converges.M- Ji,lwWZwtl:wu.u.od~Cf.,....r'lc"In elop•• ~y:= r... -ltI'

ALLAN: PRECISION CLOCKS AND OSCILLATORS651TABLE II'ClassicalClassicalTypical Noise Types Standard Standarda Name Deviation of x Deviation of y2 wllite-noise PM Q, ~-J' 7' ",( r)/../3 (constant) ",( 7) .J2(N + I )/3N1 fl icker-noise PM QI'1'- 2 -7' ",(7) Jln M/ln 2 - U,( 7) J2(N + 1/3No wllite-noise FM dor- I 70' 17.(70) J(M + 1)/6 ",( ~o)-I flicker-noise FM 0_1 T° undefined u,( ~) ,;r Ṅ :-:I- n -:-N:-:/(C":2:-("""N:----:-':-)=-In-:::2:-)- 2 random-walk FM 0_2'1' undefined u,(T) .JNj2• Note T is a general averaging time and ~o is tile initial averaging time (T = n7 0 • where n is an integer).Also note tllat tile last four entries in the founh column and the last two entries in lhe: fifth column go toinfinity as M or N go to infinity. M is the initial number of frequency difference measurements and N thenumber of phase or time difference measurements N = M + 1. If the spectral density is given by S,( f)'" haf a • thenG, = {2~ Y3Moh?G, = (2~Y (1.038 + 3 log, (hf.T) hi)a_I = 2 log. (2) h_1G_, = ;:;I (1111") 1 h_ 1•b Note this equality assumes use of modified u;(7)= a; (T).measure the phase difference or the time difference betweena pair of oscillators or clocks and we define Th =11k In other words. Th is the sample time period throughwhich the time or phase date are observed or aver:aged.Averaging n time or phase readings increases the sampletime window to nTh == T r Let 1', = 1I fr; then fr = fhln.Le., the software bandwidth is narrowed to fr. In otherwords, fr == fi,1 n decreases as we average more values;i.e., increase n (1' = n'To). One can therefore construct asecond difference composed of time deviations so-averagedand then define a modified a;' (T) == a; (T) that willremove the ambiguity through bandwidth variation:N-3n + I~j=l( 12)where N = M + 1, the number of time-deviation measurementsavailable from the data set. Now if a; (1') ­7/'0', then p.' == -ex - 1 (I ~ ex ~ 3) [10], [11]. ThusOy ( T) is typically employed as a subroutine to remove theambiguity if a ( T) - T- 1 . This is because the 11' == -exv- 1 relationship is valid as an asymptotic limit for largen and ex < 1 and is not valid in general; however, thereis evidence that U;(T) may be a better measure [12]. Specifically,for ex == 2 and I, 11' /2 equals - 3/2 and -1,respectively, providing a clean differentiation betweenwhite-noise PM and flicker-noise PM.If three or more independent oscillators or clocks areavailable along with time (phase) or frequency measure­'" See Appendix Note # 16*ments between them, then it is possible to estimate a variancefor each oscillator or clock. Often there is a referenceto which the rest are periodically measured at asampling rate 1I TO' If at each measurement the time orfrequency differences between the clocks are measured atnominally the same lime, then the time difference or frequencydifference can usually be estimated or calculatedbetween every possible pair in the set of oscillators orclocks. Given a series of measurements, variances sl canbe calculated on the time or frequency data between allpairs. It has been shown [13] that the individual clockvariances can be estimated using the following equations:wherea~ == _I- ( Esb - B)m - 2 j= I1 mB =-- ~ s2.,m - I i

652 IEEE TRANSACTIONS ON ULTRASONICS, FERROElECTRICS, AND FREQUENCY CONTROl. VOL. UFFC-)4. NO.6. NOVEMBER 1987TABLE 111aTypical Noise TypesNameOptimumPredictionx( T p ) nns"Time Error:Asymptotic Form2Io- I- 2while-noise PMflicker-noise PMwhite-noise FMflicker-noise FMrandom-walk FMTp • a,( Tp)/,fj- TI' • a, ( T1') ../"In-Tp/7.: 2--:-,n-T0T p ' a.• (T p )T p • a.(T p)/,J"1;"2T p • a.• (T p )constant~1~f7T~!,T p-"T p is the prediction interval.TIME AND FREQUENCY ESTIMATION AND PREDICTIONUsing a,(T), O',(T), S,(f), or S4>(f), one can characterizetypical power law processes. Once characterized,this opens the opportunity for determining optimum estimatesof values by employing the statistical theorem thatthe optimum estimate of a white-noise process is the simplemean.For example, consider the very common and very importantcase of white-noise FM typically found on the signalsfrom cesium standards, rubidium standards, and passivehydrogen masers_ The optimum estimate of thefrequency is the simple mean frequency, which is equivalentto (x" - XI)/MTo. It is still all too common withinour discipline to see our colleagues erroneously determiningthe frequency for these kinds of oscillators by calculatingthe slope from a linear least-squares fit to the timedeviations and quoting the standard deviation around thatfit as a measure of the clock performance. There are threeproblems in proceeding this way. First, the frequency estimateis not optimum in a mean-square-error sense. It isequivalent to throwing away about 20 percent of the dataand thereby increasing the cost in the case of a calibration.Second, the standard deviation diverges as the squareroot of the data length. Third, the standard deviation issignificantly dependent on the filter form, e.g., linear leastsquares, as well as the clock deviations. On the otherhand, such a filter is sometimes useful for assessing outliers.The optimum "end-point" method outlined earlierhas the risk that if either of the points is abnormal, (i .e"the model fails), the result will of course be adverselyeffected. Therefore such a filter is useful to assess whetherthere are outliers-paying special attention to the endpoints. Also, if the measurement noise exceeds the combinednoise in the clocks, then the end points will be adverselyaffected. The key message is that the end-pointmethod for estimating frequency is only optimum if thei:~~: ~:r:)r:;~~~eI~~; ~:~~h is easy to determine fromI There are other useful, and maybe not so obvious, optimumestimators appropriate for time-difference data sets.1) Given white-noise PM, the best time estimate is thesimple mean of the time deviations; the frequencyestimate then is the slope from a linear least-squaresfit to the time deviations, and the frequency drift Dis determined from a quadratic least-squares fit tothe time deviations per (l).2) Given white-noise FM, the optimum estimate of thetime is the last value; the optimum-frequency estimateis outlined in the previous paragraph, and theoptimum-frequency-drift estimate is derived from alinear least-squares fit to the frequency.3) Given random-walk FM, the current optimum timeestimate is the last value plus the last slope (clockrate) times the time since the last value; the opti·mum-frequency estimate is obtained from the lastslope of the time deviations; and the optimum-frequency-driftestimate is calculated from the meansecond difference of the time deviations. Cautionneeds to be exercised here, for typicaIly there willbe higher frequency component noise in a real datastream, such as white-noise FM, along with random-walkFM, and this can significantly contaminatethe drift estimate from a mean second difference.If random-walk FM is the predominant longtermpower-law process, which is often the case,then the effect of high-frequency noise can be reducedby calculating the second difference from thefirst, middle, and end-time deviation points of thedata set.The flicker-noise cases are significantly more complicated,though filters can be designed to approximate optimumestimation [14]-[ 16]. As the data length increaseswithout limit, time is not defined for flicker-noise PM,and frequency is not defined for fl icker-noise FM. Thishas some philosophical implications for the definitions oftime and frequency. unless some low-frequency cutofflimits exist. If significant frequency drift exists in the data,it should be optimally subtracted from the data or it willbias the long-term values of 0', ( T):DT0'.(7) = C' (14). ,,2Once the power-law spectra are deduced for a pair ofoscillators or clocks, then one can also develop an optimumpredictor. Table III gives both the optimum predictionuncertainty values for the various relevant purepower-law spectra as well as their asymptotic forms. Specialforecasting techniques must be used for optimal predictionwhen combinations of these processes are present[17]. To illustrate how these concepts relate to real devices,Fig. 10 shows a Oy( 7) diagram for some interestingTN-126

ALLAN: PRECISiON CLOCKS AND OSCILLATORS-10-i5ACTIVE H MASERRb GAS cru,...........'.....---­-16:-_--;__---;.__--:-__-+-__--!-_~I 2 3 4 6lO:; TAU (s.econds)Fig. 10. Square root of Allan variance for variety of state-of-the-an precisionoscillators including NBS-6. NBS primary frequency standard.-6-7II'sRMS TIME DEVIATION&/~f7.//'Cs653minimum O',(T) value. We would then change the temperatureto the other value and repeat the measurementwith the same criteria and note the Ayr.. between the twooptimally detennined frequency values. If these two stepsare repeated several times, an arbitrarily good precisionfor the temperature coefficient is achieved if it is linear.The uncertainty is approximately given by (J,(7 )/JP,mwhere P is the number of Ayr.. values obtained fromswitching back and forth. Knowing the characteristics ofboth the random and the systematic deviations of precisionclocks and oscillators clearly is useful to the designer,the manufacturer, the planner, and the user as wellas the vendor of these devices.The aforementioned procedures usually work well if theclocks or oscillators are in a reasonable environment. Ifthe environment is adverse, other procedures and analysismethods may have to be employed. As a general rule it isoften useful to analyze the data in the frequency domainas well as the time domain. The frequency domain is especiallyuseful if there are bright lines, i.e., sidebands tothe carrier frequency, The effect of a modulation sidebandfm on 0', (7) can be calculated, and is given by [18](15)-130~=~-~--':-3----J4'---..,:---~---:!lOG TAU p (seconds)Fig. I I. From frequency stability characterization shown in Fig. 10, optimumprediction algorithms to minimize time error can be obtained.Based on oplimum prediction procedures rms time prediclion error forprediction inlerval T p can be calculated for each oscillator shown in Fig.10 and corresponding values are plotted in Fig. II.state-of-the-art oscillators, and Fig. 11 shows the nns timeprediction errors for the same set of oscillators.CONCLUSIONIn conclusion, it is clear that classical statistics do notallow characterization of common kinds of random signalvariations found in precision oscillators. The two-sampleAllan variance provides a valuable and convergent measureof the power-law spectral-density models useful incharacterizing random deviations for most oscillators andclocks, Once characterized, we can calculate optimumtime and frequency estimates as well as predicted values.Characterizing the random variations also provides nearoptimumestimation of systematic effects, which oftencause the predominant time and frequency deviations. ForeX:lmple, if we wanted to optimally detennine the statictemperature dependence with the temperature set at twodifferent values, we would stabilize the oscillator at onetemperature and measure the frequency against a referencefor a time T"'" corresponding to the T for the nominalwhere x pp is equal to the peak-to-peak or twice the amplitudeof the time-deviation modulation.If one is trying to estimate the power-law spectral behaviorbetween a pair of oscillators or clocks using O'y (7),it is apparent from (15) that if significant modulationsidebands are present on the signal, these can seriouslycontaminate that estimate. However, if a O"y( 7) plot displaysa character as given by (15), then the amplitude andfrequency of that modulation sideband can be estimatedfrom this time-domain analysis technique. In practice, thisapproach is often used, but these modulation sidebandscan be more efficiently estimated in the frequency domain.If the measurement sampling rate 1/70 is set equalto 1m' then the modulation sideband is aliased away andhas no effect on l1y ( 7).The best rule in all analysis is to use common sense.Very often the most revealing information may be in aplot of the raw time (phase) difference or frequency-differenceresiduals after some trend has been removed. Sucha plot is usually the first thing to look at when characterizingclocks and oscillators. Caution here 'is also importantas a pure random walk on the time residuals (whiteFM) may be visually interpreted as having frequencysteps. This is especially true for flicker FM as often seenin quartz-crystal oscillators. Following the time-residualplot with a l1 y ( 7) analysis often answers the question asto whether or not such steps are statistically significant.ACKNOWLEDGMENTBecause the paper is mainly a review, the author is indebtedto a large number of people-many of whom arereflected in the references. Specifically. sincere appreciationis expressed to Dr. James A. Barnes, Mr. Dick D.IN-127

IEEE TRANSACTIONS ON ULTRASONICS. FERltOELECTIUCS. AND FREQUENCY CONTROL, VOL. UFFC-34. NO.6. NOVEMBER 1987Davis, Dr. John Vig, Dr. Donald B. Sullivan, Dr, FredL. Walls, and Dr. Marc A. Weiss for their extremelyhelpful comments and suggestions.REFERENCESII] J. A. Bames ~t aI., "Characterization offrequency stability," in IEEETrans. Instrum. M~as., vol. IM-20, p. 105, NBS Tech. Note 394,1971.[2] D. W. Allan, "Statistics of atomic frequency standards," in Proe.IEEE, vol. 54, 1966, p. 221­[3] R. Vessot, L. Mueller, and J. Vanier, "The specification of oscillatorcharacteristics from measurements made in the frequency domain, ..in Prot:. IEEE Sp~cial Issu~ on Fr~qu~ncy Stabiliry, vol. 54, Feb.1966, pp. 199-207./4] P. Lesage and C. Audoin, "Characterization and measurement of timeand frequency stability," Radio Sci., vol. 14, pp. 521-539, July/Aug.1979.[5] J. A. Barnes, "Atomic timekeeping and the statistics of precision signalgenerators," in Proe. IEEE Sp~cialiss ..~ on Fr~q .. ~ncy Stabiliry,vol. 54, Feb. 1966, pp. 207-220.(6) D. A. Howe. D. W. Allan, and J. A. Barnes, "Propenies of signalsources and measurement methods," in Proe. 351h Ann. Symp. Fr~qu~ncyControl (SFCj, May 1981, pp. 669, AI-A47.(7) D. W. Allan, "The measurement of frequency and frequency stabilityof precision oscillator," Nat. Bur. Standards, Boulder, CO, NBSTech. Note 669, May 1975.18] M. J. Lighthill, "Introduction to Fourier analysis and generalizedfunctions," in Cambridge Monographs on M~chanics and AppliedMarhemalics. G. K. Batchelor and J. W. Miles, Eds. London, England:Cambridge Univ. Press, 1964.19] J. J' Snyder, "An ultra-high resolution frequency meter," in Proc.35/h Ann. Fr~quenC)l Comrol Symp., USAERADCOM. FI. Monmouth,NJ, May 1981, pp. 464-469.[10] D. W. Allan and J. A. Barnes, "A modified 'Allan variance' withincreased oscillalor characterization ability." in Proe. 35rh Ann. Fr~q..~ncyControl Symp., USAERADCOM, Fl. Monmouth, NJ, May1981, pp. 47Q-47S.[II) P. Lesage and T. Ayi, "Characteriution of frequency stability: Analysisof the modified Allan variance and propenies of its estimate,"iEEE Trans. Iturrum. Meas., vol. IM-33, no. 4, pp. 332-336, Dec.1984.(12) L. G. Bernier, "Theoretical analysis of the modified Allan variance,"in Prot:. 41sr Ann. Fr~q..~ncy Control Symp., USAERADCOM, Ft.Monmouth, NJ, May 1987.113] J. A. Barnes, Notes from NBS 2nd Sem. Atomic Time Scale Algorithms.June 1982, Time and Frequency Division, Nat. Bur. Standards,Boulder, CO 80303.[14] J. A. Barnes and S. Jarvis, Jr.• "Efficient numerical and analog modelingof Ricker noise processes." Nal. Bur. Standards. Boulder, CO.NBS Tech. Note 604, June 1971.[15] J. A. Barnes, "The measurement of linear frequen"y drift in oscillators,"in Proc. 15th Ann. Pr~ciu Tim~ and Time inruval (PIT/)Applications and Planning Muting. Naval Res. Lab., Washington,DC, Dec. 1983, pp. SSI-S82.[16] D. W. Allan er al., "Perfonnance, modeling, and simulation of somecesium beam clocks," in Proc. 27th Ann. Symp. Fr~quent:y Conrrol,1972, p. 309, AD 771 042.[(7) D. W. Allan and H. Hellwig, "Time deviation and the predictionerror for clock specification, characterization, and application." inProc. Posirion Location and Navigarion Symp. (PUNS). 1978, p.29.[18] S. R. Stein, private communication.David W. Allan for a photograph and biography, please see page 571 ofthis TRANSACTIONS.TN-128

From: Proceedings o/the 42nd Annual Symposium on Frequency Control, 1988.EXTENDING THE RANGE AND ACCURACY OF PHASE NOISE MEASUREMENTSF.L. Walls. A.J.D. Clements. C.M. Felton. M.A. Lombardi, and M.D. VanekSummaryThis paper describes recent progress in extendinghigh accuracy measurements of phase noise inoscillators and other devices for carrier frequenciesfrom the rf to the millimeter region and Fourierfrequencies up to lOt of the carrier (or a maximum ofabout 1 GHz). A brief survey of traditionalprecision techniques for measuring phase noise isincluded as a basis for comparing their relativeperformance and limitations. The single oscillatortechniques, although conceptUally simple, reqUire aset of 5 to 10 references to adequately measure thephase noise from 1 Hz to 1 GHz from the carrier. Thetwo oscillator technique yields excellent noisefloors if, for oscillator measurements, one has acomparable or better oscillator for the referenceand, for other devices, either pairs of devices or areference oscillator with comparable or better noise.We have developed several new calibration techniqueswhich, when combined with previous two oscillatortechniques, permits one to calibrate all factorsaffecting the measurements of phase noise ofoscillator pairs to an accuracy which typicallyexceeds 1 dB and in favorable cases can approach 0.4dB. In order to illustrate this expanded twooscillator approach. measurements at 5 MHz and 10 GHzare described in detail. At 5 MHz we achievedaccuracies of about to.6 dB for phase noisemeasurements from 20 Hz to 100 kHz from the carrier.At 10 GHz we achieved an accuracy of iO.6 dB forphase noise measurements a few kHz from the carrierdegrading to about i1.5 dB, 1 GHz from the carrier.I, IntroductionThis paper describes recent progress at the NationalBureau of Standards (NBS) in extending high accuracymeasurements of phase noise in oscillators,amplifiers, frequency synthesizers, and passivecomponents at carrier frequencies from the rf to themillimeter region and Fourier frequencies up to 10'of the carrier (or a maximum of about 1 GHz). Anexamination of existing techniques for precisionphase noise measurements of oscillators[l-ll] showedthat present approaches which don't require a second"reference" oscillator have good resolution or noisefloor for Fourier frequencies extending over only 1or 2 decades. (7-10] Consequently, these approachesrequire a set of 5 to 10 references, either delaylines or high Q factor cavities, in order toadequately measure the phase noise from 1 Hz to 1 GHzfrom the carrier. Using the cavity approach wouldrequire a entire set of reference cavities for eachcarrier frequency measured. Similar considerationsalso apply to phase noise measurements of the otherdevices unless one can measure pairs of devices.The limitations of the single oscillator techniquesled us to adopt a two oscillator method for allmeasurements. This approach yields good resolutionor noise floor from essentially dc to the bandwidthof the mixer if. for oscillator measurements, one hasa comparable or better oscillator for the referenceContribution of the u.S. Government. not subject tocopyright.Time and Frequencv DivisionNational Institute of Standards and TechnologyBoulder. Colorado 80303and, for other devices, either pairs of deVices or areference oscillator ~ith comparable or better noise.The major limitations in the accuracy of the twooscillator method (which also apply to the singleoscillator methods) are the calibration of: the mixerphase-to-voltage conversion factor. the amplifiergain versus Fourier frequency, and the accuracy ofthe spectrum analyzer. ~e have developed several newcalibration techniques which, when combined withprevious techniques, allow us to address each ofthese limitations. The net result is the developmentof a complete measurement concept that permits one tocalibrate all factors affecting the measurements ofphase noise of oscillator pairs to an accuracy whichtypically exceeds 1 dB and in favorable cases canapproach 0.4 dB. For other types of devices thelimitations are similar if the noise of the referenceoscillator can be neglected. The ultimate accuracythat Can be easily achieved with this approach is nowlimited by the accuracy of the attenuators inavailable spectrum analyzers.Measurements at 5 MHz and 10 GHz are described indetail, in order to illustrate this expanded twooscillator approach. Specifically we measure themixer phase-to-voltage conversion factor multipliedby amplifier gain on all channels versus Fourierfrequency using a new ultra-wideband phase modulator.We then measure the absolute mixer conversion factormultiplied by the amplifier gain at one Fourierfrequency, the effect of the phase-lock loop on themeasured noise VOltage, and spectral density functionof the spectrum analyzers. These measurements areused to normalize the relative gains of the noisemeasurements on the various channels. At 5 MHz weachieved accuracies of about to.6 d6 for phase noisemeasurements from 20 Hz to 100 kHz from the carrier.At 10 GHz we achieved an accuracy of to.6 dB forphase noise measurements a few kHz to 500 MHz fromthe carrier. The accuracy degrades to about tl.5 dB,1 GHz from the carrier.II, Model of a Noisy SignalThe output of an oscillator can be expressed asV(t) = (Va + c(t)] sin(211'lI t + ¢(t». (1)owhere Va is the nominal peak output voltage, and 11 0is the nominal frequency of the oscillator. The timevariations of amplitude have been incorporated into£(t) and the time variations of the instantaneousfrequency, lI(t), have been incorporated into ¢(t).The instantaneous frequency isd[li>(t)]II (t) "'0 + ---- (2)The fractional frequency deviation is defined asv(t) - "0 d[¢(t) Jy(t) (3)432TN-129

Powet spectral analysis of the output signal Vet) increasingly important as users require thecombines the power in the carrier v. with the power specification of phase noise near the carrier wherein r(t) and .(t) and therefore is not a good method the phase excursions are large compared to 1 radian,to characterize .(t) or ~(t).or at Fourier frequencies which exceed the bandwidthof typical circuit elements.Since in many precision sources understanding thevariations in ~(t) or yet) is of primary importance, The above measures prOVide the most powerful (andwe will confine the following discussion todetailed) analysis for evaluating types and levels offrequency-domain measures of y(t), neglecting .(t) fundamental noise and spectral density structure inexcept in cases where it sets limits on theprecision oscillators and signal handling equipmentmeasurement of yet). The amplitude fluctuations,as it allows one to examine individual Fourier.(t), can be reduced using limiters whereas ~(t) can components of residual phase (or frequency)be reduced in some cases by the use of narrow band modulation.filters.Spectral (Fourier) analysis of yet) is oftenIII, A, Two Oscillator Method*expressed in terms of S~(f), the spectral density of Fig. 1 shows the block diagram for a typical schemephase fluctuations in units of radians squared per Hz used to measure the phase noise of a precision sourcebandwidth at Fourier frequency f from the carrier v•. using a double balanced mixer and a reference sourceAlternately 5y(f), the spectral density of fractional Fig. 2 illustrates a similar technique for measuringfrequency fluctuations in a 1 Hz bandwidth at only the phase noise added in multipliers, dividers,Fourier frequency f from the carrier 10'0 can be used amplifiers, and passive components. It is very[lJ. These are related asimportant that the substitution oscillator be at thesame drive level, impedance, and at the equivalentelectrical length from the mixer as the signal comingO

given byREFV",(f) ]2(10)[G(f)~ P,IJMeasurement of Scp(f) lor Two AmplifiersS~lf). (-- V. )' ­~ G(')K. BW(often more than 20 dB). Termination of the mixer IFport with 50 n maximizes the IF bandwidth, however,termination with reactive loads can reduce the mixernoise by - 6 dB, and increase K dby 3 to 6 dB asshown in Fig. 3 [4J. Accurate determination of K dcan be achieved by measuring the slope of the zerocrossing in volts/radian with an oscilloscope orother recording device when the two oscillators arebeating slowly. For some applications the digitizerin the spectrum analyzer can be used to measure boththe beat period and the slope in Vis at the zerocrossing. The time axis is easily calibrated sinceone beat period equals 2~ radians. The slope involts/radian is then calculated with a typicalaccuracy of 0.2 dB. Estimates of K dobtained frommeasurements of the peak to peak output voltageinduced can introduce errors as large as 6 dB inS~(f) even if the amplitude of the other harmonics ismeasured unless the phase relationship 1s also takeninto account [4]. S~(f) can be made independent ofthe accuracy of the spectrum analyzer voltagereference by comparing the level of an externally IFsignal (a pure tone is best), on the spectrumanalyzer used to measure V nwith the level recordedon the device used to measure K d.Fig. 2. Precision phase measurement system featuringself calibration to 0.4 dB accuracy from dc to 0.1 vaFourier frequency offset from carrier. This systemis suitable for measuring signal handling equipment,multipliers, dividers, frequency synthesizers, aswell as passive components [4J.S0.8r--,.--,--r--r--,O.OlIJF~ 0.7 2mW MIXER DRIVE~>­!::0.6> 0.5i=fi)Z 0.4w«)0.3WIJJi4:J:a-0.210mW MIXER DRIVE0.11 2 5 10 202 51020 50 100FREQUENCY (kHz)Fig. 3. Double-balanced mixer phase sensitivity at 5KHz as a function of Fourier frequency for variousoutput terminations. The curves on the left wereobtained with 10 mlJ drive while those on the rightwere obtained with 2 mlol drive. The data demonstratea clear choice between constant, but low sensitivityor much higher, but frequency dependent sensitivity[4J.where V",(f) is the RKS noise voltage at Fourierfrequency f from the carrier measured after IF gainG(f) in a noise bandwidth RIJ. Obviously RIJ must besmall compared to f. This is very important whereS~(f) is changing rapidly with f, e.g., S~(f) oftenvaries as f- 3 near the carrier. In Fig. 1, theoutput of the second amplifiQr foLlovins the mixercontains contributions from the phase noise of theoscillators, the noise of the mixers, and the postamplifiers for Fourier frequencies much larger thanthe phase-lock loop bandwidth. In Fig. 2, the phasenoise of the oscillator cancQls out to a high degreelkQThe noise bandwidth of the spectrum analyzer alsoneeds to be verified. This calibration procedure issufficient for small Fourier frequencies but loosesprecision and accuracy due to the problemsillustrated in Fig. 3 and the variations of amplifiergain with Fourier frequency. If measurements need tobe made at Fourier frequencies near or below thephase-lock-loop bandwidth, a probe signal can beinjected inside the phase-lock-loop and theattenuation measured versus Fourier frequency.Some care is necessary to assure that the spectrumanalyzer is not saturated by spurious signals such asthe power line frequency and its multiples.Sometimes aliasing in the spectrum analyzer is aproblem. If narrow spectral features are to bemeasured it is usually recommended that a flat topwindow function (in the spectrum analyzer) be used.In the region where the measured noise is changingrapidly with Fourier frequency, the noise bandwidthshould be much smaller than the measurementfrequency. The approximate level of the noise floorof the measurement system should be measured in orderto verify that it does not significantly bias themeasurements or, if necessary, to subtract its effectfrom the results.Typical best performance for various measurementtechniques is shown in Fig. 4. The two oscillatorapproach exceeds the performance of almost allavailable oscillators from below 0.1 MHz to over 100GHz and is generally the technique of first choicebecause of its versatility and simplicity. Figs. 10- 12 give some examples. Phaae noise measurements onpairs of signal sources can be made with an absoluteaccuracy better than 1 d! using the above calibrationprocedure. Such accuracy is not always attainablewhen the phase noise of the source exceeds that ofthe added noise of the components under test (seeFig. 2). The use of specialized high level mixerswith multiple diodes per leg increases the phase tovoltage conversion sensitivity, K d and thereforereduces the contribution of IF amplifier noise 15] asshown in Fig. 4. Phase noise measurements Cangenerally be made at Fourier frequencies fromapproximately de to 1/2 the source frequency. The434

major difficulty is designing a mixer terminations toremove the source frequency from the output signal,which would generally saturate the low noiseamplifiers following the mixer. without degrading thesignal-to-noise ratio. As mentioned earlier thephase noise spectrum is quite likely asymmetric whenf exceeds the bandwidth of the tuned circuits in thedevice under test. For example one expects that thephase noise at 1/2vo is different than the phasenoise at 3/2v o .Comparison of Noise Floorfor Different Techniquesreference source is also higher at the multipliedfrequency as shown in Section III.F below.III, B Enhanced PerfOrmance Using CorrelationTechniquesThe resolution of the many systems can be greatlyenhanced (typically 20 dB) by using correlationtechniques to separate the phase noise due to thedevice unde·r test from the noise in the mixer and IFamplifier [5, 11].For purposes of illustration, consider the schemeshown in Fig. 5. At the output of each doublebalanced mixer there is a signal which isproportional to the phase difference, A~, between thetwo oscillators and a noise term, V R• due tocontributions from the mixer and amplifier. Thevoltages at the input of each bandpass filter areVI(BP filter input) = GIA~(t) + C1V.I(t) (11)V 2 (BP filter input) = G 2 A~(t)+ C 2V. 2(t).where VRI(t) and V. 2 (t) are substantiallyuncorrelated and C I and C 2are constants. Eachbandpass filter produces a narrow band noise functionaround its center frequency f:VI(BP filter output) = G I [S.(f)]· B I ' cos [2Kft +~(t)](12)V 2 (BP filter output) G 2 [S.(f)]· B 2 ' cos [2lfft +.p(t)]Fig. 4.Curve A.The noise floor S¢(f) (resolution) oftypical double balanced mixer systems (e.g.Fig. 1 and Fig. 2) at carrier frequenciesfrom 0.1 MHz to 26 GHz. Similarperformance possible to 100 GHz [5J.Curve B. The noise floor, S~(f), for a high levelmixer [5 J .Curve C. The correlated component of S.(f) betweentwo channels using high level mixers [5].Curve D. The equivalent noise floor S.(f) of a 5 to25 MHz frequency multiplier.Curve E. Approximate phase noise floor of Fig. 8using a 500 ns delay line.Curve F. Approximate phase noise floor of Fig. 8where alms delay has been achieved byencoding the signal on an optical carrierand transmitted it across a long opticalfiber to a detector.OOM,REfMost double balanced mixers have a substantial nonlinearitythat can be exploited to make phasecomparison between the reference source and oddmultiples of the reference frequency. Some mixerseven feature internal even harmonic generation. Themeasurement block diagram looks identical to that ofin Fig. 1, except that the source under test is at anodd (even) harmonic of the reference source. Thismethod is relatively efficient (as long as theharmonics fall within the bandwidth of the mixer) formultiples up to x5 although multiples as high as 25have been used. The noise floor is apprOXimatelydegraded by the amount of reduction in the phasesensitivity of the mixer. The phase noise of theFig. 5.Correlation phase noise measurement system.where B I and B 2are the eqUivalent noise bandwidthsof filters 1 and 2 respectively. Both channels arebandpass filtered in order to help eliminate aliasingand dynamic range problems. The phases ~(t), n, (t)and n 2(t) take on all values between 0 and 2lf withequal likelihood. They vary slowly compared to 11fand are substantially uncorrelated. When these twovoltages are multiplied together and low passfiltered, only one term has finite average value.435

The output voltage is+ D.0.0 a:w0!Xi-0.5V = G(f)kdQdy(f) (V +f(t)] (15)outThis approach mixes frequency fluctuations betweenthe oscillator and reference resonance with theamplitude noise of the transmitted signal. By usingamplitude control (e.g. by processing to normalizethe data), one can reduce the effect of amplitudenoise. [5] The measured noise at the detector isthen related to the phase fluctuation of thereference resonance byfor f< (16)2Q.-.Fig. 7. Differential frequency discriminator using apair of high-Q resonators. In this approach thephase noise of the source tends to cancel out.436IN-133

The phase noise in the source can cancel out by 20 to40 dB depending on the similarity of resonantfrequencies Q's and the transmission properties ofthe two resonators. This approach was first used todemonstrate that the inherent frequency stability ofprecision quartz resonators exceeds the performanceof most quartz crystal controlled oscillators [7].If only one resonance is used, the output includesthe phase fluctuations of both the source and theresonator. The calibration is accomplished bystepping the frequency of the source and measuringthe output voltage, i.e., bV = G(f)K1(f) bV1' Fromthis measurement the phase spectrum can be calculatedasVN (f)- (18)f K 1 (f)[ r(B:] .Fig. 8A shows one method of implementing thisapproach at X-band. The cavity has a loaded qualityfactor of order 25,000. Fig. 8B shows the measuredfrequency discriminator curve. Note that K1(f) isconstant for f

either increase or decrease the phase sensitivity ofthe mixer system. Fig, 4 shows the noise of aPhase Noise of Quartz Oscillators-80N:r: -90 OlN 5 MHzNU -100 .IN 100 MHzltlL--110"lD 5 MHz.,...0- -120-130Q)L- x_>

detail elsewhere [12]. The low pass filter sectionis used in order not to saturate the amplifiers withthe carrier feedthrough signal from the mixer. Onedc amplifier is used for phase noise measurementsfrom dc to 100 kHz from the carrier. One acamplifier is used for phase noise measurements from50 kHz to 32 MHz. This range is well matched for oneof our spectrum analyzers. The wideband ac amplifierhas a bandwidth of 50 kHz to 1.3 GHz and is used whenthe desired measurement bandwidth exceeds 32 MHz.The wide bandwidth spectrum analyzer also provides aconvenient way to observe the gross features of theoutput phase noise and to identify any major spuriousoutputs if present. In order to obtain the mostaccurate measurement of the phase noise it is,however, necessary to measure the amplitude of thefirst IF signal at about 21 MHz in order to avoid thevariations in gains of the log amplifiers withvarious enVironmental factors.the beat period and pretrigerred at about -2V inorder to accurately determine the slope of the outputin volts per radian. This calculation, shown in eq.(23) below, is typically accurate ± 2% or 0.2 dB.K = Volts/Second)(Period/(2~) (23)3) The two sources to be measured are phase lockedtogether with sufficient bandwidth that the phaseexcursions at the mixer are less than 0.1 radian.The necessary phase lock gain is calculated using anestimate for the noise of the oscillators and thetuning rate for the reference oscillator. This isthen verified by noting the peak to peak excursionsof the dc amplifier and using the measured conversionsensitivity measured in step 2 above. If the peakphase excursions are in excess of 0.1 radian, thenthe phase lock loop bandwidth is increased (ifpossible) in order to satisfy this condition.4) The modulator is driven by a reference frequency(typically at +7 to +10 dBm) which steps through theFourier frequencies of interest and the detected RMSvoltage recorded on the appropriate spectrumanalyzer. This approach accurately yields therelative gains of each amplifier and its respectivespectrum analyzer since it automatically accounts forthe effect of the phase lock loop and residualfrequency pulling as well as the termination of themixer and the variations in gain of the variousamplifiers with Fourier frequency. The amplitude ofthe phase modulation on the carrier is constant inamplitude to better than ± 1.5 dB (typically ±0.2dBfor f < 500 MHz) for reference frequencies dc toabout 10' of the carrier frequency or a maximum of 1GHz. Initial measurements of the prototypemodulator are shown in Fig. 14.Fig. 13. Generalized block diagram of the new NBSphase noise measurement systems. The phase noise ofcarrier frequencies from 1 MHz to 100 GHz can bemeasured by varying the components in the phaseshifters and mixers. Dedicated measurement systemscovering 5 MHz to 50 GHz are described in the text.IV. B Measurement Sequence1) The output power of the two sources to bemeasured is typically set to between +5 and + 13 dBmat the mixer. This takes into account the insertionloss of the phase modulator. If the oscillatorsdon't posses sufficient internal isolation to preventunwanted frequency pulling, isolators (or isolationamplifiers) are generally inserted between thesources and the mixer.2) The absolute sensitivity of the mixer and the dcamplifier for converting small changes in phase tovoltage changes is determined in a way similar to thetraditional method, namely by allowing the twooscillators to slowly beat. The output of the dcamplifier is recorded by the digitizer in the FFTconnected to the dc amplifier in order to accuratelydetermine the period of the beat. In test sets C andD an additional 50 MHz digitizer is used to averageand record the beat frequency. The time scale of thedigitizer is then expanded to approximately 10' ofThis measurement is then combined with themeasurement of the absolute mixer sensitivitymultiplied by the gain of the dc amplifier describedin step 2 above. The absolute gain of all theamplifiers shown in Fig. 13 can generally bedetermined to an accuracy of ± 0.3 dB (1.5 dB forFourier frequencies from 500 MHz to 1 GHz) over theFourier frequencies of interest.5) Next the spectral density function of the FFT isverified. The level of the noise determined by theFFT for the input of the noise source amplifiersequentially shorted to ground, connected to groundthrough a 100 kO metal film resister, and connectedto ground through 200 pF, is then recorded. Fromthese data one can determine the inherent noisevoltage and noise current of the noise sourceamplifier plus the FFT as well as the noise of theresistor to about ± 0.25 dB which is the statedaccuracy of the FFT. This primary calibration of theFFT can be carried out from about 20 Hz to over 50kHz. Above 50 kHz, the noise gain of the amplifierwe used contributes a significant amount of noise.With some compensation the noise is flat to within±0.2 dB from 20 Hz to 100 kHz. Next the noise sourceis switched into the output of the mixer and therelative noise spectrum of all the spectrum analyzersis calibrated by knowing their relative gains. Thisprocedure verifies the voltage references and noisebandwidths of the various spectrum analyzers.6) The noise voltage is recorded on the threespectrum analyzers over the Fourier frequencies ofinterest, generally the same one used in step 4439TN-136

above. The measured noise voltages are scaled usingthe measured gains and the spectral density of phasenoise calculated. The overlapping ranges of thevarious spectrum analyzers allows one the opportunityto compare the measurements on the three spectrumanalyzers, Typically one can obtain S.(f) of theoscillator pair to an accuracy of about ± 0.6 dB atFourier frequency above 100 Hz. The agreement withrepeat measurements is often of order ± 0.2 dE asshown in Fig. 15.7) The noise floor of the system is determined bydriving both sides of the measurement system with oneoscillator haVing a similar power and impedance levelas that used in these measurements. If the noisefloor is within 13 dB of that measured in step 7above, corrections are made to the measurement datato remove the bias generated by the noise floor.Measurements of the amplitude accuracy of the phasemodulation side bands generated by the prototypephase modulators are summarized in Fig. 15. The samemodulator was used for carrier frequencies from 5 to300 MHz. The error in the phase modulation amplitudeis less than 0.5 dB for modulation frequencies fromdc to 10' of the carrier. The 10 GHz modulator alsomaintains an accuracy of better than 0.5 dB out to500 KHz from the carrier. At 1 GHz the modulationamplitude is 1.5 dB high. Once this is measured itcan be taken into account in the calibrationprocedure. The 45 GHz modulator results shown inFig. 15 should be attainable over the entire WR22wavequide bandWidth.The performance that can be obtained with thismeasurement technique is illustrated by actual phasenoise data on oscillator pairs shown in Figs. 15 and16. Typical accuracies are ±0.6 dB with a noisefloor at about - 175 dB relative to 1 radian 2 /Hz.The corrections applied to the raw data at 10 GHz areshown in Fig. 17. At low frequencies the effect ofthe phase-locked loop is apparent while at the higherfrequencies the roll-off of the amplifiers areimportant. These effects have been emphasized herein order to examine the ability of the calibrationprocess to correct for instrumentation gainvariations.V, ConclusionWe have analyzed several traditional approaches tomaking phase measurements and found that they alllacked some element necessary for making phase noisemeasurements from essentially dc out to 10, of thecarrier frequency with good phase noise floors and anaccuracy of order 1 dB. By combining several of thetechniques and adding a phase modulator which isexceptionally flat from dc to about lOt of thecarrier frequency, we have been able to achieveexcellent phase noise floors, bandwidths of at leastlOt of the carrier, and accuracies of order ±0.6 dB.1.5m Ll 1.0.!:...g 0.5wc:.Q oI--__.::=----------=:iii"5-g -0.5:ECDtil~ -1.011.-1.5!::--1-_~~-I-_...J--;!;-..l_-L-:_.L---l~:I::-....L._.L-~0,001 0.01 0.1 1 10 100Modulation Frequency (I) in % of Carrier FrequencyFig. 14 Measurement of the amplitude error ofmodulation signal versus Fourier frequencies f. forthese prototype phase modulators. Curves labeled 5,100, and 300 KHz were obtained with the modulatorused in 5 to 1,300 KHz test set. The curve labeled10 GHz was obtained with the modulator for the 2 to26 GHz test set. The curve labeled 45 GHz wasobtained with the WR22 test set.N -136I-­NU~2 -138IV.... 11dBal'Uc::- ":9­--140tf)Two Oscillator Calibration Test @ 5MHz• System Ax System BA System B + M410 Hz 100 Hz 1kHzI...og Fourier Frequency (HzlFig. 15 Demonstration of calibration accuracy fortwo oscillator concept. The curve labeled System Ashows the measured phase noise of a pair of 5 MHzOSCillators using the test set shown in Fig. 13. Thecurve labeled System B shows the measured phase noiseof the same pair of 5 MHz oscillators using a totallyseparate measurement system with the oscillators heldin phase quadrature with the measurement test set ofA. The curve labeled System B + ~/4 shows the phasenoise of the same pair of oscillators using test setB with and extra cable length of ~/4 inserted intoeach signal path. The agreement between the threecurves is in the worse case ± 0.15 dB.TN-137440

NJ:",--0~.8~..(/)-110-1:20-130-140m -150-0cc -160-170-180Phase Noise at X Band2 3 4 567Lng Fourier Frequency (Hz)--_ .. ­.. -~.--4 5 6 7 aI-Dg Fourier Frequency (Hz).. - .8 9Fig. 16 Phase noise measurement on a pair of 10.6GHz sources using the new NBS measurement technique.a::I"0 0£:tJjc:.2 -513~00c:0:;:;~.0~0-103Calibration CorrectionFig. 17 Correction factor applied to the measurementdata made to obtain the results of Fig. 16.ACknowled~ement~The authors are grateful to many colleagues,especially David W. Allan, James C. Bergquist, AndreaDeMarchi, David J. Glaze, James E. Gray, David A.Howe, John P. Lowe, Samuel R. Stein and Charles Stonefor many fruitful discussions on this topic and theCalibration Coordination Group for the funding toimprove the accuracy and bandwidth of phase noisemetrology.References1. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J.Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen,Jr., W. L. Smith R. L. Sydnor, R. F. C. Vessot, G. M.Winkler, Characterization of Frequency Stability,Proc. IEEE Trans. on I &M 22, 105-120 (1971).3. D.W. Allan, H. Hellwig, P. Kartaschoff, J.Vanier, J. Vig, G.M.R. Winkler, and N.F. Yannoni,Standard Terminology for Fundamental Frequency andTime Metrology, to be published in the Proe. of the42nd Symposium on Frequency Control, Baltimore. MD.June 1-4, 1988.4. F. L. Walls, and S. R. Stein, AccurateMeasurements of Spectral Density of Phase Noise inDevices, Proc. of 31st SFC, 335-343, (1977).(National Technical Information Service, SillsBuilding, 5825 Port Royal Road, Springfield, VA22161).5. F. L. Walls, S. R. Stein, J. E. Gray, and D. J.Glaze, Design Considerations in State-of-the-ArtSignal Processing and Phase Noise MeasurementSystems, Proc. 30th Ann. SFC, 269-274 (1976).(National Technical Information Service, SillsBuilding, 5285 Port Royal Road. Springfield, VA22161).6. R. L.. Barger, M. S. Soren. and J. L. Hall,Frequency Stabilization of a cw Dye Laser, Appl.Phys. Lett. 22. 573 (1973).7. F. L. Walls and A. E. Wainwright, Measurement ofthe Short-Term Stability of Quartz Crystal Resonatorsand the Implications for Crystal Oscillator Designand Applications, IEEE Trans. on I & M~, 15-20(1975).8. A. S. Risley, J. H. Shoaf, and J. R. Ashley,Frequency Stabilization of X-Band Sources for Use inFrequency Synthesis into the Infrared, IEEE Trans. onI & M, n, 187-195 (1974).9. J. R. Ashley, T. A. Barley, and G. J. Rast, TheMeasurement of Noise in Microwave Transmitters, IEEETrans. on Microwave Theory and Techniques, SpecialIssue on Low Noise Technology, (1977).10. A. L. Lance. ~. D. Seal, F. G. Mendoza, and N. W.Hudson, Automating Phase Noise Measurements in theFrequency Domain, Proc. 31st Ann. Symp. On Freq.Control, 347-358 (1977).11. A. L. Lance and W. D. Seal, Phase Noise and AMNoise Measurements in the Frequency Domain atMillimeter Wave Frequencies, from Infrared andMillimeter Waves, Ken Bucton Ed., Academic Press, NY1985.12. F. L. Walls and A. DeMarchi, RF Spectrum of aSignal After Frequency Multiplication Measurement andComparison with a Simple Calculation, IEEE Trans. onI & M~, 210-217 (1975).13. F.L. Walls, A New Phase Modulator for WidebandPhase Noise Measurement Systems, to be submiCted toIEEE Transactions on Ultrasonics, Ferroelectrics andFrequency Concrol.2. J. H. Shoaf, D. Halford, and A. S. Risley,Frequency Stability Specifications and Measurement,NBS Technical Note 632, (1973). Document availablefrom US Government printing office. Order SD at#CI3.46:632.441TN-13B

42nd Annual Frequency Control Symposium - 1988STANDARD TEIUlINOLOGY FORFt.iND....'{ENTAl. FREQUENCY AND TIItE !tETROLOGY *David Allan, National 8ureau of Standards, 80ulder, CO 80303Helmut Hellwig, National 8ureau of Standards,Caithersburg, HD 20899Peter Kartaschoff, Swiss PTT, R&D, CM 3000 8ern 29, SwitzerlandJacques Vanier, National Researc~ Council, Ottava, Canada KIA OR6John Vig. U.S. Army Electronics Technology and Devices Laboratory,Fort Monmouth, NJ 07703Gernot M.R. ~inkler, U.S. Naval Observatory, Vashington, DC 20390Nicholas F. Yannoni, Rome Air Development Center, Hanscom AFh,8edford, MA 017311. Introduction S7 (f) of yet)Techniques to characterize and to measure the frequencyS. (f) of ;(t)and phase instabilities in frequency and time devicesS; (f) of ;(t)and in received radio signals are of fundamen~al impor­tance to all manufacturers and users of frequency ands. (f) of x(t) .time technology.These s~ectral densities are related by theIn 1964, a subcommittee on frequency stability vas form­ equations:ed vithin t~e Institute of Electrical and ElectronicsEngineers (1££E) Standards Committee 14 and, later (in1966), in the Technical Committee on Frequency and Timevithin the Society of Instrumentation and Measurement(SIM), to prepare an IEEE standard on frequency stability.In 1969, this subcommittee completed a documentproposing definitions for measures on frequency andS; (f)phase stabilities (8arnes, et al., 1971). These recom­mended measures of instabilities in frequency generatorshave gained general acceptance among frequency and time ----&---=2 S.(f) .users throughout the world.(h"O)In this paper, measures in the time and in the frequencyA device or signal shall be characterized by a plotdomains are revieved. The particular choice as to which of spectral density vs. Fourier frequency or bydomain is usad depends on the application. Hovever, the tabulating disc,ete values or by equivalent meansusers are reminded that conversions using mathematical such as a statement of pover laves) (Appendix I).formulations (see Appendix 1) from one domain to theother can present problems.According to the conventional definition(Kartaschoff, 1978) of ~(f) (pronounced ·scriptMost of the major manufacturers now specify instability ell"). ~(f) is the ratio of the power in one sidecharacteris~icsof their standards in terms of theseband due to phase modulation by noise (for 41Hzrecommended measures. This paper thus defines andbandwidth) to the total signal power (carrier plusfo~alizes the general practice of more than a decade. sidebands), that is,2. Measures of frequency and Phase Instability!l(f) = ~ densicv, one phase-noise lIIlC11latioo. s idebar41!zTow siglol. ~rFrequency and phase instabilities shall be measured interms of the instantaneous, normalized frequency depar­ The conventional definition of ~(f) is related toture y(e) from the nominal frequency VJ and/or by phase S. (f) bydepa=tur. ~(:.), in radians, from the nominal phase 2~vJta.s follotJs:~(f) .. ~S. (f)y( :)x(t)only if the mean squared phase deviation,

is ~he preferred measure, since, unambiguously, it ft et + r:always cen be measured.y(t·)dt'.tob. Iime-pomal0:For fairly simple models, regression analysis canIn the ~ime domain, frequency inscability shall beprovide efficient escimaees of the TIE (Draper anddefined by the cwo-sample deviation ~,(r) which isSmieh. 1966: CcrR, 1986). In general, ehere arethe square root of the cwo-sample variance o,1(r),many estimators possible for any staciseical quanti~y.Ideally, ve would like an efficiene andThis variance, c,J(r). has no dead-time between thefrequency samples and is also called the Allanunbiased es~imator. Using the time domain measurevariance. For che sampling time r. ve WTlte:a,2(r) defined in (b), the follOWing estima~e ofvhere~ + r; --;- fie y(~)d~ =~x k+lThe symbol < > denotes an infinite time average.In practice, the requirement of infinite timeaverage is never fulfilled; the use of ~he foregoingcerms shall be permitted for finite timeaverages. The ~ and ~+ are time residual measurementsmade aE 7K ana ~ 1 = ~+r. k=1.2.3 ...••and l/r is the nominal fixed sampting ra~e whichgives zero dead time between frequency measuremencs."Residuai" implies the kno~ syseematiceffects have been removed.If dead cime exists becween the frequency departuremeasurements and this is ignored in the computationof c,(r), resul~ing instabilicy values vill bebiased (except for vhite frequency noise). Some ofthe biases have been seudied and some correctiontables published [Barnes, 1969; Lesage. 1983;Barnes and Allan, 19881. Therefore. the term a,(r)shall not be used to describe such biased measurements.Raeher. if biased instability measures aremade, the info~ation in the references should beused to repore an unbiased eseima~e.If che inieial sampiing race is specified as l/ro ,then it has been shovn~hat. in general, we mayobeain a more efficienc estimaee of a,(r) usingvha~ is called "overlapping eseimaees." Thisescimaee is obeained by computingN-2mLi=lwhere N is che number of original time residualmeasuremenes spaced by r.(N=~+l, where M is ~henumber of originai fr~quency measur~ments of sampleei:e r.) and T " mT•.From ~he above equaeion, We see thae a,1(r) actsiike a second-difference operator on the timedeviation residuals--providing a stationary measureof the scochascic behavior eVen for nonscacionaryprocesses. Additio~al variances, which may ~e usedco describe frequency instabilities, are deflned inA?pendix I I.c. ~L9ck-Time Prediceionthe scandard deviation (R~S) of TIE and ies associacedsyseemacic depar~ure due to 4 linearfrequency drifc (or ics uncertainty) can be U3ed topredict a probable ti=e interval error of a clocksynchronized at t"t.=O and left free running thereafter:1l(t)2]1tR.'!STtE "t [2 t2 + a 2 + a 2 (r t) + (--2-)est 4 Yo y twhere "a" is the normalized linear frequency drifeper unic of cime (aging) or the uncertainty in thedrift estimace, a, the two-sample devia~ion ofthe initial freque~cy adjustment, a,(r) the cwosampledeViation describing ehe random frequencyinseabiliey of the clock ae t=T, and x(t.) i5 theinieial synchroniza~ion uncertainey. The ehirdterm in the brackees provides an opeimum and unbiasedescimate (under the condition of an optimum(R~) prediction meehod) in the cases of whitenoise FM and/or random walk FM. The third term istoo optimistic, by about a faccor of 1.4, forflicker noise FM, and coo pessimistic, by about afactor of 3, for whice noise P~.This estimaee is a useful and fairly simple approximaeion.In general. a more complete erroranalysis becomes difficult; if carried out. such ananalysis needs to inclUde the methods of time prediction,the uncertainties of the clock parameters,using ehe confidence limits of measurements definedbelow. che detailed clock noise models, syseematiceffaces. etc.4. Confidence Limits of ~easurementsAn eseimaee for Oy(T) can be made from a finiee daea setwieh Mmeasuremenes of Yj as follows:M-la ( r)I ---l.- - 2y 2(~-l) L (Yj +1- Yj)j=lor, if ehe data are time rudings It; :M-l1I~a/r) L (X j+ 2- 2xj + l + x j2r2(~-1))j=l>,2IojIThe 68 percene confidence in~erval (or error bar), I.,for Gaussian noise of a particular value a,(r) obtainedfrOm a finite number of .ampl~s can be estimated asfo!.lo"s:The variation of the time difference be~~een a realclock And an ideal uniform ~ime scale, also knovnas ei~e intervai error, TIE, observed over a timelnter-I.11 starting a: time t. And ending at t.+tshall be defined as:420wher.. :to~alnumber of data poln~s used in the@'f1':lala~e. Ian ineeger as defined in A?pendix r,z "1 = 0.99,TN-140

"'0 .. 0.87,"'-1 .. 0.77,"'-a" 0.75.As an example of the Gaussian model vith K~lOO, Q - -1(flickar frequency noise) and Oy(' % 1 second) .. 10- 13 ,ve Illay writll:vhich gives:Oy(' .. 1 second) .. (1 ± a.08) x 10- 12 •If M i. SMall, then the plus and minus confidencll intervalsbecome asymmetric and the "'. coefficients are notvalid: however, these confidence intervals can be calculated(Lesage and Audoin, 1973).If "overlapping" estimates are used, as outlined above,chen the confidence interval of the escimate can beshown to be less than or equal to I. as given above(Howe, Allan, Sarnes, 1981).5. Rc;ommendations for CbaIacterizini Qr RepQrtini~easurcments of frequencv and Phase Instabilitiesa. Nonrandom phenomena should be recognized, forexample:oooany observed time dependency of the Statisticalmeasures should be stated;the method of modeling systematicbehavior should be specified (for example.an estimate of the linear frequencydrift vas obtained from thecoefficients of a linear least-squaresregression to M frequency measurements,each with a specified averaging or sampletime. and measurement bandvidth f h);the environmental sensitivities should bestated (for example, che dependence offrequency and/or phase on temperature,magnetic field, barometric pres.ure,vibration, etc.):b. Relevant measurement or specificationparameters should be given:aa6. Referencuthe enviro~ent during measurement;if a passive alement. such as a cryatalfiltar, i. being mea.ured in contrast toa frequency and/or time generator.8arnes, J.A., Tables of bias functions, B 1end 8,. forvariances based on finite samples of procosses withpower law spectral densieie., N8S, Washington, DC,Tech. Note 375, (Jan. 1969).Barnes, J.A. and Allan, D.W .. ·Variances 8ased on Datawith Dead Time Between the Measurements: Theoryand Table.," NBS Tech Note 1318 (1988).Same., J.A., Chi, A.R., Cutlor. L.S., Healey, O.J.,Leeson, D.S., McGunigal, T.E., Mullen, J.A.,Smith,~.L., Sydnor, R., Vessot, R.F. and Winkler,C.M.R., Characterization of frequency stability,IEEE Trans. Instr. and Meas., Vol. ~, 105-120,(May 1971).cela Report 898, Dubrovnik, (1986).Draper, N.R. and Smith, H., Applied Regression Analy.i.,John Wiley and Sons,(1966).Howe, D. A., Allan, D. W., and Barnes, J.A., Propertiesof signal sources and measurement methods. Proc.35th Annual Symposium on Frequency Control, 669­117, (May 1981).Karcaschoff, P., Frequency and Time, Academic Press, NewYork (1978).Lesage, P., and Audoin, C. Characterization of frequencystability: uncertainty due to the finiee number ofmeasurements. IEEE Trans. Instr. and Meas., Vol.ll1:ll. 157-161, (June 1913).Lesage, P., Characteri~ation of Frequency Stability:Sias due to the juxtaposition of time intervalmeasurements. IEEE Trans. on Inser. and Meas, ~12..204-207, (1983).Scein. S. R, Frequency and Time: Their measurement andcharacterization. Precision Frequency Control,Vol. Z, Academic Press, ~ew York, 191-232, (1985).oaaoooothe method of ~easurements:the characteristics of the referencesignal;the nominal signal frequency va:the measurement .•ystem bandwidth f o andthe corresponding lov pass filterresponse;the total measurement time and the numberof measurem~nCs M;the calculation techniques (for example,detail. of the windo~ f'~ction ~henestimacing power spectral densities fromtime domain data, or the assumptions~bo~t effects of dead-time when estimatingthe tva-sample deviation eyer»~;the confidence of the estimate (or errorbar) and ies statistical probability(.e.-s.- "t:.hree.-.s:lgma. II );APP;"'fOIX I1. Pover-Law Speceral pensi=iesPower-law spectral densities are often employed as reasonabl~a~d accurate mod~Ls of ~he random f~uc~ua:ionsin precision oscillators. In practice. these randomfluctuations can often be re?resented by the sum of fiveindependent noise processes, and hence:{+2L ha~cr--2for 0 < f < f h~here h~'s are con5tan~$, Q'S are incegers. a~d f b isthe high frequency cut-off of a iow pa.s filter. Highfrequency divergence is eliminated by the res~riceionson f in this equation. The identification and characterizationof the five noise processes are given inTabie 1, and sho~ in Fig. 1.4211N-141

2. Copy.r,ion Beeween Frequency and Time Dom.inThe operacion of the count.r, averaglng che frequencyfor a cime v, may be thoughc of a. a filtering op.r.­tlon. Th. transfer function, H(f), of thls equivalentfllter i. chen che Fuurier transform of the Impulseresponse of the filcer. The Cime domain frequencyinsc.bility Is chen given bya2 (!i. T, r) ,. f 5, (f) I H( f) 1 2 df,oowhere 5,(f) is the spectr.l density of frequency fluctuations.III i. the m••sur,m,nt raCe (T-v is the deadtime between measurements). In the Case of the two­sample variance IH(f)1 2 i. 2(.in '.vf)/(wvf)2. The tvo­sample vari.nce can chus be computed fromfh2 I 5 (f) ~ df.Y(wvf)2Specifically, for the power l.w model given, the timedomain measure also follows a power law.TABLE 1 - The functional characteristics of the independent noise processesused in modeling frequency instability of oscillatorsSlope characteristics of log log plotFrequency domainTillie-domainTABLE 2Descr.iption of ~7 (f) or S; (f) orNoise Process 5, (f) 5,. (f) u; (r) t:t 7(r) Mod.a 7(v)aIp p/2.JAJA~andom Valk Freq Modulation -2 -4 1 1/2 1Flicker Frequency Modulation -1 -3 0 0 0~ite Frequency Modulation 0 -2 -1 -1/2 -1~licker Phase Modulation 1 -1 -2 -1 -2~bite Phase Modulation 2 0 -2 -1 -357 (f)(2lrf)2(2lrIl0)25;(f) -h.. £"' er;(r) - Irll'SjI (f) lI~h.. £"' - 2 11; h.. f# (ft • a-2) tr 7(r) - I r I" f 21 h £",-25 z (f) .. 47 .. 4;2 h.. f# Mod. ""7 (r) - I r IIll'- Translation of frequency instability measures from spectr.l densities infrequency domain to variances in tillle dOlllain and vice versa (For 2lrf br » 1)tr;(v) ..!Desertp tlon of nois. proc.ss 57 (f) .. SjI (f) ::IỊ Randolll Vdk FrequencyAI f2 5 7(f)lv 1 Hr-1tr~(v»)£""1 II~ (v-1 t:t: (r») £""'IIModulationOiFlicker Frequency Modulation o 3Blf 57 (f) I r Hv cr; (r») r 1 4-(r o tr; (r») rIi~oite Frequency Modulation CI fO 57 (f) I r- 4-(r1 1 Hr 1 tr;(r)]f O tr;(r»)£""2iI,'Flicker Phase Modulation DI r 1 57 (f) I v- % %[r1 t:t;(r)]f 1 !:i--(r1 rr; (V») £"" 1I ;!:L( 2£1: 2 5 7(£)lr"1 Hv 2 a; (r») f1.1.01 ::e Phas, Modulation £ r er;(r»)fOi41r 2A .. C .. 1/2 £ .. ~T1.038 + 3 10g~(2,..fbr)B .. 21og.2 D ..4;l422IIN-142

L + h o 2,ADDITIONAL VARIANCES THAT HAY BE USEP TO PESCRIBEFREOOENcy INSWILITIU IN TIlE TItlE ool1llIN1. Modified Allan or Modified Tw0-$4mple Vari.nce~,2ll.lThis implicitly assumes that the random driving mechanismfor each term is independent of the others. Inaddition, there is the implicit assumption that themechanism is valid over all Fourier frequencies, whichmay not always be true.The values of h$ are characteristic models of oscillatorfrequency noise. For integer values (as Often seems tobe the case for reasonable models), ~ = -a - I,for -3 ~ a ~ I, and ~ ~ -2 for Q ~ 1, where 0,2(,)_r».Table 2 gives the coefficients of the translation amongche frequency stability measures from time domain tofrequency domain and from frequency domain to timedomain.The slope characteristics of the five independent noiseprocesses are plotted in the frequency and time domains1n Fig. 1 (log-log scale).Log Sy (t)f--~-2 I I I f 2/I I I V'\, f- II If l 17I'\.J I IfO I 1/1I I I I I ILog FOUrier Frequency, fInstead of the use of 0 2(,), a "Modified Variance" Mod0,2(,) may be used to characterize frequency iostabilities(Stein, 1985; Allan, 1987). It has the property ofyielding different dependence on r for white phase noiseand flicker phase noise. The dependenc. for Hod 0,(')is ,-1/2 and ,-1 respectively. Mod ~,2(,) is defineda.!:Mod ~2(,) '" 2y 2rwhere N is the original number of time measurementsspaced by,. and, '" m,. the sample time of choice(N=M+l). A device or signal shall be characterized by aplot of a,(') or 0,1(,) or Mod ar(,) or Kod 0,1(,) vs.sampling time '. or by tabulating discrece values or byequivalent means such as a statement of power laws(Appendix I).2. Other VariancesSeveral other variances have been introduced by workersin this field. In pareicular, before the introductionof the ~.o-sample variance, it was standard practice touse che sample variance, SI, defined as2sLog S¢(t)i\ I I I I I I I\f-4\ I I I I\I I I I I I~ I I I Ij ~ fl-31 I II 1\1 I I II I ~ f-~ I I I, I 1\ I I III I I \ f-1 II I~ I I fO I II , I , I I I I I ILog Fourier Frequency, fIn practico it may be obeained from a set of measurementsof the frequency of the oscillator as2 1 Ns = L (y - y)2N i"'l iThe sample variance diverges for some types of noiseand, therefore, is not generally useful.Ocher variances based on the structure function approachcan also be defined (Lindsey and Chi, 1976). Forexample. there are the Hadamard variance. the threesamplevariance and the high pass variance (Rutman1978). They are occasionally used in research andscientific works for specific purposes. such asdifferentiating betveen different t;rpes of noise and fordealing vi,h sys~ematics and sidebands in the speccrum.App;:mrx IIILeg cr y (T),,,I -1IT, I I I I I I r1"\,1 T-;-~ I I I I ITtn I~I 1,.-0 I IIXII , I I I IfLog Sample Time, 1"Slope characteristics of the fiveindependent noise I:rocesses.FrGtll>: 1423Allan. D. ~., Statistics of atomic frequency srandards.Proc. IEEE, Vol. ~.22l-230, (Feb 1966).Allan, D. ~ .. et al., Perfo~ance, modeling. and simula~i~naf ~om. ~.slum be4m clQC~. r~oc. 27~hAnnual Symposium on Frequency Control, 334-346,(1973) .Allan, D. W., The measurement of frequency and frequencystability of precision oscillators. P~~c. 6ehAnnual Precise Time and Time Incerval ••anningMe.:ing. 109-142, (Dec 1974).TN-143

All.n, D. Y., .nd D...s, H., Picosecond tim. diff.r.nc.••••ur•••nt .y.te•. froc. of the 29th AnnualSympo.iua on Frequency Control. 404-411,(Kay 1975).Allan. D. W.• and Hellwig, H. W.• Tise Devi.tion andTim. Prediction Error for Clock Specific.tion.Ch.r.ct.rization, .nd Application, Proc. IEEE Posi­tion Location and Navigation Symposium. 29-36,(1978).Allan, D. W., .nd Barnes, S. A., A aodified "All.nVarianc.- with incr.as.d oscillator ch.r.ct.riza­tion ability. Proc. 35th Annual Symposium on Fre­quency Control. 470-476, (Kay 1981).AIl.n, D. Y., Tim. and Frequ.ncy (Time Domain) Char­act.rization, Estimation and Prediction of Preci­sion Clocks .nd Oscillators. IEEE Trans. UFFC Vol.~, 647-654 (Nov 1987).Atkinson. Y. K.• Fey, L.• and Newman, J .• Spectruatanalysis of extremely lov-frequency variations ofquartz oscillators. Proc. IEEE. Vol. ~, 379-380,(Feb 1963).S.bitch, D., and Oliverio, J., Phase noise of v.riouso.cill.tors .t very low Fourier frequencie.. froc.28th Annual Symposium on Frequency Control, 150­159, (1974).Sames, J. A., Models for the interpretation offrequency stability measurements. NSS Tech. Note683, (Aug 1976).Sames, J. A., Jones, R. R., Tryon, P. V., and Allan,O.Y .• Noise models for atomic clocks. Proc. 14thAnnual Precise Time and Time Interval PlanningMeeting, 295-307, (Dec 1982).Baugh, R. A., Frequency modulation .nalysis with theHadamard v.riance. Proc. 25th Annual Symposium onFrequency control, 222-225, (1971).Bernier, L.G., Linear Prediction of the Non-St.tionaryClock Error Function, Proc. 2nd European Frequencyand Ti~e Forum, NeuchAtel (1988).Bl.ckman, R. B., and Tukey, J. M., The measurement ofpow.r spectra, Dover Publication, Inc., New York,NY, (1959).8lair. B. E., Time and frequency: Th.ory and fundam.nt.ls.NBS Monograph No. 140, US GovernmentPrinting Offic., Washington, DC 20402. (May 1974).Boileau, E., and Picinbono, 8., Statistical study ofphase fluctuations and oscillator stability. IEEETrans. Instr. and Meas .• Vol. Z2. 66-75,(liar 1976).Brandenberger, R., Hadom, F., Halford, 0., and Shoaf,J. H., High quality quart& crystal oscillators:frequency-domain and time-domain stability. Proc.25th Annual Symposium on Frequency Control. 226­230, (1971).Chi, A. R., Th. mechanics of translation of frequencystability measures becveen frequency and timedomainmeasurements. Proc. 9th Annual Precis. Timeand Ti2e Interval Planning Meeting, (Dec 1977).Cutler, L. S., and Searle, C. L.• Some aspects of thetheor/ and measure=.nt of frequency fluctuations infr.q~ney standards, Pro~. IEEE, Vol. ~, 136-154,(F.b 1966).424De Prins. J., and Corn.lissen, G., Analyse spectral•discrete. Eurocon (Lausanne, Swit%erland), (Oct1971) .De Prins, J., De.comet, G.• Gord:i, M., and Tuine, J.,Frequency-domain int.rpretation of oscillator phasestability. IEEE Trans. Instr.•nd Heas .• Vol. ~ll, 251-261. (Dec 1969).Fisch.r, H. C., Fr.quency stability measureMentprocedure.. Proc. 8th Annual Precise Time and TimeInterval Planning K.eting, 575-618. (Dec 1976).Greetiball. C.A., Initiali%ing a Flicker Nois. Generator.IEEE Trans. Instr. and Meas .• Vol. ~, 222-224(Jun. 1986).Groslambert, S., Oliv.r, M.• and Uebersfeld, S.,Spectral and short-t.rm stability measurements.IEEE Trans. Instr.•nd Heas .• Vol. ~, 518-521.(Dec 1974).Halford, D., A general mechanical model for IfI spectraldensity noise with special reference to flickernoise I/lf\. Proc. IEEE Vol. ~, 251-258, (Mar1968).IEEE. Special issue on time and frequency.Vol. iQ. (May 1972).IEEE-NASA, Proceedings of Symposium on short-termfrequency stability. NASA Publication SP 80.(1964) .Proc. IEEE,Jones, R. M., and Tryon, P. V., Estimating time fromatomic clocks. NBS Journal of Research, Vol. ~,November 17-24. (Jan-Feb 1983).Kartaschoff, P., Computer Simulation of the ConventionalClock Model. IEEE Trans. Instr. and Heas., Vol. ~21, 193-197 (Sept 1979).Kroupa, V. F., Frequency stability: Fundamentals andMeasurement. IEEE Press. IEEE Selected ReprintSeries. Prepared under the sponsorship of the IEEEInstrumentation and Heasurement Society, 77-80,(1984).Lesage P., and AUdoin, C., Correction to: Characterizationof frequency stability: Uncertainty due tofinite number of measurements. IEEE Trans. Instr.and Heas .• Vol. ~, 103. (Mar 1974).Lesage. P.. and Audoin, F., A time domain method formeasurement of the spectral density of fraquencyfluctuations at low Fourier frequencies. Proc.29th Annual Symposium on Frequency Control, ,94­403, (May 1975).Lesage, P.. and Audoin, C., Correction to: Charactertzationof frequency stability: Uncertainty due tothe finite number of measurements. I~EE Trans.Instr. and Heas., Vol. ~, 270, (Sept 1976).Lesage, P.. and Ayi, T., Characteri%ation of frequencystability: Analysis of modified Allan variance andproperties of its estimate. IEEE Trans. on Instr.and Meas., Vol. ~, 332-336. (Dec 1934).Lindsey, W. C., and Chie, C. M., Theory of oscillatorinstability based upon structure functions. Proc.IEEE, Vol. ~, 1662-1666, (Dec 1976).Mandetbrot, B., Some noises with Ilf spectrum; a bridgebaevaan direct current an4 white noIse, IEEE .rans.Inf. Theory, Vol. l.I.:.l1. 289-298. (Apr 1967).TN-l44

Keyer. D. C., A test set for the accurate meAsurements IHnkler, C. K. R.• Ilall, lL C.• and Percival, D. a., Theof phAse noise on high-quality aignal sources.U.S. Naval Ob.arvatory clock time raCarence and theIEEl Trans. lnatr. and ne.s., Vol. ~, 215-227. p.rfomance oC a samp1a of atolaie eloelu. Ketro­(Nov 1970). 10gia, Vol. ~, 126-134, (Oct 1970).Percival. O. 8 .• A heuristic model of long-term atomic Yoshimura, K., Characterization of fraquancy atability:clock behavior. Proc. 30th Annual Symposium on Uncertainty due to the autocorr.lation of tha fre­Frequency Control, (June 1976).quency fluctuations. IEEE Trans. Instr. and Keas.,1&.21. 1-7. (MAr 1978).Ricci, O. W.• and Peregrina. L., Phase noise measurementusing a high resolution counter with on-line dataprocessing. Proc. 30th Annual Symposium onFrequency Control, (June 1976).The authors are wembers of the Technical Committee TC-).Time and Frequency. of the IEEE Instrumentation andRutman. J., Comment on characterization of frequency MeAsurement Society. This paper is part of this Commitstability.IEEE Trans. lostr. and Meas., Vol. ~ tee's effort to develop an IEEE Standard. To this end.ll, 85, (Feb 1972).the IEEE authorized a Standards Coordinating Committee.sec 21. and the development of the IEEE Standard underRutman. J .. Characterization of frequency stability: A PAR-P·l139. sec 21 is formed out of the 16M. UFFC andtransfer function approach and its application to KTT Societies; tha authors are also members of sec 21.measurements via filtering of phase noise. IEEETrans. llUtr. and Meas., Vol. I.I1.:21.. 40-48. (Mar1974) .Rutman, J., Characterization of phase and frequencyinstabilities in precision frequency sources:Fifteen years of progress. Proc. IEEE. Vol. ~.1048-1075, (Sept 1978).Rutman, J., and Sauvage, C., Heasurement of frequencystability in the time and frequency domatns viafiltering of phAse noise. IEEE Trans. Instr. andHeas .• Vol. ~, 515-518. (Dec 1974).Rutman, J., and Uebersfeld, J., A model for flickerfrequency noise of oscillators. Proc. IEEE, Vol.iQ., 233-235, (Feb 1972).Sauvage. C., and Rutman, J., Analyse spectrale du bruitde frequence des oscillateurs par la variance deHadamard. Ann. des Telecom .• Vol. 1!. 304-314.(July-Aug 1973).Vanier. J., and Tetu, H., Time domain measurement offrequency stability: A tutorial approach, Proc.10th Annual Precise Time and Time Inteeval PlanningHeeting, 247-291, (1978).Vessot, R., Frequency and Time Standards. Methods ofExperimental Physics, Academic Press, New York,198-227, (1976).Vessot, R., Mueller, L., and Vanier. J., The specifica.ionof OScillator characteristics from measurementsmade in the frequency domain. Proe. IEEE,Vol. ~, 199-207 (Feb 1955).Von Neumann, J., Kent, R. H., Bellinson. H. R., andHa:t, B. 1., The mean square successive difference,Ann. ~ch. Stat.• Vol. 11, 153-162, (1941).Walls. F.L., and Allan, D.W., Heasurements of FrequencyStability, Proc. IEEE, Special Issue on Radio Measurementsand Standards, Vol. ~, No.1, pp. 162­168 (Jan 1986).Walls, F. L., Stein, S. R.. Gray. J. E.. and Glaze,D. J., Design considerations In state-of-the-artsignal processing and phase noise measurementsvstems. Proc. 30th Annual Symposium on FrequencyC~ntrol, (June 1976).Winkler, G. M. R., A brief review of frequency stabilitymeasures. Proe. 8th Annual Precise Ttme and TimeInterval Planning Meeting, 489-528, U.S. Navala.~ ••~eh LabQr.co~, waShington, D.C .. (Dec 1976).425 IN-145

105Characterization of Frequency Stability *JAMES A. BARNES, SENIOR MEMBER, IEEE, ANDREW R. CHI, SENIOR MEMBER, IEEE, LEONARD S. CUTLER,MEMBER, IEEE, DANIEL J. HEALEY, MEMBER, IEEE, DAVID B. LEESON, SENIOR MEMBER, IEEE, THOMAS E.McGUNIGAL, MEMBER, IEEE, JAMES A. MULLEN, JR., SENIOR MEMBER, IEEE, WARREN L. SMITH, SENIORMEMBER, IEEE, RICHARD L. SYDNOR, MEMBER, IEEE, ROBERT F. C. VESSOT, AND GERNOT M. R. WINKLER,MEMBER, IEEEAbstTact-Consider a signal generator whose instantaneousoutput voltage Va) may be written asVet) = [V o + E(t)] sin [21O'0t + ~(t)]where V o and '0 are the nominal amplitude and frequency, respectively,of the output. Provided that .(1) and ,;,Ct) = (d.,,/(dt)are sufficiently small for all time t, one may define the fractionalilutantaneous frequency deviation from nominal by the relationyet)E t,i>(t) .21TPoA proposed definition for the measure of frequency stability isthe spectral density Sif) of the function yet) where the spectrumis considered to be one sided on a per hertz basis.An alternative definition for the measure of stability is theinfinite time average of the sample variance of two adjacent averagesof y(t); that is, if1 j'H'fi~ = - yet) dtT "where T is the averaging period, tl>+I = h + T, k = 0, 1,2 ... ,to isarbitrary, and T is the time interval between the beginnings oftwo Il'UCceSllive measurements of average frequency; then thelII'!1COud measure of stability iswhere ( ) denotes infinite time average and where T - T.In practice, data records are of finite length and the InfiniteManuscript received December I, 1970. The author!! of thiBpaper are membel'll of the Subcommittee on Frequency Stability ofthe Technical Committee on Frequency and Time of the IEEEGroup on Instrumentation and Me8BUrement.J. A. Barnes iB with the Time and Frequency Division, Inatilutefor Basic Standards, NBS, Boulder, Colo. S0302.A. R. Chi is with the Head Timing Systell18 Section, NASAGodda.rd Space Flight Center, Greenbelt, Md. 20771.L. S. Cutler iB with the Physical Research Laboratory, Hewlett­Packard Company, Palo Alto, Calif. 94304.D. J. Healey iB with the Aerospace Division, WestinghouseElectric Corporation, Baltimore, Md. 21203.D. B. Leeson iB with California Microwave, Inc., SunnyvalEo,Calif. 94086.T. E. McGunigal is with the RF and Quantum TechniquesSection, NASA Goddard Space Flight Center, Greenbelt, Md.21715.J. A. Mullen iB with the Research Divi8i.on, Raytheon Company,Waltham, Mass. 02154.W. L. Smith is with Bell Telephone Laboratories, Allentown,PlI..18108.R. L. Sydnor is with the Telecommunications Department,Jet Propulsion Laboratoryl Pasadena, Calif. 91103.G. M. R. Winkler is With the Time Service Division, U. S.Nllval Observatory, Washington, D. C. 20390.R. F. C. Vessot is with the Smitheonian Astrophysical Observatory,Cambridge, Mass. 02138.:It See Appendix Note # 18time averages implied in the definitions are nomWly not avaiLt.blejthus estimates for the two measures must be used. E8timates ofS.(f) would be obtained from suitable averages either in the timedomain or the frequency domain. An obvious estimate for "';CT) isParameters of the measuring system and estimating procedureare of critical importance in the specification of frequency stability.In practice, one should experimentally establish confidence limitsfor an estimate of frequency stability by repeated trials.BI(N, r, ~),B,(r, ~)f Ef.I,get)h..W/2Ti, i, k, m, nMNrGLOSSARY 01

106SS.(J)S.(J)TuVet)V.V.(l)I/(t)v'(t)z(t):te,)Yet)z..(I)- 1)An intermediate term used inderiving (23). The definition of Sis given by (64).One-sided (power) spectral densityon a per hertz ba.'lis of the pure realfunction get). The dimensions ofS.U) are the dimensions of g\t)/f.A definition for the measure of frequencystability. One-sided (power)spectral density of y(t) on a perhertz basis. The dimensions ofS.(/) are Hz-I.Time interval between the beginningsof two successive measurementsof average frequency.Time variable.An arbitrary fixed instant of time.The time coordinate of the beginningof the kth measurement ofaverage frequency. By definition,t..., = t. + T, k = 0, 1, 2 .. '.Dummy variable of integration;14 == rt1'.Instantaneous output voltage ofsignal generator. See (2).Nominal peak amplitude of signalgenerator output. See (2).Inatanta.neous voltage of referencesignaL See (40).Peak amplitude of reference signal.See (40).Voltage output of ideal productdetector.LoW-P888 filtered output of productdetector.Real function of time related to thephase of the signal V(O by x(1) ;;lI[",(t»)/(211'"1'0).A predicted value for z(t).Fractional frequency offset of Vet)from the nominal frequency. See (7).Average fraction&l frequency offsetduring the kth measurement interval.See (9).The sample average of N successivevalues of fJ•• See (76).Nondeterministic (noise) functionwith (power) spectral density givenby (25).Exponent of f for a power-lawspectral density.Positive real constant.The Kronecker 8 function definedI, if r = 1by 8.(r - 1) iii •{ 0, otherwISe.Amplitude fluctuations of signal.See (2).!Ell:B TRANSACTIONS ON INSTRUMENT.'TlON .\>;D ME.'SUREMENr, MAY 1971JJ­ Exponent of r. See (29)..,(t)Instantaneous frequency of V(t).Defined by(".:(N, T, r)T4>(t)'I'(t)1f:(T, 1')w == 2rf.,(t) == 2 1 d.... at 4>(t).Nominal (constant) frequency ofV(t).The Fourier transform of n(t).Sample variance of N averagesof y(t), each of duration r, andspaced every T units of time.See (10).Average value of the sample variance".~(N, T, T).A second choice of the definition forthe measure of frequency stability.Defined by ".:(r) == (".:(N = 2,T = T,1'».Time stability measure defined by".;(r) == T'''':(T).Duration of averaging period ofy(t) to obtain y•. See (9).Instantaneous phase of V(t). Definedby 4>(t) =: 2rvot + \p(t).Instantaneous phase fiuctuatiorulabout the ideal phase 2....,ot. See (2).Mean-square time error for Dopplerradar. See (82).Angular Fourier frequency variable.L INTRODUCTIONTHE measurement of frequency and fluctuations infrequency has received such great attention forso many years that it is surprising that the conceptof frequency stability does not have a universallyaccepted definition. At least part of the reason has beenthat some uses are most readily described in the frequencydomain and other uses in the time domain, aswell as in combinations of the two. This situation isfurther complicated by the fact that only recently havenoise models been presented that both adequately describeperformance and allow a translation between thetime and frequency domains. Indeed, only recently hasit been recognized that there caD be a wide discrepancybetween commonly used time domain measures themselves.Following the NASA-IEEE Symposium on Short­Term Stability in 1964 and the Special Issue on FrequencyStability in the PRocEEDINGS OF THE IEEE,February 1966, it now seems reasonable to propose adefinition of frequency stability. The present paper ispresented as technical background for an eventual IEEEstandard definition.This paper attempts to present (as concisely as practical)adequate, self-consistent definitions of frequencystability_ Since more thAn one definition of frequencystability is presented, an important part of this paper

B'R:sES et ai.: ClHR.,CTY.RIZHIO:-r 01" F"REQt;E»/CY 8HBlLlTY(perhaps the most important part) deals with translationsamong the suggested definitions of frequency stability.The applicability of these definitions to the morecommon noise models is demonstrated.Consistent with an attempt to be concise, the referencescited have been selected on the basis of being ofmost value to the reader rather than on the basis ofbeing exhaustive. An exhaustive reference list coveringthe subject of frequency stability would itself be avoluminous publication.Almost any signal generator is influenced to some extentby its environment. Thus observed frequency instabilitiesmay be traced, for example, tA:> changes inambient temperature, supply voltages, magnetic field,barometric pressure, humidity, physical vibration, oreven output loading, to mention the more obvious. Whilethese environmental influences may be extremely importantfor many applications, the definition of frequencystability presented here is independent of thesecausal factors. In effect, we cannot hope to present anexhaustive list of environmental factors and a prescriptionfor handling each even though, in some cases, theseenvironmental factors may be by far the most important.Given a particular signal generator in a particularenvironment, one can obtain its frequency stabilitywith the measures presented below, but one should notthen expect an accurate prediction of frequency stabilityin a new environment.It is natural to expect any definition of stability toinvolve various statistical considerations such as stationarity,ergodicity, average, variance, spectral density,etc. There often exist fundamental difficulties in rigorousattempts to bring these concepts intA:> the laboratory. Itis worth considering, specifically, the concept of stationaritysince it is a concept at the root of many statisticaldiscussions.A random process is mathematically defined as stationaryif every translation of the time coordinate mapsthe ensemble onto itself. As a necessary condition, if onelooks at the ensemble at one instant of time t, the distributionin values within the ensemble is exactly thesame as at any other instant of time f. This is not toimply that the elements of the ensemble are constantin time, but, as one element changes value with time,other elements of the ensemble assume the previous values.Looking at it in another way, by observing theensemble at some instant of time, one can deduce noinformation as to when the particular instant was chosen.This same sort of invariance of the jomt distributionholds for any set of times tt, ~,107"', t,. and its translationtt + 1", tz + 1", "', t.. + T.It is apparent that any ensemble that has a finitepast as well as a finite future cannot be stationary, andthis neatly excludes the real world and anything ofpractical interest. The concept of stationarity does violenceto concepts of causality since we implicitly feelthat current perfonnance (Le., the applicability of stationarystatistics) cannot be logically dependent uponfuture events (i.e., if the process is terminated some timein the distant future). Also, the verification of stationaritywould involve hypothetical measurements that arenot experimentally feasible, and therefore the concept ofstationarity is not directly relevant to experimentation.Actually the utility of statistics is in the formationof idealized models that relWJna.bly describe significantobservables of real systems. One may, for example, considera hypothetical ensemble of noises with certainproperties (such as stationarity) as a model for a particularreal device. If a model is to be acceptable, itshould have at least two properties: first, the modelshould be tractable; that is, one should be able tA:l easilyarrive at estimates for the elements of the models; andsecond, the model should be consistent with observablesderived from the real device that it is simulating.Notice that one does not need tA:> know that the devicewas selected from a stationary ensemble, but only thatthe observables derived from the device are consistentwith, say, elements of a hypothetically stationary ensemble.Notice also that the actual model used maydepend upon how clever the experimenter-theorist is ingenerating models.It is worth noting, however, that while some textson statistics give "tests for stationarity," these tests arealmost always inadequate. Typically, these tests determineonly if there is a substantial fraction of thenoise power in Fourier frequencies whose periods are ofthe same order as the data length or longer. While thismay be very important, it is not logically essential tothe concept of stationarity. If a nonstationary modelactually becomes common, it will almost surely be becauseit is useful or convenient and not because theprocess is "actually nonstationary." Indeed, the phrase"actually nonstationary" appears to have no meaningin an operational sense. In short, stationarity (or nonstationarity)is a property of models, not a property ofdata [1].Fortunately, many statistical models exist that adequatelydescribe most present-day signal generators;many of these models are considered below. It is obviousthat one cannot guarantee that all signal generators areadequately described by these models, but the authorsdo feel they are adequate for the description of mostsignal generators presently encountered.II. STATEMENT OF THE PRoBLEMTo be useful, a measure of frequency stability mustallow one to predict performance of signal generatorsused in a wide variety of situations as well as allowone to make meaningful relative comparisons amongsignal generators. One must be able to predict performancein devices that may most easily be described eitherin the time domain, or in the frequency domain, or ina combination of the two. This prediction of performancemay involve actual distribution functions, and thusTN-148

ID8second moment measures (such as power spectra andvariances) are not totally adequate.Two common types of equipment used to evaluate theperformance of a frequency source are (analog) spectrumanalyzers (frequency domain) and digital electroniccounters (time domain). On occasion the digital counterdata are converted to power spectra by computers. Onemust realize that any piece of equipment simultaneouslyhas certain aspects most easily described in the timedomain and other aspects most easily described in thefrequency domain. For example, an electronic counterhas a high-frequency limitation, an experimental spectraare determined with finite time averages.Research has established that ordinary oscillatms demonstratenoise, which appears to be a superposition ofcausally generated signals and random nondeterministicnoises. The random noises include thermal noise, shotnoise, noises of undetermined origin (such as flickernoise), and integrals of these noises.One might well expect that for the more general casesone would need to use a nonstationary model (not stationaryeven in the wide sense, i.e., the covariance sense).Nonstationarity would, however, introduce significant difficultiesin the passage between the frequency and timedomains. It is interesting to note that, so far, experimentershave seldom found a nonstationary (covariance)model useful in describing actual oscillators.In what follows, an attempt has been made to separategeneral statements that hold for any noise or perturbationfrom the statements that apply only to specific models.It is important that these distinctions be kept inmind.III. BACKGROUND AND DEFINITIONSTo discuss the concept of frequency stability immediatelyimplies that frequency can change with time andthus one is not considering Fourier frequencies (at leastat this point). 'The conventional definition of instantaneous(angular) frequency is the time rate of change ofphase; that is(1)where cI>(t) is the instantaneous phase of the oscillator.This paper uses the convention that time-dependentfrequencies of oscillators are denoted by vet) (cycle frequency,hertz), and Fourier frequencies are denoted by"" (angular frequency) or f (cycle frequency, hertz) where'" == Zrrf. In order for (1) to have meaning, the phase cI>(t)must be a well-defined function. This restriction immediatelyeliminates some "nonsinusoidal" signals such asa pure random uncorrelated ("white") noise. For mostreal signal generators, the concept of phase is reasonablyamenable to an operational definition and this restrictionis not serious.Of great importance to this paper is the concept ofspectral density, Sg (I). The notation Sp (I) is to representthe one-sided spectral density of the Ipure real Ifunction g (t) on 0. per hertz basis; that is, the tot aI"power" or mean-square value of g (t) is given by1~ S,(f) df·Since the spectral density is such an important conceptto what follows, it is worthwhile to present someimportant references on spectrum estimation. There aremany references on the estimation of spectra from datarecords, but worthy of special note are [2 J- [5 J.IV. DEFINITION OF MEASURES OF FREQUENCY STABILITYA. General(SECOND-MoMENT TYPE)Consider a signal generator whose instantaneou" outputvoltage V (t) may be written asV(t) = [Vo + t(t)J sin [211"vot + 'I'(t)] (2)where V o and I'll are the nominal amplitude and frequency,respectively, of the output and it is assumedthatand(3)I.~(t) I« I (+)1-""'0for substantially all time t. Making use of (l) and (2)one sees thatand4>(t)vet) = Vo + .; .j>(t)._11"Equations (3) and (4) are essential in order that 'I'(t)may be defined conveniently and unambiguously (seemeasurement section).Since (4) must he valid even to speak of an instantaneousfrequency, there is no real need to distinguishstability measures from instability measures. That is,any fractional frequency stability measure will be farfrom unity, and the chance of confusion is slight. It istrue that in a very strict sense people usually measureinstability and speak of stability. Because the chances ofconfusion are so slight, the authors have chosen to continuein the custom of measuring "instability" and speakingof stability (a number always much less than unity).Of significant interest to many people is the radio frequency(RF) spectral density S\.(f). This is of directconcern in spectroscopy and radar. However, this is nota. good primary measure of frrquency stability for tworeasons. First, fluctuations in the :lmplitude d t) contributedirectly to 8 1 , (n ; and second, for many cases '\'hell(5)TN-149

e4.RNES et al.: CH."R.ACTERrZ.4.TIO~· OF rnEQt;E~CY ~i."'BILITY109d t) is insignificant, the RF spectrum 81' (I) IS notuniquely related to the frequency fluctuations [6],B. General:First Defim'tion of the Measure of FrequencyStability-Frequency DomainBy definition, lety(t) == ~(t) , (7)~lfVowhere I"(t) and "0 are as in (2). Thus y(t) is the instantaneousfractional frequency deviation from the nominalfrequency ..o. A proposed definition of frequencystability is the spectral density S~ (I) of the instantaneousfractional frequency fluctuations y (t). The function S.. (f)has the dimensions of Hz-l.One can show [7J that if SI/'(I) is the spectral densityof the phase fluctuations, thenS.(f) = C~J S~(f)= c:rrS~(f). (8)Thus a knowledge of the spectral density of the phasefluctuations ~(f) allows a knowledge of the spectraldensity of the frequency fluctuations SII (I), the first definitionof frequency stability. Of course, SII (I) cannotbe perfectly measured-this is the case for any physicalquantity; useful estimates of SII (I) are, however, easilyobtainable.C. General: Second DefinitiDn of the MelUltlre ()f FrequencyStability-Timeof 17:(1").DomainThe second definition is ba.sed on the sample variance ofthe fractional frequency fluctuations. In order to presentthis measure of frequency stability, define fi. by therelationwhere t h1 = t. + T, k = 0, 1, 2, .. , , T is the repetitioninterval for measurements of duration 1', and to is arbitrary.Conventional frequency counters measure the number ofcycles in a period 1'; that is, they measure "01'(1 + fh).When l' is 1 s they count the number of "0(1 + fi.).The second mea.aure of frequency stability, then, isdefined in analogy to the sample variance by the relationIN ( 1 N )2)(17:(N, T, 1'» == N _ 1 ~ Y. - N t; fi~ , (10). First,of course, one cannot practically let N approach infinityand, second, it is known that some actual noise processescontain substantial fractions of the total noise power inthe Fourier frequency range below one cycle per year.In order to improve comparability of data, it is importantto specify particular N and T. For the preferred definitionwe recommend choosing N = 2 and T = 'T (i.e., no deadtime between measurements). Writing (17:(N =2, T=1", 1'»as 17:(1'), the Allan variance [8], the proposed measure offrequency stability in the time domain may be written as(11)for T = 'T.Of course, the experimental estimate of 17:(1') must beobtained from finite samples of data, and one can neverobtain perfect confidence in the estimate; the true timeaverage is not realizable in 11 real situation. One estimatesI7:H from a finite number (say, m)' of values of 17:(2, 1', 1')and averages to obtain an estimate of 0":(1'). Appendix Ishows that the ensemble average of 17:(2, 'T, 1') is convergent(i.e., as m -+ co) even for noise processes that do not haveconvergent (O":(N, 1', 1'» as N -+ . Therefore, 17:(1') hasgreater utility as an idealization than does (.,.:( ex>, 1', 1'»even though both involve 88BUmptions of infinite averages.In effect, increasing N causes U:(N, T, 1') to become moresensitive to the low-frequency components of S.U). Inpractice, one must distinguish between an experimentalestimate of a quantity (say, of 0":(1"» and its idealizedvalue. It is reasonable to believe that extensions to theconcept of statistical ("quality") control [9Jmay proveuseful here. One should, of course, specify the actualnumber m of independent samples used for an estimateIn summary, therefore, S.(f) is the proposed measure of(instantaneous) frequency stability in the (Fourier)frequency domain and ":(1') is the proposed measure offrequency stability in the time domain.D. DUtributionsIt is natural that people first become involved withsecond moment measures of statistical quantities and onlylater with actual distributions. This is certainly true withfrequency stability. While one can specify the argumentof a distribution function to be, say (jjHl - fil), it makessense to postpone such a specfication until a real use hasmaterialized for a particular distribution function. Thispaper does not attempt to specify a preferred distributionfunction for frequency fluctuations.E. Treatment of Systematic VaMations1) General: The definition of frequency stability 17:(1')in the time domain is useful for many situations. However,some oscillators, for example, exhibit an aging or almostlinear drift of frequency with time. For some applications,this trend may be calculated and should be removed (8)before estimating 0-:(1').In general, a systematic trend is perfectly deterministic(i.e., predictable) while the noise is nondeterministic.Consider 9. function g(t), which may be written in the formTN-ISO

110 lEER TR.~NSAcrlO:'IS ON I:'lSTRl:ME:iHTlON .~:iD MEASl:REMENT, MAY 1971get) = e(t) + net) (12)where e(t) is some deterministic function of time and net),the noise, is a nondeterministic function of time. We willdefine e(t) to be the systematic tTend to the function get).A problem of significance here is to determine when andin what sense e(t) is measurable.2) Specific Case-LineaT Drift: As an example, if weconsider a typical quartz crystal oscillator whose fractionalfrequency deviation is yeo, we may letdget) = dt y(t). (13)With these conditions, c{t) is the drift rate of the oscillator(e.g., Urlo/day) and '1 (t) is related to the frequency"noise" of the oscillator by a time derivative.One sees that the time average of get) becomes1 J,.+t 1 f,·+tT I. get) dt = c, + If.. net) dt (14)where c(t) = CJ. is assumed to be the constant drift rateof the oscillator. In order for Cl to be an observable,it is natural to expect the average of the noise term tovanish, that is, converge to zero.It is instructive to assume [8], [10] that in additionto a linear drift, the oscillator is perturbed by a flickernoise, i.e.,0< f S fl(15). f> flwhere h_ 1 is a constant (see Section V-A-2) and thus,S.(f) = {(21f/h_d , (16)0,for the oscillator we are considering. With these assumptions,it is seen thatand thatI J,.+rlim -T net) dt = K(O) = 0 (17)T'-_ t.~~ {Variance [f {.+r net) dtJ} = 0 (18)where «U) is the fourier transform of net). Since &(0)= 0, «(0) must also vanish both in probability and inmean square. Thus, not only does net) average to zero,but one may obtain arbitrarily good confidence on theresult by longer averages.Having shown that one can reliably estimate the driftrate Ct of this (common) oscillator, it is instructive toattempt to fit a straight line to the frequency aging.That is, letand thusget) = yet) (19)g(t) = Co + c, (t - to) + n'(t) (20)where Co is the frequency intercept at t = to and Cl isthe drift rate previously determined. A problem ariseshere becauseandS.,(1) S.(1) (21)~~ {variance [f t'+T n'(t) dtJ} = '" (22)for the noise model we have assumed. This follows fromthe fact that the (infinite N) variance of a flicker noiseprocess is infinite [7], [8], [10]. Thus, ~ cannot bemeasured with any realistic precision, at least, in anabsolute sense.We may interpret these results as follows. After experimentingwith the oscillator for a period of time onecan fit an empirical equation to y (t) of the formyet) = Co + tel + n'(t),where n' (t) is nondetenninistic. At some later time it ispossible to reevaluate the coefficients ~ and Cl' Accordingto what has been said, the drift rate Cl should bereproducible to within the confidence estimates of theexperiment regardless of when it is reevaluated. For Co,however, this is not true. In fact, the more one attemptsto evaluate Co, the larger the fluctuations are in theresult.Depending on the spectral density of the noise term,it may be possible to predict future measurements of~ and to place realistic confidence limits on the prediction[11]. For the case considered here, however, theseconfidence limits tend to infinity when the predictioninterval is increased. Thus, in a certain sense, Co i~"measurable" but it is not in statistical control (to usethe language of the quality control engineer r91).V. TRANSLATIONS AMONG FREQUENCY STABILITYMEASURESA. Frequeru;y Domain to Time Domain1) General: It is of value to define r = T/T; that is,T is the ratio of the time interval between successivemeasurements to the duration of the averaging period.Cutler has shown (see Appendix I) that(u:(N, T, T» *= N 1~ df Sen [sin' (r/T)} {I _ sin' (rrfNr)}.(N - 1) 0 • (1ff")' N' sin' (rrfr)(23)Equation (23) in principle allows one to calculate thetime-domain stability (u:(N, T, r» from the frequencydomainstability S.(j).2) Specific Model: A model that has been found useful[8], [1O]-[13J consists of a set of five independentnoise processes z.. (t), n = -2, -1,0,1,2, such that... See Appendix Note 11 19TN-151

MII.NES et al.. CHAIl.ACfERlZATlO:-; OF FR.EQUENCY STABILITYYet) = z-,(t) + LI(t) + zo(t) + Zl(t) + z,(t) (24)and the spectral density of z~is given byS.,(f) = {h.r , 0 ~ f ~ f. (25)0, f > f.,n = -2, -1,0,1,2,where the h,. are constants. Thus, SI/ (I) becomes(26)for 0 :::; f :::; f. and S.(I) is aBSumed to be negligible beyondthis range. In effect, each z~ contributes to both S.(I) and(.,.:(N, T, T» independently of the other ~. The contributionsof the z~ to (Cl':(N, T, T» are tabulated inAppendix H.Any electronic device has a finite bandwidth and thiscertainly applies to frequency-measuring equipment also.For fractional frequency fluctuations Yet) whose spectraldensity varies asS.(I) '" r, a ~ -1 (27)for the higher Fourier components, one sees (fromAppendix I) that (a:(N, T, r» may depend 011 the exactshape of the frequency cutoff. This is true because asubstantial fraction of the noise "power" may be in thesehigher Fourier components. AB a simplifying assumption,this paper assumes a sharp cutoff in noise "power" at thefrequency f. for the noise models. It is apparent from thetables of Appendix H that the time domain measure offrequency stability may depend on f. in a very importantway, and, in some practical cases, the actual shape of thefrequency cutoff may be very important [7J. On theother hand, there are many practical measurementswhere the value of f. has little or no effect. Good practice,however, dictates that the system noise bandwidth f.should be specified with any results.In actual practice, the model of (24)-(26) seems to fitalmost all real frequency sources. Typically, only two orthree of the h-eoefficients are actually significant for areal device and the others can be neglected. Because ofits applicability, this model is used in much of whatfollows. Since the z. are assumed to be independent noises,it is normally sufficient to compute the effects for ageneral z. and recognize that the superposition can beaccomplished by simple additions for their contributionsto S.U) or (O':(N, T, r».B. Time Domain to Frequency Domain1) General: For general (O':(N, T, T» no simple prescriptionis available for translation into the frequencydomain. For this rellBOn, one might prefer SM(f) as ageneral measure of frequency stability. This is especiallytrue for theoretical work.2) Specifu; Mode/.: Equations (24)-(26) form a realisticmodel that fits the random nondeterministic noises foundon most signal generators. Obviously, if this is a goodmodel, then the tabl~ in Appendix II may be used(in reverse) to trallBlate into the frequency domain.Allan [8J and Vessot [12] showed that if111o :::; f :::; f.S.(f) = Jhof" , (28)10, f > f.where a is a constant, then(29)for N and r = T/ T held constant. The constant p. isrelatedto a by the mapping shown l in Fig. 1. II (28)and (29) hold over a reasonable range for a signal generator,then (28) can be substituted into (23) and evaluatedto determine the constant h" from measurements of(.,.:(N, T, T». It should be noted that the model of (28)and (29) may be easily extended. to a superposition ofsimilar noises as in (26).C. TranslatWns Among tk Time-Domain Meamru1) General: Since (O':(N, T, T» is a function of N, T,and T (for some types of noise f. is also important), it isverydesirable to he able to translate among differentsets of N, T, and T (f. held constant). This is, however,.not possible in general.2) SpecifU; Model: It is useful to restrict considerationto a case described by (28) and (29-). Superpositions ofindependent noises with different power-law types ofspectral densities (Le., different CIt) can also be treated bythis technique, e.g., (26). One· may define two "biasfunction.~I" B. and B, by the relations [13JandB (N ) - (O':(N, T, T» (30)­I ,r, p. = (0':(2, T, r»_ (0':(2, T, T»B ,( r, p. ) = ( '((31)Cl'. 2, T, T»where r == T/ T and p. is related to a by the mapping ofFig. 1. In words, B I is the ratio of the average varianceforN samples to the average variance for two samples(everything else held constant), while B, is the ratio ofthe average variance with dead time between measurements(r ~ 1) to that of no dead time (r = 1 and withN = 2 and T held constant). These functions are tabulatedin [13]. Figs. 2 and 3 show a computer plot ofBI(N, r = 1,14) and B,(r, p.).Suppose one, has an experimental estimate of (Cl':(N.,T., TI» and its spectral type is known, Le., (28) and (29)form a good model and p. is known. Suppose also that onewishes to know the variance at some other set of measurementparameters N" T 2, T,. An unbiased estima.te ot(0': (N" T" T,» may be calculated by1 It should be noted that in Allan [8), the exponent a correspondsto the spectrum of phtUle fluctuations while variances.are take~ over .average frequencll fluctuations. In the presentpaper, a IS Identical to the exponent a + 2 in [8).1N-152

112u·2-2Fig. 1. I'- -a mapping.-~Fig. 3. Bias function B,(T, p.).ferent parameter, such as in the use of an o"cilJator inDoppler radar measurements or in clocks..(tr:(N2,T., T.» = (::YFig. 2. Function B,(N, T = I, 1'-).. [BI(N., r., /-l)B.(r., I-')J< '(N T » (32)B.(N" T" I-')B.(r" 1-') tr. I, I, TI ,where Tl = Tt/·f'I and r2 = TdT2.3) General: While it is true that the concept of thebias functions B 1 and B 2 could be extended to otherprocesses besides those with the power-law types ofspectral densities, this generalization has not been done.Indeed, spectra of the form given in (28) [or superpositionsof such spectra as in (26)] seem to be themost common types of nondeterministic noises encounteredin signal generators and associated equipment. For-Qther types of fluctuations (such as causally generatedperturbations), translations must be handled on an in­.dividual basis.VI. ApPLICATIONS OF STABILITY MEASURESObviously, if one of the stability measures is exactlythe important parameter in the use of a signal generator,-the stability measure's application is trivial. Some non­.trivial applications arise when one is interested in a dif­A. Doppler Radar1) General: From its transmitted signal, a DopplNradar receives from a moving target a frequency-shiftedreturn signal in the presence of other large signals. The"elarge signals can include clutter (ground return) andtransmitter leakage into the receiver (spillover). Instabilitiesof radar signals result in noise energy on theclutter return, on spillover, and on local oscillators inthe equipment.The limitations of subclutter Yisibility (SCV) rejertionsdue to the radar signals themselves are related tothe RF power spectral density Sv (f). The quantity typicallyreferred to is the carrier-to-noise ratio and can bemathematically approximated by the quantityThe effects of coherence of target return and otherradar parameters are amply considered in the literature[14]-[17].2) Special Case: Because FM effects generally predominateover AM effects, this carrier-to-noise ratio isapproximately given by [6](33)for many signal sources provided If - vol is sufficientlygreater than zero. (The factor of t arises from the factthat Sip (f) is a one-sided spectrum.) Thus, if f - Vll isTN-IS3

:l frequcllcy ~eparatioll froll1 the carrier, the carrier-tonoiseratio at that point is approximatelyB. Clock Ermrs1) Gel/eral: A clock is a device that counts the cyclesof a periodic phenomenon. Thus, the reading error x(t)of a clock run from the signal given by (2) is'P(t).r(l) = -'J­-?l'Vo(35)and the dimensions of .r(t) are seconds.If this clock is a secondary standard, then one couldhave available some past history of x(t), the time errorre)ati"e to the standard clock. It often occurs that oneis interested in predicting; the clock error x(t) for somefuture date, say t" + T, where to is the present date.Obviously, this is a problem in pure prediction and canhe handlrd by cOll\"entional methods [3].2) Special Case: Although one could handle the predictionof clock errors by the rigorous methods of predictiontheory, it is more common to use simpler predictionmethods [10], [11]. In particular, one often predicts a clockerror for the future by adding to the present error acorrection· that is derived from the current rate of gain(or loss) of time. That is, the predicted error £(t o + T)is related to the past history of x(t) byiLlY(t). One of the most common techniques is a heterodyneor beat frequency technique. In this method, the signalfrom the oscillator to be tested is mixed with a referencesignal of almost the same frequency as the test oscillatorin order that one is left with a lower average frequencyfor analysis without reducing the frequency (or phase)fluctuations themselves. Following Vessot et al. [18],consider an ideal reference oscillator whose output signalisV,(t) = Yo, sin 211"1' 0 t (40)and a second oscillat

114i - .. --- 1 Lonpfittet'tUi..II,ILow Pa••Fltter1---""'v'CtiIEEg TR.

MRNES et al.: CHARACTERIZATION OF FREQGENCY ST.,BILITYFrom the analog voltage one may use analog spectrumanalyzers to determine S~ (f), the frequency stability. Byconverting to digital data, other analyses are possibleon a computer.F. Common Hazards1) Errors Caused by Signal-Processing Equipment: Theintent of most frequency stability measurements is toevaluate the source and not the measuring equipment.Thus, one must know the performance of the measuringsystem. Of obvious importance are such aspects of themeasuring equipment as noise level, dynamic range,resolution (dead time), and frequency range.It has been pointed out that the noise bandwidth fl isvery essential for the mathematical convergence of certainexpressions. Insofar as one wants to measure the signalsource, one must know that the measuring system is notlimiting the frequency response. At the very least, onemust recognize that the frequency limit of the measuringsystem may be a very important, implicit parameter foreither .,.:(t) or S.(j). Indeed, one must account for anydeviations of the measuring system form ideality such asa "nonflat" frequency response of the spectrum analyzeritself.Almost any electronic circuit that processes a signalwill, to some extent, convert amplitude fluctuations at theinput terminals into phase fluctuations at the output.Thus, AM noise at the input will cause a time-varyingphase (or FM noise) at the output. This can impose importantconstraints on limiters and automatic gain control(AGe) circuits when good frequency stability is needed.Similarly, this imposes constraints on equipment used forfrequency stability measurements.2) Analog Spectrum Analyzers (Frequency Domain) :Typical analog spectrum analyzers are very similar indesign to radio receivers of the superheterodyne type, andthus certain design features are quite similar. For example,image rejection (related to predetection bandwidth)is very important. Similarly, the actual shape of theanalyzer's frequency window is important since this affectsspectral resolution. As with receivers, dynamicrange can be critical for the analysis of weak signals inthe presence of substantial power in relatively narrowbandwidths (e.g., 60 Hz).The slewing rate of the analyzer must be consistentwith the analyzer's frequency window and the post-detectionbandwidth. If one has a frequency window of 1 Hz,one cannot reliably estimate the intensity of a brightline unless the slewing rate is much slower than 1 Hz/s.Additional post~detection filtering will further reduce themaximum usable slewing rate.S) Spectral Density Estimation from Time DomainData: It is beyond the scope of this paper to present acomprehensive list of hazards for spectral density estimation;one should consult the literature [2]-[5J. There115are a few points, however, which are worthy of specialnotice: a) data aliasing (similar to predetection bandwidthproblems); b) spectral resolution; and c) COnfidenceof the estimate.4) Variances of Frequency Fluci:ual.ions "':(T): It is notuncommon to have discrete frequency modulation of asource such as that associated with the power supplyfrequencies. The existence of discrete frequencies in S.(j)can cause .,.:(.,.) to be a very rapidly changing functionof T. An interesting situation results when T is an exactmultiple of the period of the modulation frequency (e.g.,one makes T = 1 s and there exists 6O-Hz frequencymodulation on the signal). In this situation, .,.:(T = 1 s)can be very optimistic relative to values with slightlydifferent values of T.One also must be concerned with the convergenceproperties of "':(T) since not all noise processes will havefinite limits to the estimates of "':(T) (see Appendix I).One must be as critically aware of any "dead time" in themeasurement process as of the system bandwidth.5) SigruJ Source and Loading: In measuring frequencystability one should specify the exact location in thecircuit from which the signal is obtained and the natureof the load used. It is obvious that the transfer characteristicsof the device being specified will depend on the loadand that the measured frequency stability might beaffected. If the load itself is not constant during themeasurements, one expects large effects on frequencystability.8) Confidence of the Estim.a.u: As with any measurementin science, one wants to know the confidence to assign tonumerical results. Thus, when one measures S.U) or ,,:(T),it is important to know the accuracies of these estimates.a) The Allan Variance: It is apparent that a singlesample variance u:(4, T, T) does not have good confidence,but, by averaging many independent samples, one canimprove the accuracy of the estimate greatly. There is akey point in this statement, "independent samples." Forthis argument to be true, it is important that one samplevariance be independent of the next. Since cr:(2, 1", T) isrelated to the first difference of the frequency (11),it is sufficient that the noise perturbing Yet) have "independentincrements," i.e., that y(t) be a random walk.In other words, it is sufficient that S.(j) r-J ,-2 for lowfrequencies. One can show that for noise processes thatare more divergent at low frequencies than r J , it isdifficult (or impossible) to gain good confidence onestimates of o?(T). For noise processes that are lessdivergent than r J , no problem exists.It is worth noting that if we were interested in.,.:(N = CD, 1", T), then the limit noise would becomeS.(j) r-J f instead of ,-2 as it is for 0-:(2, 1", .,.). Since mostreal signal generators possess low-frequency divergentnoises, (cr:(2, T, T» is more useful than cr:(N = co, T, T).Although the sample variances ,,:(2, T, 1") will not benormally distributed, the variance of the average of 11'1.TN-156

116independent (nonoverlapping) samples of CT: (2, r, r)(i.e., the variance of the Allan variance) will decrease as11m provided the conditions on low-frequency divergenceare met. For sufficiently large m, the distribution of them sample averages of 0-:(2, r, r) will tend toward normal(central limit theorem). It is thus possible to estimateconfidence interva.la based on the normal distribution.As always, one Dl8.y be interested in r values approachingthe limits of available data. Clearly, when one isinterested in r values of the order of a year, one is severelylimited in the size of m, the number of samples of 0-:(2, T, 1").Unfortunately, there seems to be no substitute for manysamples and one extends T at the expense of confidence inthe results. "Truth in packaging" dictates that the samplesiu m be stated with the re8Ults.b) Spectral Density: As before, one is referred to theliterature for discussions of spectrum estiD18.tion {2]-{5].It is worth pointing out, however, that for S.(j) there arebasically two different types of averaging that can beemployed: sample averaging of independent estiD18.tesof 8.00, and frequency averaging where the resolutionbandwidth is made much greater than the reciprocal datalength.VIII. CoNCLUSIONSA good measure of frequency stability is the spectraldensity S.(j) of fractional frequency fluctuations yet).An alternative is the expected variance of N sampleaverages of 1I(t) taken over a duration T. With the beginningof successive sample periods spaced every T unitsof time, the variance is denoted by 11:(N, T, T). Thestability measure, then, is the expected value of manymeasurements of U:(N, T, T) with N = 2 and T = T;that is, 11:(T). For all real experiments one has a finitebandwidth. In general, the time domain measure offrequency stability U:(T) is dependent on the noise bandwidthof the system. Thus, there are four importantparameters to the time domain measure of frequencystability.N Number of sample averages (N = 2 for preferredmeasure).T Repetition time for successive /.lB.mple averages(T = ., for preferred measure).1" Duration of each sample average.f. System noise bandwidth.Translations among the various stability measures forcommon noise types are possible, but there are significantre8BOnB for choosing N = 2 and T = T for the preferredmeasure of frequency stability in the time domain. Thismeasure, the Allan variance, (N = 2) has been referencedby {12J, [20]-{22] and more.Although S.(j) appears to be a. function of the singlevariable f, actual experimental estimation proceduresfor the spectral density involve a great many parameters.Indeed, its experimental estimation can be at least asinvolved 8B the estimation of .,.:(..).ApPEl'DIX IWe want to derive (23) in the text. Starting from (101we have(CT~(N, 1', r»)IN ( 1 N )')== . ( -- L ii. - - L y.N - 1'_1 N ._11 {N 1 N N }= N - 1 L (y~> - " L L (y,ii;>.",-1 lV ,-I ,-1= _1__, {tf"" dt U f".,(N - 1).. ._1 " I.lit t fdt' (y(t')y(t"»- l ., dt" f'i., dl' (Y(fl)Y(t"»} (56)N ._1 /-1 I, "where (9) has been used. Now(y(t')y(t") > = R.(l' - t") (57)where R.(..) is the autocorrelation function of y(t) a.ndis the Fourier transform of S. (f), the power spectraldensity of yet). Equation (57) is true provided thaty(t) is stationary (at least in the wide or covariancesense), and that the average exists. If we assume thepower spectral density of y (t), S" (f) has low and highfrequency cutoffs !I and IA (if necessary) so that1~ S,(1) dfexists, then if y is a random variable, the average doesexist and we may safely asSume stationarity.In practice, the high-frequency cutoff fA is alwayspresent either in the device being measured or in themeasuring equipment itself. When the high-frequencycutoff is necessary for convergence of integrals of S" (f)(or is too low in frequency), the stability measure willdepend on IA. The latter case can occur when the measuringequipment is too narrow-band. In fact, a usefultype of spectral analysis may be done by varying IApurposefully {I8].The low-frequency cutoff II may be taken to be muchsmaller than the reciprocal of the longest time of interest.The results of calculations as well as measurementswill be meaningful if they are independent of II as IIapproaches zero. The range of exponents in power lawspectral densities for which this is true will be discussedand are given in Fig. 1.To continue, the derivation requires the Fourier transformrelationships between the autocorrelation functionand the power spectral densityS.(f) = 41~ R.(r) cos 27C"jr d..R,(1") = 1- S.(f) cos 27flr df.Using (58) and (57) in (56) gives(58)TN-IS7

B\R~f:S eL al.: CIHRACTERIZ.,TIOI< OF FREQVENCY STABILiTY( 0, and so we may pullout the term for k = 0 separately and writeS = 2(~ cos kx ~ I) + ~ 1= 2('I: (N - k) cos kX) + N. (65)t-.0- I-l0 0 0 0 0 0;0.... 0 0 0,:0 0'. 0 0, ,'0 0.0', 0,, ,.i,.,I, ..0'5> 0 0:, , II0 0'.....0 0',, ,0 0 0""'~:-4 0 0 0 0 0 0Fig. 5. Region of summation for i and k for N = 4.This may be written as1 d] N-I .S = N + 2 Re [ N - "7 -d L e'h (66)tXt_Iwhere Re[U] means the real part of U and d/dxis the differential operator. The series is a simple geometricseries and may be summed easily, giving1 d ] is iNS}S = N + 2 Re N _ _ _ e - e.{[ i dx 1 - en= N + 2 Re {I - eiNs -.- N(I - e iS )}4sm 2 x/2sm 2 Nx/2= sin~ x/2 .000000000b00011,(67)Combining everything we get, after some rearrangement,(.Therefore (11:(N, T, T» is convergent for power lawspectral densities, Sij) = h.!", without any low- or highfrequencycutoffs for -3 < a < 1. Using (69) for powerlaw spectral densities we find(11~{N, T. 1'» = T-·-1h.C". -3 < a < 1Jl ;; -a - 1TN-158

llSandC"=-N(.V -l)7l"o~1r~ d 0 sin' U {I, 0 U U.• 0U_ sin' .\"ru }.,-. . ••• sm ru(70)This is the basis for the plot in Fig. 1 in the text of f.lversus a. For a ~ 1 we must include the high-frequencycutoff fA.For loT = 2 and r = 1 the results are particularly simple.We have( 2(2 '» -a-1h 2 1~ d a-2'« (71)a. ."','" = T " -;;;: U U SID U11" 0for power law spectral densities. For N = 2 and generalr we get= ';-1~ du S.(.!£..)...1fT 0 7l:'Tcos 2u(r+ 1)+cos 2u(r-l)1- cos 2u- cos 2ru + 2 2= ~ 1~ du S.(.!£..) sin' 1.1 ~in2 ru.(72)11"'" n 1I"T UThe first form. in (72) is particularly simple and is alsouseful for r = 1 in place of (71).Let us discuss the case for

8IR:-IES et al.: CH.'R.'CTER[Z.

120ACK!W9;"LEllGMENTThe authors are particularly indebted to D. W. All8Jl,Dr. D. Halford, Dr. S. Jar\'is, and Dr. J. J. Filliben ofthe National Bureau of Standards. The authors are alsoindebted to Mrs. Carol Wright for preparing the manyrevised copies of this paper.REFERENCES[1] E. T. Ja)-nes, "Information theory IUld .tatinica1 mechanips,"PhY3. Rev., vol. lOS, !leC. 15, Oct. 1957, pp. 171-190.[2] C. BlDgham, M. D. Godfrey, IUld J. W. Tukey, "Moderntechniques of power spectrum estimation," IEEE Tram.Audio ElectrOC01.l&t., vol. AU-15, June 1967, pp. 56-66.[3] R. B. Bladanan, Linear DilUl ST/l.Qothif'l{} and Prediction in.Thecrv and Practice. Reading, MflBS.: Addison-Wesley,1965.[4] R. B. BlackmlUl and J. W. Tukey, The Measu.rement ofPOlDer Spectra. New York: Donr, 1958.[5] E. O. Brigham and R. E. Morrow, "The fast Fouriertransform," IEEE Spectrum, vol. 4, Dec. 1967, pp. 63-70.[6] E. J. Baghdady, R. D. Lincoln, IUld B. D. Nelin. "Shorttermfrequency stability: theory, mellSUrement, and sta.tus,"Proc. IEEE-NASA Slimp. on Short-Term Frequency Stability(NASA SP-80) , ~o,. 1964, pp. 65-87; also in PTC/c.IEEE, "01. 53, JuJy 1965, pp. 70+-722 and Proc. IEEE (CorrespJ,vol 53, Dec. 1965. pp. 2110-2111.[7) L. Cutler and C. Searle. "Some aspecUl of the theory andmellSUrement of freQUoenc~' fluctuatioll8 in frequency standards."Proe. IEEE, vol. 54, Feb. 1966, pp. 136-154.[8] D. W. AllBll, "Statistics of atomic frequency standards,"Proc. IEEE, vol. 54, Feb. 1966, pp. 221-230.[9] N. A. Shews.rt, Economic Control of Q1Ullity of M emufacturedProduct. Princeton, N. J.: "Van ~ostrand, 1931,p.l46.[IO] J. A. Barnes, "Atomic timekeeping and the statistics ofprecision signal generatom," Proc. IEEE, "01. 54, Feb. 1966,pp. 207-220.[11] J. A. Barnea and D. W. Allan, "An approach to the predictionof coordinated uni"er:u.l time," Frequ.en.c'V, vol.5, :So".-Dec. 196i, pp. 15-20.[12] R. F. C. Vesset et 01., "An intercom~arison of hydrogenand cesium frequency artande..rda," IEEE Tram. ImtTli.m.Me08., vol. IM-15, Dee. 1966, pp. 165-175.[13] J. A. Barnes, "Twles of hiu functioll8, B, and Bt. forvariances hued on finite samples of processes with powerlaw spectral densities," NBS, Wuhington, D. C., Tech. Kote375, Jan. 1969.[14] D. B. Leeson and G. F. Johnson, "Short-term stability fora Dopp,ler radar: requirements, measurements, IUld techniQues,, Proc. IEEE, vol. MJ.Feb. 1966,_ pp. 244-248.[15] W. K. Saunders in Radar nandbook, M. I. Skolnik, Ed.New York: McGraw Bill, 1970, ch. 16.[16] R. S. Raven, "Requirements on muter oscillators for coherentradar," Proe. IEEE, vol. 54, Feb. 1966, pp. 23;-243.[Ii] D. B. Lee!lOD, "A simple model of feedback oscillator noisespectrum," Proc. IEEE (Lett.), "01. 54, Feb. 1966, pp. 329­330.[18] R. F. C. Yessot, L. Mueller, and J. "Vanier. "The specificationof oscillal.or cheracteristics from measurements I:ladein the frequen~y domain," Proc. IEEE, \'01. 54, Feb. 1966.pp. 199-207.[19] Floyd ~f. Gardner, Phascloe/: Techniques. :S-ew York:Wile\·. 1966.[20] C. Menoud, J. Racine. and P. Kartaschoff. "Atomic hydro­@:en maser work al L.5-.R.H .. ~euchalel. Switzerland." Proc.£lst Ann. Symp. Frequency Control, Apr. 1%7. pp. 543-567.[21l A. G. Mungall, D. Morris, H. Daams, and R. Bailey. "Atomichydrogen maser de"elopmen~ at the ~atioDal ResearchCouncil of Canada." Metrologuz, "01. 4. July 1968, pp. 87-94[22] R. F. C. Vessot, "Atomic h)'drogen masel>. IUl introductionand progress report," Hewlett-Packard J., "01. 20, Oct.1968, pp. 15-20.Reprinted by permission fromIEEE TRA:'\SACTIO:'\S 0:\l~STRCME:\TATIO:'\ A~D ME.-\SCRDIE'\T\'01. V,f·20. ~o. 2. !\fa\ 1971Copyright ® 19i1. hy the Institute of E;leclrical ~nd Electronics Engineers. IncPRINTED I~ THE L.SA.

142 Rep. 580--2Reprinted, with permIssIOn, from Report 580 of the International Radio ConsultativeCommittee (C.c.I.R.), pp. 142-150, 1986.CHARACTERIZATION OF FREQUENCY AND l-HASE NOISE(Study Programme 3BI7)*(1974-1978-1986)l. IntrodudfollTechniques to characterize and to measure the frequency and phase instabilities in frequency generatorsand received radio signals are of fundamental importance to users of frequency and time standards.In 1964 a subcommittee on frequency stability was formed, within the Institute of Electrical and ElectronicEngineers (IEEE) Standards Committee 14 and later (in 1966) in the Technical Committee on Frequency andTime within the Society of Instrumentation and Measurement (SIM), to prepare an IEEE standard on frequencystability. In 1969, this subcommittee completed a document proposing definitions for measures on frequency andphase stabilities. These recommended measures of stabilities in frequency generators have gained generalacceptance among frequency and time users throughout the world. Some of the major manufacturers now specifystability characteristics of their standards in terms of these recommended measures.Models of the instabilities may include both stationary and non-stationary random processes as well assystematic processes. Concerning the apparently random processes, considerable progress has been made[IEEE-NASA, 1964; IEEE, 1972] in characterizing these processes with reasonable statistical models. In contrast,the presence of systematic changes of frequencies such as drifts should not be modelled statistically, but should bedescribed in some reasonable analytic way as measured with respect to an adequate reference standard, e.g., linearregression to determine a model for linear frequency drift. The separation between systematic and random partshowever is not always easy or obvious. The systematic effects generally become predominant in the long term, andthus it is extremely important to specify them in order to give a full characterization of a signal's stability. ThisReport presents some methods of characterizing the random processes and some important types of systematicprocesses.Since then, additional significant work has been accomplished. For example, Baugh [1971] illustrated theproperties of the Hadamard variance - a time-domain method of estimating discrete frequency modulationsidebands - particularly appropriate for Fourier frequencies less than about 10 Hz; a mathematical analysis ofthis technique has been made by Sauvage and Rutman [1973]; Rutman [1972] has suggested some alternativetime-domain measures while still giving general support to the subcommittee's recommendations; De Prins et al.[1969] and De Prins and Cornelissen, [1971] have proposed alternatives for the measure of frequency stability inthe frequency domain with specific emphasis on sample averages of discrete spectra. A National Bureau ofStandards Monograph ,devotes Chapter 8 to the "Statistics of time and frequency data analysis" [Blair, 1974]. Thischapter contains some measurement methods, and applications of both frequency-domain and time-domainmeasures of frequency/phase instabilities. It also describes methods of conversion among various time-domainmeasures of frequency stability, as well as conversion relationships from frequency-domain measures to timedomainmeasures and vice versa. The effect of a finite number of measurements on the accuracy with which thetwo-sample variance is determined has been specified [Lesage and Audoin, 1973, 1974 and 1976;Yoshimura, 1978]. Box-Jenkins-type models have been applied for the interpretation of frequency stabilitymeasurements [Bames, 1976; Percival, 1976] and reviewed by Winkler [1976].Lindsey and Chie [1976] have generalized the r.m.s. fractional frequency deviation and the two-samplevariance in the sense of providing a larger class of time-domain oscillator stability measures. They have developedmeasures which characterize the random time-domain phase stability and the frequency stability of an oscillator'ssignal by the use of Kolmogorov structure functions. These measures are connected to the frequency-domainstability measure Sy(f) via the Mellin transform. In this theory, polynominal type drifts are included and sometheoretical convergence problems due to power-law type spectra are alleviated. They also show the closerelationship of these measures to the r.m.s. fractional deviation (Cutler and Searle, 1966] and to the two-samplevariance [Allan, 1966J. And finally, they show that other members from the set of stability measures developed areimportant in specifying performance and writing system specifications for applications such as radar, communications,and tracking system engineering work.Other forms of limited sample variances have been discussed (Baugh, 1971; Lesage and Audoin, 1975;Boileau and Picinbono, 1976] and a review of the classical and new approaches has been published[Rutman, 1978~* See Appendix Note # 23 TN-162

Rep. 580-2 143Frequency and phase instabilities may be characterized by random processes that can be representedstatistically in either the Fourier frequency domain or in the time domain [Blackman and Tukey, 1959). Theinstantaneous, normalized frequency departure ye'l from the nominal frequency Vo is related to the instantaneousphasefluctuation rp(l) about the nominal phase 2ltVoI by:y(l) __1_ drp(l)Ijl(I)2ltVo dl 2ltVo(I)xC') ... rp(l)2ltVowhere X(I) is the phase variation expressed in units of time.1. Fourier frequency domainIn the Fourier frequency domain, frequency stability may be defined by several one-sided (the Fourierfrequency ranges from 0 to 00) spectral densities such as:S,(f) of y(I), S"I(f) of l{l(1), S.,(f) of 1jl(1), Sx(f) of x(t), etc.These spectral densities are related by the equations:pS,(f) ... 2 S,(f) (2)Vo(3)Sx(f).- (2~o)2 S

i44 Rep. 580-2If the initial sampling rate is specified as lito. then it has been shown [Howe et al., 1981] that in generalone may obtain a more efficient estimate of 0,(1:) using what is called "overlapping estimates". This estimate isobtained by computing equation (7).o~ (1:) -1N-2m1: (Xi+2m - 2Xi+m + Xi)2 (7)2(N - 2m) 1: 2 i-Iwhere N is the number of original time departure measurements spaced by To. (N - M + I, where M is thenumber of original frequency measurements of sample time, 1:0) and 1: - m'to. The corresponding confidenceintervals [Howe et 01., 1981], discussed in § 6, are smaller than those obtained by using equation (12), and theestimate is still un biased.If dead time exists between the frequency departure measurements and this is ignored in computingequation (6), it has been shown that the resulting stability values (which are no longer the Allan variances). will bebiased (except for the white frequency noise) as the frequency measurements are regrouped to estimate the stabilityfor m1:o (m > 1). This bias has been studied and some tables for its correction published [Barnes, 1969;Lesage, 1983].A plot of 0,(1:) versus 1: for a frequency standard typically shows a behaviour consisting of elements asshown in Fig. 1. The first part, with 0,(1:) - C 1l2 (white frequency noise) and/or 0,(1:) - 1:- 1 (white or flickerphase noise) reflects the fundamental noise properties of the standard. In the case where 0,(1:) - 1:- 1 , it is notpractical to decide whether the oscillator is perturbed by white phase noise or by flicker phase noise. Alternativetechniques are suggested below. This is a limitation of the usefulness of 0,(1:) when one wishes to study the natureof the existing noise sources in the oscillator. A frequency·domain analysis is typically more adequate for Fourierfrequencies greater than about J Hz. This 't- I and/or 1:- 112 law continues with increasing averaging time until theso-called flicker "floor" is reached, where 0,(1:) is independent of the averaging time 'to This behaviour is found inalmost all frequency standards; it depends on the particular frequency standard and is not fully understood in itsphysical basis. Examples of probable causes for the flicker "floor" are power supply voltage fluctuations, magneticfield fluctuations, changes in components of the standard, and microwave power changes. Finally the curve showsa deterioration of the stability with increasing averaging time. This occurs typically at times ranging from hours todays, depending on the particular kind of standard.A "modified Allan variance". MOD o~(1:), has been 'developed [Allan and Barnes, 1981] which has theproperty of yielding different dependences on 1: for white phase noise and flicker phase noise. The dependencesfor MOD 0,(1:) are 1:- 3/2 and 1:- 1 respectively. The relationships between 0,(1:) and MOD 0,(1:) are also explainedin [Allan and Barnes, 1981; IEEE 1981, Lesage and Ayi, 1984]. MOD 0,(1:) is estimated using the followingequation:N-3m+11 m+.~-. 1 ]MOD o~ (1:) - [ LI (Xi+2m - 2xi+ '" + Xi) 2 (8)2r m 2 (N - 3m + 1) '-jwhere N is the original number of time measurements spaced by 1:0, and 1: - m1:o the sample time of choice.Properties and confidence of the estimate are discussed in Lesage and Ayi [1984]. Jones and Tryon [1983J andBarnes el al. [1982] have developed maximum likelihood methods of estimating 0.(1:) for the specific models ofwhite frequency noise and random walk frequency noise, which has been shown to be a good model forobservation times longer than a few seconds for caesium beam standard's.4. Conversion between frequency and time domainsIn general, if the spectral density of the normalized frequency fluctuations S,(f) is known, the two-samplevariance can be computed [Barnes et al., 1971; Rutman, 1972):ff"42( ) (f) sin 1[1:/ df (9)uy L = 2 .sy (1[1:/FoTN-164

Rep. 580-2 145r f\ \:\.\ f":\.\r\,'.'\~\.~,-.~......... ,-.'i-.. "Fourier frequency ­'Y ('1")r "IT-''\."T"'\."­"""­ T'V' "./~ T' /'Sampling time -FIGURE 1 -Slope characteriftics ofthe five independent noiu processes(log scale)Specifically, for the power law model given by equation (5), the time-domain measure also follows thepower law as derived by Cutler from equations (5) and (9).Note. - The factor 1.038 in the fourth term of equation (to) is different from the value given in most previouspublications.IN-165

146 Rep. 580-2The values of ha are characteristics of oscillator frequency noise. One may note for integer values (as oftenseems to be the case) that 11 = -a - 1, for -3 $ ex $ 1, and 11'" -2 for ex ~ I where OHt) - "[I'.These conversions have been verified experimentally [Brandenberger et 01., 1971] and by computation[Chi, 1977J. Table II gives the coefficients of the translation among the frequency stability measures from timedomain to frequency domain and from frequency domain to time domain.The slope characteristics of the five independent noise processes are plotted in the frequency and timedomains in Fig. 1 (log log scale).S. Measurement techniquesThe spectral density of phase Iluctuations S",{f) may be approximately measured using a phase-locke4loop and a low frequency wave analyzer [Meyer, 1970; Walls et al., 1976J. A double-balanced mixer is used as thephase detector in a lightly coupled phase lock loop. The measuring system uses available state-of-the-art electroniccomponents; also a very high quality oscillator is used as the reference. For very low Fourier frequencies (wellbelow I Hz), digital techniques have been used [Atkinson et 01., 1963; De Prins et 01., 1969; Babitch andOliverio, 1974]. New methods of measuring time (phase) and frequency stabilities have been introduced withpicosecond time precision [Allan and Daams, 1975J, and of measuring the Fourier frequencies of phase noise with30 dB more sensitivity than the previous state of the art [Walls et al., 1976J.Several measurement systems using frequency counters have been used to determine time-domain stabilitywith or without measurement dead time [Allan, 1974; Allan and Daams, 1975J. A system without any counter hasalso been developed [Rutman, 1974; Rutman and Sauvage, 1974J. Frequency measurements without dead time canbe made by sampling time intervals instead of measuring frequency directly. Problems encountered when deadtime exists between adjacent frequency measurements have also been discussed and solutions recommended[Blair. 1974; Allan and Daams, 1975; Ricci and Peregrino. 1976]. Discrete spectra have been measured byGroslambert et al. [1974].6. Confidence limits of time domain measurementsA method of data acquisition is to measure time variations Xi at intervals to. Then Oy{t) can be estimatedfor any t -n"[o (n is any positive integer) since one may use those Xi values for which j is equal to nk. Anestimate for Oy{"[) can be made from a data set with M measurements of Yi as follows:(ll)or equivalent(12)Thus. one can ascertain the dependence of Oy{"[) as a function of"[ from a single data set in a very simple way.For a given data set, M of course decreases as n increases.To estimate the confidence interval or error bar for a Gaussian type of noise of a particular value Oy{t)obtained from a finite number of samples [Lesage and Audoin. 1973] have shown that:Confidence Interval fa "'" Oy{"[) . Ka' M-1 /2 for M > 10 (13)where:M: total number of data points used in the estimate,a: as defined in the previous section,K2 ... KI "" 0.99.KO ,.. 0.87,IC_I 0.77.K_2 0.75.TN-166

Rep. 580-2147Oy(t -As an example of the Gaussian model with M - 100, a - -1 (flicker frequency noise) and1 second) _ 10- 12 , one may write:(14)which gives:o,(t - 1 second) - (l ± 0.08) x 10- 12 (15)A modified estimation procedure including dead-time between pairs of measurements has also beendeveloped [Yoshimura, 1978], showing the influence of frequency fluctuations auto-correlation.7. ConclusionThe statistical methods for describing frequency and phase instability and the corresponding power lawspectral density model described are sufficient for describing oscillator instability on the short term. Equation (9)shows that the spectral density can be unambiguously transformed into the time-domain measure. The converse isnot true in all cases but is true for the power law spectra often used to model precision oscillators.Non-random variations are not covered by the model described. These can be either periodic ormonotonic. Periodic variations are to be analyzed by means of known methods of harmonic analysis. Monotonicvariations are described by linear or higher order drift terms.TABLE I -The junctional characteristics offi¥e independent noise processesfor frequency instability of oscillatorsSlope characteristics of log log plotDescription of noise processS,(f)Frequency-domaineS.,(f) or Sx(f)Time-domainea2 (t)" (t)(J. ~.(J.-2 II. 11./2Random walle frequency -2 -41~Aicker frequency -I -300White frequency 0 -2 -I-\I,Aicker phase 1-I -2-IWhite phase2 0 -2 -Ia2 (t) - I t I~" (t) - I t 1p!2TN-167

148 Rep. SSO-2TABLE II - Translalion offrequency slability measures from spectral densities infrequency domain 10 variance in time domain and vice versa (for 21Cfi, 'C ;. 1)Description of noise process a} (t) - S, (f) - S,,(f) -Random walk frequencyA [fl S, (Jj]t l~ [C I a; (t)]f- lv l - 0[C I a; (t)]f- 4AFlicker frequencyB [f s, (f)/r-~ [r- a; (t)]f- 1V; [r- a;. (t)]f-JWhite frequencyC [.fIS, (f)/c l~ ItI a;. (t)].fIv l 0- [t a; l (t)]f- lCFlicker phaseD[f- I S,(f)]c l..!.. [tlcrl(t)]pD ,.v~ [r a;. (t)]f­ 'White phaseE [f-lS. (f)/c l~ ka;(t)kv l 0- [t l a; (t)]rED _ 1.038 + 3 log. (211.fi,t)6 411. 2B-210g.23fi,E­ 411.2C - 1/2REFERENCESALLA.N, D. W. [February, 1966J StatistiC!! of atomic frequency standards. Proc. IEEE, Vol. 54, 221·230.ALLAN, D. W. [December, 1974] The measurement of frequency and frequency stability of precision oscillators. Proc. 6thAnnual Precise Time and Time Interval (PlTI) Planning Meeting NASA/DOD (US Naval Research Laboratory,Washington, DC), 109-142.ALLAN, D. W. and BARNES, J. A. [May, 1981J A modified KAlIan Variance" with increased oscillator characterization ability.Proc. 35th Annual Symposium on Frequency Control. Philadelphia, PA, USA (US Army Electronics Command,Ft. Monmouth, NJ 07703), 470-476 (Electronic Industries Association, Washington, DC 20006, USA).ALLAN, D. W. and DAAMS, H. [May, 1975] Picosecond time difference measurement system. Proc. of the 29th AnnualSymposium on Frequency Control, 404-411.ATKINSON, W. K., FEY, L. and NEWMAN, J. [February, 1963] Spectrum analysis of extremely low-frequency variations ofquartz oscillators. Proc. IEEE, Vol. 51, 2, 379.BABITCH, D. and OLIVERIO, J. [1974] Phase noise of various oscillators at very low Fourier frequencies. Proc. of the 28thAnnual Symposium on Frequency Control, 150-159.BARNES, J. A. [January, 1969] Tables of bias functions, B 1 and B 2 , for variances based on finite samples of processes' withpower law spectral densities. Tech. Note No. 375, National Bureau of Standards, Washington, DC, USA.BARNES, J. A. (August, 1976J Models for the interpretation of frequency stability measurements. NBS Technical Note 683.BARNES, J. A., CHI, A. R., CUTLER, L. S., HEALEY, D. J., LEESON, D. B., McGUNIGAL, T. E., MULLEN, J. A.,SMITH, W. L., SYDNOR, R., VESSOT, R. F. and WINKLER, G. M. R. [May, 1971] Characterization of frequencystability. IEEE Trans. Inslr. Meas., Vol. 1M-20, 105-120.BARNES, J. A., JONES, R. H., TRYON, P. V. and ALLAN, D. W. [December, 1982] Noise models for atomic clocks. Proc.14th Annual Precise Time and Time Interval (PITI) Applications and Planning Meeting, Washington, DC, USA,295-307.BAUGH, R. A. (1971] Frequency modulation analysis with the Hadamard variance. Proc. 25th Annual Symposium on FrequencyControl, 222-225.BLACKMAN, R. B. and TUKEY. J. M. (1959) The measurement of power spectra. (Dover Publication, Inc., New York, N.Y.).TN-168

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Proc. of the 30th Annual Symposium on Frequency Control.RUTMAN, J. [February, 1972J Comment on characterization of frequency stability. IEEE Trans. Ins". Meas., Vol. IM-21, 2, 85.RUTMAN, J. [March, 1974J Characterization of frequency stability: a transfer function approach and its application tomeasurements via filtering of phase noise. IEEE Trans. Insrr. Meas., Vol. (M-23, I, 40-48.RUTMAN, J. [September, 1978J Characterization of phase and frequency instabilities in precision frequency sources: fifteenyears of progress. Proc. IEEE, Vol. 66, 9, 1048-1075.RUTMAN, J. and SAUVAGE, G. [December, 1974J Measurement of frequency stability in the time and frequency domains viafiltering of phase noise. IEEE Trans. Ins/r. Meas., VoL (M-23, 4, 515, 518.SAUVAGE, G. and RUTMAN, J. [July· August, 1973] Analyse spectrale du bruit de frequence des oscillateurs par la variance deHadamard. Ann. des Telecom., Vol. 28, 7-8, 304-314.VON NEUMANN, J., KENT, R. H., BELLINSON, H. R. and HART, B. L [1941] The mean square successive difference. Ann.Math. S/ar., U, 153-162.WALLS, F. L., STEIN, S. R., GRAY, J. E. and GLAZE, D. J. [June, 1976] Design considerations in state-of-the·art signalprocessing and phase noise measurement systems. Proc. of the 30th Annual Symposium on Frequency Control.WINKLER, G. M. R. [December, 1976) A brief review of frequency stability measures. Proc. 8th Annual Precise Time and TimeInterval (PTTI) Applications and Planning Meeting (US Naval Research Laboratory, Washington, DC), 489-528.YOSHIMURA, K. [March, 1978] Characterization of frequency stability: Uncertainty due to the autocorrelation of the frequencyfluctuations. IEEE Trans. Ins/r. Meas., IM-27 I, 1-7.TN-169

ISO Rep. 580-2, 738-2BIBLIOGRAPHYALLAN, D. W., ~t at. (1913) Performance, modelling, and simulation of some caesium beam clocks. Proc. 21th AnnualSymposium on Frequency Control, 334-346.FISCHER, M. C. (December, 1916] Frequency stability measurement procedures. "Proc. 8th Annual Precise Time and TimeInterval (PTTI) Applications and Planning Meeting (US Naval Research Laboratory, Washington, DC), 515~18.HALFORD, D. (March, 1968J A general mechanical model for 1/1« spectral density noise with special reference to flicker noise1/1/1. hoc. IEEE (Corres.), Vol. 56,3, 2SI·2S8.MANDELBROT, B. [April, 1961] Some noises with II/spectrum, a bridge between direct current and white noise. IEEE TrailS.In! 17I~ory, Vol. IT·l3, 2, 289·298.RUTMAN, J. and UEBERSFELD, J. [February, 1972] A model for flicker frequency noise of oscillators. hoc. IEEE, Vol. 60, 1,233-235.VANIER, J. and TETU, M. (1918] Time domain measurement of frequency stability: a tutorial approach. Proc. 10th AnnualPrecise Time and Time Interval (PTTI) Applications and Planning Meeting, Washington, DC, USA, 247·291.VESSOT, R. (1916] Frequency and Time Standards. Chap. 5.4 Methods of Experimental Physics, Academic Press, 198·221.VESSOT, R., MUELLER, L. and VANIER, J. (February, 1966] The specification of oscillator characteristics from measurementsmade in the frequency domain. hoc. IEEE, Vol. 54, 2, 199-201.WINKLER, G. M. R., HALL. R. G. and PERCIVAL, D. B. (October, 1970] The US Naval Observatory clock time reference andthe performance of a sample of atomic clocks. Ml'lrotogia, Vol. 6, .., 126-134.TN-170

Reprinted, with permission, from P. Lesage and C Audoin in Radio Science, VoL 14,No.4, pp. 521-539, 1979, Copyright © by the American Geophysical Union.Characterization and measurement of time and frequency stabilityP. Lesage and C. A udoin*Laboratoire de f'Horloge Atomique, Equipe de Recherche du CNRS, Universili Paris-Sud, Orsay, France(Received February I, 1979.)The roles which spectral density of fractional frequency fluctuations, two-sample variance, andpower spectra play in different pans of the electromagnetic spectrum are introduced. Their relationshipis discussed. Data acquisition in the frequency and the time domain is considered, and examplesare given throughout the spectrum. Recently proposed methods for the characterization ,of a singlehigh-quality frequency source are briefly described. Possible difficulties and limitations in theinterpretation of measurement results are specified, mostly in the presence of a dead time betweenmeasurements. The link between past developments in the field, such as two-sample variance andspectral analysis from time domain measurement, and recently introduced structure functions isemphasized.I. INTRODUCTIONProgress in the characterization of time andfrequency stability has been initiated owing to thework of the various authors of papers deliveredat the IEEE-NASA Symposium on Short TermFrequency Stability [1964] and of articles publishedin a special issue of the Proceedings of the IEEE[1966J. Presently widespread defmitions of frequencystability have been given by Barnes et al.(l97IJ. Many of the most important articles onthe subject of time and frequency have been gatheredin the NBS Monograph 140 [1974]. Since thattime, many papers have been published whichoutline different aspects of the field. Owing to theextent of the subject, they will be only partlyreviewed here. We will emphasize recentlyproposed principles of measurements and recentdevelopments in the time domain characterizationof frequency stability. The subject of time predictionand modeling as well as its use for estimationofthe spectrum offrequency fluctuations [Percival,1978J are beyond the scope of this paper. Recentreviews which outline several different aspects ofthe field of time and frequency characterizationhave been published [Barnes, 1976; Winkler, 1976;Barnes, 1977; Rutman, 1978; Kartaschoff, 1978].2. DEFINITIONS: MODEL OF FREQUENCYFLUCTUAnONSThe instantaneous output voltage of a frequencygenerator can be written asCopyriahl Cl 1979 by the Americon Geophysical UDioa.V(I) = [Vo + AV(t») cos [2'1Tvol + '9(1»where V o and V o are constants which represent thenominal amplitude and frequency, respectively.tJ. V(t) and cp(t) denote time-dependent voltage andphase variations.Fractional amplitude fluctuations are defmed by(I)E(t) = AV(t)! V o (2)A power spectral density of fractional amplitudefluctuations S.if) can be introduced if amplitudefluctuations are random and stationary in the widesense. Usually, for high-quality frequency sources,one hasIE(t)1 « I (3)and amplitude fluctuations are neglected. However,it is known that amplitude fluctuations can beconverted into phase fluctuations in electronic circuitsused for frequency metrology [Barillet andAudoin, 1976; Bava et al., 1977a] and that theymay perturb measurement of phase fluctuations[Brendel et al., 1977]. It is then likely that amplitudefluctuations will become the subject of more detailedanalysis in the future.According to the conventional defmition of instantaneousfrequency we haveV(I) = "0 + (l /2'1T)cP(l) (4)In a stable frequency generator the conditionIcP(t)1I2'1Tvo « I (5)is generally satisfied.We will use the following notations [Barnes et'" See Appendix Note # 24 521TN-171

522 LESAGE AND AUDOIN01., 1971J:S,I']'P(/)Ii> (I)x(1) =-- and y(/) =-- (6)2'11"11 02'11"v owhere x(t) and y(/) are the fractional phase andfrequency fluctuations, respectively. The quantityx(t) represents the fluctuation in the time definedby the generator considered as a clock.At first, we will make the following assumptions:I. The quantities x(!) and y(!) are random functionsof time with zero mean values, which impliesthat systematic trends are removed [Barnes el 01.,1971 J . They might be due to ageing or to imperfectdecoupling from environmental changes such astemperature, pressure, acceleration, or voltage.Characterization of drifts will be considered insection 8.2. The statistical properties of the stable frequencygenerators are described by a model whichis stationary of order 2. This point has been fullydiscussed in the literature [Barnes et 01., 1971;Boileau and Picinbono, 1976; Barnes, 1976]. Thisassumption allows one to derive useful results andto define simple data processing for the characterizationof frequency stability.Actual experimental practice shows that, besideslong-term frequency drifts, the frequency of ahigh-quality frequency source can be perturbed bya superposition of independent noise processes,which can be adequately represented by randomfluctuations having the following 'one-sided powerspectral density of fractional frequency fluctuations:Sy (f) = 2:2 11..1'" (7)Q--2Sy (f) is depicted in Figure I. Its dimensions areHz-I. Lower values of (I( may be present in thespectral density of frequency fluctuations. Theyhave not been clearly identified yet because ofexperimental difflculties related to very long termdata acquisition and to control of experimentalconditions for long times. Moreover, the relatednoise processes may be difficult to distinguish fromsystematic drifts.Finite duration of measurements introduces alow-frequency cutoff which prevents one fromobtaining in{ormation at Fourier frequencies smaller«.. -1Fig. I. Asymptotic log-log plot os Sy(f) for commonly encounterednoise processes.than 1/9, approximately, where 9 is the total durationof the measurement [Cutler and Searle, 1966].Alternatively, this made it possible to invok.e physicalarguments to remove some possible mathematicaldifficulties related to the divergence of S" (f)as/- 0 for (I( < U.Furthermore, high pass fI1tering is always presentin the measuring instruments or in the frequencygenerator to be characterized. It insures convergenceconditions at the higher-frequency side ofthe power spectra for a > O.The spectral density of fractional phase fluctuationsis also often considered. From (6), one canwrite, at least formally,The dimensions of S,,{f) are S2 Hz-I. Similarly,the spectral density of phase fluctuations ",(t) issuch asIt is expressed in (rad)2 Hz-I.The quantity 2(f) [Halford et al., 1973] issometimes considered to characterize phase fluctuations.If phase fluctuations at frequencies >/are small compared with I rad, one has, .(8)(9).Y(f) = }S..(f) (10)where S (f) is the spectral density of phase fluctuationsof..the frequency generator considered.. Thedefmition of 2{f) implies a connection with theradio frequency spectrum, and its use is not recommended.Since the class of noise processes for which y(l)is stationary is broader than that for which theIN-I72

TIME AND FREQUENCY523TABLE I. Designation of noise processesQ Designation Class of Stationarity2 white noise of phase stationary phasefluctuationsflicker noise of phase stationary frrst-orderphase incrementso white noise of frequency stationary frrst-orderphase increments-I flicker noise of frequency stationary second-orderphase increments-2 random will of frequency stationary second-orderphase incrementsphase is stationary [Boileau and Picinbono, 1976],S y(f) should preferably be used in mathematicalanalysis. However, it is true that many experimentalsetups transduce q>(t) into voltage fluctuations andallow one to experimentally determine an estimateof its power spectral density S.{f).Table I shows the designations of the noiseprocesses considered. It also indicates the classof stationarity to which they pertain, as will bejustified later on.Sy(f) is one of the recommended definitions offrequency stability [Barnes et 01., 1971]. It givesthe widest information on frequency deviations yet)within the limits stated previously.3. NOISE PROCESSES TN FREQUENCY GENERATORSThe white phase noise (0: = 2) predominates forI large enough. It is the result of the additive thermal(for the lower part of the electromagneticspectrum, including microwaves) or quantum (foroptical frequencies) noise which is unavoidablysuperimposed on the signal generated in the oscillator[Cutler and Searle, 1966]. It leads to a one-sidedspectral density Sy(f) of the form Fk1J2/11~P orFhvoP III~P, depending on the frequency range,where k is Boltzmann's constant, h is Planck'sconstant, T is the absolute temperature, F is thenoise figure of the components under consideration,and P is the power delivered by atoms.The flicker phase noise (0: = I) is generated mainlyin transistors, where this noise modulates the current[Halford et 01., 1968; Healey, 1972]. The theoryof this noise is not yet very well understood.Diffusion processes across junctions of semiconductordevices may produce this noise.White noise of frequency (a = 0) is present inoscillators. It is the result of the noise perturbationin the generation of the oscillation which is dueto white noise within the bandwidth of the frequency-determiningelement of the oscillator [Blaquiere,19530, b]. It is often masked by other typesof noise but has been observed in lasers [Siegmanand Arrathoon, 1968] and more recently in masers[ Vessot et 01., 1977]. The one-sided spectral densityof fractional frequency fluctuations is then kTI PQ 2of h 11 01PQ 2 depending on the frequency range, asstated above. Q is the quality factor of the frequency-determiningelement.White noise of frequency is typical of passivefrequency standards such as cesium beam tube andrubidium cell devices as well as stabilized lasers.It is related to the shot noise in the detection ofthe resonance to which an oscillator is slaved [Cutlerand Searle, 1966].Flicker noise of frequency and the random walk:frequency noise for which a = -I and -2, respectively,are sources of limitation in the long-termfrequency stability of frequency sources. They areobserved in active devices as well as passive ones.For instance, flicker and random walk: frequencynoises have been observed in quartz crystal resonators[Wainwright et 01., 1974] and rubidium masers[Vanier et 01., 1977]. The origin is not well understoodyet. It might be connected, in the first case,with fluctuations in the phonon energy density[Musha, 1975].Figures 3 and 5 show, for the purpose of illustration,Sy(f) for a hydrogen maser for IO- s sIs3 Hz [Vessot et al., 1977] and for an iodinestabilizedHe-Ne laser for 10- 2 sIs 100 Hz [Cerezel 01., 1978]. In both cases, Sy(f) is derived fromthe results of time domain frequency measurements10- 11lO_140, IT)10. 11 T [oJ10- 1 10 1 lO4 10 1Fig. 2. Frequency stability. characterized by the root meansquare of the two-sample variance of fractional frequencyfluctuation. of a hydrogen maser. The part of the graph with• slope of -t is typical of white noise of frequency.TN-l73

524 LESAGE AND AUDOINKl- 1410- 11 \SlfJ [Hd/10- 410-11,'------­f [Hz]Fig. 3. Spectral density of fractional frequency fluctuationsof a hydrogen maser. The parts of the graph with slopes 0and 2 coincide with theoretical expectations.(see section 6.4), as shown in Figures 2 and 4 butis very close to theoretical limits specified above.It is worth pointing out here that in systems wherea frequency source is frequency slaved to a frequencyreference or phase locked to another frequencygenerator the different kinds of noise involvedare mtered in the system [Cutler and Searle,1966; C. Audoin, unpublished manuscript, 1976].In these cases, at the output of the system, onecan rmd noise contributions pertaining to the model(7) but appearing on the Fourier frequency scalein an order different than that shown in FigureI. This is depicted in Figures 6 and 7 for the caseof a cesium beam frequency standard consistingofa good quartz crystal oscillator which is frequencycontrolled by a cesium beam tube resonator.The model for the frequency fluctuations is moreuseful if the noise processes can be assumed tobe gaussian ones (in particular, momenta of allorders can then be expressed with the help ofmomenta of second order). The deviation of thefrequency being the result of a number of elementaryperturbations, this assumption seems a reasonableone. Furthermore, the normal distribution ofFig. 5. Spectral density of fractional frequency fluctuationsof a He-Ne iodine-stabilized laser. The solid line representsthe spectral density of fractional frequency fluctuations correspondingto experimental results, and the dotted line representsthe expected value of Sy(f).j, the mean value of frequency fluctuationsaveraged over time interval T as dermed in (16),has been experimentally checked for Q = 2, I, 0,and -I [Lesage and Audoin, 1973, 1977]. Thisis shown in Figure 8 for white noise of frequency,for instance.4. MEASUREMENtS IN THE FREQUENCY DOMAIN *Measurement of power spectral density of freq'Jencyand phase fluctuations can be performedin the frequency domain for Fourier frequenciesgreater than a few 10- 3 Hz owing to the availabilityof good low-frequency spectrum analyzers.4.1. Use of a frequency discriminatorFrequency discriminators are of current use tocharacterize radio frequency and microwave generators.A resonant device such as a tuned circuitor a microwave cavity acts as a transducer whichT [oJFig. 4. Frequency stability, characterized by the root meansquare of the two-sample variance of fractional frequencyfluctuations, of a He-Ne iodine-stabilized laser.~n f~~........-:-.--'--:1---L---,L..---..L-=--410-' 10- 01 KlFig. 6. Spectral density of fractional frequency fluctuations ofa good quartz crystal oscillator.* See Appendix Note # 6TN-174

TIME AND FREQUENCY525V(v)10- 11vFig. 7. Spectral density of fractional frequency fluctuationsof the same quartz crystal oscillator as in Figure 6 but frequencycontrolled by a Cs beam tube resonance.transforms frequency to voltage fluctuations. Thismethod can be applied to optical frequency sourcestoo, as shown in Figures 9 and 10. Here the sourceis, for instance, a CW dye laser, and the frequencyselective device is a Fabry-Perot etalon. The secondlight pass allows one to compensate for the effectsof amplitude fluctuations and to adjust to a nullthe mean value of the output voltage. The slopeof this frequency discriminator equals I V MHz-.,typically, with a good Fabry-Perot etalon in thevisible.c._ullti.. P ro• I H "I!V V.9 -/5.9.8.'.6 /'.sV/kf~VIt,-YI.A""- ~ 40 60 80 1m 1'$.3..2.15L"' ""-R~'" Ii1-"-"'-with T = (T.). Solid linescorrespond to the normal distribution of the same width.Fig. 8. Distribution of counting time results for white frequencynoise (cesium beam frequency standards, T = 10 s)in Galtonian coordinates. Circles represent the cumulative probabilitycorresponding to IT. - TIFig. 9. Principle of frequency to voltage transfer in a frequencydiscriminator.4.2. Use of a phase detectorThis technique is well suited for the study offrequency sources in the radio frequency domain0.2 MHz < Vo < 500 MHz, in a range where verylow noise balanced-diode mixers which utilizeSchottky barrier diodes are available. This techniquehas mainly been promoted by the NationalBureau of Standards [Shoaf, 1971; Walls and Stein,1977].Figure II shows the principle of the determinationof the phase fluctuations in frequency multipliers,for instance. The two frequency multipliers aredriven by the same source. A phase shifter isadjusted in order to satisfy the quadrature condition.One then basl'(t) = D [ep, (t) - ep2(t)] (II) *where D is a constant and 'P, and 'P2 are the phasefluctuations introduced in the devices under test.It is assumed that the mixer is properly used toallow a balance of the phase and amplitude fluctuationsof the frequency source.This technique is often used to characterize phasefluctuations of two separate frequency sources ofthe same frequency. The quadrature condition isF. P. elllon dirrerenti.1Source~i : ~ A";;".PhotocellsFig. 10. Principle of frequency noise analysis of a dye laser.* See Appendix Note # 25IN-175

526LESAGE AND AUDOINQuartz ClJstalFrequelll;fSourceFrequenc,MultiplierI nPhaseShi rterr-------,Frequenc,ultiplierHl-------JI noutputFig. II. Principle of measurement of phase noise with a high-quality balanced mixer used as a phase comparator.insured by phase locking the reference oscillator,number 2 of Figure 12, to the oscillator under test.Fluctuations of the output voltage u(t) at frequenciesflargerthan the frequency cutoff of the phaseloop are proportional to phase fluctuations ofoscillator number l. On the contrary, componentsof u(t) at frequencies smaller than the above frequencycutoff are representative of frequency fluctuationsof oscillator number I.The requirement of having a reference oscillatorof the same quality as the oscillator to be testedmay be inconvenient. It has recently been shownthat the phase noise of a single oscillator can bemeasured by using the mixer technique, but witha delay line [Lance et 01., 1977]. Figure 13 showsa schematic of the setup. The signal from thefrequency source is split into two channels. Thereference channel includes a phase shifter for thepurpose of adjustment. It feeds one of the mixerinputs. The other channel delays the; signal beforeit is applied to the second mixer input. It can beseen that the power spectra density of the mixeroutput is proportional to (2Trf't d) 2 S

TIME AND FREQUENCY527Fig. 13. Principle of phase noise measurement of a single.frequency source with a delay line.p "" (T· Av) -DO(12)* and Av~ denote the frequency fluctuations of theNo ~ttempt has been made, to our knowledge, to two sources and AV 1 the frequency fluctuations ofspecify p more accurately for the different noise the beat note, one hasprocesses which can be encountered in frequencymetrology of stable sources.5. MEASUREMENTS IN THE TIME DOMAINTime and frequency counting techniques are wellknown [Cutler and Searle, 1966J. They are theeasiest to implement to provide information on thelow-frequency content (f:s I Hz) of the powerspectra of fractional frequency fluctuations.5. I. The beat frequency methodA beat note at frequency VI is obtained fromtwo frequency sources under test, with frequenciesVo and v~, respectively, such that V :=.: v~. If AvooY. __ Av. __ Vo IAvo _ AV,~ I(13)VI VI Vo VoThe fractional frequency fluctuations of the beatnote are then proportional to those of the frequenciesVa and v~ but multiplied by the factor va/v.which is much larger than unity.With stable generators at frequencies lower thanapproximately 100 GHz the frequency fluctuationsare small enough that the beat note can be at lowfrequency. The counter is then used as a periodmeter, and a high precision in the measurementis achievable.Optical frequency standards show larger frequencyfluctuations in absolute value. For instance,a laser stal?ilized at 500 THz (A = 0.6 IJ.m) witha fractional frequency stability of I X 10- 13 exhibitsfrequency fluctuations of 50 Hz. They can be easilymeasured if the beat note is at 50 MHz, say, whenthe counter is used as a frequency meter. In thecase of iodine-stabilized He-Ne lasers the beat noteis easily obtained by locking the two lasers todifferent hypetfme components of the considerediodine transition. Otherwise, the frequency offsettechnique is used [Barger and Hall, 1969J.Fig. 14. Principle 'of phase noise measurement of a single high_quality frequency source with a correlator.5.2. The time difference methodThe time difference method [Allan and Dooms,1915J must be used with time standards whichdeliver pulses as time scale marks. Distant timecomparison and synchronization by TV pulses, lightpulses, Loran-C pulses, for instance, pertain to thiscategory. It provides information on the relativephases of the two clocks under test.Time interval measurements being very precise* See Appendix Note # 26TN-I77

528 LESAGE AND AUDOINMinrFrequent,requent,SDurce ~-(Xll-4-I SourcenOln02Fig. 15. Principle of the time difference method.(a precision of 10-100 ns is typical, depending onthe class of the counter), they are also used withc.w. frequency generators, as shown in Figure 15.This method is then well suited to the case in whichthe frequency sources under test have the samenominal frequency v o ' e.g., atomic frequency standards.An auxiliary frequency source, such as afrequency synthesizer, with frequency v ~ allowsone to obtain, at the output of the mixers, twobeat notes at the desired frequency VI = Ivo ­v~l. After amplification the zero crossing of oneof the beat notes starts the time interval counter,and the zero crossing of the other beat note stopsiL One has(14)where x I is the fractional phase fluctuation of thebeat note and X o and x~ that of the two frequencystandards. For instance, with V o = 5 MHz, VI =0.5 Hz, and a precision in the time interval measurementof 0.1 ~s a precision of 10 -14 S at the nominalfrequency Vo is achieved.6. CHARACTERIZATION OF FREQUENCY STABILITYIN THE TIME DOMAIN6.1. Significance of experimental dataIt is well established that measurement in thetime domain with an electronic counter samplesphase increments and gives A.'P(t k) dermed asA,.

TIME AND FREQUENCY 529Forf such thatwe haveTrfT< I (20)N(N + I)IH(fW"", (TrIT)' (21)3Equation (21) shows that for ftnite N the integralin (18) will converge at the lower limit for n =-I and n = -2, as well as for a = 2, I, andO. One sees that very low frequency componentsof S/f) are best eliminated for small values ofN.6.3. Variance of time-averagedfrequencyfluctuationsWhen N goes to inftnity, (cr:(oo, 'T, 'T» becomes 1 2W/,T

530 LESAGE AND AUOOINsuch as c:r;(T) = kiT .... This is specified as followsfor 27T/ c T > I:sIJ.Q--2 21 20 1-I 0-2 -IFor Q = -I the c:ry(r) graph is a horizontal line,which justifies the designation 'flicker floor' forthat part of the graph.TUsually, under experimental conditions the relalL~ ~ ~ ~~_ "TTtion 27T/cT > I is satisfied. The noise processes n-' 10" Il 100which perturb the oscillation can then be identifiedfrom a Uy(T) graph if it is assumed that the afore­ Fig. 17. The solid line represents the variation of "_,

TIME AND FREQUENCY5311974] critically depends on the precision of thep(p - 1)/2 frequency comparisons which can beperformed by arranging oscillators in pairs. Furthermore,the uncertainty in the determination ofO'y(T) translates directly into the uncertainty indetermining the h", coefficients if the frequencygenerator is perturbed by noise processes modeledby (7).The precision in the estimation of time domainmeasurements of frequency stability has been consideredby several authors [Tausworthe, 1972; Lesageand Audoin, 1973; Yoshimura. 1978]. It hasbeen determined for most of the experimentalsituations which can be encountered in the twosamplevariance characterization of frequencystability, with or without dead time (P. Lesage andC. Audoin, private communication, 1978).Calculation of the expectation value of the twosamplevariance according to (25) requires an inImitenumber of data. But, in practice, only m countingresults are available, and one calculates the estimatedaverage of the two-sample variance as follows:1 ... -1 _t).~(2, T, T, m) = L (Yhl - Y1)2 (29)2(m - I) k-IOne can easily show that the expectation valueof ()~(2, T, T, m) equals the averaged two-samplevariance with dead time. Thus the rmite numberof measurements does not introduce bias in theestimation of the two-sample variance.The estimated averaged two-sample variance(EATSV) being a random function of m, we needto characterize the uncertainty~ in the estimation.We thus introduce the variance of the EATSV,according to the common understanding of avariance. We set0'2 [b~(2, T, T. m»)= ([~;(2, T. T, m) - (0:(2, T, T») 2) (30)With the expression (29) of the EATSV we get0'2 [b~(2, T, T, m))withI]2("'.1 ",-I= [ 2: 2: 1l,1l/ )2(m - I) '_I /-1(31)Il, = (f,... - y,)~ - 2(

532 LESAGE AND AUDOINl.l-TEquations (27) and (28) show that the defmitionof the time domain measurement of frequencystability 0': (1") involves a mtering of Sy(f) in a linearmter. Figure 19 shows the impulse response of thisruter, which represents the sequence of measureh(t)I-iIT0 TFig. 19. Impulse response of a linear fdter which representscomputation of two-sample variance. .r..10 100Fig. 18. Variation of K. as a function of r = TIT for thecommonly encoUDlered noise processes and for hrf< T = 10.from a limited number of data and allows one todraw error bars on a frequency stability graph.For 2Tr/ c 1" :> I and m :> I we have(37)The values of K nare given as follows subject tothe condition that the dead time is negligible, i.e.,2Tr/ c (T ­ 1")

TIME AND FREQUENCY5338. SPECTRAL ANALYSIS INFERRED FROM TIMEDOMAIN MEASUREMENTSThe methods of time domain characterization offrequency stability reviewed above allow one toidentify noise process if they are described bySy(f) = 2: lI..r (38)Ol--lThis may not be the case. Furthermore, it is ofinterest to determine the power spectral densityof fractional frequency fluctuations for Fourierfrequencies lower than I Hz. In this region, timedomain measurements are the most convenient, andthe question arises as to their best use for spectralanalysis.8.1. Selective numerical jilteriT,gEquations (24) and (21) show that calculation ofthe variance of the second difference of phasefluctuations (the two-sample variance) involves amore selective filtering than calculation of thevariance ofthe first difference ofphase fluctuations.One can then consider higher-order differences[Barnes, 1966; Lesage and Audoin, 19150, hI. Thenth-order difference of phase fluctuations is denotedas "Ar.TIfl(t k ), where T and T have the samemeaning as in section 6.1. and 6.2. This nth differenceis dermed by the following recursive equation:which introduces binomial coefrlcients C~_I' Wehave,,-1"Ar.,q>(tk) = L (-I)IC~_I {q> Itt + (n -1-01 - i) T + TJ- q>[t~ + (n - 1 - i)TJ) (40)The transfer ftmction H,,(f, T, T) of the linear filterwhich represents the calculation of the varianceof the nth difference of phase fluctuations is givenby2"-1IH"U: T, T)l = -(Sin1T/T),,-1 Sin1T/T (41)1T/TIt should be pointed out that for f T ¢: I, onehas(42)Fig. 21. Impulse response of a linear filter which representscomputation of the variance of the 20th difference of phasefluctuations. C; represents binomial coefficients.Figure 21 shows the impulse response of the linearfilter which represents the calculation of thevariance of the 20th difference of phase fluctuations,and Figure 22 shows the related transferfunction. A selective ftllering is then involvedaround freqnency 1/2T.Such a variance is also known as a modifiedHadamard variance [Baugh, 1911]. The spuriousresponses at frequencies (21 + 1)/2T, where I isan integer, can be eliminated by a proper weightingIH 20 (W,T.T) Io 72VTFig. 22. Transfer function ofthe linear filter considered in Figure21.TN-183

534 LESAGE AND AUDOINof measurement results or by mtering with the helpof an analog fIlter [Graslambert, 1976].Such a technique of linear fIltering has been usedto show that good quartz crystal oscillators exhibitflicker noise of frequency for Fourier frequenciesas low as 10- 3 Hz [Lesage and Audain, 1975b].Furthermore, it is well suited to the design ofautomated measurement setups [Peregrina andRicci, 1976; Groslambert, 1977].The best use of experimental time domain datafor selective fIltering has been considered by Boileau[1976].8.2. High-pass filteringIf frequency fluctuations y(t) are mtered in anideal high-pass fIlter with transfer function Go (f. 'f) such that//.its output z(t) is such that0;,: r~ Go(f,fI)Sy(f)d!= r'0 Sy(f)df (44)JoJhEquation (44) shows that the derivative of U; is-Sy(f), and spectral analysis, and thereforecharacterization offrequency stability, are possible,in principle, by high-pass fIltering.Possible realization of the high-pass fIlter bytechniques of digital data processing have beenspecified, such as the method offInite-time varianceand the method of fInite-time frequency control.Processing of fInite-time data is aimed to properlydeal with the nonintegrable singularity of the powerspectral density at v = 0 [Boileau, 1975; Boileauand Picinbono, 1976]. The method is well suitedto the analysis of drifts or slow frequency changes.Practical use of this method has not been reportedyet.8.3. Use of the sample spectral densityIt has been shown in section 8.1. that spectralanalysis from the Hadamard variance or its modifiedforms requires a series of measurements at timeinterval T in order to specify the spectral densityat frequency 1/2T. Another point of view has beenconsidered [Boileau and Lecourtier, 1977]. Froma set of measurements of Y., sampled at frequency11T, it allows one to obtain an estimation of thespectral density for discrete values of the Fourierfrequency.9. STRUCTURE FUNCTIONS OF OSCILLATORFRACTIONAL PHASE AND FREQUENCYFLUCTUATIONSInterest in the variance of nth-order differenceof phase fluctuations was recognized early in thefIeld of time keeping (see for instance, Barnes[1966]). This can be easily understood from (43),which shows that an effIcient ftltering of lowfrequencycomponents of frequency fluctuationsis then introduced. It allows one to deal properlywith frequency drifts, which will now be considered,and poles of Sy(f) of order 2(n - 1) at the origin.It is equivalent to saying that the nth differenceofphase fluctuations allows one to consider randomprocesses with stationary nth-order phase increments.This question has been formalized by Lindseyand Chie [1976, 1971], who introduce structurefunctions of oscillator phase fluctuations. The nthorder structure function of phase fluctuations isnothing else but the variance of the nth differenceof phase fluctuations, as considered in section 8.Then, by defInition, the nth-order structure functionof fractional phase fluctuations is given byD~)(T) = EO"..1 •.•x(t.)] 2} (45)where E {.} means expectation value. The fractionalphase (or the clock reading) at time tIc is x(t,,).We assume T = T.Let us consider an oscillator, the phase If" (t)of which is of the following form except for anadditive constant:(46)where 0" is a random variable modeling the kthorderfrequency drift and If'(t) represents randomphase fluctuations. We then haveandtIcx'(t) =L d H ~ + x(t) (47)"-2 k.I-ItIcy'(t) = L dIe - + y(t) (48)>_. k!where dIe = O,,/21Tv o is the normalized drift coeffI­TN-184

TIME AND FREQUENCY535cient and x(t) and yet) have the same meaning asin the preceding sections. The notations x' and y Irefer to an oscillator with drift.It can then be shown that we haveD~a)(T) = '1'2.'1 E [d~_,J"" sm:bl(-lT/T)+ 2:b1 S if) df t = n (49)0" (2'lTf) 2and~1.. Sin:bl('lTfT)D(a)(T) = 2 210 S if) df t> n (50)• 0" (2'lTf) 2If one applies (49) to the case of an oscillatorwithout drift, one can easily show that the followingequations are satisfied:and(51)0;('1") = (l/2T2)D~2)(T) (52)This is indeed not surprising because apart frommore or less complicated mathematical formalismthe definitions of the considered variances andstructure functions are closely related, as has beenemphasized here.Relations between sample variance and structurefunctions have been given by Lindsey and Chie[1976], whereas the relation between structurefunctions and several different approaches of frequencystability characterization has been analyzedby Rutman [1917, 1978].For the generally accepted noise model definedby (7) the 'I' dependence of higher-orde.r structurefunctions is the same as the two-sample variance,as shown in Table 4 [Lindsey and Chie, 1978b].As stated above, structure functions of order nallow one to consider spectral densities which varyasr at the origin with a ~ -2(n - I). For instance,for n = 3 it is possible to characterize frequencyfluctuations of an oscillator with a power spectraldensity of fractional frequency fluctuations givenby S)f) = ~~_-4 hJo. This oscillator exhibitsstationary third-order increments of phase fluctuations.In the presence of a frequency drift describedby a polynomial of degree I - I, structure functionsof degree 11 < I are meaningless: their computationyields a time-dependent result. For n = I the lthstructure function shows a long-term 'I' dependenceproportional to 2J T . This dependence disappears forn > t. Although a power spectral density of theformj-(2J-I) would also give the structure functionsa variation of the form '1'2/, this variation does notdepend on n, provided that the function is meaningful.It is then possible, at least in principle, toidentify frequency drifts and to specify their order.This is illustrated in Figure 23 according to Lindseyand Chie (l978b]. However, there are not yetexperimental proofs that such a characterizationis achievable in practice.10. POWER SPECTRAL DENSITY OF STABLEFREQUENCY SOURCESThe power emitted by a source of time-dependentvoltage vet) given by (I) is S)v)dvin the frequencyrange [v, v + dv], where S)v) is the power spectraldensity of the source. The dimensions of Sv(v) areV 2 Hz-I. The main interest of power spectraldensity, in frequency metrology, is related to highorderfrequency multiplication. We will only introducethe subject by giving the relations betweenSv(v) and S'f{f) and stating present problems inthe field.TABLE 4.Structure functions of orders 1,2, 3, and 4 for fraclional phase fluctualions of commonly encounlered noise processesfor 2Trf. T :> IS,,(f)5 ~h i-;- lid.2Tr 2 >J.TrI 3 5 35II.! - h, In (Trf.T) -2 h,m frrf.T) -;- h, In (Trf. T) -h,In(Trf.T)2Tr 2 2Tr Tr 2Tr 2-}lI o T hoT IOheT411_,T 2 1n2 20.7 "_,T 1From Lillds~ and Cllie (l978bI4_Tr 1 11T'3 -2TN-I8S

LESAGE AND AUDOIN2 - Vo )]53610- 1 2S,,(v) = V o(21T.6.v)2m\v+ (21T(Vquartz xtal 5 x 10" 4 X 10- 2710 12 3 x 10- 132 (56)H maser 1.4 x 10'He-Ne laser 4 X 10- 27 10 7 3 x 10-'10'· 3 X 10- 29 10- 2 30where t\v is the half width at half maximum ofthe power spectra defmed as10­1(f1(57)We obviously have 21Tt\V . T c= l. If oscillatorsare considered, one has h o = kTI PQ 2 in the~PTTJ(n ithr)lO-T [5]I) 10 1 I)l 1)4 10 5Fig. 23. (Solid line) Two-sample variance of an oscillatorshowill8 a linear frequency drift of 10- 10 per day and flickernoise of frequency given by Sylf) = 1.2 X 10-2> fO'. (Dolledline) The third difference of fractional phase fluctuation isindependent of the drift (according to Lindsey and Chie [1918b].Negligible amplitude noise and gaussian phasefluctuations being assumed, it is well known that10.2. White noise o/phasethe autocorrelation function of v(t) is R,,(T) givenbyPresently availaple good quartz oscillators areaffected by white noise of phase. It is easy to showfrom the deftnition (16) of Y~ ­tthat the followingR,,(T) = - cos 21TV TCXP [-t(21TV )2a2(y )]o o (53) equation is satisfIed:t2t (21TV o TO'(Yt)] 2 = R. (0) - R.. (T) (58)As ~(Yt) is only defIned for stationary phasefluctuations and for phase fluctuations with stationaryfIrst increments, the same is true for R,,(T) and of the stationary phase fluctuations r,p(t).where R ... (T) denotes the autocorrelation functiontherefore S,,(f).The expression of the one-sided S,,(v) then follows[Rutman, 1974a; Lindsey and Chie, 1978a]:10.1. White noise offrequencyv 2Jl roSu(v) = -:; e- • ) [8 (v - vo)This is the simplest to deal with. If the frequencyof the source is perturbed by a broadband whitenoise of frequency, one has S/f) = h o and ~(Yt)+ S.(v - vo) + 1S.(v). S.(v) + ...] (59)= (h o /2T). Whencewhere the asterisk denotes convolution and thebracket contains an inftnite set of multiple-convolutionproducts of S ... (v) by itself. Such an equationR.(T) =2 v~cos 21TIIOTexp(--:;:-ITI) (54)is not easily tractable. It is the reason why thewhere T cis the coherence time of the signaL Wehave(55) TABLE 5. Theoretical values of correlation time and powerspectrum linewidth for various oscillatorsThe one-sided power spectral density is then representedby a Lorentzian given byOscillator5-MHzVO' Hz hOt Hz-' T c, S 2Av, Hzradiofrequency and microwave domain and h o =h vol PQ 2 in the optical frequency domain, whereP is the power delivered by the oscillator and Qthe quality factor of the frequency-determiningelement. Table 5 gives theoretical values of coherencetime and linewidth of good oscillators. Itis only intended to illustrate a comparison, oftenmade, of the spectral purity of oscillators. It mustbe pointed out that, in practice, other noise processesexist which modify these results. Even anymeaning of T cand t\v is removed if R,,(T) is notdefmed.Multiplication of the frequency by n multipliest\v and divides T cby the factor n 2 •TN-186

o~ _TIME AND FREQUENCY537approximation of small-phase fluctuations is oftenmade. If R'9(O) = If>2

538 LESAGE AND AUWINstability measuremenlS, Nat. Bur. Stand. Tuh. NOIt, 683.Bames, J. A. (1977), A review of methods of analyzing frequencystability, Prouedings of the 9th Annual Precise Time andTime Intenal Applications and Planning Meeting, pp. 61-84,Technical Information and Administrative Support Division,Goddard Space Flight Center, Greenbelt, Md.Bames, J. A., A. R. Chi, L. S. Cutler, D. J. Healey, D. B.Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R.L. Sydnor, R. f. C. Vessot, and G. M. R. Winkler (1971),Characterization offrequency stability, IEEE Trans. Instrum.Meas., IM-20(2), 105-120.Baugh, R. A. (1971), frequency modulation analysis with theHadamard variance, Proceedings ofthe 25th Annual FrequencyControl Symposium, pp. 222-225, National Technical InformationService, Springfield, Va.Bava, E., G. P. Bava, A. de Marchi, and A. Godone (l977a),Measurement of static AM-PM conversion in frequency multipliers,IEEE Trans. Instrum. Meas., IM-26(1), 33-38.Bava, E., A. deMarchi, and A. Godone(I 977b), Spectral analysisof synthesized signals in the mm wavelength region, IEEETrans. Instrum. Meas., IM-26 (2), 128-132.B1aquiere, A. (1953a), EfTet du bruir de fond sur Ia frequencedes aut

TIME AND FREQUENCY539Lindsey, W. C., and C. M. Chie (l978a), Frequency multiplicationeffects on oscillator instability, IEEE Trans. Instrum.M~as.• IM-17(1), 26-28.Lindsey, W. C., and C. M. Chie (l978b), Identification ofpower-law type oscillator phase noise spectra from measure·ments, IEEE Trans. Instrum. M~as., IM-17(1), 46-53.Musha, T. (1975), Ilfresonant frequency fluctuation ofa quartzcrystal, Procudings of th~ 19th Annual Fr~quency ControlSymposium, pp. 308-310, Electronic Industries Association,Washington, D.c.National Bureau of Standards (1974), Tim~ and Fr~qu~ncy:Th~ory and Fundamentals, Monogr. 140, 459 pp., Washington,D.c.Papoulis, A. (1965), Probability Random and Stochastic Procus~s,58> pp., McGraw-Hili, New York.Percival, D. B. (1978), Estimation of the spectral density offractional frequency devistes, Proce~dings of th~ 32nd AnnualFr~quency Control Symposium, pp. 542-548, Electronic industriesAssociation, Washington, D.c.Peregrino, L., and D. W. Ricci (1976), Phase noise measurementusing a high resolution counter with on-line data processing,Procudingsofth~30th Annual Fr~qu~ncy Control Symposium,pp. 309-317, Electronic Industries Association, Washington,D.c.IEEE (1966), Special Issue on Frequency Stability, Proc. IEEE,54(2).Rutman, J. (19740), Relations between spectral purity andfrequency stability, Procudings ofth~ 18th Annual Frequ~ncyControl Symposium, pp. 160--165, Electronic Industries Association,Washington, D.c.Rutman, J. (l974b), Characterization of frequency stability: Atransfer function approach and its application to measurementsvia filtering of phase noise, IEEE Trans. Instrum. M~as.,IM-13(\), 40-48.Rutman, J. (1977), Oscillator specifications: A review of classicaland new ideas, Proce~dings of th~ 31st Annual Fr~qu~ncyControl Symposium, pp. 291-301, Electronic Industries Association,Washington, D.C.Rutman, J. (1978), Characterization of phase and frequencyinstabilities in precision frequency sources: Fifteen years ofprogress, Proc. IEEE, 66(9), 1048-1075.Rutman, J., and G. Sauvage (1974), Measurement of frequencystability in time lIUId frequency domains via fLltering of phase.noise, IEEE Trans. Instrum. M~as., IM-13(4), 515-518.Siegman, A. E., and R. Arrathoon (1968), Observation ofquantum phase noise in a laser oscillator, Phys. R~v. un.,10(17), 901-902.Shoaf, J. H. (1971), Specification and measurement offrequ~ncystability, N BS R~p. 9794, U.S. Government Printing Office,Washington, D.c.Tausworthe, R. C. (1972), Convergence of oscillator spectralestimators for counted·frequency measurements, IEEE Trans.Commun., COM·1O(2), 214-217.Vanier, J., M. Tetu, and R. Brousseau (1977), Frequency andtime domain stability of the RbI? maser and related oscillators:A progress report, Proceedings of th~ 31st Annual Fuqu~ncyCor,trol Symposium, pp. 344-346, Electronic Industries Association,Washington, D.C.Vessot, R. F. c., M. W. Levine, and E. M. Mattison (1977),Comparison of theoretical and observed maser stability limitationsdue to thermal noise and the prospect of improvementby low temperature operation, Procudings of the 9th AnnualPr~cis~ Tim~ and Tim~ Inttrval Applications and PlanningMuting, pp. 549-566, Technical Information and AdministrativeSupport Division, Goddard Space Flight Center, Greenbelt,Md.Wainwright, A. E., F. L. Walls, and W. D. McCoa (1974).Direct measurements of the inherent frequency stability ofquartz crystal resonators, Procudings of th~ 18th AnnualFr~qu~ncy Control Symposium, pp. 177-180, Electronic industriesAssociation, Washington, D.c.Walls, F. L., and A. de Marchi (1975), RF spectrum of a signalafter frequency multiplication: Measurement and comparisonwith a simple calculation. IEEE Trans. Instrum. M~as.,IM-14(3), 210-217.Walls, F. L., and S. R. Stein (1977), Accurate measurementsof spectral density of phase noise in devices, Proce~dingsofth~ 31st Annual Fr~qu~ncy Control Symposium, pp. 335-343,Electronic Industries Association, Washington, D.C.Wiley, R. G. (1977), A direct time-domain measure of frequencystability: The modified Allan variance, IEEE Trans. Instrum.M~as.• IM-26(l), 38-41.Winkler, G. M. R. (1976), A brief review of frequency stabilitymeasures, Proceedings of th~ 8th Annual Precis~ Ti~ andTim~ Inttrval Applications and Planning Muting, pp. 489­527, Technical Information and Administrative Support Division,Goddard Space Flight Center, Greenbelt, Md.Yoshimura, K. (1978), Characterization of frequency stability:Uncertainty due to the autocorrelation of the frequencyfluctuations, IEEE Trans. Instrum. Mtas., IM-27(1), 1-7.TN-189

Copyright Ie) 1984 Academic Press. Reprinted, with permission, from Infrared andMillimeter Waves, Vol. 11, pp. 239-289, 1984.INFRARED AND MILUMETER WAVES. VOL. IICHAPTER 7Phase Noise and AM Noise Measurementsin the Frequency DomainAlqie L. Lance, Wendell D. Seal, and Frederik LahaarTRW Operations and Support GroupOne Space ParkRedondo Beach. CaliforniaI. [NTRODUCTION ~39[I. FUNDAMENTAL CONCEPTS 240A. Noise Sidebands 243B. Spectral Densl/Y 243C. Spectral Densities of Phase Fluctuations in theFrequency Dnmain 246D. Modulation Theory and Spectral Density Relationships 248E. Noise Processes ~51F. Integrated Phase Noise 252G. AM Noise in the Frequency Domain 254III. PHASE-NOISE MEASUREMENTS USING THEIV.TWO-OSCILLATOR TECHNIQUE ~55A. Two Noisy Oscillators ~57B. Automated Phase-Noise Measurements Using theTwo-Oscli/aror Technique 258C. Calibration and Measurements Using theTwo-Osctllator System 259SINGLE-OSCILLATOR PHASE-NOISE MEASUReMeNTSYSTeMS A:'

240 A. L. LANCE, W. D. SEAL, AND F. LABAARThe term frequency stability encompasses the concepts of random noise,intended and incidental modulation, and any other fluctuations of the outputfrequency of a device. In general, frequency stability is the degree to whichan oscillating source produces the same frequency value throughout aspecified period of time. It is implicit in this general definition of frequencystability that the stability of a given frequency decreases if anything excepta perfect sine function is the signal wave shape.Phase noise is the term most widely used to describe the characteristicrandomness of frequency stability. The term spectral purity refers to theratio of signal power to phase-noise sideband power. Measurements of phasenoise and AM noise are performed in the frequency domain using a spectrumanalyzer that provides a frequency window following the detector (doublebalancedmixer). Frequency stability can also be measured in the timedomain with a gated counter that provides a time window following thedetector.Long-term stability is usually expressed in terms of parts per million perhour, day, week, month, or year. This stability represents phenomenacaused by the aging process of circuit elements and of the material used inthe frequency-determining element. Short-term stability relates to frequencychanges of less than a few seconds duration about the nominal frequency.Automated measurement systems have been developed for measuring thecombined phase noise of two signal sources (the two-oscillator technique)and a single signal source (the single-oscillator technique), as reported byLance et al. (1977) and Seal and Lance (1981). When two source signalsare applied in quadrature to a phase-sensitive detector (double-balancedmixer), the voltage fluctuations analogous to phasefiucruations are measuredat the detector output. The single-oscillator measurement system is usuallydesigned using a frequency cavity or a delay line as an FM discriminator.Voltage fluctuations analogous to frequency fluctuations are measured atthe detector output.The integrated phase noise can be calculated for any selected range ofFourier frequencies. A representation of fluctuations in the frequencydomain is called spectral density graph. This graph is the distribution ofpower variance versus frequency.II.Fundamental ConceptsIn this presentation we shall attempt to conform to the definitions.symbols, and terminology set forth by Barnes et al. (1970). The Greekletter v represents frequency for carrier-related measures. Modulationrelatedfrequencies are designated f. If the carrier is considered as dc, thefrequencies measured with respect to the carrier are referred to as baseband,offset from the carrier, modulation, noise, or Fourier frequencies.TN-l91

7. PHASE NOISE AND AM NOISE MEASUREMENTS 241I'4-----1 CYCLE OF "0----1>1.,IIIIIIITIME tI"270I0IANGULAR FREQUENCY,,,1------ ...PERIOD TOla)FIG. I Sine wave characteristics: (a) voltage V changes with time r as (b) amplitudechanges with phase angle

242 A. L. LANCE, W. D. SEAL, AND F. LABAARThe instantaneous phase 4>(e) of V(e) for the noisy signal is4>(t) = 2nvot + 4>(t), (6)where 1>(e) is the instantaneous phase fluctuation (lbout the ideal phase2nv o r of Eq. (4).The simplified illustration in Fig. 1 shows the sine-wave signal perturbedfor a short instant by noise. In the perturbed area, the .1v and .1e relationshipscorrespond to other frequencies, as shown by the dashed-line waveforms.In this sense, frequency variations (phase noise) occur for a giveninstant within the cycle.The instantaneous output voltage V(t) of a signal generator or oscillatoris nowV(e) = [Vo + t:(e)] sin[2n1'oe + 4>(t)], (7)where V o and 1'0 are the nominal amplitude and frequency, respectively, ande(t) and 4>(t) are the instantaneous amplitude and phase fluctuations of thesignal.It is assumed in Eq. (7) thatEquation (7) can also be expressed as¢(e)and ~ ~ 1 for all (e), ¢(t) = d4>ldt. (8)1'0V(T) = [Vo +

7. PHASE NOISE AND AM NOISE MEASUREMENTS243where y(c) is the instantaneous fractional frequency deviation from thenominal frequency Vo 'A. NOISE SIDEBANDSNoise sidebands can be thought of as arising from a composite of lowfrequencysignals. Each of these signals modulate the carrier-producingcomponents in both sidebands separated by the modulation frequency, asillustrated in Fig. 2. The signal is represented by a pair of symmetricalsidebands (pure AM) and a pair of antisymmetrical sidebands (pure FM).The basis of measurement is that when noise modulation indices aresmall, correlation noise can be neglected. Two signals are uncorrelaced iftheir phase and amplitudes have different time distributions so that they donot cancel in a phase detector. The separation of the AM and FM componentsare illustrated as a modulation phenomenon in Fig. 3. Amplitude fluctuationscan be measured with a simple detector such as a crystal. Phase or frequencyfluctuations can be detected with a discriminator. Frequency modulation(FM) noise or rms frequency deviation can also be measured with an amplitude(AM) detection system after the FM variations are converted toAM variations, as shown in Fig. 3a. The FM-AM conversion is obtainedby applying two signals in phase quadrature (90°) at the inputs to a balancedmixer (detector). This is illustrated in Fig. 3 by the 90° phase advances ofthe carrier.B. SPECTRAL DENSITYStability in the frequency domain is commonly specified in terms ofspectraldensities. There are several different, but closely related, spectral densitiesthat are relevant to the specification and measurement of stability of thefrequency, phase, period, amplitude, and power of signals. Concise, tutorialCARRIERCARRIERVoCARRIERv v v y, V nIf -Yo VnRADIAN FREQUENCY RADIAN FREQUENCY RADIAN FREQUENCYlal Ib) (c)FIG.2 (a) Carrier and single upper sideband signals; (b) symmetrical sidebands (pureAM); (0) an antisymmetrioal pair of sidebands (pure FM).IN-194

244 A. L. LANCE, W. D. SEAL, AND F. LABAARL..FM VARIATIONlUPPERI +V nSIDEBAND IUPPERSIDEBANDAM MEASURED~-- - -­ -., WITH AM DETECTORLOWERSIDEBAND- V nV o(CARRIERIlal(b\L ~~·NAV~RTED-'1- TO FM -,CARRIER~ UPPER SIDEBAND--(T'OW," 5

7. PHASE NOISE AND AM NOISE MEASUREMENTS245Voga:w~0..FIG.4A power density plot.The spectral density is the distribution of total variance over frequency. Theunits of power spectral density are power per hertz; therefore. a plot of powerspectral density obtained from amplitude (voltage) measurements requiresthat the voltage measurements be squared.The spectral density of power versus frequency, shown in Fig. 4, is a twosidedspectral density because the range of Fourier frequencies f is fromminus infinity to plus infinity.The notation 5if) represents the two-sided spectral density of fluctationsof any specified time-dependent quantity get). Because the frequency bandis defined by the two limit frequencies of minus infinity and plus infinity,the total mean-square f1uctilation of that quantity is defined by+OOGsideband = _ 00 5if) df. (13)fTwo-sided spectral densities are useful mainly in pure mathematical analysisinvolving Fourier transformations.Similarly, for the one-sided spectral density,J+OOGsideband = 0 5 9 (/) df. (14)The two-sided and one-sided spectral densities are related as follows:592 df = 2 {+oo S92 df = {... co 5 dj, (15)91f+oo-00 Jo Jowhere g I indicates one-sided and g2 two-sided spectral densities. It is notedthat the one-sided density is twice as large as the corresponding two-sided

246 A. L. LANCE, W. D. SEAL, AND F. LABAARspectral density. The terminology for single-sideband versus double-sidebandsignals is totally distinct from the one-sided spectral density versus twosidedspectral terminology. They are totally different concepts. The definitionsand concepts of spectral density are set forth in NBS Technical Note632 (Shoaf et al., 1973).C. SPECTRAL DENSITIES OF PHASE FLUCTUATIONS IN THEFREQUENCY DOMAn,; *The spectral density 5 y (/) of the instantaneous fractional frequencyfluctuations y(t) is defined as a measure of frequency stability, as set forthby Barnes et at. (1970). Sif) is the one-sided spectral density ofIrequencyfluctuatiOns on a "per hertz" basis, i.e., the dimensionality is Hz - 1. Therange of Fourier frequency 1 is from ~ero to infinity. S6'(/)' in hertz squaredper hertz, is the spectral density of/requency fluctuations bv. It is calculatedas5 (/) = -. (bvrms)2 (16)6\" bandwidth used in the measurement of bv rmsThe range of the Fourier frequency 1 is from zero to infinity.The spectral density of phaseflucruations is a normalized frequency domainmeasure of phase fluctuation sidebands. S6"'(/)' in radians squared perhertz. is the one-sided spectral density of the phase fluctuations on a "perhertz ,. basis:5 (/) = &Prms (17)6 bandwidth used in the measurement of bt/>rms'The power spectral densities ofphase and frequency fluctuation are related byS6",(f) = (v~!f)S}'(/). (18)The range of the Fourier frequency 1 is from zero to infinity.5 m (f). in radians squared Hertz squared per hertz is the spectral densityof angular frequency fluctuations 150:The defined spectral densities have the following relationships:(19)S6,(f) = v6S/f) = (l/2n)2S 6o (f) = 12S6"'(/); (20)S6"'(/) = (l/W)2S 60 (f) = (vo!f)25y(/) = [5 6 .(/)/1 2 ]. (21)Note that Eq. (20) is hertz squared per hertz, whereas Eq. (21) is in radianssquared per hertz.The term 5.;rfP(V), in watts per hertz, is the spectral density of the (square.. See Appendix Note # 30TN-197

7. PHASE ~OISE AND AM NOISE MEASUREMENTS247root of) radio frequency power P. The power of a signal is dispersed over thefrequency spectrum owing to noise, instability, and modulation. This conceptis similar to the concept of spectral density of voltage fluctuationsSbv(f)· Typically, Sbv(f) is more convenient for characterizing a basebandsignal where voltage, rather than power, is relevant. S.;rfP(v) is typicallymore convenient for characterizing the dispersion of the signal power inthe vicinity of the nominal carrier frequency Va' To relate the two spectraldensities, it is necessary to specify the impedance associated with the signal.A definition of frequency stability that relates the actual sideband powerof phase fluctuations with respect to the carrier power level, discussed byGlaze (1970), is called !I!(f). For a signal with PM and with no AM, !I!(f)is the normalized version of S.;rfP(v), with its frequency parameter f referencedto the signal's average frequency Va as the origin such that f equalsV - Va' If the signal also has AM, !I!(f) is the normalized version of thoseportions of S.;rfP(v) that are phase-modulation sidebands.Because I is the Fourier frequency difference (v - va), the range of risfrom minus Va to plus infinity. Since !I!(f) is a normalized density (phasenoise sideband power),(22) *!I!(f) is defined as the ratio of the power in one sideband, referred to theinput carrier frequency on a per hertz of bandwidth spectral density basis,to the total signal power, at Fourier frequency difference f from the carrier,per one device. It is a normalized frequency domain measure of phase fluctuationsidebands, expressed in decibels relative to the carrier per hertz:power density (one phase modulation sideband)!I!(f) = . . (23)carner powerFor the types of signals under consideration, by definition the two phasenoisesidebands (lower sideband and upper sideband, at - f and f from Va,respectively) of a signal are approximately coherent with each other, andthey are of approximately equal intensity.It was previously show that the measurement of phase fluctuations (phasenoise) required driving a double-balanced mixer with two signals in phasequadrature so the FM-AM conversion resulted in voltage fluctuations atthe mixer output that were analogous to the phase fluctuations. The operationof the mixer when it is driven at quadrature is such that the amplitudesof the two phase sidebands are added linearly in the output of the mixer,resulting in four times as much power in the output as would be present ifonly one of the phase sidebands were allowed to contribute to the output., See Appendix Note # 31TN-198

248 A. L. LANCE, W. D. SEAL, AND F. LABAARofthe mixer. Hence, for If I < v o , and considering only the phase modulationportion of the spectral density of the (square root of) power. we obtainand, using the definition of f.e(f),SW(jfl)/{Vrms)2 ~ 4[SJif'P(vo + f)]/(Ptot) (24)f.e(f) == [S,;;rp(l'o + f)]/(P1oJ ~ 1S6q,(1 f I). (25)Therefore, for the condition that the phase fluctuations occurring at rates(f) and faster are small compared to one radian, a good approximation inradians squared per hertz for one unit isIf the small angle condition is not met, Bessel-function algebra must be usedto relate !E(f) to SM,(f)·The NBS-defined spectral density is usually expressed in decibels relativeto the carrier per hertz and is calculated for one unit asIr is very important to note that the theory, definitions, and equations previouslyset forth relate to a single device.(26)(27)D. MODULATION THEORY AND SPECTRAL DENSITYRELATIONSHIPSApplying a sinusoidal frequency modulation fm to a sinusoidal carrierfrequency l'O produces a wave that is sinusoidally advanced and retardedin phase as a function of times. The instantaneous voltage is expressed as,V(t) = Va sin(2nvot + ~¢ sin 2nfmf), (28)where ~¢ is the peak phase deviation caused by the modulation signal.The first term inside the parentheses represents the linearly progressingphase of the carrier. The second term is the phase variation (advancing andretarded) from the linearly progressing wave. The effects of modulation canbe expressed as residual fm noise or as single-sideband phase noise. Formodulation by a single sinusoidal signal, the peak-frequency deviation ofthe carrier (l'O) is~vo = ~¢'fm'(29)~¢ = ~vo/fm, (30)where Im is the modulation frequency. This ratio of peak frequency deviationto modulation frequency is called modulation index m so that ~¢ = m andm = ~vo/fm' (31 )TN-199

7. PHASE NOISE AND AM NOISE MEASUREMENTS 249The frequency spectrum of the modulated carrier contains frequencycomponents (sidebands) other than the carrier. Fbr small values or modulationindex (m ~ 1). as is the case with random phase noise. only the carrierand first upper and lower sidebands are significantly high in energy. The.ratio of the amplitude of either single sideband to the amplitude of thecarrier isV.bfV o = m12. (32)This ratio is expressed in decibels below the carrier and is referred to as dBcfor the given bandwidth B:v.JV o = 20 10g(mJ2) = 20 log(~vo!2fm)= 10 log(m/2)2 = 10 log(~vo;2fm):' (33)If the frequency deviation is given in terms of its rms value. thenEquation (33) now becomesA A'= UVOh/'1UVrms_. (34)V.bJVO = ff(f) = 20 log(~vrms/V2fm)= 10 log(~vrms/v2fm)2.(35)The ratio of single sideband to carrier power in decibels (carrier) per hertz isand, in decibels relative to one squared radian per hertz.(36)(37)The interrelationships of modulation index. peak frequency deviation.rms frequency. and spectral density of phase fluctuations can be found fromthe following:tm = !i.vo/2fm = ~vrms/-J1fm' (38)= 10 exp(.ff(f)j 10) = 1SM(f): (39)orandm = ~voj fm = 2~vrms/,j2j;"= 2y! 10 exp(.ff(f)/IO) = 2v!1S~ep(f). (41 )The basic relationships are plotted in Fig. 5.TN-200

3.16 x 10- 4 r- 10- 77010- 4 1- ~ 10-8'"lJ'"~3.16 x 10- 5 ~ 10-9-en""10-5 enz 10 10axj::w0 3.16x 10- 6 10- 11f­~ uZ ::::>...JaLLj:: 10-6 10- 12

7. PHASE :'-iOrSE AND AM NOISE MEASUREMENTS 251E. NOISE PROCESSESThe spectral density plot of a typical oscillator's output is usually a combinationof different noise processes. It is very useful and meaningful tocategorize these processes because the first job in evaluating a spectraldensity plot is to determine which type of noise exists for the particularrange of Fourier frequencies.The two basic categories are the discrete-frequency noise and the power-lawnoise process. Discrete-frequency noise is a type of noise in which there isa dominant observable probability, i.e., deterministic in that they can usuallybe related to the mean frequency, power-line frequency, vibration frequencies,or ac magnetic fields, or to Fourier components of the nominal frequency.Discrete-frequency noise is illustrated in the frequency domain plot of Fig. 6.These frequencies can have their own spectral density plots, which can bedefined as noise on noise.Power-law noise processes are types of noise that produce a certain slopeon the one-sided spectral density plot. They are characterized by theirdependence on frequency. The spectral density plot of a typical oscillatoroutput is usually a combination of the various power-law processes.In general, we can classify the power-law noise processes into five categories.These five processes are illustrated in Fig. 5, which can be referred to withrespect to the following description of each process.(I) Random walk FM (random walk of frequency). The plot goes down as1/[4. This noise is usually very close to the carrier and is difficult to measure.It is usually related to the oscillator's physical environment (mechanicalshock, vibration, temperature, or other environmental effects).VoFrequencyFIG.6A basic discrete-frequency signal display.TN-202

252 A. L. LANCE, W. D. SEAL, AND F. LABAAR(2) Flicker FM (flicker of frequency). The plot goes down as l/f3. Thisnoise is typically related to the physical resonance mechanism of the activeoscillator or the design or choice of parts used for the electronic or powersupply, or even environmental properties. The time domain frequencystability over extended periods is constant. In high-quality oscillators, thisnoise may be marked by white FM (l/f2) or flicker phase modulation¢>M (1j). It may be masked by drift in low-quality oscillators.(3) White FM (white frequency, random walk of phase). The plot goesdown as I/f2. A common type of noise found in passive-resonator frequencystandards. Cesium and rubidium frequency standards have white FM noisecharacteristic because the oscillator (usually quartz) is locked to the resonancefeature of these devices. This noise gets better as a function of timeuntil it (usually) becomes flicker FM (I/f3) noise.(4) Flicker ¢>M (flicker modulation of phase). The plot goes down asl/f This noise may relate to the physical resonance mechanism in an oscillator.It is common in the highest-quality oscillators. This noise can beintroduced by noisy electronics-amplifiers necessary to bring the signalamplitude up to a usable level-and frequency multipliers. This noise canbe reduced by careful design and by hand-selecting all components.(5) White 4JM (white phase). White phase noise plot is flat f O , Broadbandphase noise is generally produced in the same way as flicker ¢>M (I/f). Latestages of amplification are usually responsible. This noise can be kept lowby careful selection ofcomponents and by narrow-band filtering at the outputThe power-law processes are illustrated in Fig. 5.F. INTEGRATED PHASE NOISEThe integrated phase noise is a measure of the phase-noise contribution(rms radians, rms degrees) over a designated range of Fourier frequencies.The integration is a process of summation that must be performed on themeasured spectral density within the actual IF bandwidth (B) used in themeasurement of b4Jrms' Therefore, the spectral density S6t1>(f) must be unnormalizedto the particular bandwidth used in the measurement. DefineSu(f), in radians squared, as the unnormalized spectral density:SuCf) = 2[10 exp(Sf(f) + 10 log B)/IO). (42)Then, the integrated phase noise over the band ofFourier frequencies (fl tofn)where measurements are performed using a constant JF bandwidth, in radianssquared, is(43)if·TN-203

or, in rms radians,7. PHASE !'rOISE AND AM NOISE MEASUREMENTS 253and the integrated phase noise in rms degrees is calculated asSB(360/21l). (45)The integrated phase noise in decibels relative to the carrier is calculated asS8 = 10 10g(tS~). (46)The previous calculations correspond to the illustration in Fig. 7, whichincludes two bandwidths (Bl and 82) over two ranges of Fourier frequencies.In the measurement program, different IF bandwidths are used as setforth by Lance et ai. (1977). The total integrated phase noise over the differentranges of Fourier frequencies, which are measured at constant bandwidthsas illustrated, is calculated in rms radians as follows:,I 2 2 (5 )2 (47)5 B'ol = V (SBI) + (SB2) + ... + Bn'where it is recalled that the summation is performed in terms of radianssquared.sFIG. 7 Integrated phase noise over Fourier frequency ranges at which measurementsperformed using conSlant bandwidth.TN-204

254 A. L. LANCE. W. D. SEAL, AND F. LABAARG. AM NOISE IN THE FREQUENCY DOMAINThe spectral density of AM fluctuations of a signal follows the samegeneral derivation previously given for the spectral density of phase f1uctations.Amplitude fluctuations & of the signal under test produces voltagefluctuations bA at the output of the mixer. Interpretating the mean-squarefluctuations & and 1> in spectal density fashion, we obtain S6,(f), the spectraldensity of amplitude fluctuations bE: of a signal in volts squared per hertz:(48)The term m(j) is the normalized version of the amplitude modulation (AM)portion of SJrf p( v), with its frequency parameter f referenced to the signal'saverage frequency vo, taken as the origin such that the difference frequencyf equals ~' - ~'o' The range of Fourier frequency difference f is from minusVo to plus infinity.Theterm men is defined as the ratio ofthe spectral density ofone amplitudemodulatedsideband to the total signal power, at Fourier frequency differencef from the signal's average frequency Va, for a single specified signal or device.The dimensionality is per hertz. !f(f) and m(f) are similar functions; theformer is a measure of phase-modulated (PM) sidebands, the later is a correspondingmeasure of amplitude-modulated (AM) sidebands. We introducethe symbol m(f) to have useful terminology for the important concept ofnormalized AM sideband power.For the types of signals under consideration, by definition the two amplitude-fluctuationsidebands (lower sideband and upper sideband, at - ff from \'0' respectively) of a signal are coherent with each other. Also, theyare of equal intensity. The operation of the mixer when it is driven at colinearphase is such that the amplitudes of the two AM sidebands are added linearlyin the output of the mixer, resulting in four times as much power in the outputas would be present if only one of the AM sidebands were allowed to contributeto the output of the mixer. Hence, for If I < VO,and, using the definitionwe find, in decibels (carrier) per hertz,(49)(50)m(f) = (l/2V6)S.ll f I). (51)TN-205

i. PHASE :-JOISE AND AM NOISE MEASUREMENTS 255III. Phase-Noise Measurements Using the Two-Oscillator *TechniqueA functional block diagram of the two-oscillator system for measuringphase noise is shown in Fig. 8. NBS has performed phase noise measurementssince 196i using this basic system. The signal level and sideband levels canbe measured in terms of voltage or power. The low-pass filter prevents localoscillator leakage power from overloading the spectrum analyzer whenbaseband measurements are performed at the Fourier (offset) frequenciesof interest. Leakage signals will interfere with autoranging and with thedynamic range of the spectrum analyzer.The low-noise, high-gain preamplifier provides additional system sensitivityby amplyfying the noise signals to be measured. Also, because spectrumanalyzers usually have high values of noise figure, this amplifier is very desirable.As an example, if the high-gain preamplifier had a noise figure of3 dB and the spectrum analyzer had a noise figure of 18 dB, the system sensitivityat this point has been improved by 15 dB. The overall system sensitivitywould not necessarily be improved 15 dB in all cases, because the limitingsensitivity could have been imposed by a noisy mixer.NOISY SIGNAL(THIS IS THE QUANTITYTO BE MEASURED)SMALLFLUCTUATIONSNOISE ONLYMIXERLOW-NOISEAMPLIFIERFIG. 8 The two-oscillator technique for measuring phase noise. Small fluctuations fromnominal voltage are equivalent to phase variations. The phase shifter adjusts the two signalsto quadrature in the mIxer, which cancels carriers and converts phase noise to fluctuating devoltage... See Appendix Note # 6TN-206

256 A. L. LANCE, W. D. SEAL, AND F. LABAARAssume that the reference oscillator is perfect (no phase noise), and that itcan be adjusted in frequency. Also, assume that both oscillators are extremelystable. so that phase quadrature can be maintained without the use of anexternal phase-locked loop or reference. The double-balanced mixer actsas a phase detector so that when two input signals are identical in frequencyand are in phase quadrature, the output is a small fluctuating voltage. Thisrepresents the phase-modulated (PM) sideband component of the signalbecause, due to the quadrature of the signals at the mixer input, the mixerconverts the amplitude-modulated (AM) sideband components to FM, andat the same time it converts the PM sideband components to AM. TheseAM components can be detected with an amplitude detector, as shown inFig. 3.If the two oscillator signals applied to the double-balanced mixer of Fig. 8are slightly out of zero beat, a slow sinusoidal voltage with a peak-to-peakvoltage V plP can be measured at the mixer output. If these same signalsare returned to zero beat and adjusted for phase quadrature, the output ofthe mixer is a small fluctuating voltage (bv) centered at zero volts. If thefluctuating voltage is small compared to t V pIP' the phase quadrature conditionis being closely maintained and the"small angle" condition is beingmet. Phase fluctuations in radians between the test and reference signals(phases) areb¢ = b(¢, - ¢r)' (52)These phase fluctuations produce voltage fluctuations at the output of themixer.bv = t V plPb¢, (53)where phase angles are in radian measure and sin b¢ = b¢ for small b¢(b¢ ~ I rad). Solving for b¢, squaring both sides, and taking a time averagegives«(b¢)2) = 4«bv)2)/(V p,p )2, (54)where the angle brackets represent the time average.For the sinusoidal beat signal,(V pIP)2 = 8(J~ms)2. (55)The mean-square fluctuations of phase b¢ and voltage bv interpreted in aspectral density fashion gives the following in radians squared per hertz:Sd¢(f) = Sdt.(f)/2(VrmY. (56)Here, Sdt.(f), in volts squared per hertz, is the spectral density of the voltagefluctuations at the mixer output. Because the spectrum analyzer measuresTN-207

7. PHASE NOISE AND AM NOISE MEASUREMENTS 257rms voltage, the noise voltage is in units of volts per square root hertz. whichmeans volts per square root bandwidth. Therefore,Shv(f) = [JVrm,,,,,/B] = (t5v rm ,)2,B, (57)where B is the noise power bandwidth used in the measurement.Because it was assumed that the reference oscillator did not contribute anynoise, the voltage fluctuations V rms represent the oscillator under test, andthe spectral density of the phase fluctuations in terms ofthe voltage measurementsperformed with the spectrum analyzer, in radians squared per hertz, isEquation (46) is sometimes expressed asS•.p(f) = H(l5v rms)2/B{Vrms)2]. (58)5h4>(f) = Shv(f)/K 2 , (59)where K is the calibration factor in volts per radian. For sinusoidal beatsignals. the peak voltage of the signal equals the slope of the zero crossingin volts per radian. Therefore, (V p )2 = 2(V.ms)2, which is the same as thedenominator in Eq. (56).The term S'(f) = 20 log(bvrms) - 20 log(V.ms) - 1010g(B) - J. (60)A correction of 2.5 is required for the tracking spectrum analyzer used inthese measurement systems. !f(f) differs by 3 dB and is expressed in decibels(carrier) per hertz as!f(f) = 20 log(bvrms) - 20 10g(V.ms) - 10 10g(B) - 6. (61)A. Two NOISY OSCILLATORSThe measurement system of Fig. 6 yields the output noise from bothoscillators. If the reference oscillator is superior in performance as assumedin the previous discussions, then one obtains a direct measure of the noisecharacteristics of the oscillator under test.If the reference and test oscillators are the same type, a useful approximationis to assume that the measured noise power is twice that associatedwith one noisy oscillator. This approximation is in error by no more than3 dB for the noisier oscillator, even if one oscillator is the major source ofnoise. The equation for the spectral density of measured phase fluctuationsin radians squared per hertz isSh

258 A. L. LANCE, W. D. SEAL, AND F. LABAARThe measured value is therefore divided by two to obtain the value for thesingle oscillator. A determination of the noise ofeach oscillator can be madeif one has three oscillators that can be measured in all pair combinations.The phase noise of each source I, 2, and 3 is calculated as follows:Y1U) = 10 log[100g'"U)ilO + IOg'IJ(J);JO - IOg'2J(J)/IO)]. (63):f2(f) = 10 !og[fOOg'I2(JI;iO + l~2J(J)/IO - lOY1J(/)!lo)], (64)y 3 U) = 10 ]og[1{lOYIJ(J)/IO + JOY2J(J)IIO - 10g'12(/jllO)]. (65)B. AUTOMATED PHASE-NoISE MEASUREMENTS USING THETWO-OSCILLATOR TECHNIQUEThe automated phase-noise measurement system is shown in Fig. 9. It iscontrolled by a programmable calculator. Each step of the calibration andmeasurement sequence is included in the program. The software programcontrols frequency slection, bandwidth settings, settling time, amplituderanging, measurements, calculations, graphics, and data plotting. Normally,the system is used to obtain a direct plot ofY(f). The integrated phase noisecan be calculated for any selected range of Fourier frequencies.A quasi-continuous plot of phase noise performance .!f(f) is obtainedby performing measurements at Fourier frequencies separated by the IFbandwidth of the spectrum analyzer used during the measurement. Plots ofother defined parameters can be obtained and plotted as desired.The IF bandwidth settings for the Fourier (offset) frequency-rangeselections are shown in the follOWing tabulation:IF Fourier IF Fourierbandwidth frequency bandwidth frequency(Hz) (kHz) (kHz) (kHz)3 0.001-0.4 I 40-10010 0.4-1 3 100--40030 1--4 10 400-1300100 4--10200 10--40The particular range of Fourier frequencies is limited by the particularspectrum analyzer used in the system. A fast Fourier analyzer (FFT) is alsoincorporated in the system to measure phase noise from submillihertz to25 kHz.High-quality sources can be measured without multiplication to enhancethe phase noise prior to downconverting and measuring at baseband frequencies.The measurements are not completely automated because thecalibration sequence requires several manual operations.1N-209

7. PHASE NOISE AND AM NOISE MEASUREMENTS 259OSCILLATOR UNDER TESTĪIIIDOUBLE­':E:E;E:C;-'BALANCED' I FREQUENCylMIXER__ ---1: REFERENCE OSCILLATOR, , ILI ,1 IL

260 A. L. LANCE, W. D. SEAL, AND F. LABAARo-10dB -20-30FREQUENCYFIG. 10Plot of automated noise-power bandwidth.The spectrum analyzer power output is recorded for each frequency settingover the range, as illustrated in Fig. 10. The 4o-dB level and the 100 incrementsin frequency are not the minimum permissible values. The recordedcan be plotted for each IF bandwidth, as illustrated, and the noise-powerbandwidth is calculated in hertz as(P + P + P + ... + plOD) ~fnoise power bandwidth = 12 3 d' ,(66)peak power rea mgwhere fj,fis the frequency increment in hertz and the peak power is the maximummeasured point obtained during the measurements. All power valuesare in watts.2 Setting the Carrier Power Reference LevelRecall from Section III.6 that for sinusoidal signals the peak voltage ofthe signal equals the slope of the zero crossing, in volts per radian. A frequencyoffset is established. and the peak power of the difference frequency is measuredas the carrier-power reference level; this establishes the calibrationfactor of the mixer in volts per radian.Because the precision IF attenuator is used in the calibration process, onemust be aware that the impedance looking back into the mixer should be50 n. Also, the mixer output signal should be sinusoidal. Fischer (1978)discussed the mixer as the " critical element" in the measurement system.It is advisable to drive the mixer so that the sinusoidal signal is obtainedat the mixer output. In most of the TR W systems. the mixer drive levels are10 dBm for the reference signal and about zero dBm for the unit under test.1N-211

7. PHASE ~OISE AND AM NOISE MEASVREMENTS 261System sensitivity can be increased by driving the mixer with high-levelsIgnals that lower the mixer output impedance to a few ohms. This ~resentsa problem in establishing the calibration factor of the mixer, because itmight be necessary to calibrate the mixer for different fourier frequencyranges.The equation sensitivity = slope = beat-note amplitude does not holdif the output of the mixer is not a sine wave. The Hewlett-Packard 3047automated phase noise measurement system allows accurate calibrationof the phase-detector sensitivity even with high-level inputs by using thederivative of the Fourier representation of the signal (the fundamental andits harmonics). The slope at 4J ;;:; 0 radians is given byA sin 4J - B sin 34J + C sin 54J = A cos 4J - 3B cos 34J + 5C cos 54J= A - 3B + 5C + .... (67)Referring to Fig. 9, the carrier-power reference level is obtained as follows.(1) The precision IF step attenuator is set to a high value to preventoverloading the spectrum analyzer (assume 50 dB as our example).(2) The reference and test signals at the mixer inputs are set to approximately10 dBm and 0 dBm, as previously discussed.(3) If the frequency of one of the oscillators can be adjusted, adjust itsfrequency for an IF output frequency in the range of 10 to 20 kHz. If neitheroscillator is adjustable, replace the oscillator under test with one that canbe adjusted as required and that can be set to the identical power level of theoscillator under test.(4) The resulting IF power level is measured by the spectrum analyzer,and the measured value is corrected for the attentuator setting, which wasassumed to be 50 dB. The correction is necessary because this attenuatorwill be set to its zero decibel indication during the measurements of noisepower. Assuming a spectrum analyzer reading of -40 dBm, the carrierpowerreference level is calculated ascarrier power reference level = 50 dB - 40 dBm = 10 dBm. (68)3. Phase Quadrature of the Mixer Input SignalsAfter the carrier-power reference has been established, the oscillator undertest and the reference oscillator are tuned to the same frequency, and theoriginal reference levels that were used during calibration are reestablished.The quadrature adjustment depends on the type of system used, Threepossibilities, illustrated in Fig. 9, are described here.(1) If the oscillators are very stable, have high-resolution tuning, and arenot phase-locked, the frequency of one oscillator is adjusted for zero deTN-212

262 A. L. LANCE, W, D. SEAL, AND F, LABAARvoltage output of the mixer as indicated by the sensitive oscilloscope.Note: Experience has shown that the quadrature setting is not critical if thesources have low AM noise characteristics. As an example, experimentsperformed using two HP 3335 synthesizers showed that degradation of thephase-noise measurement became noticeable with a phase-quadratureoffset of 16 degrees.(2) If the common reference frequency is used, as illustrated in Fig. 9, thenit is necessary to include a phase shifter in the line between one of the oscillatorsand the mixer (preferably between the attenuator and mixer). Thephase shifter is adjusted to obtain and maintain zero volts dc at the mixeroutput. A correction for a nonzero dc value can be applied as exemplified bythe HP 3047 automated phase-noise measurement system.(3) If one oscillator is phase-locked using a phase-locked loop, as showndotted in on Fig. 9, the frequency of the unit under test is adjusted for zerodc output of the mixer as indicated on the oscilloscope.A phase-locked loop is a feedback system whose function is to force avoltage-controlled oscillator (VeO) to be coherent with a certain frequency,i.e., it is highly correlated in both frequency and phase. The phase detectoris a mixer circuit that mixes the input signal with the yeO signal. The mixeroutput is Vi ± v o ' when the loop is locked, the YCO duplicates the inputfrequency so that the difference frequency is zero, and the output is a dcvoltage proportional to the phase difference. The low-pass filter removesthe sum frequency component but passes the dc component to control theYeo. The time constant of the loop can be adjusted as needed by varyingamplifier gain and RC filtering within the loop.A loose phase-locked loop is characterized by the following.(I) The correction voltage varies as phase (in the short term) and phasevariations are therefore observed directly.(2) The bandwidth of the servo response is small compared with theFourier frequency to be measured.(3) The response time is very slow.A right phase-locked loop is characterized by the following.( I) The correction voltage of the servo loop varies as frequency.(2) The bandwidth of the servo response is relatively large.(3) The response time is much smaller than the smallest time interval rat which measurements are performed.Figure 1I shows the phase-noise characteristics of the H.P. 86408 synthesizermeasured at 512 MHz. The phase-locked-loop attenuation characteristicsextend to 10 k Hz. The internal-oscillator-source characteristics areplotted at Fourier frequencies beyond the loop-bandwidth cutoff at 10 kHz.TN-213

7. PHASE NOISE AND AM NOISE MEASUREMENTS 263-80-100;g -120CD ";" -140'-'-160_ PHASE-LOCKED _LOOP512-MHzOSCILLATOR-180'-----.....----'-----......----'------'10 103 10 4 105FOURIER FREQUENCY (Hz)FIG. II Phase-locked-loop characteristics of the H.P. 8640B signal generator. showingthe normalized phase-noise sideband power spectral density.4. Measurements, Calculations, and Data PlotsThe measurement sequence is automated except for the case where manualadjustments are required to maintain phase quadrature of the signals. Afterphase quadrature of the signals into the mixer is established, the {F attenuatoris returned to the zero-decibel reference setting. This attenuator isset to a high value [assumed to be 50 dB in Eq. (65)] to prevent saturationof the spectrum analyzer during the calibration process.The automated measurements are executed, and the direct measurementand data plot of fi'(f) is obtained in decibels (carrier) per hertz using theequationfi'(f) = - [carrier power level - (noise power level - 6 + 2.5- 10 log B-3)]. (69)The noise power (dBm) is measured relative to the carrier-power level(dBm), and the remaining terms of the equation represent corrections thatmust be applied because of the type of measurement and the characteristicsof the measurement equipment, as follows.(l) The measurement of noise sidebands with the signals in phasequadrature requires the -6-dB correction that is noted in Eq. (69).(2) The nonlinearity of the spectrum analyzer's logarithmic IF amplifierresults in compression of the noise peaks which, when average-detected,require the 2.5-dB correction.(3) The bandwidth correction is required because the spectrum analyzermeasurements of random or white noise are a function of the particularbandwidth used in the measurement.TN-214

264 A. L. LANCE, W. D. SEAL, AND F. LABAAR(4) The - 3-dB correction is required because this is a direct measure of5e(f) of two oscillators, assuming that the oscillators are of a similar typeand that the noise contribution is the same for each oscillator. If one oscillatoris sufficiently superior to the other, this correction is not required.Other defined spectral densities can be calculated and plotted as desired.The plotted or stored value of the spectral density of phase fluctuationsin decibels relative to one square radian (dBc rad 2 /Hz) is calculated asThe spectral density of phase fluctuations, in radians squared per hertz, iscalculated asThe spectral density of frequency fluctuations, in hertz squared per hertz, iswhere S';dJ(F) is in decibels with respect to 1 radian.5. System Noise Floor Verification(70)(71)Sb'.(f) = j 2 Sb¢(f). (72)A plot of the system noise floor (sensitivity) is obtained by repeating theautomated measurement procedures with the system modified as shown inFig. 12. Accurate measurements can be obtained using the configurationshown in Fig. 12a. The reference source supplies 10 dBm to one side of themixer and 0 dBm to the other mixer input through equal path lengths;phase quadrature is maintained with the phase shifter.0-------50- n6}----: \.::J TERMINATIONIIPHASESHIFTER-10 dBmI"\....)-------'la!{blFIG. 12 System configurations for measuring the system noise floor (sensitiVitY): (a)configuration used for accurate measurements: (b) alternate configuration sometimes used.TN-215

7. PHASE NOISE AND AM NOISE MEASUREMENTS 265The configuration shown in Fig. 12b is sometimes used and does notgreatly degrade the noise floor because the reference signal of 10 dBm islarger Ihan Ihe signal frequency. See Sections IV.B and IV.C.4 for additionaldiscussions related to system sensitivity and recommended system eva·luation.Proper selection of drive and output termination of the double-balancedmixer can result in improvement by (5 to 25 dB in the performance of phasenoisemeasurements, as discussed by Walls et al. (1976). The beat frequencybetween the two oscillators can be a sine wave. as previously mentioned,with proper low drive levels. This requires a proper terminating impedancefor the mixer. With high drive levels, the mixer output waveform will beclipped. The slope of the clipped waveform at the zero crossings, illustratedby Walls et al. (1976), is twice the slope of the sine wave and therefore improvesthe noise floor sensitivity by 6 dB, i.e., the output signal, proportionalto the phase fluctuations. increases with drive level. This condition of clippingrequires characterization over the Fourier frequency range, as previouslymentioned for the Hewlett-Packard 3047 phase noise measurement system.An amplifier can be used to increase the mixer drive levels for devices thathave insufficient output power to drive the double-balanced mixers.Lower noise floors can be achieved using high-level mixers when availabledrive levels are sufficient. A step-up transformer can be used to increasethe mixer drive voltage because the signal and noise power increase in thesame ratio, and the spectral density of phase of the device under test is unchanged,but the noise floor of the measurement system is reduced.Walls et al. (1976) used a correlation technique that consisted primarilyof two phase-noise measurement systems. At TRW the technique is used asshown in Fig. 13. The cross spectrum is obtained with the fast Fouriertransform (FFT) analyzer that performs the product of the Fourier trans-·form of one signal and the complex conjugate of the Fourier transform ofDUAL­CHANNELFFTFIG. 13Cross-spectrum measurement using the two-oscillator technique.TN-216

266A. L. LANCE, W. D. SEAL, AND F. LABAARthe second signal. This cross spectrum, which is a phase-sensitive characteristic,gives a phase and amplitude sensitivity measure directly, A signalto-noiseenhancement greater than 20 dB can be achieved.Ifthe double-balanced phase noise measurement system does not providea noise floor sufficient for measuring a high-quality source. frequencymultiplier chains can be used if their inherent noise is 10-20 dB below themeasurement system noise. In frequency multipli

7. PHASE NOISE AND AM NOISE MEASUREMENTS 267The following equation is used to correct for noise-floor contributionPnr, in dBc(Hz, if desired or necessary:2'(f)(corrected) = -2'(f) + 10 10g[P!f(fJ - Pnr] (74)P!f(f)The correction for noise-floor contribution can also be obtained by usingthe measurement of S"cU) of Eq. (57). Measurement of S"cU) of the oscillatorplus floor is obtained, then S",,(f) is obtained for the noise floor only. Then,S"v(f) I = S"v(f) I - S"vC!) I (75)cor (osc + nO ofFigure 14a shows a phase noise plot of a very high-quality (5-MHz)quartz oscillator, measured by the two-oscillator technique. The sharppeaks below 1000 Hz represent the 60-Hz line frequency of the power supplyand its harmonics and are not part of the oscillator phase noise. Figure 14bshows measurements to 0.02 Hz of the carrier at a frequency of 20 MHz.IV. Single-Oscillator Phase-Noise Measurement Systems and *TechniquesThe phase-noise measurements of a single-oscillator are based on themeasurement of frequency fluctuations using discriminator techniques. Thepractical discriminator acts as a filter with finite bandwidth that suppressesthe carrier and the sidebands on both sides of the carrier. The ideal carriersuppressionfilter would provide infinite attentuation of the carrier andzero attenutation of all other frequencies. The effective Q of the practicaldiscriminator determines how much the signals are attenuated.Frequency discrimination at very high frequencies (VHF) has been obtainedusing slope detectors and ratio detectors, by use of lumped circuitelements of inductance and capacitance. At ultrahigh frequencies (UHF)between the VHF and microwave regions, measurements can be performedby beating, or heterodyning, the UHF signal with a local oscillator to obtaina VHF signal that is analyzed with a discriminator in the VHF frequencyrange. Those techniques provide a means for rejecting residual amplitudemodulated(AM) noise on the signal under test. The VHF discriminatorsusually employ a limiter or ratio detector.Ashley et ai. (1968) and Ondria (1968) have discussed the microwavecavity discriminator that rejects AM noise, suppresses the carrier so that theinput level can be increased, and provides a high discriminated output toimprove the signal-to-noise floor ratio. The delay line used as an FM discriminatorhas been discussed by Tykulsky (1966), Halford (1975), and* See Appendix Note # 6TN-218

268A. L. LANCE, W. D. SEAL, AND F. LABAARTUNERVARIABLESHORTCIRCUITDELAY LINE(.1 (01OSCILLATORUNDER TEST---+'\.-J-------{DI RECTIONALCOUPLERDELAY LINEMIXERLOW-NOISEAMPLIFIERDIGITALSIGNALANALYZERIFFTI(c)FIG. 15 Single-oscillator phase-noise measurement techniques: (a) cavity discriminator;(b) reflective-type delay-line disenmmator; (e) one-way delay line.Ashley et al. (1968). Ashley et al. (1968) proposed the reflective-type delaylinediscriminator shown in Fig. ISb. The cavity can also be used to replacedelay line. The one-way delay line shown in Fig. 15c is implemented in theTRW measurement systems. The theory and applications set forth in thissection are based on a system of this particular type.A. THE DELAY LINE AS AN FM DISCRIMINATORI. The Single-Oscillator Measurement SystemThe single-oscillator signal is split into two channels in the system shownin Fig. 15. One channel is called the nondelay or reference channel. It is alsoreferred to as the local-oscillator (LO) channel because the signal in thischannel drives the mixer at the prescribed impedance level (the usual LOdrive). The signal in the second channel arrives at the mixer through a delayline. The two signals are adjusted for phase quadrature with the phaseshifter, and the output of the mixer is a fluctuating voltage, analogous to thefrequency fluctuations of the source, centered on approximately zero dcvolts.The delay line yields a phase shift by the time the signal arrives at thebalanced mixer. The phase shIft depends on the instantaneous frequency ofTN-219

7. PHASE NOISE AND AM NOISE MEASUREMENTS 269the signal. The presence of frequency modulation (FM) on the signal givesrise to differential phase modulation (PM) at the output of the differentialdelay and its associated (nondelay) reference line. This relationship is linearif the delay Ld is nondispersive. This is the property that allows the delay lineto be used as an FM discriminator. In general, the conversion factors are afunction of the delay (Ld) and the fourier frequency j but not of the carrierfrequency.The differential phase shift of the nominal frequency Yo caused by the delayline iswhere Ld is the time delay.The phase fluctuations at the mixer are related to the frequency fluctuations(at the rate f) byThe spectral density relationships areandThen,(76)b(/J = 2rrLd bv(J). (77)5 6 (J)1. = (2rrLd)2 5 60 (J)1 (78)mu.crosc(79)5 6 (J)1 = (2rcjLd)2 5 6 4>U)! (80)dimoscwhere the subscript dIm indicates delay-line method. From Eq. (56), thespectral density of phase for the two-oscillator technique, in radians squaredper hertz, is(81)becauseandper hertz.TN·220

270 A. L. LANCE, W. D. SEAL, AND F. LABAARThe sensitivity (noise floor) of the two-oscillator measurement systemincludes the thermal and shot noise of the mixer and the noise of the basebandpreamplifier (referred to its input). This noise floor is measured withthe oscillator under test inoperative. The measurement system sensitivity ofthe two-oscillator system, on a per hertz density basis (dBc/Hz) iswhere bL: n is the rms noise voltage measured in a one-hertz bandwidth.The two-oscillator system therefore yields the output noise from bothoscillators. If the reference oscillator is superior in performance, as assumedin the previous discussions, then one obtains a direct measure of the noisecharacteristics of the oscillator under test. If the reference and test oscillatorsare the same type, a useful approximation is to assume that the measurednoise power is twice that associated with one noisy oscillator. This approximationis in error by no more than 3 dB for the noisier oscillator. Substitutingin Eq. (80) and using the relationships in Eq. (56), we have, per hertz,(83)!t'(f)1 = 2[(6V rm Y/(VPIP)2](2njr d )2 (84)dImExamination of this equation reveals the following.(I) The term in the brackets represents the two-oscillator response.Note that this term represents the noise floor oj the two-oscillator method.Therefore, adoption of the delay-line method results in a higher noise by thefactor (2rr.jr d )2 when compared with the two-oscillator measurement method.The sensitivity (noise floor) for delay lines with different values of time delayare illustrated in Fig. 17.(2) Equation (84) also indicates that the measured value of !t'(f) isperiodic in w = 2rrf This is shown in Fig. 21. The first null in the responsesis at the Fourier frequency j = l/rd' The periodicity indicates that the calibrationrange of the discriminator is limited and that valid measurementsoccur only in the indicated range. as verified by the discriminator slope shownin Fig. 16. (See Fig. 23.)(3) The maximum value of (2rrjr d )2 can be greater than unity (it is 4 atj = 1'2r d )· This 6-dB advantage is utilized in the noise-floor measurement.However, it is beyond the valid calibration range of the delay-line system.The 6-dB advantage is offset by the line attenuation at microwave frequencies,as discussed by Halford (1975).The delay-line discriminator system has been analyzed in terms ofa powerlimitedsystem (a particular idealized system in which the choice of poweroscillator voltage. the attenuator of the delay line, and the conversion lossTN-221

7. PHASE NOISE AND AM NOISE MEASUREMENTS271of the mixer are limited by the capability of the mixer) by Tykulsky (1966),Halford (1975), and Ashley et al. (1977). For this particular case. Eq. (83)indicates that an increase in the length of the delay line (to increase 'd fordecorrelation of Fourier frequencies closer to the carrier) results in anincrease in attenuation of the line, which causes a corresponding decreasein V ptp ' The optimum length occurs where'd is such that the decrease inV ptp is approximately compensated by the increase in (2nf'd)' i.e., where~ 2nf'd = O. (85)d'd V plPThis condition occurs where the attenuation of the delay line is 1 Np(8.686 dB). However, when the system is not power limited, the attenuationof the delay line is not limited, because the input power to the delay line canbe adjusted to maintain V ptp at the desired value. The optimum delay-linelength is determined at a particular selectable frequency. However, sincethe attenuation varies slowly (approximately proportional to the square rootof frequency), this characteristic allows near-optimum operation over aconsiderable frequency range without appreciable degradation in themeasurements.A practical view of the time delay ('d) and Fourier-frequency functionalrelationship can be obtained by reviewing the basic concepts of the dualchanneltime-delay measurement system discussed by Lance (1964). If thedifferential delay between the two channels is zero, there is no phase differenceat the detector output when a swept-frequency cw signal is appliedto the system. Figure 16 shows the detected output interference display whena swept-frequency cw signal (zero to 4 MHz) is applied to a system that has1+----- 360" ~ IoTd = 1500 nsec:f = 2 MHzf = 4 MHzFIG. 16 Swept-frequency interference display at the output of a dual-ehannel systemwith a differential delay of 500 nsee.TN-222

272A. L. LANCE, W. D. SEAL, AND F. LABAARa differential delay of 500 nsec between the two channels. The signal amplitudesare assumed to be almost equal, thus producing the familiar voltagestanding-wavepattern or interference display. Because this is a two-channelsystem, there is a null every 360°, as shown.2. System Sensitivity (Noise Floor) When Using theDifferential Delay-Line TechniqueHalford (1975) has shown that the sensitivity (noise floor) of the singleoscillatordifferential delay-line technique is reduced relative to the twooscillatortechniques. The sensitivity is modified by the factorFor WT d = 2ndTd ~ 1 a good approximation is(86)where 8 is the phase delay of the differential delay line evaluated at thefrequency.r Figure 17 shows the relative sensitivity (noise floor) of the twooscillatortechnique and the single-oscillator technique with differentdelay-line lengths. The f - 2 slope is noted at Fourier frequencies beyond-20-40-60~ -80co "-100:s0::00-'.~-...._~u.. -120UJVl0 -140z-160520 nsec}260 n!eC65nsec-180TWO OSCILLATORS-19010 10 3 10 4 10 5FOUR IER FREQUENCYSING LEOSCillATORFIG. 17 Relative sensitivity (noise floor) of single-oscillator and two-oscillator phasenoisemeasurement systems.TN-223

7. PHASE NOISE AND AM NOISE MEASUREMENTS 273about I kHz. For Fourier frequencies closer to the carrier, the slope is f - 3.i.e.. the sum of the f - 2 slope of Eq. (87) and the f - I flicker noise.Phase-locked sources have phase-noise characteristics that cannot bemeasured at close-in Fourier frequencies using this basic system. Therelative sensitivity of the system can be improved by using a dual (twochannel)delay-line system and performing cross-spectrum analysis, whichwill be presented in this chapter.Labaar (1982) developed the delay-line rf bridge configuration shown inFig. 18. At microwave frequencies where a high-gain amplifier is available.suppression of the carrier by the rf bridge allows amplification of the noisegoing into the mixer. A relative sensitivity improvement of 35 dB has beenobtained without difficulty. The limitations of the technique depend on theavailable rf power and the carrier suppression by the bridge. Naturally, ifthe rf input to the bridge is high one must use the technique with adequateprecautions to prevent mixer damage that can occur by an accidental bridgeunbalance. Labaar (1982) indicated the added advantage of using the rfbridge carrier-suppression technique when attempting to measure phasenoise close to the carrier when AM noise is present. Figure 19 shows thePHASESHIFTERDELAY-LINErfBRIDGEDELAY LINE--------------------­POWERSPLITTERMIXERPHASESHIFTERFIG. 18 Carrier suppression using an rf bridge to increase relative sensitivity. (CourtesyInstrument Society of America.)IN-224

274 A. L. LANCE, W. D. SEAL, AND F. LABAARSIN h, r II:>AM LEAKAGEFrequency (Hz)FIG. 19Phase detector output (AM-PM crossover); T, delay time.mixer output for phase (PM) and amplitude (AM) noise in the singleoscillatordelay-line FM discriminator system. It is noted that the phasenoise and AM noise intersect and that the AM will therefore limit the measurementaccuracy near the carrier. Even though AM noise is much lowerthan phase noise in most sources, and even though the AM is normallysuppressed about 20 dB, there is still AM at the mixer output. This outputis AM leakage and is caused by the finite isolation between the mixer ports.The two-oscillator technique does not experience this problem to thisextent because the phase noise and AM noise maintain their relative relationshipsat the mixer output independent of the offset frequency from thecarrier.B. CALIBRATION AND MEASUREMENTS USING TIlE DELAYLINE AS AN FM DISCRIMINATORThe block diagram of a practical single-oscillator phase noise measurementsystem is shown in Fig. 20. The signals in the delay-line channel of thesystem experience the one-way delay of the line. With adequate sourcepower, the system is not limited to the optimum I Np (8.686 dB) previouslydiscussed for a power-limited system. Measurements are performed usingthe following operational procedures.(1) Measure the tracking spectrum analyzer IF bandwidths as set forthin Section III.C.I.(2) Establish the system power levels (Section IV.B.l).(3) Establish the discriminator calil'>ration factor (Section IV.B.2).(4) Measure and plot the oscillator characteristics in the automaticsystem used (Section IV.B.3).(5) Measure the system noise floor (sensitivity) (Section IV.B.4).TN-225

7. PHASE NOISE AND AM NOISE MEASUREMENTS 275SIGNALGENERATOR(FREQUENCY­MODULATIONCAPABILITYI1. TO AUTOMATIC MEASUREMENTSYSTEM2. HP 5420A DIGITAL SIGNAL ANALYZE R(SUBMILLIHERT2 TO 25 kHZ I3. HP 5390A FREQUENCY STABILITYANAL YZER (0.01 Hz TO 10 kHzlHIGH GAINLOW NOISEAMPFIG. 20 Single-oscillator phase noise measurement system using the delay line as anFM discriminator. (From Lance et al., 1977a.)I. System Power LevelsThe system power levels are set using attenuators, as shown in Fig. 20.Because the characteristic impedance of attenuator No.4 is 50 n, mismatcherrors will occur if the mixer output impedance is not 50 n. As previouslydiscussed, the mixer drive levels are set so that the mixer output signal, asobserved during calibration, is sinusoidal. This has been accomplished inTRW systems with a reference (LO) signal level of 10 dBm and a mixerinput level of about adBm from the delay line.A power amplifier can be used to increase the source signal to the measure­ment system. This amplifier must not contribute appreciable additionalnoise to the signal.2. Discriminator CalibrationThe discriminator characteristics are measured as a function of frequencyand voltage. The hertz-per-volt sensitivity of the discriminator is definedas the calibration factor (CF). The calibration process involves measuringthe effects of intentional modulation of the source (carrier) frequency. Aknown modulation index must be obtained to calculate the calibrationfactors of the discriminator. The modulation index is obtained by usingamplitude modulation to establish the carrier-to-sideband ratio when thereis considerable instability of the source or when the source cannot be fre­quency modulated.IN-226

276 A. L. LANCE, W. D. SEAL, AND F. LABAARIt is convenient to consider the system equations and calibration techniquesin terms of frequency modulation of stable sources. If the source to bemeasured cannot be frequency modulated, it must be replaced, during thecalibration process, with a modulatable source. The calibration processwill be described using a modulatable source and a 20-kHz modulationfrequency. However, other modulation frequencies can be used. The calibrationfactor of this type discriminator has been found to be constant overthe usable Fourier frequency range, within the resolution of the measuringtechnique. The calibration factor of the discriminator is established after thesystem power levels have been set with the unit under test as the source.The discriminator calibration procedures are as follows.(1) Set attenuator No.4 (Fig. 20) to 50 dB.(2) Replace the oscillator under test with a signal generator or oscillatorthat can be frequency modulated. The power output and operating frequencyof rhe generaror must be set to the same precise frequency and amplitudevalues that the oscillator under test will present to the system during rhe measuremel1lprocess.(3) Select a modulation frequency of 20 kHz and increase the modulationuntil the carrier is reduced to the first Bessel null, as indicated on the spectrumanalyzer connected to coupler No. 1. This establishes a modulation index(m = 2.405).(4) Adjust the phase shifter for zero volts dc at the output of the mixer,as indicated on the oscilloscope connected as shown in Fig. 20. This establishesrhe quadrature condition for the two inputs to the mixer. This quadraturecondition is continuously monitored and is adjusted if necessary.(5) Tune the tracking spectrum analyzer to the modulation frequencyof 20 kHz. The power reading at this frequency is recorded in the programand is corrected for the 50-dB setting of attenuator No.4, which will be setto zero decibel indication during the automated measurements.P(dBm) = (-dBm power reading) +50 dB (88)This power level is converted to the equivalent rms voltage that the spectrumanalyzer would have read if the total signal had been applied:v'ms = jlO P11O /lOOO + R. (89)(6) The discriminator calibration factor can now be calculated becausethis power in dBm can be converted to the corresponding rms voltage usingthe following equation:v'ms = j(lOPilo/lOOO) x R, (90)where R = 50 n in this system.IN-227

7. PHASE NOISE AND AM NOISE MEASUREMENTS 277(7) The discriminator calibration factor is calculated in hertz per volt asCF = mfml...;2 v'ms = 2.405fmli2 v,ms· (91 )The modulation index m for the first Bessel null as used in this techniqueis 2.405. The modulation frequency is j~.3. .\leasurement and Data PlottingAfter the discriminator is calibrated, the modulated signal source is replacedwith the frequency source to be measured. Quadrature of the signalsinto the mixer is reestablished, attenuator No.4 (Fig. 20) is set to 0 dB, andthe measurement process can begin.The measurements, calculations, and data plotting are completely automated.The calculator program selects the Fourier frequency, performsautoranging, and sets the bandwidth, and measurements of Fourier frequencypower are performed by the tracking spectrum analyzer. Each Fourierfrequency noise-power reading P n (dBm) is converted to the correspondingrms voltage byThe rms frequency fluctuations are calculated as(92)l5v rms = V lrms X CF. (93)The spectral density of frequency fluctuations in hertz squared per hertz iscalculated aswhere B is the measured IF noise-power bandwidth of the spectrum analyzer.The spectral density of phase fluctuations in radians squared per hertz iscalculated asThe NBS-designated spectral density in decibels (carrier) per hertz is calculatedasSpectral density is plotted in real time in our program. However, the datacan be stored and the desired spectral density can be plotted in other forms.Integrated phase noise can be obtained as desired.(94)(95)(96)TN-228

278 A. L. LANCE, W. D. SEAL, AND F. LABAAR4. l"/oise Floor MeasurementsThe relative sensitivity (noise floor) of the single-oscillator measurementsystem is measured as shown in Fig. 12a for the two-oscillator technique.The delay line must be removed and equal channel lengths constructed,as in Fig. (12a). The same power levels used in the original calibration andmeasurements are reestablished, and the noise floor is measured at specificFourier frequencies, using the same calibration-measurement technique,or by repeating the automated measurement sequence.A correction for the noise floor requires a measurement of the rms voltageof the oscillator (Vlrm,) and a measurement of the noise floor rms voltage(V2rm,)' These voltages are used in the following equation to obtain thecorrected value:The value L"rm, is then used in the calculation of frequency fluctuations.If adequate memory is available, each value of V 1rm , can be stored and usedafter the other set of measurements are performed at the same Fourierfrequencies.The following technique was developed by Labaar (1982). Carrier suppressionis obtained using the rf bridge illustrated in Fig. 18. One can easilyimprove sensitivity more than 40 dB. At 2.0 and 3.0 GHz 70-dB carriersuppression was realized. In general, the improvement in sensitivity willdepend on the availability of an amplifier or adequate input power.Figure 21 shows the different noise floors in a delay-line bridge discriminator.It is good measurement discipline to always determine these noisefloors; also, the measurements, displayed in Fig. 21, give a quick understandingof the physical process involved. The first trace is obtained by terminatingthe input of the baseband spectrum analyzer. The measured outputnoise power is then a direct measure of the spectrum analyzer's noise figure(NF). The input noise is thermal noise and is usually indicated by "KTB,"which is short for "the thermal noise power at absolute temperature of Tdegrees K(elvin) per one hertz bandwidth (B). This KTB number is, at 18°C,about - 174 dBm/Hz.Figure 21a shows that trace number 1for frequencies above about 1 kHz islevel with a value of about -150 dBm = -(174-24) dBm, which means thatthe spectrum analyzer has an NF of 24 dB. At 20 Hz the NF has gone up toabout 48 dB. To improve the NF, a low-noise (NF, 2dB), low-frequency(10 Hz-1O MHz) amplifier is inserted as a preamplifier. Terminating itsinput now results in trace number 2. At the high frequency end, the measuredpower goes up by about 12-13 dB, and the amplifiers gain is 34 dB. Thismeans that the NF is improved by 34 - 12-13 = 21-22 dB, which is an(97)TN-229

7. PHASE NOISE AND AM NOISE MEASUREMENTS 279... -120IE'"~ -140-160"kTB"-1800.01 0.1 1.0 10 100 1000 10,000FREQUENCY IkHzllalDELAY­LINErf BRIDGEDISCRLO6 dBmCONV. LOSS ~NF " 7 dBIbJFIG. 21 (a) Noise contribution analysis: (b) phase noise test setup using a delay-line rfbridge discriminator (rf = 2.8 GHz; O.termlnation POints. NF. noise figure: LF.low frequen~y.(From Seal and Lance. 1981.)NF ~ 2-3 dB as expected, i.e., the first stage noise predominates. The lowfrequencyend at 20 Hz gives an NF of 26 dB, which overall is quite an improvement.[n trace number 3 the mixer is included with it's rf (signal) port terminated.It is clear from this trace that certainly up to 100 kHz, the noise generatedby the mixer diodes being" pumped" by the LO signal dominates. This caserepresents the" classic" delay-line discriminator. The last trace (number 4)includes the low-noise, high-gain rf amplifier that can be used because thecarrier is suppressed in the delay-line rf bridge discriminator, in contrastto the classic delay-line discriminator case. This trace shows that from 1 kHzon up the measured output power is flat, representing a 2-3-dB NF.At about 20-40 Hz, trace numbers 3 and 4 begin nearing their crossoverfloor. In this particular case, which is discussed in full by Labaar (1982),the measurement systems noise floor (resolution) has been improved by40 dB.Figure 22 shows plots of phase noise as measured at two frequenciesusing delay lines ofdifferent lengths. The delay line used measure at 600 MHzTN-230

280 A. L. LANCE, W. D. SEAL, AND F. LABAAR-30-40LINE HARMONICS-60-80~ ·100'"~=- -120;:;­DISCRIMINATOR NOT CALIBRATEDIN THIS FREOUENCY RANGE \RADIOSTA TlON-140·160180-190 1.--1-.--'4'-0-L..JWOO-...l---l...........J....L,-..L.-....J..~.1..1 0"-:4,.--1-.--l..............J-:,--l._...................L;,-........._J......I-'-'1Q70 1FOURIER FREOUENCY I HzlFIG. 221977a.)Phase noise of 6OO-MHz oscillator multiplied to 2.4 GHz (From Lance et al..was about 500 nsec long, as noted by the first null, i.e., the reciprocal of theFourier frequency of 2 MHz is the approximate differential time delay.Note that a shorter delay line (approximately 250 nsec differential) is usedto measure the higher frequency because the delay-line discriminatorcalibration is valid only to a Fourier frequency at approximately 35~~ ofthe Fourier frequency at which the first null occurs, if a linear transferfunction is assumed.The actual transfer function of a delay-line discriminator (classic and rfbridge types) is sinusoidal, as shown in Fig. 23a. The baseband spectrumanalyzer measures power in a finite bandwidth, and as a consequence it ispossible to measure through a transfer-function null if the noise power doesnot change substantially over a spectrum-analyzer bandwidth, The followingpower relations then hold:"J+6w;2IP(w) i"'+6W;2Pmeas(w) = lI~w P(w') dw' :::: - dw' = P(w), (98)w-6w 2 ~w w- 6w. 2TN-231

7. PHASE NOISE AND AM NOISE MEASUREMENTS 281z~t::­;)00­-801.1-100:cu:o!-120'"Y-140-160CORRECT(SINUSOIDALITRANSFERFUNCTIONCORRECTION ---"'''ril-­8REAKDOWNLINEAR TRANSFER \IFUNCTION·180FREQUENCY (kHziIblFIG. 23 Transfer functions for a delay-line rf bridge discriminator: (a) actual; (b)approximate (linear) and "correct" (sinusoidal). Phase noise: H.P. 8672A at 2.4 GHz.Figure 23b shows the results using a linear approximation and the "correct"transfer function for a delay-line rf bridge discriminator. The correcttransfer function breaks down close to the null because the signal leveldrops below the system's noise floor, as explained by Labaar (1982).Using the sinusoidal transfer function in the calculator software givescorrect results barring frequency intervals of 5 to 10 spectrum analyzer'sbandwidths (10 x 30 = 300 kHz) centered at the transfer function nulls.These particular data were selected to illustrate the characteristics of thesystem. Recall that one can easily make the noise floor 40 dB lower usingthe rf bridge shown in Fig. 18.C. DUAL DELAy-LINE DISCRIMINATORI. Phase Noise MeasurementsThe dual delay-line discriminator is shown in Fig. 24. This system wassuggested by Halford (1975) as a technique for lowering the noise floorof the delay-line phase noise measurement system. The system consists of

282 A. L. LANCE, W. D. SEAL, AND F. LABAARA,.....---., IISIGNALIGENERATOR(FREQ.-MOD.CAPABILITY)IIIADELAY LINEHEWLETT-PACKARD5420ADIGITAL SIGNALANALYZERDUAL­CHANNELdeSCOPEDELAY LINEFIG. 24 A dual delay-line phase noise measurement system. (Courtesy Instrument Societyof America.)two differential delay-line systems. The single-oscillator signal is appliedto both systems and cross-spectrum analysis is performed on the signaloutput from the two delay-line systems. Signal processing is performed withthe Hewlett-Packard 5420A digital signal analyzer. The cross spectrum isobtained by taking the product of the Fourier transform of one signal andthe complex conjugate of the Fourier transform of a second signal. It isa phase-sensitive characteristic resulting in a complex product that servesas a measurement of the relative phase of two signals. Cross spectrum givesa phase- and amplitude-sensitive measurement directly. By performingthe product Sy(f) . Sx(f)*, a certain signal-to-noise enhancement is achieved.TN-233

7. PHASE NOISE AND AM :>lOISE MEASUREMENTS 283The low-noise amplifiers preceding the digital signal analyzer are usedwhen performing measurements at Fourier frequencies from 1 Hz to 25 kHz.The amplifiers are not used when performing measurements below theFourier frequency of 1 Hz.2. Calibrating the Dual Delay-Line SystemEach delay line in the system is calibrated separately following the samebasic procedure set forth in Section IV.B. The Hewlett-Packard 5420measures the one-sided spectral density of frequency fluctuations in hertzsquared per hertz. The spectral density of phase fluctuation in radianssquared per hertz can be calculated asand(99)(l00)per hertz. The Hewlett-Packard 5420 measurement ofSJy(f) in Hz 2 /Hz must,therefore, be corrected by 1/2f2. However, the f2 correction must be enteredin terms of radian frequency (w = 2rr.j). This conversion is accomplished by5£(f) = SJy(f)(1/2f2)(4rr. 2 /4rr. 2 ) = [2rr. 2 SJy(f)]/(W)2per hertz since Eq. (100) can be stated in the following terms:[2rr. 1 SJy(f)]/4rr.2f 2.(LOllSignal-to-noise enhancement greater than 20 dB has been obtained usingthe dual-channel delay-line system.D. MILLIMETER-WAVE PHASE-NOISE MEASUREMENTSI. Spectral Density of Phase FluctuationsThe delay line used as an FM discriminator is based, in principle, on anondispersive delay line. However, a waveguide can be used as the delayline because the Fourier frequency range of interest is a small percentageof the operating bandwidth (seldom over 100 MHz), and the dispersion canbe considered negligible.The calibration and measurement are performed as set forth in SectionIV.B. The modulation index m is usually established using the carrier-tosidebandratio that uses amplitude modulation because millimeter sourcesare either unstable or cannot be modulated. The two approaches to measurementsat millimeter frequencies are shown in Figs. 25 and 26. Figure 25shows the direct measurement using a waveguide delay line. This systemoffers improved sensitivity if adequate input power is available. The rf bridgeand delay-line portion of the system differs from Fig. 18 because pre- andTN-234

284 A. L. LANCE, W. D. SEAL, AND F. LABAARSOURCE(CALIS.)PHASESHIFTER9872APLOTTEROSCILLOSCOPED9845ACALCULATORDoSPECTRUMANALYZERFIG. 25 Millimeter-wave phase noise measurements using a waveguide delay line. (FromSeal and Lance. 198 I.)post-bridge amplifiers with appropriate gain are not available, so the sensitivitycan equal the amount of carrier suppression.Figure 26 shows the use of a harmonic mixer to downconvert to theconvenient lower frequency where post·bridge amplifiers are available.The relatively low sensitivity to frequency drift that is characteristic ofdelaylinediscriminators becomes an advantage here. A separate calibrationgenerator is required, as shown in Fig. 25, and a power meter is used to assureproper power levels during the calibration process.2. SPECTRAL DENSITY OF AMPLITUDE FLUCTUATIONSAM noise measurements require equal electrical length in the two channelsthat supply the signals to the mixer. The delay line must be replaced with thenecessary length of transmission line to establish the equal·length conditionwhen the systems shown in Figs. 25 and 26 are used. The AM noise measurementsystem is calibrated and the noise measurements are performed directlyin units of power for a direct measurement of m(f) in dBc/Hz. m(f) is thespectral density of one modulation sideband divided by the total signalpower at a Fourier frequency difference f from the signal's average frequencyVo' The system calibration establishes the detection characteristics in termsof total power output at the IF port of the mixer (detector).IN-235

_~ ~A _7. PHASE NOISE AND AM NOISE MEASUREMENTS 285• 20 dBmOdBmIIII~IIIIIPHASESHIFTERDELAYLINELOW-NOISEPREAMP. TO AUTOMATICPHASE/AM NOISEMEASUREMENTSYSTEMFIG. 26 A millimeter-wave hybrid phase noise measurement system that produces IFfrequency and uses a delay-line discriminator at IF frequency. (From Seal and Lance, 1981.)The AM noise measurements are performed according to the following.(1) A known AM modulation (carrier-sideband ratio) must be establishedto calibrate this detector in terms of total power output at the IF port.The modulation must be low enough so that the sidebands are at least20dB below the carrier. This is to keep the total added power due to themodulation small enough to cause an insignificant change in the detectorcharacteristics.(2) The rf power levels are adjusted for levels of approximately 10 dBmat the reference port and 0 dBm at the test port of the mixer.IN-236

286 A. L LANCE, W. D. SEAL, AND F. LABAAR(3) Approximately 40 dB is set in the precision IF attenuator. The systemis adjusted for an out-of-phase quadrature condition.(4) The modulation frequency and power level are measured by theautomatic baseband spectrum analyzer. The total carrier-power referencelevel is measured power, plus the carrier-sideband modulation ratio, plusthe IF attenuator setting.(5) The AM modulation is removed. the IF attenuator set to 0 dB, andthe system re-checked to verify the out-of-phase quadrature (maximum dcoutput from the mixer IF port). Noise (Vn) is measured at the selected Fourierfrequencies. A direct calculation ofm(f) in dBc/Hz ismU) = [(modulation power (dBm) + carrier-sideband ratio (dB)+ IF attenuation (dB) - noise power (dBm) + 2.5 dB- 10 10g(BW)]. (102)Figure 27 illustrates the measurements of AM and phase noise of twoGUNN oscillators that were offset in frequency by I GHz. The measurementswere performed using the coaxial delay-line system.FOURIER FREQUENCY (Hz)FIG.27 Phase noise and AM noise of 40- and 41-GHz Gunn oscillators. (From Sealand Lance. 1981.)TN-237

7. PHASE NOISE AND AM NOISE MEASUREMENTS 287V. ConclusionThe fundamentals and techniques for measurement of phase noise havebeen set forth for two basic systems. The two-oscillator technique providesthe capability for measuring high-performance cw sources. The systemsensitivity is superior to the single-oscillator technique for measuring phasenoise very close to the carrier.High-stability sources such as those used in frequency standards applicationscan be measured without using phase-locked loops. However, mostmicrowave sources exhibit frequency instability that requires phase-lockedloops to maintain the necessary quadrature conditions. The characteristicsof the phase-locked loops must be evaluated to obtain the source phasenoise characteristics. Also. in principle, one must have three sources at thesame frequency to characterize a given source. If three sources are not available,one must assume that either one source is superior in performance orthat they have equal phase noise contributions.The single-oscillator technique employing the delay line as an FM discriminatorhas adequate sensitivity for measuring most microwave sources.The economic advantages of using this system include the fact that onlyone source is required, phase-locked loops are not required, system configurationis relatively inexpensive, and the system is inherently insensitive tooscillator frequency drift.The single-oscillator technique using the delay-line discriminator canbe adapted to measure the phase noise of pulsed sources. Pulsed sourceshave been measured at 94 GHz by F. Labaar at TRW. Redondo Beach,California.ACKNOWLEDGMENTSOur inItial preparations for developing a phase noise measurement capability were the resultof discussions WIth Dr. 10rg Raue of TRW. Our first phase noise development effort wasassisting in the evaluauon of phase noise measurement systems designed and developed byBill Hook of TRW (Hook, 1973). The efforts of Don Leavill ofTRW were vital in initiating themeasurement program.We are very grateful to Dr. Donald Halford of the National Bureau ofStandards in Boulder,Colorado, for his interest, consuhauons, and valuable suggestions during the development ofthe phase noise measurement systems at TRW.We appreciate the measurement cross-checks perfonned by C. Reynolds. 1. Oliverio. andH. Cole of the Hewlett-Packard Company. Dr. 1. Robert Ashley of the University of Colorado.ColoradO Springs, and G. Rast of the U.S. Army Missile Command, Redstone Arsenal,Huntsville, Alabama.REFERENCESAllen. D. W. ([966). Proc.IEEE54. 221-230.Ashley. 1. R.. Searles, C. B.. and Palka. F. M. (1968). IEEE Trans. Microwave Theory Tech.MlT-I6(9),753-760.TN-238

288 A. L. LANCE, W. D. SEAL, AND F. LABAARAshley. J. R.. Barley, T. A., and Rast, G. J. (1977). IEEE Trans. Microwat,e Theory Tech.MTI·25(4),294-318.Baghdady, E. 1.. Lincoln, R. N., and Nelin. B. D. (1965). Proc. IEEE 53,704-722.Barnes. J. A. (1969). "Tables of Bias Functions, B, and B" for Variances Based on FiniteSamples of Processes with Power Law Spectral Densities," National Bureau of StandardsTechnical Note 375.Barnes, J. A., Chie, A. R., and Cutler, L. S. (1970). "Characterization of Frequency Stability,"National Bureau of Standards Technical Note 394; also published in IEEE Trans. Instrum.Meas.IM-20(2). 105-120 (1971).Brandenberger, H., Hadorn, F., Halford, D., and Shoaf, J. H. (1971). Proc. 25th Ann. Symp.Freq. Control, Fort Monmouth, New Jersey, pp. 226-230.Culter. L. S., and Searle, C. L. (/966). Proc.IEEE54, 136-154.Fisher. M. C. (1978). "Frequency Domain Measurement Systems," Paper presented at the10th Annual Precise Time and Time Interval Applications and Planning Meeting, GoddardSpace Flight Center, Greenbelt, Maryland. *Halford. D. (1968). Proc. IEEE 5&)),251-258.Halford, D. (1975). "The Delay Line Discriminator." National Bureau of Standards TechnicalNote 10, pp. 19-38.Halford. D., Wainwright, A. E.. and Barnes, J. A. (\968). Proc. 22nd Ann. Symp. Freq. Control.Fort Monmouth. New Jersey, 340-341.Halford. D., Shoaf, J. H., and Risley. A. S. (1973). Proc. 27th Ann. Symp. Freq. Control, CherryHill. New Jersey, pp. 421-430.Hook. W. R. (1973). "Phase Noise Measurement Techniques for Sources Having ExtremelyLow Phase Noise." TRW Defense and Space Systems Group Internal Document, RedondoBeach, California.Hewlett·Packard Company (1965).., Frequency and Time Standards." Application Note 52.Hewlett·Packard Co.. Palo Alto, California.Hewlett-Packard Company (1970). "Computing Counter Applications Library," ApplicationNOles 7, 22. 27, and 29. Hewlett·Packard Co.. Palo Alto, California.Hewlett·Packard Company (1976). "Understanding and Measuring Phase Noise in the Fre­quency Domain." Application Note 207. Hewlett·Packard Co., Loveland, Colorado.Labaar. F. (1982)... New Discrimi nator Boosts Phase Noise Testing." Microwat'es 21(3), 65-69.Lance, A. L. (1964). "Introduction to Microwave Theory and Measurements." McGraw·Hill,New York.Lance. A. L.. Seal, W. D., Mendoza, F. G .. and Hudson, N. W. (\977a). Microwat'e.l. 20(6).87-lOlLance, A. L.. Seal. W. D., Mendoza, F. G.. and Hudson. N. W. (I 977b). Proc. 31st Annu. S.vmp.Freq. Control. Atlantic City. New Jersey, pp. 463-48lLance, A. L.. Seal. W. D., Halford, D.. Hudson, N.. and Mendoza. F. (1978). "Phase NoiseMeasurements UsingCross·Spectrum Analysis." Paper presented at the IEEE Conference onElectromagnetic Measurements. Ottawa, Canada.Lance. A. L.. Seal. W. D., and Labaar, F. (l982).ISA Trans. 21(4), 37-44.Meyer. D. G. (1970). IEEE Trans. IM-19(4), 215-227.Ondria. J (1968). IEEE Trans. Microwave Theory Tech. MTI-I6(9), 767-781.Ondria. J. G. (\980). IEEE·MTT·S Int. Microwat'e Symp. Dig .. pp. 24-25.Payne, J. B., III (1976). Microwat'e System News 6(2),118-128.Scherer, D. (1979)... Design PrinCIples and Measurement Low Phase Noise RF and MicrowaveSources." Hewlett·Packard Co.. Palo Alto. California.Seal. W. D.. and Lance. A. L. (1981) MicrOlI'Gt'e Sysrem News 11(7). 54-61.Shoaf. J H.. Halford, D.. and Risley. A. S (1973)... Frequency Stability Specifications andMeasurement." National Bureau of Standards Technical Note 632.• See Appendix Note # 33TN-239

7. PHASE NOISE AND AM NOISE MEASUREMENTS289Tykulsky, A. (1966). Proc. IEEE 54(2), 306.Van Duzer, V. (1965). Proc. IEEE-NASA Symp. Definition Meas. Short-Term Freq. Stability.NASA SP·80. 269-272.Walls. F. L., and Stein, S. R. (1977). Proc. 31st Ann. S.vmp. Freq. Control, Atlantic City, NewJersey, pp. 335-343.Walls. F. L., Stein. S. R., Gray, J. E.. Glaze. D. J.• and Allen. D. W. (1976). Proc. 30th Ann.Symp. Freq. Co.ntrol, Atlantic City. New Jersey, p. 269.TN-240

36th Annual Frequency Control Symposium· 1982SummaryPERFORMANCE OF AN AUTOMATED HIGH ACCURACYPHASE MEASUREMENT SYSTEMS. Stein, D. Glaze, J. Levine, J. Gray. D. Hilliard, D. HoweTime and Frequency DivisionNational Bureau of StandardsBoulder, Colorado 80303A fully automated measurement system hasbeen developed that combines many propertiespreviously realized with separate techniques.This system is an extension of the dual mixertime difference technique, and maintains itsimportant features: zero dead time, absolutephase difference measurement, very high precision,the ability to measure oscillators ofequal frequency and the ability to make me1surementsat the time of the operator's choice. Forone set of des i gn parameters, the theoreti ca1resolution is 0.2 ps, the measurement noise is 2ps rms and measurements may be made within 0.1 sof any selected time. The dual mixer techniquehas been extended by adding scalers which removethe cycle ambiguity experienced in previousreal izations. In this respect, the systemfunctions like a divider plus clock, storing theepoch of each device under test in hardware.The automation is based on the ANSI/IEEE­583 (CAMAC) interface standard. 2 Each measurementchannel consists of a mixer, zero-crossingdetector, scaler and time interval counter.Four channels fit in a double width CAMAC modulewhich in turn is installed in a standard CAMACcrate. Controllers are available to interfacewith a wide variety of computers as well as anyIEEE-488 compatible device. Two systems havebeen in operation for several months. Oneoperates 24 hours a day, taki ng data from 15clocks for the NBS time scale, and the other isused for short duration laboratory experiments.Review of the Dual MixerTime Difference TechnigueIt is advantageous to measure time di rectlyrather than time fluctuations, frequency orfrequency fl uctuationns. These measurementsconstitute a hierarchy in which the subsequentlylisted quantities may always be calculated fromthe previous ones. However, the reverse is nottrue when there are gaps in the measurements.In the past, frequency was usually not derivedfrom time measurements for short sample timesbecause time interval measurements could not beperformed with adequate precision. The dualandL. ErbER8TEC Engineering Inc.Boulder, Coloradomixer technique, illustrated in Figure I, madeit possible to realize the precision of the beatfrequency technique in time interval measurements.The signals from two oscillators (clocks) areapplied to two ports of a pair of double balancedmixers. Another signal synthesized from oneof the oscillators is applied to the remainingtwo ports of the mixer pair. The input signalsmay be represented in the usual fashionVI(t) = V 10sin [2nu 10 t + ~l(t)],V 2(t) = V 20sin [2nu 20t + ~2(t)]andVs(t) =V cos [2nu t + ~s(t)]so sowhere u = u (l-l/R) and R is a constantusually eRlled th~ heterodyne factor.areThe low passed outputs of the two mixersV BI= V BIOsin [~l(t) -~s(t)] andV B2=V S20sin [~2(t)-~s(t)] where$(t) = 2nu t + ~(t).oThe time interval counter starts at time t M whenV Rlcrosses zero in the positive directitJ"n andstops at time tN' the time of the very nextpositive zero croS'sing of V B2. Thus$l(t M) - ~s(tM) = 2Mrr and$2(t N) - $s(t N) = 2Nn whereNand Mare integers.SUbtracting the two equations in order to com­pare the phases of osc; 11 ators 1 and 2, oneobtains$2(t N)-$1(t M) =$s{t N )-$s(t M )+2(N-M)n.The phase of an asci 11 ator at time t Nmaybe written in terms of its phase at t Mand it~314TN-241

average frequency over the interval t M< tN'$(t N ) =$(t M) + 2n[v(t M;t N)](t N-t M) andwhen we apply this equation to both tjlZ and tjlSwe find~2(tM)-tjll(tM)= 2(N-M)n-2n[vS2(tM;tN)](tN-~)where vsz = vZ-v ' sSince M and N are not measureab1e with theequipment in Figure 1, the dual mixer techniquehas heretofore only been used to measure thephase difference between two oscillators moduloZn. We denote the period of the time intervalcounter time base by L and the number of countsrecorded in a measure~ent by P. Then the phasedifference between the two oscillators is givenby[~2(tM)-~1(tM)]mod2n = -Zn[vS2(tM;tN)]LcPFigure 2 illustrates the output of themeasurement system over a period of time. If ameasurement begins and ends without the timeinterval counter making a transition betweenzero and its maximum value, e.g., t < t < t

(5) All oscillators in the range of 5 MHz ± 5H~ may be compared. Other carrier frequen­Cles such as 1 MHz, 5.115 MHz, 10 MHz and10.23 101Hz are also usable. However, differentcarrier frequencies may not be mixedon the same system. The sys tem has beensuccessfully tested with an oscillator offset4.6 Hz from nominal 5MHz. Measurementswere made at intervals of 2 hours betweenwhich the sys~m had to accumulate approximately2 x 10 n. The system has also b~§ntested with an oscillator offset 4 x 10 ,and no errors were detected during a periodof 40 days.(6) All sampling times in the range of 1 secondto 16 days with a resolution of 0.1 secondare possible. Measurements may be made oncommand or in a preprogrammed sequence.(7) Measurements are synchronized precisely,i.e. at the picosecond level, with thereference clock. They may therefore besynchronized with important user systemevents, such as the switching times of aFSK or PSK system.(B)All oscillators are compared synchronouslyand all measurements are performed within amaximum interval of 0.1 second. As a result,the phase of any oscillator needs tobe i nterpo1ated to the chosen measurementtime for an interval of 0.1 second maximum.This capability, which is not present ineither si ngl e heterodyne measurement systemsor switched measurement systemseliminates a source of "measurement" errorwhich is generally much larger than thenoise induced errors. For example, interpe1ationof the PD!!e Rf a high performanceCs clock (0 - 10 II~) over a period of 3hours would' produce approximately 1.5 nsphase uncertainty. To maintain 4 ps accuracyrequ i res measurements simultaneous toO.ls.(9) There are no phase errors due to the switchingof rf signals since there is noswitching anywhere in the analog measurementsystem.(10) No appreciable phase errors are introducedwhen it is necessary to change the referenceclock since, as shown in Figure 6, thepeak error due to changes in synthesizerphase is 20 ps.(11) The measurement system is capable of measuringits own phase noise when the samesignal is app1 ied to two input ports.Figure 7 shews the phase deviations betweentwo such channels over a period of 75,000seconds and Fi gure B is the correspondi ngAllan variance plot. ~igure 9 shows thephase deviations between 2 input channelsover a period of 40 days.(12) Since the IEEE-533 (CAMAC) interface standardhas been followed for all tne customhardware, the system may be easily interfacedto almost any instrument controller.NBS has already tes ted the sys tern us i ng alarge minicomputer, a small minicomputerand a desk top calculator. Interfacesbetween IEEE-583 and IEEE-48B controllersare available and have been used successfully.(13) The system is capable of camparing a verylarge number of oscillators at a reasonablecost per device.There are also disadvantages to this measurementsystem. The most important are:(l) The camp 1exity of the hardware is greaterthan for some systems. It is possiblethat this will reduce reliability.(2) A high level of redundancy is difficult toachieve. The system design stresses size,power, convenience and cost, resulting inan increase in the number of possiblesingle point failure mechanisms compared tosome other techniques. For example, aCAMAC power supply failure will result in aloss of data for all devices being measured.(3) A SUbstantial committment is required inboth specialized hardware and software.(4) If an oscillator under test experiences aphase jump which exceeds 1 cycle, themeasurement system records a jump withincorrect absolute magnitude. As a reSUlt,it may not be applicable to signals whichare frequency modulated with discontinuousphase steps larger than 2n.ConclusionsWe have demonstrated a new phase measurementsystem with very desirable properties: Alloscll 1ators in the range of 5MHI ± 5Hz may bemeasured directly. The sampling times are onlyrestricted by the requirement that they exceedone jiecond. The noise floor is a (2,t) = 3 x10- 1 It in short term and the tin?e deviationsare less than 100 ps. All circuitry is designedas modules which allows expansion at modestcost. Compatibility with a variety of computersis insured through the use of the lEEE-5B3interface and adapters are available to permituse with an IEEE-4BB controller. The systemmakes it feasible to make completely automatedphase measurements at predetermined times onlarge numbers of atomic clocks. It's own noiseis one-hundred times less than the state-of-theartin clock performance. It will be used inthe near future to make all measurement neededto compute NBS atomic time, but it will also bevery valuable for any laboratory which usesthree or more atomic clocks.References1. O. W. Allan, "The measurement of frequency316

and frequency stability of precisionoscillators ," NBS Tech. Note 669(1975)."CAMAC instrumentation and interface standards,"Institute of Electrical and ElectronicEngineers, Inc.• 345 E. 47th St.New York. NY 10017.O. J. Glaze and S. R. Stein, "Picosecondtime difference measurements uti 1i zi ng CA­MAC based ANSI/IEEE -488 data acquisitionhardware", NBS Tech Note 1056 (in preparation).ZERO STOP ,..----,,,,,'--.... CROSSINGTIMEDETECTOR START INTERVALCOUNTERpZEROCXl--1 CROSSINGDETECTORFigure 1.Dual Mixer Time Difference Measurement SystemFigure 2.Dual Mixer DataL------1N ISTOPIZEROCROSSINGDETECTORZEROCROSSINGDETECTORV BI = VI/RSTARTL--.---...{~M. IIIISTOPSTARTPIFigure 3.Extended Dual Mixer Time Difference MeasurementSystem317 ~-244

•• • ••• -z,,(;;BZ(ll; IN)Jrl6. Z(N-H)n--- SumFigure 4.Extended Dual Mixer DataClock ~ 1Offset5MHzStartStart• Output•Clock #4PULSE DISTRIBUTIONMODULE••Start 10 MHz TIme BaseInputMEASUREMENT MODULEIStop Inputs......~--------....;::.-',6 X 4 =24 clocks maximumrequires one pulse distribution module andsix measurement modules maltlmumFigure 5. System Block Diagram1000800600X(PS) 400200 • • • • 41•••••INPUT AMPLITUDEOdB I':: 1Vrms'Or10X(PS)O~-10-20SYNTHESIZER PHASEFigure 6.Measured Time Difference vs. Input Amplitude andSynthesizer Phase318TN-245

PHASE P~OT Clook No. ~- e I"d,.... -I"'l~.,,"~.~ S~r Slopo 0' -1.g~e2Ceo-17/& R.~oy.d EtW e~ '"0)' 82-1..-11 1I IT- 7~7me SECONOSFigure 7.Raw Phase Data for Two Channels Driven from theSame Source-12iV. 1_ Clook No. 4- II-15"'1°9 T-17Figure 8.Noise Floor of Measurement System3191N-246

D. J4S,__, __' T ,- -.P$(C 1let .665.39.-9. e5 'H1lI"1J1,I!LI-5.41\-I e .77-) 3 .4'S...... ...L--L ...L -L_~_~e .99 9.75 19.59 29.25DRY$CLQU~ I< 1- 2( } .)(OFo&. .fl.4Se"'7. 19-I1AR-e2Figure 9.Raw Phase Data for Two Channels Driven from theSame Source320IN-247

BL-\5ES .-\.\D \·A.Rl.-\.\CES OF SEVERAL FFT SPECTRAL ESTI~.LHORSAS A FC='iCTION OF :-WISE TYPE A~D ~t:~1BER OF SA.\IPLESF. 1. Walls. Time and Frequency Division.:-lational Institute of Standards and Technology, Boulder. CO 80303D. B. Percival, Applied Physics Laboratory,t.;'niversity of Washington, Seattle, WA 98106 From: Proceedings of theW. R. Irelan, Irelan Electronics. 43rd Annual Symposium on412 Janet, Tahlequah, OK 74464 Frequency Control, 1989.AbstractWe theoretically and experimentally investigate the biasesand the variances of Fast Fourier transform (FFT) spectral estimateswith different windows (data tapers) when used to analyzepower·law noise types JO, 1- 1 , 1- 3 and 1- 4 . There is awide body of literature for white noise but virtually no investigationof biases and variances of spectral estimates for power-lawnoise spectra commonly seen in oscillators. amplifiers, mixers,etc. Biases (errors) in some cases exceed 3{) dB. The experimentaltechniques introduced here permit one to lUlalyze theperformance of virtually any window for any power-law noise.This makes it possible to determine the level of a particularnoise type to a specified statistical accuracy for a particularwindow.I. IntroductionFast Fourier transform (FFT) spectrum analyzers are verycommonly used to estimate the spectral density of noise. Theseinstruments often have several different windows (data tapers)available for analyzing different types of spectra. For example,in some applications spectral resolution is important; in others,the precise amplitude of a widely resolved line is important; andin still other applications, noise analysis is important. Thesediverse applications require different types of windows.We theoretically and experimentally investigate the biasesand variances of FIT spectral estimates with different windowswhen used to analyze a number of common power-law noisetypes. There is a wide body of literature for white noise but virtuallyno investigation of these effects for the types of power-lawnoise spectra commonly seen in oscillators, amplifiers, mixers,etc. Speci£cally, we pTe3ent theoretical results for the biasesassociated with two common windows - the uniform and HIUlningwindows - when applied to power-law spectra varyingas 1 0 , 1- 1 and 1- 4 . We then introduce experimental techniquesfor accurately determining the biases of lUly window anduse them to evaluate the biases of three different windows forpower-law spectra varying as r, 1- 1 , 1- 3 IUld 1- 4 • As an examplewe £nd with I-~ noise thAt the uniform window CIUl haveerrors ranging from a few dB to over 30 dB, depending on thelength of span of the f-t noise.We have also theoreticalJy investigated the varilUlces ofFFT spectral estimates with the uniform IUld Hanning windows(confidence of the estimates) as a function of the power-lawnoise type and as a function of the amount c:i data. We introduceexperimental techniques thAt lIlAke it relatively easy toindependently determine the variance of the spectral estimatefor virtually lUly window on any FFT spectrum analyzer. Thevariance that is realized on a particular instrument depends notonly on the window but on the specific implementation in bothhardware and software. We find that the variance c:i the spectraldensity estimates for white Doise, r, is very similar forthree specific windows available on one instrument and almostidentical to that obtained by standard statistical analysis. Thevariances for spectral density estimates of 1- 4 noise are only4% higher than that of 1° noise for two of the windows studied.The third window - the uniform window - does Dot vieldusable results for either 1- 3 or 1- 4 noise. ­Based on this work it is now possible to determine theminimum number of samples necessary to determine the levelof a particular noise type to a specified statistical accuracy as afunction of the window. To our knowledge this was previouslypossible only for white noise -although the traditional resultsare generally valid for noise that varied as 1- 8 , where a wasequal to or lesll than 4.II.Spectrum Analyzer BuiesThe spectrum analyzer which W!lS U!ed in the experimentalwork reported here is fairly typical of a number c:i such instrumentscurrently aV1l.ilable from various manufacturers. Thebasic measurement process generally consists of taking a stringof N. =1024 digital samples of the input wave form, which werepresent here by XI, X 1 , ..•, XN•. The basic measurementperiod was 4 IDS. This yields a sampling time t:.t = 3.90625 iJS.Associated with the FFT of a time series with S. data points.there are usually (N./2) + 1 = 513 frequencies)I, = N.ut' j =0, 1, ... , .'11./2.The fundamental frequency II is 250 Hz, and the Nyquist frequencyI N.n is 128 kHz. Since the spectrum analyzer usesan anti-aliasing filter which significantly distorts the high frequencyportion of the spectrum, the instrument only displaysthe measured spectrum for the lowest 400 nonzero frequencies,namely, II = 250 Hz, h =500 Hz, . ", 1400 = 100 kHz.The exact details of how the spectrum analyzer estimatesthe spectrum for X I, ..., X N. are unfortunately not providedin the documentation !Upplied by the manufacturer, so the followingmust be regarded only as a reasonable guess on our partas to its operation (see [1] for a good discussion on the basicideas behind a spectrum lUlalyzer; two good general referencesfor spectral lUlalysis are [2] and [4]). The sample mean,_ 1 N.Xa N LXr,• talis subtracted from each ci the samples, and each of these "demeaned"~ples is multipled by a window h, (sometimes calleda data taper) to produceThe spectral estimate,(.) -)X, = h, (X, - X .TN-248j = O. 1. .....V. /2.

is then computed using a.n FFT algorithm.The subscript "1" on 51 (/,) indicates that this is the spectralestimate formed from the first block of .v, samples. Asimilar spectral estimate 52(/ J) is then formed from the secondblock of contiguous data X N. +1. X N.+2 • ...• X 2N•. In all. thereare !Vb different spectral estimates from N. contiguous blocks.and the spectrum o.nalyzer averages thege together to formIt is the statistical properties of 5UJ) with which we are concernedin this paper.Cnfortunately some import.ant aspects of the windows arenot provided in the documentation for the instrument. One importantdetail is the manner in which the window is normalized.There are two common normalizations:andThe first of these is common in engineering applications becauseit ensures that the power in the windowed !WIlples X?) is thesame as in the original demeaned samples: the second is equivalentto the first in expectation and is computationaly moreconvenient. but it can result in small discrepancies in powerlevels. Either normalization affects only the level of the spectralestimate and not its shape.There are three windows built into the spectrum analyzerused here. The first is the uniform (rectangular. default) windowh~V) = l/JN•. The second is the Hanning data window.for which there are several slightly different definitions in theliterature. In lieu of !I~ific details, we assume the followingsymmetric definition:h < t < N.,t - cos N. ' - - 2(H) _ C(H) (1 _ 2l1'(t - 0.5)) 1(H) N.= h N.-t+l' '2 + 1:$ t :$ N,;here C(HJ is a constant which forces the normal.i.zation in Equation(2). The third window is a proprietary "fiattened peak"window, about which little specific information is available (itis evidently designed to e.ccurately measure the heights of peaksin a spectrum).III.III.A.Expected Value and Biu of Spectral E.timatelTheoretical AnalysisWe need to assume a noise modd {or the Xt's in orderto detenmne the statistical properties of 5( Ii) in Equation (1).We consider three different models, each of which is representedin terms of a Gaussian white noise process f, with mean zeroand variance 17;. The second-order properties c1l!!ach modela.re given by a spectral density function 5(·) de.lini!!d over theinterval [-1/(2At), 1/(2At)J in cycles/At. The llrst modl!!l is adiscrete parameter. white noise process (r noise):X. = E, IWd S(f) = u; At.(1)(2)The second model is a disc~ete.pa.ra.mete~. rancom·",.a.,;:.: ?roc"SS(nominally 1- 2 Doise):The third model is a discrete-parll.IIleter. r8.Ildom-ru.n process(nominally f-· noise):8.Ild S(f) = cr? 6.t .16sin l (1rf6.t)Continuous parameter versioIis of these three models have beenused extensively in the literature as models for noise commonlyseen in oscillators.For each of the three models we have derived expressionsfor E {SU)}, the expected value of 5(1). These expresslOnsdepend on the window hI, the number of samples S, in eachblock and - in the ea.se of a random-run process - the numberof blocks N b • The deta.ils behind these calculations will berl!!ported el:!i!!where [31; here we merely summarize our conclusionsfor the three models in combination with the uniform 8.IldHanning windows and N, =1024.First, for a white noise process,j = 1.2, ... ,512,when the uniform window is used. For the Hanning window.the above equality also holds to a very good approximation for2 S j :$ 511 and to within 0.8 dB for j = 1 and 512 (the latteris of no practical importance since the highest frequency indexgiven by thl!! spectrum analyzer is j = 400). These theoreticalcalculations agree with our experimental data except at 11 (seeTable 1).Second, for e. random-walk process,j = 1,2.... ,512,when the uniform window is used, i.e .. the expected value isttulce what it should be at all frequencies. This theoretical resulthas been verified by Monte Carlo simulations. but it doesnot agree with our experimental data. which shows no significantlevel shift in the estimated tpeCtrum. The source of thisdillCfepancy is currently under investigation, but it may be dueto either (a) facton in the experimental data wh.ieb effectivelymake it band-limited, random-walk noise, i.e.. its spectral shapeis marl.:ed1y different from r 2 {or, say. 0 < I < II or (b) anincorrect guess 00 our part as to how the spectral estimate isnormalized by the spectrum analyzer. For the Hanning window,we found that1.085UJ) j =1;l.48S(f.i) j = 2:E{S(!J)} = 1.l5S(/i)) = 3;l.07S(/i) ) =4;{ l.04S(jJ) j =5;SU,) 6 :$ j :$ 511 to within 3%.i.e., SUi) is essentially an unbiased. spectral estimate exceptfor the lowest few frequencies. This theoretical result has beenverified by Monte Carlo simulations and also agrees in generalwith our experimental data.Third, for a random-run process,1 $ j $ 400.TN-249

to a g

-59d8

while the Hanning ""!Odo"" yIelds1.305 2 (f) )/.V b, 1==1:2.205 2 (/) )IN" j == 2:1.315 2 (/j )IN" j =: 3;1.155 2 (/) )/N" ] =: 4;1.095 2 (/))/N" J =: 5:1.065 2 (!i )/N" j =: 6;1.045 2 (/j )iN" j =: 7;5'2U))IN" 8 .$ j :s: 511 to within 3%.Except for the few lowest frequenci~. the results for the Hanningwindow agree with our experimental results and with standardstatistical theory; however. the factor of five in the variancefor the uniform window disagrees with our experiments and withstandard theory (although it has been verified by Monte Carlotechniques). The cause of this di!lcrepancy is under investigation,but we think it is due to the band-limited nature of theexperimental data.Third, for a random-IUD process, the variance computationsBJ'E not useful since the variance is domillated by the factthat the expected value of the sample variance for each block ofsampl~ increases with time. The agreement which we found betweenstandard statistical theory and our experimental resultson the liN. rate of decrease of variance is undoubtedly dueto the band-limited nature of the experimental data. We will. a.ttempt to verify these conciusions in the future using MonteCarlo techniques.-7]de(V)'0"e/OIVA.IJlIATHexample we have cho~n to take .vb = 1000 blocks of :be va,.,;ouspower-law noise types examined m IlLS above and compare thevalue of the spectral estimate with that obtained from .\"6 = 32blocks (see Figure 4). Since the variance of the 1000 block datais about 32 times smaller than that of the 32 block data. it canserve as an accurate estimate of the "true value. ~ Let 5 1000 (f) )represent this quantity at the loth cha.n.nel (frequency). By subtractingthe 1000 block data from the 32 block data at the l-thcha.n.nel, we then have one estimate of the error for the 32 blockdata; by repeating this procedure over N c different cha.n.nelsand N r clifi'erent repliCAtions, we can obtain accurate estimatesof the variance for the 32 block data. Let 5 32 i (f) ) represent thespectral estimate for the 32 block data at the j-th cha.n.nel andthe i-th repliCAtion. To compensate for the variation in the levelof the spectral estimates with cha.n.nel, it is aecessary to dividethe error at the j-th cha.n.ne1 by the "true value~ 5 1000 (/)), Themean square fractional error of the 32 block data for the noisetype under study is given byIt is assumed that all channels with biM - as indicated in Table1 - have been excluded in the summation CfIIer j. It is alsoimportant that the changes in the spectral density not exceedthe dynamic range of the digitizer because under this conditionthe quantization errors - in addition to causing biases in thespectral estimates as discussed earlier - can lead to situationswhere the variance does not improve as N 6 increases_ These Vll.!­ues can be sealed to any number of blocks N, if care is taken toavoid these quantization errors. Upper and lower approximate67% confidence limits for SUj) - the true spectral density atchannel j - using Hanning, uniform and the proprietary 'ilattenedpeak" windows for N 6 approximately independent blocksare given bys;~;~. L~=""H""Z--------------;-S"'TC=-., -1'""OO::::-:OOO=-:Ha-:-lFigure 4. Comparison of the spectral estimate of /-4 powerlawnoise with 1000 samples with that obtained with 32 samples.The text expwIUl how these two curves are used to obtainthe fractional RMS confidence of the spectral estimate for 32samples.IV.B.Experimental DeterminationThe following procedure can be used to experimentally determinethe VlUiance of the spectral estimates of virtually anytype of noise spectrum with any type of window for a particularinstrument. Since the spectral density of interest is in generalnonwhite, we must determine both the "true value" and a wayto nonnalize the fractional error of the estimate lila a functionof the number of SIlmples. This can be done by making IJlIelof the above th~retical ane.lysis that shows that the varianceshould decrease as the square root of the number al samplesIsince they are approximately statistically indepf!Ildf!Ilt (in fut,lexa.ctly 90 in the ca.ses of white and random-waJ.k noise). As anwhere S(Ii) is the spectral estimate given by Equation (1) andV(a,N,) is the fractionalll!lriance given in Table 2 for fQ andQ = 0, ·2, -3 and -4 (these results were obtained by averagingover NrNc = 1200 channels). The variances obtained are veryclose to thoee obtained from standard statistical analysis forwhite noise, i.e.,Table 2.power lawConfidence Interval! for ITT Spectral Estimateswindownoise type uniform Hanning Battened peakr 1.02/,fN; 0.98/,fN; O.9S/v'N;/-2 l.02/..;'N; 1.04/,fN; 1.04/.;:v;r 3 unusable 1.04/ ,J'lV; 1.04/y'N;/-4 unusable 1.041,J'lV; l.04/v'NbV. CondlUioDJIWe have introduced experimental techniques to evaluatethe stati"tieal properties of FFT spectral estimates for commonnoise types found in oecillators, ampli£.ers. mixers and similarTN-252

de\·~ces. and ...·e ba-;e co~?a.re,j :::'ese ".tZl :beoretlcai :aic-..:atlODS\\"e have used th"se tecb.nJques to stud:. the bllLSes iUldVallanCe! of FFT spectral estimates using the w:uform. Hanning.and ~ propneu.ry "il.attened peu" Q"ndow. Tbe theoreticalwalysis was gre~t1y hampe~ beeaU5e the instrumentmanufacturer does not diKlose tbe exact form CIf the "flattenedpeak" window or the norm&1.izAtion procedure for tbe other win·dows. :-.reverthele5.S. we obtLined fa.i.r agreement between thetheoretical and the experimental. analysis. The vv-iances ofthe spectral estimation were "rtuallv identical to & few percentfor r to f-' noise except for tbe uniform window whichis incape.ble of measuring noise wIDch falls off futer than /-1.There was a very large difference in the biases ci the first fewchannels for the t~ W1ndows. The Hanning wiDdow shO'llledsigruficant biases in the fint 3 channels while the proprietary":flAttened peu~ window showed large biues for /-4 noille evenup to channel 13. The Ha.n.n.ing W'indow therefore yield, usdulinformation over three times w,der frequency range tba..o theproprieta,r:' "'flattened pea.k" W'indow. In the particulAr instrumentstudied. the proprietary ":flattened peal" window is thebest choice for estimating the height of &. DlUTOW band source,while the Hanning wmdow is by !Ar the best choice for spectnJlUlaJYSIS of common DOlse types found in OlIcillaton. ampliiiers,mixers. etc. We have also shown that the 6i% conndence levelsfor spectral estimation a.s a function of the number of contiguousnonoverlappmg blocks .....~, is approxima.teiy given byfor wIDte noise (/0) and byS =':>m - ( 1±~ 104). \/.\for DoiSt' t~~ f- 2 to f-·. This a.g-ree5 to within 4% of tbatfound by standard statistical analyStS for ~IDte noise. r sm~this data one can now detet1IllDe the number of sa.mpies nec·e!SAJ:)' to estimate - to a ~lVen level of statistical uncrrtamtv- the spectrum of the v&rious nOIse types commooly found ~oscillators, a.mplifiers. mix101"-:171_1Kfl BAa

Proc. 35th Ann. Freq. Control Symposium, USAERADC0I1, Ft. ~Ionmouth, UJ 07703, May 1981A MODIFIED "ALLAN VARIANCE" WITH INCREASEDOSCILLATOR CHARACTERIZATION ABILITYDavid W.Allan and James A. BarnesTime and Frequency DivisionNational Bureau of StandardsBoulder, Colorado 80303Summarythe same, Le., - t-2. It is not at all uncommonHer~tofore, the "Allan Variance," cr 2(t), hasbecome the de facto standard for measuring oscillatorinstabil ity in the time-domain. Often osci 11 ators for t of the order of one second andfor white PM and flick.er PM to occur in precisionoscill ator frequency instabil iti es are resonabJ,ymodel able with a power law spectrum: Sshorter. The modified Allan variance, as developedin this paper, depends as t- 2 for or ::: +1 and as(f) ~ ,...,where y is the normal i zed frequency. y f is theFourier frequency, and or is a constant over somet-range of Fourier frequencies. It has been shown3 for or ::: +2. This yields a clear distinction inthat for power law spectrum cr 2(t) ~ t~, and that the time domain between these heretofore somewhat~ ::: -or-1 for -3

The "Allan Variance" 1s defined as:U (1) - a(a) X[12]'{~hl-a+l- It\ -a+l] (10)(5)where the brackets "< >" denote infinite timeaverage. Using equation (4), one may write:It has been shown that typically 0 2(1)Y 1 2varies as 1 1l , and that 1.1 = -a-1 for -3 ~ a ~ +1. 'Hence, we see one of the dimensions of usefulnessof 0y2(t); 1.e., ascertaini ng the dependence on tallows an estimate of a (the power law spectraltype of noise). However, if a ~ +1, then 1.1 =-2,and the 1 dependence becomes somewhat ambiguous asto the type of noise in this region.It is interestingto note that in the region a ~ +1, cry2(1)is bandwi dth (fh) dependent; 1. e., the bandwi dthof the measurement system will affect the value ofo (1), and furthermore, one may use the bandwidthy 3dependence to determine the value of a (see alsoAppendix Ref. Z).Development of the Modified Allan VarianceOne may also write Oy2(1) in terms of ageneralized autocovariance function:whereand wherethe classical autocovariance function of x(t).(7)(8)(9)Using the Fourier transforms of general ized functions,one may determine the coefficients relatingthe power spectral density to o/(t). Ref. 1gives these relationships.It is of interest tonote that U (1) has the following approximate formxin the region a ~ + 1 (see Appendi.: R.. f. 2):Hence, one notes that by changing the reciprocalbandwidth as well as 1, one affects o/(t) insimilar ways, depending on the value of (Y, Fromthis, one should be able to deduce the value of a,since the bandwith dependence becomes stronger fora moving positive from +1, and the 1 dependencebecomes stronger as a moves negative from +1.Onecan change the bandwidth in the hardware or in thesoftware. In the past, it has typically been donein the hardware. 3 James Snyder 4 has shown that itis relatively easy to change the bandwidth in thedata processing by a clever technique and we havefollowed his lead.a newIn particular, we have chosenvariance analysis scheme which coincideswith the Allan variance at the minimum sampletime, 1 , (i.e., minimum data spacing), but whichochanges the bandwidth in the software as thesample time, 1, is changed.Each reading of the time deviation, xi' hasassociated with it an intrinsic nominal (hardware)measurement system bandwi dth, f h' Defi net = _1 . and similarly we may define a softwareh 2nf'bandwidt~, f = fhln, which is lin times narrowersthan the hardware bandwidth. This software bandwidth can be real i zed by averagi ng n adjacentx.'s·, 's1 = n1 where 1 =l/f. We have definedh' s sa modified Allan variance which allows the reciprocalsoftware bandwidth to change linearly with theJsamp 1e time, 1:where 1 = nl 'oG for n = 1.have formed2x . + x.) ] ~ (11)1+n 1 JEq. 11 clearly coincides with Eq.One can see that, in general, wea second difference of three timereadi ngs with each of the three be i ng an average,of n of the x.'s (with non-overlapping averages).As n increases, the (software) bandwidth decreasesand this bandwidth varies just as f s: fh/n.For a fi ni te data set of N I readi ngs of xi(i ~ 1 to N), we may write an estimate:471IN-255

N-3n+lEj=1Mod Oy2 (t) =0 2(t) {.! + .,.:-=1=-- .,,.Y n n2 4n-a+1 _ (2n)-a+1(12) n-1L (n-i)' [_6(a+l + 4(n+i)-«+1 (15)i=1-(2n+i)-a+l + 4(n-i)-a+l - (2n-i)-a+~}Eq. 12 is easy to program, but takes more time toSince we know that a 2(t) is well behaved in thiscompute than for ay

The UxCt) function for flicker noise PH isextremely complicated and has not been developed,but one can arrive at an empirical value for it.The Ux(t) function is derivable for the otherpower law spectral processes. Table 2 gives therelationships between the time domain measureHod Gy2 (t) and its power law spectral counterpart,given in Eq. 1. Also listed in the right handcolumn of Table 2 are the asymptotic valuesof R(n):Ho1n Typ.M....TABLE 2at")Mod a/(t) C~nt n' •3 f hWhIte PPI +2 hacth2'~1.038 -+ 31n(w t)Flic:t.r PM +1hE~iricalhI'(20»"­R(n)""'ft. FN 0 hO'~ fJcac:t 0.5Fl1c:ker FM -1 h. Z1n(2) • R(n) [_pci r1 CI.'; Exa.c:t o. G74IA"atlablea.ndo. Val k FH -2 h_ ' ( 2n tt • R(n) £~it'icAl; Exact 0.824ZAllailabl.It is clear from Table 2 that Mod 0 y2(t) isvery useful for white PM and flicker noise PH, butfor a < +1 the conventional Allan variance, Oy2(t),gives both an easier-to-interpret and an easier-to·calculate measure of stability.It is interesting to make a graph of a versusIJ for both the ordinary Allan variance and themodified Allan variance. Shown in Fig. 2.is sucha graph. This graph allows one to determine powerlaw spectra for non-interger as well as intergervalues of C1. The dashed 1ine for the modifiedAllan variance has been intentionally moved to the1eft in Fig. 2 because for small va 1ues of n thevalue of ~ will appear to be slightly more negativethat for O/(t), even though for large n, theyboth approach the same slope (i.e., the samevalues of IJ). In fact, in the asymtotic 1imit,the equation relating IJ and C1 for the modifiedAllan variance isa = -IJ -I, for -3 < a < +3 . (17)'" See Appendix Note # 34*The value of IJ = -4 for a = +3 was verified empiricallywith simulated data, and it appears thatfor a > +3, IJ remains at -4.A direct application for using the modifiedAllan variance recently arose in the analysis ofatomic clock data as received from a GlobalPositioning System (GPS) sate11 ite. We wereinterested in knowing the short-term characteristicsof the newly developed, high-accuracy NBS/GPSreceiver, as well as the propagation fluctuations.Fig. 3 shows both 0 2(t) and Mod ayy2(t) for comparison.Using Mod 0 2(1), ywe can tell that thefundamental limiting noise process involved ih thesystem is white noise PH with the exciting resultthat averaging for four minutes can allow one toascertain time difference to better than onenanosecond excluding other systematic effects.ConclusionWe have developed a supplemental measure, the"Modified Allan Variance" (Mod 0/(1»), which hasvery useful properties when analyzing oscillatoror signal stability in the presence of white noisephase modulation or flicker noise phase modulation.It also works reasonably well as a stabilitymeasure for other commonly occuring noise processesin precision oscillators.We would recommend that for most time domainanalysis, 0 2(1) should be the first choice. Ifya 2(1) depends on t as t- 1 , then the modi fi edyAllan variance can be used as a substitute to helpremove the ambiguity as to the noise processes.~cknow1edgmentsThe authors are deeply grateful for stimulatingdiscussions with Dr. James J. Snyder, andfor the useful comments of Dr. Robert Kamper andMr. David A. Howe.References1. James A. Barnes, Andrew R. Chi, Leonard S.Cutler, Daniel J. Healey, David B. Leeson,Tomas E. HcGunical, James A. MUllen, Jr.,Warren L. Smith,' Richard L. Sydnor, Robert F.C. Vessot, and Gernot M. R. Winkler, Proc.IEEE Trans. Instrum. Meas. 1M-20, 105 (1971);also pub1 ished as NBS Tech. Note 394 (1971).2. David W. Allan, Proc. IEEE 54, 221 (1966).3. R. F. C. Vessot, Proc. 1EEE'54, 199 (1966).4. James J. Snyder, Proc. 35th Annual Symp. onFreq. Control (1981), to be pub1 ished.473TN-257

(a)1SIMULATED NOISE(b)1-10- 2- --10- 2 ...."""b ~o -" o -3010 " 0 10:2 :210-4 ........ -'- ---1 10- 4L..- -'- --'1 10 10 2 1 10 10 2SAMPLE TIME, TSAMPLE TIME, T(c)1 1--(cO-....- -1 -""" -1 FlICKER FMt)'10 ~10o:2 10-2 :2 -2 L..-__---l --l'-------'------'10 . 21 10 10 2 1 10 10SAMPLE TIME, TSAMPLE TIME, T10j:- b'1o(e)oRANDOM WALK FMo:2 10-1L..­ -'­ --'110SAMPLE TIME, TFig la-e. Mod aiL) using Eq. 12 was calculated for different sample times forindependently generated and simulated noise processes. which werewhite phase noise. flicker phase noise. white frequency noise.fl i cker frequency noi se, and random walk frequency noi se, respecti ve ly. Mod a (L) was computed for 199 data poi nts in each case.yOne sees the excellent fit to the theory for white phase noise andflicker phase noise. an important new contribution in the ability tocharacterize oscillators having these noise processes.00474TN-258

332IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM·33, NO.4, DECEMBER 1984Copyright Cl 1984 IEEE. Reprinted, with permission, from IEEE Transactions onInstrumentation and Measurements, Vol. IM-33, No.4, pp. 332-336, December 1984.Characterization of Frequency Stability: Analysis ofthe Modified Allan Variance and Properties ofIts EstimatePAUL LESAGE A~D THEOPHANE AYIAbstrtlct-An analytical expression for the modified ABan variance isgiven for each component of the model usually considered to describethe frequency or phase fluctuations in frequency staJndards. The relationbetween the Allan variance and the modified Allan variance isspecified and compared with that of 5 previously published analysis.The uncertainty on the estimate of the modified Allan variance calculatedfrom a futite set of measurement data is discussed.Manuscript received December 8, 1983; revised February 17, 1984.The authors are with the Laboratoire de I'Horloge Atomique, Equipede Recherche du CNRS, associee al'Universite Paris-Sud, 91405 Orsay,France.TN-259

IEEE TRANSACTIONS ON INSTRUMENTATlON AND MEASUREMENT. VOL. 'IM·33. NO. 4. DECEMBER 1984333L INTRODUCTIONMany works [11-[ 5 J have been devoted to the characterizationof the frequency stability of ultrastable frequency sourcesand have shown that the frequency noise of a generator can beeasily characterized by means of the "two-sample variance"[2] of frequency fluctuations, which is also known as the"Allan variance" [2) in the special case where the dead timebetween samples is zero.An algorithm for frequency measurements has been developedby J. J. Snyder [6 J, [7]. It increases the resolution offrequency meters, in the presence of white phase noise. It hasbeen considered in detail by D. W. Allan and 1. A. Barnes [8].They have defined a function called the "modified Allanvariance" and they have analyzed its properties for the commonlyencountered components of phase or frequency fluctuations[3]. For that purpose, the authors of [81 have expressedthe modified Allan variance in terms of the autocorrelation ofthe phase fluctuations. For each noise component, they havecomputed the modified Allan variance and deduced an empiricalexpression for the ratio between the modified Allan varianceand the Allan variance.In this paper, we show that the analytical expression of thisratio can be obtained directly, even for the noise componentsfor which the autocorrelation of phase functions is not definedfrom the mathematical point of view. We give the theoreticalexpressions and compare them with those published in [8).The precision of the estimate of the modified Allan varianceis discussed and results related to white phase and white frequencynoises are presented.II. BACKGROUND AND DEFINITIONSIn the time domain, the characterization of frequency stabilityis currently achieved by means of the two-sample variance(2) (at(2, T, T» of fractional frequency fluctuations. It isdefined as(0;(2, T, T» = i «Ybi - J!I

.., .."w~~l.'U";) UN INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO.4, DECEMBER 1984TABLE IANALVTlCAL EXPRESSION FOR THE MODIFIED ALLAN VARIANCE WITHI?'ICoNDITION 21'j;rO»NOI SE TYPEIY.2Mod 0 (elyWelTE P M23 h f2 c8 n~ .,2fLlC~ERP M1hI [n-I 2~3n~n(2nf r) + L (n - k) {4 ~n (~ ­4 'TT n'"1 c k=l k-2I) - tn 4n 1 _I) }]'7WH I TE F M0ho n 2 + 1z: x 27*FLICKER f M- 12h'l~n2 r4n 2 • 3n + 1 I n-I~l +-r.-::- x [(n-k) x 2n 2 n ~n2 k=l+ ~ (k +n)(k - 2n)~n(k +n) +j (k - n)(k + 2nl~nik- (n'2k l tn ik'1IJ}J{n [(k+2n)~n(k+2n)-(k'2nl~n(2n-k~- n;+ 3k 2 tnk - ,[(n + 2k)tn(k +7)RANDOM WALK F M- 233 1 14O+Br!+~In order to illustrate (10), variations with time of the shiftedfunctions ht(t - iTo) and of the impulse response hn(t) arerepresented in Fig. 3(a) and (b), respectively, for n =10.For n '" 1, the Allan variance and the modified one are equal.We haveMod 0; (T) =0; (T). (11)One can express (9) in terms of the spectral density Sy (f).We have{i= 12 4Mod O;(T) = 222' n -2 Sy(f) sin (rrfnTo) dfntrT 0 fn-I fCC 1+ 2 L (n - k) -2 Sy (f) cos (2rr[kTo)k=1 0 f(a). sin 4 (TffnTo)df}. (12)It should be noted that the integrals involved in (12) are con·vergent for each noise component. The analytical expressionfor the modified Allan variance can therefore be deduceddirectly from this equation.In the following, it is assumed that the condition 2trf c To » Iis fulfilled. This means that the hardware bandwidth of themeasurement system must be much larger than the referenceclock frequency.We have calculated the modified Allan variance for each(b)noise component. ReSUlts are reported in Table L It appears Fig. 3. (a) The impulse response h n (r), associated with the modifiedthat the analytical expression for Mod 0; (T) is relatively simple Allan variance calculation, represented as a sum of n shifted impulseexcept for flicker phase and flicker frequency noises where it response hI (r). It is assumed n = 10. (b) Variations with time ofis given as a finite sum of functions depending on n. In order the impulse response hn(t), in the special case where n = 10.to compare the Allan variance with the modified one, we., See Appendix Note # 35TN-261

IEEE TRANSACTIONS ON INSTRUYlENTATION AND MEASUREMENT, VOL. IM-33, NO, 4, DECEMBER 1984335TABLE IIA~Al YTiCAl EXPRESSIONS A'D ASY\fPTOTICAl VAlUS FOR R(n)(Results Are Valid within Condition 2rrfcro » I)exR( n)1im R( n)n - :c21.n01I~2[ .n3

336IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-B, NO.4, DECEMBER 1984The EMA V is a random function of m. Its calculation requiresan observatlOn tIme of duration 3m1.We consider f, the fractional deviation of the EMAV relativeto the modified Allan variance defined as follows:f= Mod &;(1) - Mod a~(1)Moda~(r) (16)The standard deviation a(f) off dermes the relative uncertaintyon the measurement of the modified Allan variance due to thefinite number of averaging cycles. We have 'a(e) = M d 2 ( ) {a 2 [Mod a;(1)1p/2 (I 7)o 0y 12 A2where a [Mod ay(r)J denotes the true variance of the EMAVsuch asa 2 [Mod &;(1)J = {(Mod a;(r)J 2 )- [Mod 0;. (r») 2. (18)We assume that the fluctuations y(t) are normally distributed[1O]. One can therefore express ([Mod a; (1')] 2) asm 2 ([Mod 0;(1')]2)= (m 2 + 2m) [Mod a~(r)J2+ 4 mil (m - P){2 1: (n - i)In o}2 +nI (19)pc I1= Iwhere In are integrals which depend on n and on the noise process.We have281l'21'2 n In = I" ~ S (f)cos 61l'np[roo .fX {6 cos 21f[TO i - 4 cos 21l'[ToU + n)- 4 cos 21l'[roU - n) + cos 21l'.f100 + 2n)+ cos 21l'[TO(i - 2n)} df. (20)For each noise component, the expression for a(e) can bededuced from the calculation of integrals involved in (20).These expressions are generally lengthy and complicated exceptfor white phase and white frequency noise modulations, whereintegrals In equal zero. We have limited the present analysisto these two noise components. We get for a(l:)2a(e)=-, for 0: = 2 and O. (21)mWe now compare (21) with previously published results relatedto the estimate of the Allan variance [5 J. For a given timeobservation of duration 3m1', it can be easily deduced from(5) that the relative uncertainty on the estimate of the Allanvariance varies asymptotically as 1.14 m- 1/2 and 1.0 m -112 for0: = 2 and 0, respectively. For these two noise components, theuncertainty on the EMA V is larger than the uncertainty on theestimated Allan variance, but of the same order of magnitude.V. CONCLUSIONWe have calculated the analytical expression for the modifiedAllan variance for each component of the model usuallyconsidered to characterize random frequency fluctuations inprecision oscillators. These expressions have been comparedwith previously published results and the link between theAllan variance and the modified Allan variance has beenspecified.The uncertainty on the estimate of the modified Allan vari­0018-9456/84/0900-0336$01.00 © 1984 IEEEance h~s been studied and numerical values have been reportedfor white phase and white frequency noise modulations.In conclusion, the modified Allan variance appears to be wellsuited for removing the ambiguity between white and flickerphase noise modulation. Nevertheless, the calculation of themodified Allan variance requires signal processing which iscomplicated, compared to the Allan variance. In the presenceof white or flicker phase noise, the Allan variance cannot beeasily deduced from the modified Allan variance. Furthermore,for a given source exhibiting differen t noise components,the determination of the Allan variance from the modifiedone is difficult to perform. For most of time-domain measurements,the use of the Allan variance is preferred.ACKNOWLEDGMENTThe authors would like to express their thanks to Dr. ClaudeAudoin for constructive discussions and valuable comments onthe manuscript.REFERENCES(1) D. W. Allan, "Statistics of atomic frequency standards," Proc.IEEE, voL 54, pp. 221-230, Feb. 1966.[2J J. A. Barnes et 01., "Characterization of frequency stability,"IEEE Trans. Instrum. Meat., vol. 1M-20, pp. 105-120, May 1971.[3] L. S. Cutler and C. L. Searle, "Some aspects of the theory andmeasurement of frequency fluctuations in frequency standards,"Proc. IEEE. vol. 54, pp. 136-154, Feb. 1966.[4) J. Rutman, "Characterization of phase and frequency instabilitiesin precision frequency sources: Fifteen years of progress," Proc.IEEE, vol. 66, pp. 1048-1075, Sept 1978.[5 J P. Lesage and C.· Audoin, "Effect of dead-time on the estimationof the two-sample variance," IEEE Trans. Insrrum. Meas .. volIM-28, pp. 6-10, Mar. 1979.16) J. J. Snyder, "Algorithm for fast digital analysis of interferencefringes," AppL Opt., vol 19, pp. 1223-1225, Apr. 1980.(7] -, "An ultra-high resolution frequency meter," in Proc. 35thAnnu. Symp. Frequency ConTrol (Fort Monmouth, NJ), 1981,pp.464-469.[8] D. W. Allan and J. A. Barnes, "A modified Allan variance withincreased oscillator characterization ability," in Proc. 35th Annu.Symp. Frequency ConTrol (Fort Monmouth, NJ), 1981, pp. 470­474.(9) B. Picinbono, "Processus de diffusion et stationnarite," C.R.Acad. Sci., vol 271, pp. 661-664, Oct 1970.(10) P. Lesage and C. Audoin, "CharacteriZation of frequency stability:Uncertainty due to the finite number of measurements,"IEEE Trans. Instrum. Meas., vol IM·22, pp. 157-161. June 1973.

From: Proceedings of the 15th Annual PITI Meeting, 1983.THE MEASUREMENT OF LINEAR FREQUENCY DRIFT IN OSCILLATORSJames A. BarnesAcstron, Inc., Austin, TexasABSTRACTA linear drift in frequency is an important element in moststochastic models of oscillator performance. Quartz crystaloscillators often have drifts in excess of a part in ten tothe tenth power per day. Even commercial cesium beam devicesoften show drifts of a few parts in ten to the thirteenth peryear. There are many ways to estimate the drift rates fromdata samples (e.g., regress the phase on a quadratic; regressthe frequency on a linear; compute the simple mean of thefirst difference of frequency; use Kalman filters with adrift term as one element in the state vector; and others).Although most of these estimators are unbiased, they vary inefficiency (i.e., confidence intervals). Further, the estimationof confidence intervals using the standard analysisof variance (typically associated with the specific estimationtechnique) can give amazingly optimistic results. Thesource of these problems is not an error in, say, the regressionstechniques, but rather the problems arise fromcorrelations within the residuals. That is, the oscillatormodel is often not consistent with constraints on the analysistechnique or, in other words, some specific analysistechniques are often inappropriate for the task at hand.The appropriateness of a specific analysis technique is criticallydependent on the oscillator model and can often bechecked with a simple "whiteness" test on the residuals.Following a brief review of linear regression techniques,the paper prOVides guidelines for appropriate drift estimationfor various oscillator models, including estimation ofrealistic confidence intervals for the drift.I. INTRODUCTIONAlmost all oscillators display a superposition of random and deterministicvariations in frequency and phase. The most typical model used is[1]:X(t)(1)*where X(t) is the time (phase) error of the oscillator (or clock) relati~e tosome standard; a, b. and Dr are constants for the particular clock; and ~(t) isthe random part. X(t) is a random variable by virtue of its dependence on ~(t).* See Appendix Note # 36551IN-264

Even though one cannot predict future values of X(t) exactly, there are oftensignificant autocorrelations within the random parts of the model. These correlationsallow forecasts which can significantly reduce clock errors. Errorsin each element of the model (Eq. 1) contribute their own uncertainties to theprediction. These time uncertainties depend on the duration of the forecastinterval, T, as shown below in Table 1:TABLE 1.GROWTH OF TIME ERRORSMODEL ELEMENTNAMECLOCKPARAMETERRMS TIMEERRORInitial Time Error a ConstantInitial Freq Error b - TFrequency DriftRandom VariationsDrq,(t)*The growth of time uncertainties due to the random component, q,(t).can have various time dependencies. The three-halves power-lawshowo.here is a "worst case" model. [2]One of the most significant points provided by Table 1 is that eventually, thelinear drift term in the model over-powers all other uncertainties for sufficientlylong forecast intervals! While one can certainly measure (i.e., estimate)the drift coefficient, Dr. and make corrections, there must always remainsome uncertainty in the value used. That is, the effect of a drift correctionbased on a measurement of Dr, is to reduce (hopefully!) the magnitude of thedrift error, but not remove it. Thus, even with drift corrections. the driftterm eventually dominates all time uncertainties in the model.As with any random process, one wants not only the point estimate of a parameter,but one also wants the confidence interval. For example, one might behappy to know that a particular value (e.g., clock time error) can be estimatedwithout bias. he may still want to know how large an error range he shouldexpect. Clearly, an error in the drift estimate (see Eq. 1) leads directly toa time error and hence the drift confidence interval leads directly to a confidenceinterval for the forecast time.II. LEAST SQUARES REGRESSION OF PHASE ON A QUADRATICA conventional least squares regression of oscillator phase data on a quadraticfunction reveals a great deal about the general problems. A slight modificationof Eq. 1 provides a conventional model used in regression analysis[3]:X(t) = a + b·t + c o t 2 + q,(t) (2)552

where c = Dr/2. In regression analysis, it is customary to use the symbol "y"as the dependent variable and "X" as the independent variable. This is in conflictwith usage in time and frequency where "X" and "y" (time error and frequencyerror, respectively) are dependent on a coarse measure of time, t, theindependent variable. This paper will follow the time and frequency custom eventhough this may cause Some confusion in the use of regression analysis text books.The model given by Eq. 2 is complete if the random component, ~(t), is a whitenoise (i.e., random, uncorrelated, normally distributed, zero mean, and finitevariance).III. EXAMPLEOne must emphasize here that ALL results regarding parameter error magnitudesand their distributions are totally dependent on the adequacy of the model. Aprimary source of errors is often autocorrelation of the residuals (contrary tothe explicit model assumptions). While simple visual inspection of the residualsis often sufficient to recognize the autocorrelation problem, "whitenesstests" can be more objective and precise.This section analyzes a set of 94 hourly values of the time difference betweentwo oscillators. Figure 1 is a plot of the time difference (measured in microseconds)between the two oscillators. The general curve of the data along withthe general expectation of frequency drift in crystal oscillators leads one totry the quadratic behavior (Model #1; models #2 and #3 discussed below). Whileit is not common to find white phase noise on oscillators at levels indicated onthe plot, that assumption will be made temporarily. The results of the regressionare summarized in a conventional Analysis of Variance, Table 2.TABLE 2.ANALYSIS OF VARIANCE QUADRATIC FIT TO PHASE(Units: seconds squared)SOURCE SUM OF SQUARES d.f. MEAN SQUARERegression 2.32E-9 3Residuals 4.26E-12 91 4.68E-14Total 2.329E-9 94 2.48E-11Coefficient of simple determination 0.99713Parameters:AaAbAc1.10795E-5 (seconds) t-ratio 161.98= 1.4034E-10 (sec/sec) t-ratio 152.02= -3.7534E-16 (sec/sec2) t-ratio -143.51553IN-266

(J)Q)t)a:::::l I::Q)0 l-l..., :x: Q)lLJ~r4-l4-l-r-!'t:1Q)(,:) fJlZIIIz..c:p..Z:::la::.--lt;rl-.-I~--~=-------~-o+'X ~ 0:­554TN-267

'" Dr 22 = -7.507E-16 (about -6.5E-ll per day)Std. Error = O.05231E-16(Note:a " is the value estimated for the a-parameter. etc.)The Analysis of Variance. Table 2. above suggests an impressive fit of the datato a quadratic function. with 99.71% of the variationl in the data "explained"by the regression. The estimated drift coefficient. Dr. is -7.507E-16 (sec/sec2)or about -6.5E-ll per day --- 143 times the indicated standard error of theestimate. However, Figure 2. a plot of the residuals. reveals significant autocorrelationseven visually and without sensitive tests. (The autocorrelationscan be recognized by the essentially smooth variations in the plot. See Fig. 5as an example of a more nearly white data set.) It is true that the regressionreduced the peak-to-peak deviations from about 18 microseconds to less than onemicrosecond. It is also true that the drift rate is an unbiased estimate of theactual drift rate, but the model assumptions are NOT consistent with the autocorrelationvisible in Fig. 2. This means that the confidence intervals for theparameters are not reliable. In fact, the analysis to follow will show just howextremely optimistic these intervals really are.At this point we can consider at least two other simple analysis schemes whichmight provide more realistic estimates of the drift rate and its variance. Eachof the two analysis schemes has its own implicit model; they are:(2) Regress the beat frequency on a straight line.(Model: Linear frequency drift and white FM.)(3) Remove a simple average from the second difference of the phase.(Model: Linear frequency drift and random walk FM.)Continuing with scheme 2. above, the (average) frequency. yet). is the firstdifference of the phase data divided by the time interval between successivedata points. The regression model is:( 3)*where £(t) [~(t + TO) - ~(t)]/TO. Following standard regression procedures asbefore. the results are summarized in another Analysis of Variance Table. Table3.TABLE 3.ANALYSIS OF VARIANCE LINEAR FIT TO FREQUENCY(Units: sec 2 /sec 2 )SOURCE SUM OF SQUARES d.L MEAN SQUARERegression 1.879E-18 2Residuals 2.084E-20 91 2.29E-22Total 1.899E-18 93 2.042E-20(Note:Takingnumberthe first differences of the original dataof data points from 94 to 93.)set reduces the'" See Appendix Note # 375551N-268

5.12)(IO-7SECo'"0\x(t)- 4 .61 x10 ­ 7 SECo 94RUNNING TIME (HOURS)~IV0­\0 Fig. 2 - Residuals from quadratic fit to phase

Coefficient of simple determination 0.9605Parameters:.....Dr1.049E-lO (sec/sec) t-ratl0 3.32-7.63SE-16Std. Error(sec/sec2) t-ratio -4.700.1624E-16 (sec!sec2)While the drift rate estimates for the two regressions are comparable in value(-7.507E-16 and -7.635E-16). the standard errors of the drift estimates havegone from O.052E-16 to 0.162E-16 (a factor of 3). The linear regression's coefficientof simple determination is 96.05% compared to 99.17% for the quadraticfit. Figure 3 shows the residuals from the linear fit and they appear morenearly white. A cumulative periodogram[4] is a more objective test of whiteness,however. The periodogram, Fig. 4, does not find the residuals acceptableat all.IV.DRIFT AND RANDOM WALK FMIn the absence of noise, the second difference of the phase would be a constant,Dr· T 0 2 • If one assumes that the second difference of the noise part is white.then one has the classic problem of estimating a constant (the drift term), inthe presence of white noise (the second difference of the phase noise). Ofcourse. the optimum estimate of the drift term is just the simple mean of thesecond difference divided by T o 2 • The results are summarized below, Table 4:TABLE 4.SIMPLE MEAN OF SECOND DIFFERENCE PHASE"Simple mean Dr -6.709E-16 t-ratio -2.45Degrees of Freedom 91Standard Deviation s = 26.2405E-16Standard Deviation of the Mean2.7358E-16Figure 5 shows the second difference of the phase after the mean was subtracted.Visually. the data appear reasonably white. and the periodogram, Fig. 6. cannotreject the null hypothesis of whiteness. Now the standard error of the driftterm is 2.735E-16. 52 times larger than that computed for the quadratic fit!Indeed. the estimated drift term is only 2.45 times its standard error.v. SUMMARY OF TESTSIThe analyses reported above were all performed on a single data set. In orderIto verify any conclusions. all three analyses used above were performed a totallof four times on four different data sets from the same pair of oscillators.ITable 5 summarizes the results:557TN-270

4.99x 10- 11 o I I \+1-\/ I I II I 1\ I f I \ 11 \. II II 1 'I I I IU'1

100%CUMULATIVE PERIODOGRAM ­~50%(,11(,11\090% CONFIDENCEINTERVALSoo .1 .2FREOUENCY (CYC/HR).3 .4 .SI~IN'-..lFig. 4 - First diff - linear

8.16)( 10-15Sec/ Sec 2o I { \ I H+-I-i \ I I '" I \ II I f--kJ-\/-il I I I ',I '........ f I I I , ',: I 'I / I II I II !0101o-7.59)1;10- 15 o 92RUNNING TIME (HOURS)~ Fig. 5 - Second diff of phase(".)

100%CUMULATIVE PER I ODOGRAM50%U'10'1......90% CONFIDENCEINTERVALSo .1 .2 .3FREOUENCY (CYC/HR),4 .5~IV~ Fig. 6 - Second diff - avg.

TABLE 5. SUMMARY OF DRIFT ESTIMATES(Units of 1.E-16 sec/sec 2 )ESTIMATIONPROCEDURE& MODELDRIITESTIMATECOMPUTEDSTANDARDERRORPASSWHITENESSTESTS?Quad Fit(White PMand Drift)-7. 507-8.746.0523.0493NoNo-6.479 .0645 No-6.468 .0880 No1st DifferenceLinear Fit(White FMand Drift)-7.635-8.558-6.443.162.206.192NoNoNo-6.253 .295 NoSecond DifferenceLess Mean(Random WalkFM and Drif t)-6.710-7.462-6.8702.7369.3353.424YesNoYes-6.412 3.543 YesOne can calculate the sample means and variances of the drift estimates for eachof the three procedures listed in Table 5, and compare these "external" estimateswith those values listed in the table under "Computed Standard Error," the"internal" estimates. Of course the sample size is small and we do not expecthigh precision in the results, but some conclusions can be drawn. The comparisonsare shown in Table 6.It is clear that the quadratic fit to the data displays a very optimistic internalestimate for the standard deviation of the drift rate. Other conclusionsare not so clear cut, but still some things can be said. Considering Table 5,the "2nd Diff - Mean" residuals passed the whiteness test three times out offour. The external estimate of the drift standard deviation lies between theinternal estimates based on the first and second differences. Since the oscillatorsunder test were crystal oscillators, one expects flicker FM to be presentat some level. One also expects the flicker FM behavior to lie between white FMand random walk FM. This may be the explanation of the observed standard deviations,noted in Table 6.562TN-275

TABLE 6.STANDARD DEVIATIONSPROCEDURE(Model)EXTERNAL ESTIMATE(Std. Dev. of DriftEstimates from Col./'2, Table 5)INTERNAL ESTIMATE(RMS Computed Std.Dev. Col. /13,Table 5)Quad Fit (White PM) 1.08 0.0651st DiH -(White FM)Lin1.08 0.2032nd DiH - Mean(Rand Wlk PM)0.44 5.45VI.DISCUSSIONIn all three of the analysis procedures used above, more parameters than justthe frequency drift rate were estimated. Indeed, this is generally the case.The estimated parameters included the drift rate, the variance of the random(white) noise component, and other parameters appropriate to the specific model(e.g., the initial frequency offset for the first two models). If these otherparameters could be known precisely by some other means, then methods exist toexploit this knowledge and get even better estimates of the drift rate. Thereal problems, however, seem to require the estimate of several parameters inaddition to the drift rate, and it is not appropriate to just ignore unknownmodel parameters.To this point, we have considered only three, rather ideal oscillator models,and seldom does one encounter such simplicity. Typical models for commercialcesium beam frequency standards include white FM, random walks FM, and frequencydrift. Unfortunately, none of the three estimation routines discussed above areappropriate to such a model. This problem has been solved in some of the recentwork of Jones and Tryon[5J,[6J. Their estimation routines are based on KalmanFilters and maximum likelihood estimators and these methods are appropriate forthe more complex models. For details, the reader is referred to the works ofJones and Tryon.Still left untreated are the models which, in addition to drift and other noises,incorporate flicker noises, either in PM or FM or both. In principle, themethods of Jones and Tryon could be applied to Kalman Filters which incorporateempirical flicker models[7J. To the author's knowledg~, however, no such analyseshave been reported.VII.CONCLUSIONSThere are two primary conclusions to be drawn:(1) The estimation of the linear frequency drift rates of oscillatorsand the inclusion of realistic confidence intervals for these563

estimatesused and,are critically dependent on the adequacy of the modelhence the adequacy of the analysis procedures.(2) The estimation of the drift rate must be carried along with theestimation of any and all other model parameters which are notknown precisely from other considerations (e.g., initial frequencyand time offsets, phase noise types, etc.)More and more, scientists and engineers require clocks which can be relied on tomaintain accuracy relative to some master clock. Not only is it important toknow that on the average the clock runs well, but it is essential to have somemeasure of time imprecision as the clock ages. For example, the uncertaintiesmight be expressed as, say,. 90% certain that the clock will be within 5 microsecondsof the master two weeks after synchronization. Such measures are whatstatisticians call "interval estimates" (in contrast to point estimates) andtheir estimations require interval estimates of the clock's model parameters.Clearly, the parameter estimation routines must be reliable and based on soundmeasurement practices. Some inappropriate estimation routines can be appliedto clocks and oscillators and give dangerously optimistic forecasts ofperformance.564TN-277

APPENDIX AREGRESSION ANALYSIS(Equally Spaced Data)We begin with the continuous model equation:X(t) = a + b·t + c·t 2 + ~(t) (AI)We assume that the data is in the form of discrete readings of the dependentvariable X(t) at the regular intervals given by:Equation (AI) can then be written in the obvious form:for n = 1, 2, 3 ••• , N.Next, we define the matrices:1 1 11 2 4(A2)(A3)N1 3 9x =N21 N XNT 1 0 00 To 00 0 T 20NLXu L Xn Sx1 1N(NT)'.! To L X n n T •L X n n T • Snx1 1NNNT 0 2L Xu 0 2 LXn n 21 1NSnnx565TN-278

where four quantities must be calculated from the data:Sx ...NLn=1x nNLn=lSnxNIn=1X n nSxxN= L Xn Zn=1DefineaB = bcWith these definitions, Eq. A3 can be rewritten in the matrix form:---X N T B + E: (A4)and the coefficients, B, which minimize the squared errors are given by:a '"'" B = = T • ( N'N )-1 • ( N'X ) (A5)'" cThe advantage of evenly spaced data for these regressions is that, with a bit ofalgebra, the matrix, ( N'N )-1, can be written down in closed form:A B C( N'N )-1 B D E 1 / G (A6)whereC E F* A :=: 3 [3 (N + 1) + 2JB = -18 (2N + 1)C = 30D 12 (2N + 1) (BN + 11) / [(N + 1) (N + 2)]E -180 / (N + 2)F = 180 / {(N + 1) (N + 2)J* See Appendix Note #40566TN-279

andG == N (N - 1) (N - 2)Also, the inverse of T is just:=The complete solution for the regression parameters can be summarized as follows:There are four quantities which must calculate from the data:NSx I X n Snnx = I X n n2n=1 n=1NNSnx I X n n Sxx = I X 2n=l n=1Nfor n == 1, 2, 3, "" N. Based on these four quantities, the regression parametersare calculated from the seven following equations:n'" a (A Sx + 'a B Snx + '0 2 C Snnx) / G'"2b (B Sx + '0 D Sox + '0 E Snnx) / (G '0)'" 2 2c '" (C Sx + '0 E Sox + '0 F Snnx) / (G '0 )"'2°2 '"'" - To c Snnx) / (N - 3)= (Sxx a Sx'" 2 = 0 2 A / G°a;b 2 0 2 D / (G '0 2 )'" 2°c 8 2 F / (G T 04)561TN-280

where the coefficients A. B, C, etc., are given by:A 3 [3N (N + 1) + 2]B = -18 (2N + 1)C 30D 12 (2N + 1) (8N + 11) I [(N + 1) (N + 2)]E -180 I (N + 2)F = 180 I [(N + 1) (N + 2)]andG N (N - 1) (N - 2).In matrix form, the error variance for forecast values is:where Nk' = [1 no no] and TO no is the date for the forecast point, Xk' Thatis, no = N + K and K is the number of lags past the last data point at lag N.A568TN-28I

APPENDIX BREGRESSIONS ON LINEAR AND CUBIC FUNCTIONSThe matrixes (N'N)-1 for the linear fit and cubic fit, which correspond to Eq.A6 in Appendix A are as follows:For the linear fit:(N'N)-1[: :] lIDwhereA 2 (2N + 1)B -6C 12 I (N + 1)andD = N (N - 1)For the cubic fit:A B C D(N'N)-1BCEFFHGI1 I KD G I JwhereA 8 (2N + 1) (N + N + 3)B = -20 (6N2 + 6N + 5)C 120 (2N + 1)D -140EFG200 (6N4 + 27 N3 + 42 N2 + 30 N + 11) I L-300 (N + 1) (3N + 5) ON + 2) I L280 (6N2 + lSN + 11) I L569TN-282

360 (2N + 1) (9N + 13) I LH ..I .. -4200 (N + 1) I LJ2800 I LandK = N (N - 1) (N - 2) (N - 3)L (N + 1) (N + 2) (N + 3).The restrictions on these equations are that the data is evenly spaced beginningwith n = 1 to n = N. and no missing values. For error estimates (and theirdistributions) to be valid. the residuals must be random. uncorrelated, (i.e.,white) •570TN-283

REFERENCES1. D.W. Allan, "The Measurement of Frequency and Frequency Stability of PrecisionOscillators," Proc. 6th PTTI Planning Meeting, p109 (1975).2. J.A. Barnes, et al., "Characterization of Frequency Stability," Proc IEEETrans. on lnst. and Meas., lM-20, pl0S (1971).3. NoR. Draper and H. Smith, "Applied Regression Analysis," John Wiley andSons, N. Yo, (1966).4. GoE.P. Box and G.M. Jenkins, "Time Series Analysis," Holden-Day, SanFrancisco, p294, (1970).s. P.V. Tryon and R.H. Jones, "Estimation of Parameters in Models for CesiumBeam Atomic Clocks," NBS Journal of Research, Vol. 88, No.1, Jan-Feb. 1983.6. R.H. Jones and P.V. Tryon, "Estimating Time from Atomic Clocks," NBS Journalof Research, Vol. 88, No.1, Jan-Feb. 1983.7. J.A. Barnes and S. Jarvis, Jr., "Efficient Numerical and Analog Modeling ofFlicker Noise Processes," NBS Tech. Note 604, (1971).571TN-284

----~-~Q)C)(J) ~Q)a::::>HQ)0 4-l::I:4-l"..1wro:EQ)[J) I-m..c:Cl:z p.- zZ:::> r-la:tJ1"..1~~-~--~-0+>-X572TN-285

5.12 )( 10 - 7 SEC

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FREQUENCY DRIFT AND WHITE PMSTANDARD DEVIATIONS32tn'IcoFORCASTUNCERTAINTYFIT UNCERTAINTY(BACKCAST)(FORECAST)N'0-~ -40 -20o 20 40

FREQUENCY DRIFT AND WHITESTANDARD DEVIATIONSPM:32(J1.......1.0(BACKCAST)(FORECAST)~tv\0tv-10-5I~0DATA5.110

QUESTIONS AND ANSWERSMR. ALLAN:We have found the second difference drift estimator to be useful forthe random walk FM process. One has to be careful when you apply it,because if you use overlapping second differences, for example, ifyou have a cesium beam, and you have white noise FM out to severaldays and then random walk PM beyond that point, and you are makinghourly or 2 hour measurements, if you use the mean of the seconddifferences, you can show that all the middle terms cancel, and infact you are looking at the frequency at the beginning of the run andthe frequency at the end of the run to compute the frequency drift andthat's very poor.DR. BARNES:My comment would be: There again the problem is in the model, and notin the arithmatic. The models applied here were 3 very simple models,very simple, simpler than you will run into in life. It was pure randomwalk plus a drift or pure frequency noise plus a drift, or pure randomwalk of frequency noise plus a drift and it did not approach at all anyof the noise complex models where you would have both white frequency andrandom walk frequency and a drift. That's got to be handled separately,I'm not even totally sure how to perform it in all cases at this point.MR. McCASKILL:I would like to know if you would comment on the value of the sample timeand the reason why, of course, is that the mean second difference doesdepend on the sample time? For instance, if you wanted to estimate theaging rate or change in linear change of frequency, what value of sampletime would you use in order to make that correction. So, really, thequestion is, how does the sample time enter into your calculations?DR. BARNES:At least I will try to answer in part. I don't know in all cases, I'msure, but if you have a complex noise process where you have at shortterm different noise behavior than in long term, it may benefit you totake a longer sampling time and effectively not look at the short term,and then one of the simple models might apply. I honestly haven'tlooked in great detail at how to choose the sample time. It is aninteresting question.580TN-293

MR. McCASKILL:Well, let me go further, because we had the benefit of being out atNBS and talking with Dr. Allan earlier and he suggested that we usethe mean second difference, and the only problem is if we want tocalculate or correct for our aging rate at a tau of five days or tendays, and you calculate the mean second difference, you come up withexactly the Allan Variance. What appears is that in order to come upwith the number for the aging rate, you have to calculate the meansecond difference using a sample time whenever you take your differencesof longer than, let's say, ten days sample time. So we use, of course,the regression model, but we use on the order of two or three weeks inorder to calculate an aging rate correction for, say, something like therubidium in the NAVSTAR 3 clock. It looks like in order to come upwith a valid value for the Allan Variance of five or ten day sampletime you have to calculate the aging rate at a longer, maybe two orthree times longer sample time.DR. BARNES:MR. ALLAN:Dave Allan, do you think you can answer that?Not to go into details, but if you assume that in the longer term youhave random walk frequency modulation as the predominant noise processin a clock, which seems to be true for rubidium, cesium and hydrogen,you can do a very simple thing. You can take the full data lengthand take the time at the beginning, the time in the middle and thetime in the end and construct a second difference, and that's yourdrift.There is still the issue of the confidence interval on that. If youreally want to verify your confidence interval, you have to have enoughdata to do a regression. You need enough data to test to be sure themodel is good.DR. WINKLER:That argument is fourteen years old, because we have been critizedhere. I still believe that for a practical case where you depend onmeasurements which are contaminated and maybe even contaminated byarbitrarily large errors if they are digital. These, theoreticaladvantages that you have outlined may not be as important as thebenefits which you get when you make a least square regression. Inthis case you can immediately identify the wrong data. Otherwise, ifyou put the data into an algorithm you may not know how much your datais contaminated. So for practical applications, the first model eventhough theoretically it is poor, it still gives reasonable estimatesof the drift and you have residuals which let you identify wrong phasevalues immediately.581TN-294

DR. BARNES:I think that's true and you may have different reasons to do regressionanalysis, if your purpose is to measure a drift and understand theconfidence intervals, then I think what has been presented is reasonable,if you have as your purpose to look--to see if there are indications offunny behavior in a curve that has such strong curvature or drift thatyou can't get it on graph paper without doing that, I think it is a veryreasonable thing to do. I think looking at the data is one of thehealthiest things any analyst can do.582TN-295

NIST Technical Note 1318, 1990.VARIANCES BASED ON DATA WITH DEAD TIME BETWEEN THE MEASUREMENTSJames A. BarnesAustron, Inc.Boulder, Colorado 80301andDavid W. AllanTime and Frequency DivisionNational Institute of Standards and TechnologyBoulder, Colorado 80303The accepted definition of frequency stability in the time domain isthe two-sample variance (or Allan variance). It is based on themeasurement of average frequencies over adjacent time intervals,with no "dead time" between the intervals. The primary advantagesof the Allan variance are that (1) it is convergent for manyencountered noise models for which the conventional variance isdivergent; (2) it can distinguish between many important anddifferent spectral noise types; (3) the two-sample approach relatesto many practical implementations; for example, the rms change of anoscillator's frequency from one period to the next; and (4) Allanvariances can be easily estimated at integer multiples of the sampleinterval.In 1974 a table of bias functions which related variance estimateswith various configurations of number of samples and dead time tothe Allan variance was published [1]. The tables were based onnoises with pure power-law spectral densities.Often situations occur that unavoidably have dead time betweenmeasurements, but still the conventional variances are notconvergent. Some of these applications are outside of the time-andfrequencyfield. Also, the dead times are often distributedthroughout a given average, and this distributed dead time is nottreated in the 1974 tables.This paper reviews the bias functions Bl(N,r,~), and B2(r,~) andintroduces a new bias function, B3(2,M,r,~), to handle the commonlyoccurring cases of the effect of distributed dead time on thecomputed variances. Some convenient and easy-to-interpretasymptotic limits are reported. A set of tables for the biasfunctions are included at the end of this paper.Key words: Allan variance; bias functions; data sampling and deadtime; dead time between the measurement; definition of frequencystability; distributed dead time; two-sample variance1TN-296

1. IntroductionThe sample mean and variance indicate respectively the approximate magnitudeof a quantity and its uncertainty. For many situations a continuous functionof time is sampled, or measured, at fairly regular intervals. Sampling is notalways instantaneous. It takes a finite time and provides an "averagereading." If the underlying process (or noise) is random and uncorrelated intime, then the fluctuations are said to be "white" noise. In this situation,the sample mean and variance calculated by the conventional formulas,m1N1N-lNI Yn'n=l(1)provide the needed information. The "bar" over the y in eq (1) above denotesthe average over a finite time interval. In time and frequency work, y isdefined as the average fractional (or normalized) frequency deviation fromnominal over an interval r and at some specified measurement time. As inscience generally, the physical model determines the appropriate mathematicalmodel. For the white noise model, the sample mean and variance are themainstays of most analyses.Although white noise is a common model for many physical processes, moregeneral noise models are being identified and used. In precise time andfrequency measurement, for example, there are two quantities of greatinterest: instantaneous frequency and phase. These two quantities bydefinition are exactly related by a differential. (We are NOT consideringFourier frequencies at this point.) That is, the instantaneous frequency isthe time rate of change of phase. Thus, if we were employing a model of whitefrequency-modulation (white FM) noise, then the phase noise is the integral ofthe white FM noise, commonly called a Brownian motion or random walk.Therefore, depending on whether we are currently interested in phase orfrequency, the sample mean and variance mayor may not be appropriate.2TN-297

By definition, white noise has a power spectral density (PSD) that is constantwith Fourier frequency. Since random walk noise is the integral of whitenoise, the power spectral density of a random walk varies as 1/f 2 (where f isthe Fourier frequency) [2]. We encounter noise models whose power spectraldensities are various power laws of their Fourier frequencies. Flicker noiseis very common and is defined as a noise whose power spectral density variesas l/f over a relevant spectral range. If an oscillator's instantaneousfrequency is well modeled by flicker noise, then its phase would be theintegral of the flicker noise. It would have a PSD which varied as 1/f 3 .Noise models whose PSD's are power laws of the Fourier frequency but notinteger exponents are possible as well but not as common. This paperconsiders power-law PSD's of a quantity yet); yet) is a continuous samplefunction which can be measured at regular intervals. For noises whose PSD'svary as f~ with a < -1 at low frequencies, the conventional sample mean andvariance given in eq (1) do not converge as N gets large [2, 3]. This lack ofconvergence renders the sample mean and variance ineffective and oftenmisleading in some situations.Although the sample mean and variance have limitations, other time-domainstatistics can be convergent and quite useful. The quantities that weconsider in this paper depend significantly on the details of the samplingprocedures. Indeed, each sampling scheme has its own bias, and this is themotivation for the bias functions discussed in this paper.2. The Allan VarianceRecognizing that for particular types of noise, the conventional samplevariance fails to converge as the number of samples, N, grows, Allan suggestedthat we set N = 2 and average many of these two-sample variances to get aconvergent and stable measure of the spread of the quantity in question [3].This is what has come to be called the Allan variance.3~-298

More specifically let us consider a sample function of time as indicated infigure 1. A measurement consists of averaging yet) over the interval r. Thenext measurement begins at a time T after the beginning of the previousmeasurement interval. There is no logical reason why T must be as large as ror larger--if T < r, then the second measurement begins before the first iscompleted, which is unusual but possible. When T = r, there is no dead timebetween measurements.The accepted definition of the Allan variance is the expected value of a twosamplevariance with no dead time between successive measurements.symbols, the Allan variance is given byIn(2)where there is no dead time between the two sample averages for the Allanvariance and the E['] denotes the expectation operator.3. The Bias Function B 1(N,r,p)Define N to be the number of sample averages of yet) used in eq (1) toestimate a sample variance (N = 2 for an Allan variance). Also define r to bethe ratio of T to r (r = 1 when there is no dead time between measurements).The parameter ~ is related to the exponent of the power law of the PSD of theprocess yet). If a is the exponent in the power-law spectrum for yet), thenthe Allan variance varies as r raised to the ~ power, where a and ~ arerelated as shown in figure 2 [2-4]. We can use estimates of ~ to infer 0, thespectral type. The ambiguity in a for ~ = -2 has been resolved by using amodified a;(r) [5-7].Often data cannot be taken without dead time between sample averages, and itis useful to consider other than two-sample variances.bias function Bl(N,r,~) by the ratio,We will define thea 2 (N,T,r)a 2 (2,T,r)(3)4TN-299

where aZ(N,T,r) is the expected sample variance given in eq (1) and based on Nmeasurements at intervals T and averaged over a time rand r = Tlr. In words,B1(N,r,p) is the ratio of the expected variance for N measurements to theexpected variance for two samples (everything else held constant). Thevariances on the right in eq (3) depend implicitly on the noise type eventhough p or 0 are not shown as independent variables. The noise-typeparameter, p, is shown as an independent variable for all of the biasfunctions in this paper, because the values of the ratio of these variancesexplicitly depend on p as will be derived later in the paper. Allan showedthat if Nand r are held constant, then the 0, p relationship shown in figure2 is the same; that is, we can still infer the spectral type from the rdependence using the equation 0 = -p-1, -2 ~ p < 2 [3].4. The Bias Function B 2 (r,p)The bias function Bz(r,p) is defined in [1] by the relation,a Z (2, T, r) a Z B (2,T,r)z (r, p) (4)a Z (2, r, r) a; (r)In words, Bz(r,p) is the ratio of the expected two-sample variance with deadtime to that without dead time (with N = 2 and r the same for both variances).A plot of the Bz(r,p) function is shown in figure 3. The bias functions B 1and B z represent biases relative to N = 2 rather than infinity; that is, theratio of the N sample variance (with or without dead time) to the Allanvariance and the ratio of the two-sample dead-time variance to the Allanvariance respectively.5. The Bias Function B 3(N,M,r,p)Consider the case where a great many measurements are available with dead timebetween each pair of measurements (To> ro). The measurements are averagedover the time interval ro' the spacing between the beginning of onemeasurement to the next is To, and it may not be convenient to retake thedata. We might want to estimate the Allan variance at, say, multiples M of5TN-300

the averaging time To. If we average groups of the measurements of yet), thenthe dead times between the original measurements are distributed periodicallythroughout the new average measurements (see figure 4). DefineM+i-11M I Yn' (5)n=iwhere Yi are the raw or original measurements based on dead time TO-TO.Also define the two-sample variance with distributed dead time as(6)with T = M7 0and T MT o .We can now define B 3as the ratio of the N-sample variance with distributeddead time to the N-sample variance with dead time accumulated at the end as infigure 1:a 2 (N,M,T,T)B 3 (N,M,r,J-L) (7)a 2 (N, T, T)Although B 3(N,M,r,J-L) is defined for general N, the tables in the Appendixconfine treatment to the case where N = 2. There is little value in extendingthe tables to include general N. Though the variances on the right in eq (7)depend explicitly on N, T and T, the ratio B 3(N,M,r,J-L) depends on the ratior = T/7, and on J-L as developed later in this paper.In words, B 3(2,M,r,J-L) is the ratio of the expected two-sample variance withperiodically distributed dead time, as shown in figure 4, to the expected twosamplevariance with all the dead time grouped together as shown in figure 1.Both the numerator and the denominator have the same total averaging time anddead time, but they are apportioned differently. The product B 2 (r,J-L) •B 3(2,M,r,J-L) is the distributed dead-time variance over the Allan variance fora particular T, 7, M and J-L.6TN-30l

Some useful asymptotic forms of B 3 can be found. In the case of large M andM> r, we may write that~ 1 :$3/l4 in(2)2 in(r) + 3'/l O.:$ 2,(8)One simple and important conclusion from these two equations is that for thecases of flicker FM noise and random-walk FM noise, the r~ dependence forlarge r is the same whether or not there is periodically distributed deadtime. The values of the variances differ only by a constant, and in thelatter case the constant is 1. This conclusion is also true for white FMnoise, and in this case the constant is also 1.In the cases r » 1 and -2 :$ /l :$ -1, we may write for the asymptotic behaviorof B 3-/l-1 , (9)as was determined empirically. In this region of power-law spectrum the B 3function has an Madependence for an fa spectrum.6. The Bias FunctionsThe bias functions can be written fairly simply by first defining thefunction,F(A) (10)The bias functions becomeB 1(N,r,,u) ==N-11 + In=11 +N-nN(N-1)~ F(r)F(nr)( 11)7TN-302

1 + liF(r)B z (r ,I!) = (12)2(1-2"')as given in [lJ, and2M + M·F(Mr) -M-1I (M-n)[2F(nr)-F«M+n)r) -F«M-n)r)Jn=l( 13)as indicated in the appendix.For ~ = 0, eqs (11), (12), and (13) are the indeterminate form % and must beevaluated by l'Hopital's rule. Special attention must also be given whenexpressions of the form 0° arise. We verified a random sampling of the tableentries using noise simulation and Monte Carlo techniques. No errors weredetected. The results in this paper differ some from those in [8], whichsuggests that there may be some mistakes. Tables for the three bias functionsare listed at the end of the paper (note that the computer print-out did nothave a symbol for Greek mu = IL).7. Examples of the Use of the Bias FunctionsThe spectral type, that is, the value of IL, may be inferred by varying T, thesample time. However, another useful way of determining the value of I! is byusing B1(N,r,~) as follows: calculate an estimate of o;(N,T,T) and o;(2,T,T)and hence B1(N,r,~); then use the tables to infer the value of IL.Suppose one has an experimental value for o;(N 1 , T 1 , T 1 ) and its spectraltype is known, that is, p is known.Suppose also that one wishes to know thevariance at some other set of measurement parameters, N z ' T z ' T Z 'estimate of o;(Nz,Tz,T Z ) may be calculated by the equation:An unbiasedo~ (Nz , T 2 ' T Z ) (14)where r 18TN-303

Since the time-domain definition for frequency stability is the Allanvariance, it behooves us, where possible, to relate other variances to theAllan variance. If we have an N-sample variance on data with dead-time T-rand we know the power-law spectral type (the value of ~), then we may writea; (r) (15)If we have an N-sample variance where each data entry is an average of Msamples with distributed dead time, then we may write(16)8. ConclusionFor some important power-law spectral density models often used incharacterizing precision oscillators (Sy(f) - f Q , Q = -2, -1, 0, +1, +2), wehave studied the effects on variances when there is dead time between thefrequency samples, and the frequency samples are averaged to increase theintegration time. Since dead time between measurements is a common problemthroughout metrology, the analysis here has broader applicability than just totime and frequency. Specifically, this kind of analysis has been used withgage blocks and standard volt cells--showing that the classical variance maybe non-convergent in some cases [9].Heretofore, the Allan variance has been shown to have some convenienttheoretical properties in relation to power-law spectra as the integration orsample time is varied (if Gy 2 (r) - r~, then Q = -~ -1, -2 < ~ 5 2). Sinceoy(r), by definition, is estimated from data with no dead time, the sample orintegration time can be unambiguously changed to investigate the r dependence.From our analysis, we have concluded that for the asymptotic limit of severalsamples being averaged with dead time present in the data, the r dependence ofthe variances is the same. The Q = -~ -1 relationship still remains valid forwhite FM noise (~= -1, Q = 0), flicker FM noise (~= 0, Q = -1), and for9TN-304

andom-walk FM noise (JJ = +1, Q = -2). The asymptotic limit is approached asthe product of number of samples averaged and the initial data sample time,TO' becomes larger than the dead time (M > r). The variances so obtaineddiffer only by a constant, which can be calculated as given in this paper.A knowledge of the appropriate power-law spectral model is required totranslate a distributed dead-time variance to the corresponding value of theAllan variance. In principle, the power-law spectral model can be estimatedfrom the T~ dependence, using the variance analysis on the data as outlinedabove.9. References[1] J.A. Barnes, "Tables of Bias Functions, B 1and B z ' for Variances Based onFinite Samples of Processes with Power Law Spectral Densities," NBS Tech.Note 375 (1969).[2] J.A. Barnes, "Atomic Timekeeping and the Statistics of PrecisionSignal Generation," IEEE Proc. 54, No.2, pp. 207-220, Feb. 1966.[3] D.W. Allan, "Statistics of Atomic Frequency Standards, "IEEE Proc.54, No.2, pp. 221-230, Feb. 1966.[4] RFC Vessot, L. Mueller, and J. Vanier, "The Specification ofOscillator Characteristics from Measurements Made in the FrequencyDomain," IEEE Proc. 54, No.2, pp. 199-207, Feb. 1966.[5J D.W. Allan and J.A. Barnes, "A Modified "Allan Variance" WithIncreased Oscillator Characterization Ability," Proc. 35th AnnualFrequency Control Symposium, USAERADCOM, Ft. Monmouth, NJ, May 1981,pp. 470-475.[6] D.W. Allan, "Time and Frequency (Time-Domain) Characterization,Estimation, and Prediction of Precision Clocks and Oscillators," IEEETransactions on UFFC, November 1987.[7] P. Lesage and T. Ayi, "Characterization of Frequency Stability: Analysisof the Modified Allan Variance and Properties of its Estimate," IEEETrans. Instrum. Meas., IM-33, no. 4, pp. 332-336, Dec. 1984.[8] N.D. Faulkner and E.V.I. Mestre, "Time-Domain Analysis of Frequency10TN-305

Stability Using Non-zero Dead-Time Counter Techniques," IEEE Trans. onInstrum. & Meas., IM-34, 144-151 (1985).[9] D.W. Allan, "Should the Classical Variance Be Used as a Basic Measure inStandards Metrology?" IEEE Trans. on Instrumentation and Measurement,IM-36, 646-654, 1987.11TN-306

Table 1.Table of some bias function identitiesB 1 (2,r,lt)B 1 (N,r,2)B 1 (N, 1 , 1)B 1 (N,1,1t)* B 1 (N,1,1t)1(N(N+l»/6= N/2= (N(1-N~»/[(2(N-l)(1-2~)] for 1t~0= N 1n(N)/[2(N-l) 1n(2)] for 1t=0= 1 for J.t < 0N-1== [2/(N(N-1)] I (N-n) . IT' for J.t > 0n==lB 1(N, r, -1)B 1 (N, r, -2)B l(0 ,It)B l(1 ,It)lifr~l== 1 if r ~ 1 or 0== 0== 1B l(r, 2)B 2 (r, 1)B l (r, ~ 1)B l(r, - 2)B 3(2 ,M, 1 , f-£)B 3 (2 , M, r , - 2)B 3 (2, r, iL)B 3(2 ,M, r , 2 )B 3(2 ,M, r , - 1 )(3r - 1)/2 if r ~ 1==r if O::;;r::;; 1== 1 if r ~ 10 if r==O== 1 if r==l== 2/3 otherwise1== M== 1== 1== 1 for r ~ 1-~_._----* See Appendix Note #4112TN-307

I!I­I.....I~I.....cIQ)IEQ) I~I~(J)ro,.... Iw Q) - --I~IItMeasurementInterval~wo001WO MEASUREMENTS OF A SET WITH DEAD TIMEFigure 1. Illustration of two fractional frequency samples with dead time, T-T. between the samples.This is two of a set of adjatent frequency measurements, each averaged over an interval T, neededto calculate a two-sample variance from a data set.

Oy2(T) rv T'"3 (lSy(f) rv faWhite PM ,.. "21Flicker PMWhite FMJJ-t1 2 3a = -J.I'-1Flicker FM -1Mod. O/(T} rv T""Random FM -2WalkFlicker FM -3WalkFigure 2. A plot of the relationship between the frequency-domain power-lawspectral-density exponent Q and the time-domain two-sample Allan varianceexponent p (0 = -p-I, -2 ~ p < 2 and Q ~ 1 for p = -2), Also shown is thesimilar relationship between 0 and the modified Allan variance with exponenton r of p' (0 = -p'-I, -4 ~ p' < 2), The pointing arrows indicate the mualpharelationship (0 vs. p or ~') for which the particular variance applies.14TN·309

-2,THE BIAS FUNCTION, 8 2 (r,p.)Figure 3.A three dimensional plot of the bias function B2(r,~), wherer = T/r, and the dead time is T - T. The "fin" at r = 1 and ~ = -2 approacheszero width as the measurement bandwidth approaches infinity (see appendix ref.[3]) .15TN-3i0

+=;'- >.2cQ)~;:,(/)t-'0- roQ)~___ L'_YII, I-t ,_TO I'11 It+-f' o~Illustrations with M =4Distributed Dead Time = 4 • (To -TO),--..J- 1I I II II I II ,II I II I II 'IIIIII I I't-...L - - T -+,I I II I I II I I II I I II I I I/Yi+M-------1I " 'I " " ,. " " II II ... t, ITWO AVERAGES WITH FOUR MEASUREMENTSEACH FROM A SET WITH DEAD TIMEFigure 4. An illustration of determining two averages, each taken over four measurements. In this~ illustration the original data have a sample time r o and a dead time To - roo The two averages each have aIW...... distributed sample time 4ro = r, and a distributed dead time 4(To - To) = T - r, since in this case M : 4 .......These two averages are two of a set similar averages with distributed sample times and dead times needed tocalculate a two-sample distributed sample-time and dead-time variance -- the numerator for the bias functionB 3 • The denominator for B 3 is a two-sample variance with no distributed sample time or dead time but with T =MToand T : Mroand dead time T - T.

AppendixWith reference to figure 1, the frequency sampling window has an equivalentphase sampling window.The intent is to evaluate the variance, S(M), of thesampled phase function in terms of the phase autocorrelation function, R(r).The process here is to correctly account for terms and cross-terms coming fromsquaring and averaging the samples for each M. The B 3(2,M,r,p.) function canthen be obtained from the relation,S(M)S(l) .W'+2for appropriate M, r, and p.. The denominator is just the two-sample variancewith dead time for MT and Mr (in accordance with the definition ofB 3 (2,M,r,p.». The factors common to the numerator and denominator are ignoredin the following.For MI, the variance S(l) is justS(l)4·R(O) - 4'R(r) - 4·R(T) + 2·R(T+r) + 2R(T-r),where use has been made of the definition of the autocorrelation function,R(T)E(¢>(t)· ¢>(t+T)].It is convenient to define a function G(T) asG(T)2·R(T) - R(T+r) - R(T-r).Similarly, S(2) can now be written in the form,S(2) = 8'R(O) - 8·R(r) + 2'G(T) - 4'G(2T) - 2·G(3T).Following this procedure, we can verify that the general S(M) is just17TN-312

SCM)4-M-R(O) - 4-M-R(r) - 2 oMoG(MT)M-l+ 2 I (M-n)[2 o G(nT) - G«M+n)T) - G«M-n)T)] 0n=1Following the work of Barnes and Allan [2,3], we can define the function U(r)by the relation,U(r) = 2 oR(O) - 2-R(r),and also defineF(nr) = G(nT)/U(r),where r = T/r_the form,The function U(r) for power-law power spectral densities hasU(r)which yieldsF(nr)Finally, the working relation can be written asB 3 (2,M,r,p)20M + M-F(Mr)M-l-I (M-n) [2 oF(nr) - F«M+n)r)n=1[2 + F(r)] 0MJL+2- F«M-n)r)]18TN-313

8lIN,r,lll) for r = .011\1\ N= ~ 816 32 M 128 2S6 512 102~ INF....--..................... _...------_..... - ..---- --- _... -_ ... -_ .......--..--..------_ ... - ..._-- ----------..--------------- - ......... ------ _..--------- ---_ .....--------­-2 1.000f+OO 1.000E+OO 1.000E+OO 1.000£+00 I.OOOE+OO I.OOOE+00 \.OOOE+OO I.OOOE+OO I. OOOE+OO I. 000f+00-1.8 1.001E+OO 1.210E+OO I. 3bOf+00 1.S45£+OO I. m£+OO 2. 06'9E+OO 2.349£+00 2. ~ S6E+00 2.~9~+OO 2.512£+00-1.6 I. 199£+00 I. ~87E+OO I. 893E+00 2.~SSE+OO 3. ;mE+00 ~. 440E+00 5.505E+OO 6.002£+00 6.1~+OO 6. 309E+00-I.~ 1.328£+00 1.8S6E+OO 2.688E+OO 3. 991E+OO 6.OS4E+OO 9. 500E+00 1.282E+01 1.~S6E+01 1.532£+01 1.S8SE+01-1.2 I. 482E+OO 2.W>E+OO 3.BBOE+OO 6.599£+00 1.1~5£+O1 2.026E+OI 2.9~7E+01 3.4$bE+01 3.7S4f:+01 3. 980E+01-I I.667E+OO 3.000E+OO 5.667E+OO I. IOOE+OI 2. 167E+OI ~.25SE+OI 6. 628E+OI 8. 19OE+OI 9.064E+O\ 1.000E+02-.8 I. IJ:8.4E+00 3.86OE+OO 8.303E+OO 1.828E+01 ~. ~1E+OI 8. 691E+OI I. 443E+02 1.86BE+02 2. I37E+02 2.519E+02-.6 2. 134E+{I() 4.959£+00 1.205£+01 2. 977E+01 7. 296E+01 I.697E+02 2.99SE+02 ~.076E+02 ~.8SIE+02 6.~23E+02-.4 2.0\07E+OO 6. 279E+OO I. 703E+Ol 4.655E+01 1.2~7E+02 3.106E+02 5.814£+02 8. 357E+02 1.~3E+03 1.715£+03-.2 2.677E+OO 7.714£+00 2.296£+01 6.836E+01 1.976E+02 5.22OE+02 1.03bE+03 1.58OE+03 2. 08-4E+03 5. 58IE+030 2.912E+OO 9.075E+OO 2.909E+01 9.282E+01 2.S57E+02 7.95IE+02 I.672E+03 2.719E+03 3. 826E+03.2 3.089E+OO 1.01BE+01 3.44S£+01 1.162£+02 3. 763E+02 1.097E+03 2.446£+03 4. 263E+03 6. 453E+03.4 3.203E+OO 1.095f+01 3. 857E+01 1. 354E+02 4. 572E+02 1.39OE+03 3. 287E+03 6. I72E+03 1.014£+04.6 3.268E+OO I. 143E+01 4. 133E+01 I. 495f+02 5.22IE+02 1. M9E+03 ~.I~2E+03 8.~I9E+03 1.5J4£+04.8 3.302£+00 1.170E+OI 4.303E+01 1.59OE+02 5. 708E+02 I.869E+03 ~.991E+03 1.104E+04 2.189£+041 3.319£+00 1.185E+O\ 4. 4Q3E+0 I 1.6S3E+02 6.066E+02 2.05SE+03 5. 842E+03 I. ~12E+04 3.109£+041.2 3. 327E+00 1.192£+01 ~. 46IE+OI 1.693E+02 6. 33OE+02 2.215£+03 6.717E+03 I.782E+04 4.3B3E+041.4 3.33Of+OO 1. I96E+OI ~.~94E+01 I. 72OE+02 6.53OE+02 2. 359E+03 7.MOE+03 2.234E+04 6.169£+041.6 3.332E+00 1.198£+01 4.514E+01 I.738E+02 6.688£+02 2. 493E+03 8.b37E+03 2. 793E+04 8. 697E+041.8 3. 333E+1)1) 1.199£+01 4.526E+OI I. 750E+02 6.819£+02 2. 623E+03 9.737E+03 3.~93E+04 1.23IE+052 3. 333E+OO 1.2OOE+OI 4.533E+01 1. 76OE+02 6.933£+02 2.752£+03 1.097E+04 4. 37BE+04 1.749£+05BICN,r,lIIU) for r = .03Mu \ N= 4 8 16 32 64 128 2S6 512 1024 INF------------------------------------------------------.-------------------------------------------------------------------­-2 1.000E+OO 1.000E+OO 1.000E+00 I.OOOE+OO 1.000E+OO I.OOOE+OO I.OOOE+OO I.OOOE+OO 1.000E+OO 1.000E+OO-I. 8 I. 09IE+OO 1.211E+OO 1.366E+OO 1.573£+00 I. 827E+00 I. 9SOE+OO 1.995£+00 2.010E+OO 2.015E+OO 2. 01 6E+OO-1.6 1.199£+00 I. 49\E+00 1.910E+OO 2. 532E+0

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E. APPENDIX - Notes and ErrataThe notes listed below are included to bring attention to notation and other problems. The pagenumber provides reference to the location of the problem or comment.1. page TN-4AM noise is especially important in the measurement of the (residual) phase noise addedby amplifiers and other signal handling components. Often the driving source for themeasurements is a frequency synthesizer that has phase and/or amplitude noise that iscomparable or larger than the added noise of the component under test. Most measurementsystems are configured so that the phase noise of the source cancels out to a largedegree (at high Fourier frequencies decorrelation effects limit the cancellation). In suchmeasurement systems, the AM to PM conversion factor and the AM noise of the sourcemay then set the noise floor. Discussion of the effect of AM noise and AM to PMconversion factors on the accuracy and precision of phase noise measurements is found in"Residual Phase Noise Measurements of vhf, uhf, and Microwave Components" by G. K.Montress, T. E. Parker, and M. J. Loboda Proc. 43rd Annual Symposium on FrequencyControl, pp. 349-359 (1989) and in "Accuracy Model for Phase Noise Measurements by F.L. Walls, C M. Felton, and A. J. D. Clements, 21st Annual Precision Time and TimeInterval Meeting (1989). The notation in these papers as well as that in other parts of theliterature differs from that given below. Our notation below is drawn from a modest levelof consensus among individuals responsible for setting standards. We expect that it willgradually be adopted within standards committees. The following comments are directedspecificalJy at the specification of noise performance.The total power spectral density in a signal can be approximated by expanding eq 12-5 ofpaper B.2 (by Stein) and extending it to include the spectral density of relative amplitudefluctuations, Sa(f} The double-sideband density written in single-sideband form is given bySv(j)

The more general expression is important only very near the carrier and in certain typesof frequency multiplication. The single-sideband, amplitude noise, normalized to the totalsignal power is given simply by Sif) for 0 :S f:S 00. The measurement of added phase andamplitude noise for amplifiers and other signal handling components should specify thesignal level since the AM noise level and the contribution due to AM-to-PM conversiondepend on the signal level.2. page TN-6For a direct measurement, time accuracy only has meaning when the phase of the timebaseoscillator of the frequency counter is known with respect to some time standard.Either it is phase locked or is calibrated with respect to that standard at the time ofmeasurement. The phase of the time-base oscillator can then be measured with respectto the phase of the frequency standard being calibrated (accounting for cable delays, etc.).Except for the cycle ambiguity of the carrier, the phase of the frequency standard beingmeasured can carry time information and have time accuracy. This technique is notcommon, but is very useful and eliminates divider noise that typically occurs in going from5 or 10 Mhz to 1 pulse per second. Caution must be exercised to assure that the phasepoint measured in a sine wave is at a reproducible voltage and impedance so that the cycleambiguity is an exact integer.3. page TN-35A second-order servo loop provides substantially enhanced performance. See, for example,F.L. Walls and S.R. Stein, "Servo techniques in oscillators and measurement systems," NBSTech. Note 692 (1976).4. page TN-35This error can be identified and corrected using the phase modulation scheme describedin paper BA on page TN-B6.5. page TN-36Low-noise DC amplifiers have been substantially improved since publication of this paper.6. pages TN-37, TN-91. TN-BO, TN-174, TN-206 and TN-218The reader is reminded that the discussions of frequency-domain measurements assumeincoherent noise processes. Often the phase noise spectrum of a signal will contain brightspectral features (spurious lines) other than the carrier. Frequency-domain measurementsare often useful in identifying such features. But if these spurious lines are narrowcompared to the measurement bandwidth, statistical measures such as Sq\(f) and 2(f) arenot appropriate. It is better to specify the phase deviation in terms of the rms value ofthe phase deviations, rms(rms radians), without reference to bandwidth. This specificationin rms radians can be related to the Allan variance (see note # 8 below).7. page TN-51Humidity is often an important environmental factor. See, for example, J.E. Gray, H.E.MacWan and D.W. Allan, "The Effect of humidity on commercial cesium beam atomicclocks," 42ndAnnual Symp. on Frequency Control, pp. 514-518 (1988) and F.L. WaUs, "TheTN-337

Influence of Pressure and Humidity on the Medium and Long-Term Frequency Stabilityof Quartz Oscillators," 42nd Annual Symp. on Frequency Control, pp. 279-283 (1988).8. page TN-74For a bright line (one which is narrow compared with the measurement bandwidth), thesolution of eq 12-27 simplifies towhere ¢rmsis the rms value of the phase deviations. The above relationship may be usefulwhere one is trying to determine the effect of a bright line in the time domain. If thebright line is the dominant factor, the plot of ayCr) versus r has strong sin 2 (1rfr) oscillationsand it can be ambiguous. In that case, it is better to provide a specification in termsof

15. page TN-124In eq 9 the subscript, k - n, should read k + n. That is, the equation should readk+Il-1-'I' 1 ~ -'1'0 x k + 1l - XJcYk - - LJ Yi ­n i-k ~16. page TN-125The quantity a~ 1) is now commonly known as moda~ 1). This latter form has beenrecently adopted by IEEE as the standard terminology.17. page TN-139IEEE Std 1139-1988, IEEE Standard Definitions of Physical Quantities for FundamentalFrequency and Time Metrology, is an almost exact replica of this paper (D.1). The paperwas published during the latter period of the development of the standard. The onlysubstantial difference is that, wherever it occurs, the word "departure" in the paper isreplaced in the IEEE standard by "deviation."18. page TN-146This widely cited paper provided the de facto standards for terminology and oscillatorcharacterization until the recent adoption of the IEEE standard presented in paper C.l(page TN-139). For terminology, the latest IEEE standard should always take precedence.This paper (C.2) is fairly consistent with the IEEE standard. One exception is that, in thispaper, N denotes the number of frequency measurements. The symbol M in the IEEEstandard is the same as N in this paper. In the standard, the equation relating M and NisM=N-1.19. page TN-151Equation (23) can be substantially simplified as shown, for example, by Stein (page TN-74,eq (12-27)), which is..o~(.) - 2 fStj,(f)sin4(1tft)d/.(1tV .)2 0 o20. page TN-154In eq 36 the T outside the brackets should be a 1. The equation should read21. page TN-160The 2 expressions listed as eq (95) are in error. They should read-h-1.2[3 + 2lnr - 1/(6r 2 )] r>1-h_1~[3 - 2lnr] r< 1.TN-339

22. page TN-160In eqs (101), (102), (103), (104), and (105), a Greek '1 was mistakenly replaced by thenumber 2. In each of these equations the quantity [2 + In(21l'f h T)] should be replaced bythe quantity b + In(21l'f h T)]. '1, Euler's constant, has the value 0.5772156649 .23. page TN-162This paper is included in this collection because it presents the internationally acceptedterminology and definitions. There are no substantial inconsistencies with the new IEEEstandard (paper C1, page TN-139), but the latest IEEE standard should is considered tobe the most up-to-date authority. A new version of this international report should beissued by the CCIR in 1990.24. page TN-171The definitions for symbols used in this paper are fairly consistent with those adopted inthe recent IEEE standard (C1). One exception is that, in this paper, N denotes thenumber of frequency measurements. The symbol M in the IEEE standard is the same asN in this paper. Another is that, while this paper uses J.L as the exponent of r in describingthe power-law noise processes, the paper adopts the opposite sign convention for J.L.vet) is used where many other papers use Vet) for instantaneous voltage. Some confusionis generated when this small v is typeset in the equations to look almost identical to theGreek v, a symbol which is used exclusively to represent frequency. For example, Inequation (1) the left-hand quantity is voltage, whereas the v(t) and va in equation (4) areclearly frequencies. Finally, the authors of this paper, in equation (2), define E(t) as thenormalized amplitude fluctuations, a very sound choice, but the reader should note thatmost other papers have not normaljzed it.25. page TN-175For consistency with figure 12 and the text, vet), the left-hand member of eq (11) shouldprobably be u(t).26. page TN-I77Walls, Percival and Irelan (DA) have recently addressed the more accurate specificationof the quantity p in eq (12).27. page TN-179It is important to note that the expressions in Table 2 in this paper are derived assuminguse of a single-pole filter. The calculations can also be done using an ideal (infinitelysharp) filter. The solutions in these two limits are useful because they define the range ofpractical values (using n-pole filters) for the expressions. Table I in this section is anexpansion of Table 2 of Lesage and Audoin providing both the single-pole results as wellas the results for an infinitely sharp filter. There are discrepencies in several of the coefficientsbetween terms in Table 2 in the paper and those in Table I on the next page.28. page TN-180Barnes and Allan (paper D.B) have recently completed further analysis of the effect ofdead time on measurements.TN-340

Table 1. Asymptotic forms of a~( r) for various power-law types and two filter types. Note: wh./2'ff = f his the measurementsystem bandwidth, often called the high-frequency cutoff. In =loge.Name of Noise ex S/O a/(T)w hr»l wh. r »l Whr <

29. page TN-180If the ratio of T/ T is constant and greater than 1 (the usual case), the problem describedis eliminated. However, in taking data for a plot of Oy( T) versus T, it is difficult toachieve this in the hardware and not possible to do it with software processing alone. Forfurther discussion see paper D.8 (page TN-296).30. page TN-197The most recent definitions and concepts for spectral density are given in a new IEEEstandard (paper CIon page TN-B8). This new standard should be consulted as the latestauthority on definitions and terminology.31. page TN-198The newly accepted definition of ~(f) is given in paper Cl. This new definition, ~(f) ==Y2Sif), was always valid for Fourier frequencies far from the carrier. It has now beenextended to cover all frequencies.32. page TN-217Equation (73) should read 20 log (final frequency/original frequency).33. page TN-239On page TN-198 the authors refer to a paper by Glaze (1970). The reference, apparentlylost in printing, is: Glaze, D.J. (1970). "Improvements in Atomic Beam Frequency Standardsat the National Bureau of Standards, "IEEE Trans. Instrum. Meas. IM-19(3), 156-160.34. page TN-257There are two errors in Table 2. Under R(n) the first entry should be l/n rather than 1.The second item in the same column is not single valued (1), but takes on different valuesfor different measurement bandwidths. The reader is referred to section A.6 (page TN-9)for a discussion of this topic.35. pages TN-261 and TN-262Subsequent work on moda~T) and R(n) is reported in section A.6 (page TN-9) of thisreport. There are some differences between the results in A.6 and the ones reported inTables I and II and Figure 4 in this paper.36. page TN-264To be consistent with other papers in the literature, ¢(t) in eq (1) should probably bewritten as xl(t).37. page TN-268The term e(t) in eq (3) is normally used to represent the amplitude fluctuations in theoutput voltage of an oscillator. This term might be better designated Y1(t).TN-342

38. Page TN-30There are two errors in eq 6.6. The equation for Flicker PM is missing a square root inthe exponential. It should readFlicker PM= (1 N-l I (2n+I)(N-l)jV,df . . exp n n .2n 4In the equation for Flicker PM, the numerator of the upper term should read 2(N - 2)2instead of 2(N - 2). The equation should readFlicker FM d.f ""2(N-2)2 for n:= 12.3N-4.9'SN 2 for n ~ 24n(N +3n)'39. Page TN-85There are errors in two of the terms of Table 12-4. The small n in the denominator of thefirst logarithmic term for flicker phase should be an ill, and the whole quantity in thesquare bracket should have an exponent of 1/2 . The equation should readFlicker phase exp[IO( ~~ 1 ) IO( (2m + 11(N - 1) )rThe (N - 2) in the flicker frequency term should be replaced by (N - 2)2 so that df is2(N- 2)2Flicker frequency for m:= 1 .2.3N -4.940. Page TN-279There is a factor N missing in the first expression under eq A6. It should readA = 3[3N(N+l)+2] .41. Page TN-307The fifth identity for BI, BI(N, 1,/-,) = 1 for /-' < 0 and = [2/N(N - 1)]..... for /-' > 0,should be deleted. The identity for /-' < 0 is incorrect and that for jJ. > 0 is superfluousconsidering the identity above it which covers all cases for jJ. *- O.TN-343

NIST-114A(REV. 3-89)4. TITLE AND SUBTITLEU.S. DEPARTMENT OF COMMERCENATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGYBIBLIOGRAPHIC DATA SHEET1. PUBLICATION OR REPORT NUMBERNIST/TN-13372. PERFORMING ORGANIZATION REPORT NUMBER3. PUBLICATION DATEMarch 1990Characterization of Clocks and Oscillators5. AUTHOR(S)D.B. Sullivan, D. W. Allan, D. A. Howe, and F.L. Walls, Editors6. PERFORMING ORGANIZATION (IF JOiNT OR OTHER THAN NIST, SEE INSTRUCTIONS) 7. CONTRACT/GRANT NUMBERU.S. DEPARTMENT OF COMMERCENATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGYGAiTHERSBURG, MD 20899 8. TYPE OF REPORT AND PERIOD COVERED9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)10. SUPPLEMENTARY NOTESn DOCUMENT DESCRIBES A COMPUTER PROGRAM; SF-185, FiPS SOFTWARE SUMMARY, IS ATTACHED.11. ABSTRACT (A 2OO-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNiFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBLIOGRAPHY ORLITERATURE SURVEY, MENTION IT HERE.)This is a collection of published papers assembled as a reference for those involvedin characterizing and specifying high-performance clocks and oscillators. Itis an interim replacement for NBS Monograph 140, Time and Frequency: Theoryand Fwzdamentals, an older volume of papers edited by Byron E. Blair. Thiscurrent volume includes tutorial papers, papers on standards and definitions, anda collection of papers detailing specific measurement and analysis techniques.The discussion in the introduction to the volume provides a guide to the contentof the papers, and tables and graphs provide further help in organizing methodsdescribed in the papers.12. KEY WORDS 16 TO 12 ENTRIES; ALPHABETICAL ORDER: CAPITALIZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)Key words: Allan variance, clocks, frequency, oscillators, phase noise, spectraldensity, time, two-sample variance.13. AVAILABILITYr-­..lLUNLIMITEDFOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).r-­-XORDER FROM SUPERINTIONOIONT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,WASHINGTON, DC 20402.ORDER FROM NATIONAL TECHNiCAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.ELECTRONIC FORM14. NUMBER OF PRINTED PAGES35215. PRICE*U.S. GOVERNMENT PRINTING OFFiCE 1998-0-773-022/40121