NIST Technical Note 1337: Characterization of Clocks and Oscillators

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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
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Characterization ofClocks and Oscil
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PREFACEFor many years following its
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0 • • • • • • • •
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TOPICAL INDEX TO ALL PAPERSThe foll
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of the wide range of measurement si
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The first of these (D.l) by Lesage
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Table 1. Guide to Selection of Meas
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10- 1410- 15c-een10- 1610- 1710- 18
Page 19 and 20: Table 2. Ratio of mod a~('r) to a~(
Page 21 and 22: of the frequency measured in this m
Page 23 and 24: From: Proceedings of the 35th Annua
Page 25 and 26: where V0 ;: nomi na I peak vo Itage
Page 27 and 28: ports of the pair of double balance
Page 29 and 30: fluctuations prior to the bias box
Page 31 and 32: up an interesting hierarchy of kind
Page 33 and 34: is determi ned by the measurement s
Page 35 and 36: Since each average of the fractiona
Page 37 and 38: Thus. with 9~ probability the calcu
Page 39 and 40: in fact. Also for n=l the "experime
Page 41 and 42: 1n tenns of the typical performance
Page 43 and 44: and eq (1.1) we get2m>(t) = ~T(t) =
Page 45 and 46: varactor control in th@ reference o
Page 47: V nn5(45 Hz) = 100 nV por root hort
Page 50 and 51: We can use the convolution theorem
Page 52 and 53: though the concept of harmonics in
Page 54 and 55: and wz(t) is now the si!npled versi
Page 56 and 57: 10.7 Time Domain-Frequency Domain T
Page 58 and 59: o/(t). In the table, the left colum
Page 60 and 61: Weean make the following general re
Page 62 and 63: -flO• r.1OSP1 t(f2-,~ 1Ci~-/'10 f
Page 64 and 65: frequency extent of the analysis ba
Page 66 and 67: 28. P. Kartaschoff and J. Barnes, "
Page 68 and 69: _--_._~--~II..gIIIIII1.......oIIIII
Page 72 and 73: 12 FREQUENCY AND TIME MEASUREMENT 1
Page 74 and 75: 12 FREQUENCY AND TIME MEASUREMENT19
Page 80 and 81: 12 FREQUENCY AND TIME MEASUREMENT 2
Page 82 and 83: 12 FREOU::NCY AND TIME MEASUREMENT2
Page 90 and 91: 12 FREQUENCY AND TIME MEASUREMENTI~
Page 94 and 95: 12 FREOUENCV AND TIME MEASUREMENT21
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Page 107 and 108: 228 SAMUEL R. STEIN9.3 GHzOSCILLATO
Page 109 and 110: 230 SAMUEL R. STEINThe combination
Page 111 and 112: 232 SAMUEl R STEINHowever, the spec
Page 113 and 114: 400 CHAPTER BIBLIOGRAPHIESAllan. D.
Page 115 and 116: 402 CHAPTER BIBLIOGRAPHIESBaugh. R.
Page 117 and 118: 404 CHAPTER BIBLIOGRAPHIESConway. A
Page 119 and 120: 406 CHAPTER BIBLIOGRAPHIESGardner.
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408 CHAPTER BIBLIOGRAPHIESJackisch.
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410 CHAPTER BIBLIOGRAPHIESlesage. P
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412 CHAPTER BIBLIOGRAPHIESPaul. F.
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414 CHAPTER BIBLIOGRAPHIESSorden. J
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416 CHAPTER BIBLIOGRAPHIESYoshimura
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648IEEE TRANSACTIONS ON ULTRASONICS
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650 IEEE TRANSACTIONS ON ULTRASONIC
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652 IEEE TRANSACTIONS ON ULTRASONIC
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IEEE TRANSACTIONS ON ULTRASONICS. F
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Powet spectral analysis of the outp
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major difficulty is designing a mix
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The phase noise in the source can c
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detail elsewhere [12]. The low pass
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NJ:",--0~.8~..(/)-110-1:20-130-140m
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is ~he preferred measure, since, un
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2. Copy.r,ion Beeween Frequency and
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All.n, D. Y., .nd D...s, H., Picose
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105Characterization of Frequency St
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B'R:sES et ai.: ClHR.,CTY.RIZHIO:-r
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e4.RNES et al.: CH."R.ACTERrZ.4.TIO
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MII.NES et al.. CHAIl.ACfERlZATlO:-
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:l frequcllcy ~eparatioll froll1 th
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MRNES et al.: CHARACTERIZATION OF F
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B\R~f:S eL al.: CIHRACTERIZ.,TIOI<
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8IR:-IES et al.: CH.'R.'CTER[Z.
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142 Rep. 580--2Reprinted, with perm
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i44 Rep. 580-2If the initial sampli
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146 Rep. 580-2The values of ha are
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148 Rep. SSO-2TABLE II - Translalio
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ISO Rep. 580-2, 738-2BIBLIOGRAPHYAL
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522 LESAGE AND AUDOIN01., 1971J:S,I
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524 LESAGE AND AUDOINKl- 1410- 11 \
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526LESAGE AND AUDOINQuartz ClJstalF
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528 LESAGE AND AUDOINMinrFrequent,r
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530 LESAGE AND AUOOINsuch as c:r;(T
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532 LESAGE AND AUDOINl.l-TEquations
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534 LESAGE AND AUDOINof measurement
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LESAGE AND AUDOIN2 - Vo )]53610- 1
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538 LESAGE AND AUWINstability measu
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Copyright Ie) 1984 Academic Press.
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7. PHASE NOISE AND AM NOISE MEASURE
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7. PHASE ~OISE AND AM NOISE MEASURE
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7. PHASE :'-iOrSE AND AM NOISE MEAS
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or, in rms radians,7. PHASE !'rOISE
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i. PHASE :-JOISE AND AM NOISE MEASU
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7. PHASE ~OISE AND AM NOISE MEASVRE
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7. PHASE NOISE AND AM :>lOISE MEASU
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_~ ~A _7. PHASE NOISE AND AM NOISE
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average frequency over the interval
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and frequency stability of precisio
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PHASE P~OT Clook No. ~- e I"d,....
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BL-\5ES .-\.\D \·A.Rl.-\.\CES OF S
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to a g
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while the Hanning ""!Odo"" yIelds1.
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Proc. 35th Ann. Freq. Control Sympo
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N-3n+lEj=1Mod Oy2 (t) =0 2(t) {.! +
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(a)1SIMULATED NOISE(b)1-10- 2- --10
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IEEE TRANSACTIONS ON INSTRUMENTATlO
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IEEE TRANSACTIONS ON INSTRUYlENTATI
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From: Proceedings of the 15th Annua
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where c = Dr/2. In regression analy
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'" Dr 22 = -7.507E-16 (about -6.5E-
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Coefficient of simple determination
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100%CUMULATIVE PERIODOGRAM ~50%(,
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100%CUMULATIVE PER I ODOGRAM50%U'10
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TABLE 6.STANDARD DEVIATIONSPROCEDUR
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APPENDIX AREGRESSION ANALYSIS(Equal
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andG == N (N - 1) (N - 2)Also, the
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APPENDIX BREGRESSIONS ON LINEAR AND
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REFERENCES1. D.W. Allan, "The Measu
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5.12 )( 10 - 7 SEC
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100%CUMULATIVE PERIODOGRAM -50%U'
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100%CUMULATIVE PERIODOGRAM50%(J'l..
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FREQUENCY DRIFT AND WHITESTANDARD D
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MR. McCASKILL:Well, let me go furth
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NIST Technical Note 1318, 1990.VARI
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By definition, white noise has a po
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where aZ(N,T,r) is the expected sam
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Some useful asymptotic forms of B 3
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Since the time-domain definition fo
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Stability Using Non-zero Dead-Time
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I!II.....I~I.....cIQ)IEQ) I~I~(J)
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-2,THE BIAS FUNCTION, 8 2 (r,p.)Fig
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AppendixWith reference to figure 1,
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8lIN,r,lll) for r = .011\1\ N= ~ 81
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BUN,r,IItJI for r =/tl\ !'I= 8 16 3
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81 (N,r,lIUl for r =ItJ \ IF 4 8 16
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__ ....... ~...........__ .........
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BllN,r,kl) for r = 1024It! \ N= 4 B
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B2(r,llU)It! \ r = .0001 .0003 .001
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B3( 2, P1,I", IIU) far r '" .01~\ ~
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83(2,I'f,r,~)for r '"""'III \ 2 4 B
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B312.I'I,r,llU) for r " 4I\J \ pt::
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B3(2,I'I,r,lIJ) Fer r ~ MI\J\ PI::
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B3(2,"',r,ail for r = 1024MIl \ I'l
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E. APPENDIX - Notes and ErrataThe n
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Influence of Pressure and Humidity
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22. page TN-160In eqs (101), (102),
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29. page TN-180If the ratio of T/ T
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NIST-114A(REV. 3-89)4. TITLE AND SU
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NIST Technical Note 1337: Characterization of Clocks and Oscillators

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