Keyer. D. C., A test set for the accurate meAsurements IHnkler, C. K. R.• Ilall, lL C.• <strong>and</strong> Percival, D. a., The<strong>of</strong> phAse noise on high-quality aignal sources.U.S. Naval Ob.arvatory clock time raCarence <strong>and</strong> theIEEl Trans. lnatr. <strong>and</strong> ne.s., Vol. ~, 215-227. p.rfomance oC a samp1a <strong>of</strong> atolaie eloelu. Ketro(Nov 1970). 10gia, Vol. ~, 126-134, (Oct 1970).Percival. O. 8 .• A heuristic model <strong>of</strong> long-term atomic Yoshimura, K., <strong>Characterization</strong> <strong>of</strong> fraquancy atability:clock behavior. Proc. 30th Annual Symposium on Uncertainty due to the autocorr.lation <strong>of</strong> tha freFrequency Control, (June 1976).quency fluctuations. IEEE Trans. Instr. <strong>and</strong> Keas.,1&.21. 1-7. (MAr 1978).Ricci, O. W.• <strong>and</strong> 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 <strong>of</strong> the <strong>Technical</strong> Committee TC-).Time <strong>and</strong> Frequency. <strong>of</strong> the IEEE Instrumentation <strong>and</strong>Rutman. J., Comment on characterization <strong>of</strong> frequency MeAsurement Society. This paper is part <strong>of</strong> this Commitstability.IEEE Trans. lostr. <strong>and</strong> Meas., Vol. ~ tee's effort to develop an IEEE St<strong>and</strong>ard. To this end.ll, 85, (Feb 1972).the IEEE authorized a St<strong>and</strong>ards Coordinating Committee.sec 21. <strong>and</strong> the development <strong>of</strong> the IEEE St<strong>and</strong>ard underRutman. J .. <strong>Characterization</strong> <strong>of</strong> frequency stability: A PAR-P·l139. sec 21 is formed out <strong>of</strong> the 16M. UFFC <strong>and</strong>transfer function approach <strong>and</strong> its application to KTT Societies; tha authors are also members <strong>of</strong> sec 21.measurements via filtering <strong>of</strong> phase noise. IEEETrans. llUtr. <strong>and</strong> Meas., Vol. I.I1.:21.. 40-48. (Mar1974) .Rutman, J., <strong>Characterization</strong> <strong>of</strong> phase <strong>and</strong> frequencyinstabilities in precision frequency sources:Fifteen years <strong>of</strong> progress. Proc. IEEE. Vol. ~.1048-1075, (Sept 1978).Rutman, J., <strong>and</strong> Sauvage, C., Heasurement <strong>of</strong> frequencystability in the time <strong>and</strong> frequency domatns viafiltering <strong>of</strong> phAse noise. IEEE Trans. Instr. <strong>and</strong>Heas .• Vol. ~, 515-518. (Dec 1974).Rutman, J., <strong>and</strong> Uebersfeld, J., A model for flickerfrequency noise <strong>of</strong> oscillators. Proc. IEEE, Vol.iQ., 233-235, (Feb 1972).Sauvage. C., <strong>and</strong> 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., <strong>and</strong> Tetu, H., Time domain measurement <strong>of</strong>frequency stability: A tutorial approach, Proc.10th Annual Precise Time <strong>and</strong> Time Inteeval PlanningHeeting, 247-291, (1978).Vessot, R., Frequency <strong>and</strong> Time St<strong>and</strong>ards. Methods <strong>of</strong>Experimental Physics, Academic Press, New York,198-227, (1976).Vessot, R., Mueller, L., <strong>and</strong> Vanier. J., The specifica.ion<strong>of</strong> OScillator characteristics from measurementsmade in the frequency domain. Proe. IEEE,Vol. ~, 199-207 (Feb 1955).Von Neumann, J., Kent, R. H., Bellinson. H. R., <strong>and</strong>Ha:t, B. 1., The mean square successive difference,Ann. ~ch. Stat.• Vol. 11, 153-162, (1941).Walls. F.L., <strong>and</strong> Allan, D.W., Heasurements <strong>of</strong> FrequencyStability, Proc. IEEE, Special Issue on Radio Measurements<strong>and</strong> St<strong>and</strong>ards, Vol. ~, No.1, pp. 162168 (Jan 1986).Walls, F. L., Stein, S. R.. Gray. J. E.. <strong>and</strong> Glaze,D. J., Design considerations In state-<strong>of</strong>-the-artsignal processing <strong>and</strong> phase noise measurementsvstems. Proc. 30th Annual Symposium on FrequencyC~ntrol, (June 1976).Winkler, G. M. R., A brief review <strong>of</strong> frequency stabilitymeasures. Proe. 8th Annual Precise Ttme <strong>and</strong> TimeInterval Planning Meeting, 489-528, U.S. Navala.~ ••~eh LabQr.co~, waShington, D.C .. (Dec 1976).425 IN-145
105<strong>Characterization</strong> <strong>of</strong> 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 <strong>and</strong> '0 are the nominal amplitude <strong>and</strong> frequency, respectively,<strong>of</strong> the output. Provided that .(1) <strong>and</strong> ,;,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 <strong>of</strong> frequency stability isthe spectral density Sif) <strong>of</strong> the function yet) where the spectrumis considered to be one sided on a per hertz basis.An alternative definition for the measure <strong>of</strong> stability is theinfinite time average <strong>of</strong> the sample variance <strong>of</strong> two adjacent averages<strong>of</strong> 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, <strong>and</strong> T is the time interval between the beginnings <strong>of</strong>two Il'UCceSllive measurements <strong>of</strong> average frequency; then thelII'!1COud measure <strong>of</strong> stability iswhere ( ) denotes infinite time average <strong>and</strong> where T - T.In practice, data records are <strong>of</strong> finite length <strong>and</strong> the InfiniteManuscript received December I, 1970. The author!! <strong>of</strong> thiBpaper are membel'll <strong>of</strong> the Subcommittee on Frequency Stability <strong>of</strong>the <strong>Technical</strong> Committee on Frequency <strong>and</strong> Time <strong>of</strong> the IEEEGroup on Instrumentation <strong>and</strong> Me8BUrement.J. A. Barnes iB with the Time <strong>and</strong> Frequency Division, Inatilutefor Basic St<strong>and</strong>ards, 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 <strong>and</strong> 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 <strong>Note</strong> # 18time averages implied in the definitions are nomWly not avaiLt.blejthus estimates for the two measures must be used. E8timates <strong>of</strong>S.(f) would be obtained from suitable averages either in the timedomain or the frequency domain. An obvious estimate for "';CT) isParameters <strong>of</strong> the measuring system <strong>and</strong> estimating procedureare <strong>of</strong> critical importance in the specification <strong>of</strong> frequency stability.In practice, one should experimentally establish confidence limitsfor an estimate <strong>of</strong> frequency stability by repeated trials.BI(N, r, ~),B,(r, ~)f Ef.I,get)h..W/2Ti, i, k, m, nMNrGLOSSARY 01
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Characterization ofClocks and Oscil
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PREFACEFor many years following its
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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
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Table 2. Ratio of mod a~('r) to a~(
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of the frequency measured in this m
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From: Proceedings of the 35th Annua
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where V0 ;: nomi na I peak vo Itage
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ports of the pair of double balance
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fluctuations prior to the bias box
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up an interesting hierarchy of kind
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is determi ned by the measurement s
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Since each average of the fractiona
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Thus. with 9~ probability the calcu
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in fact. Also for n=l the "experime
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1n tenns of the typical performance
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and eq (1.1) we get2m>(t) = ~T(t) =
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varactor control in th@ reference o
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V nn5(45 Hz) = 100 nV por root hort
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We can use the convolution theorem
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though the concept of harmonics in
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and wz(t) is now the si!npled versi
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10.7 Time Domain-Frequency Domain T
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o/(t). In the table, the left colum
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Weean make the following general re
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-flO• r.1OSP1 t(f2-,~ 1Ci~-/'10 f
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frequency extent of the analysis ba
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28. P. Kartaschoff and J. Barnes, "
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_--_._~--~II..gIIIIII1.......oIIIII
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From: Precision FrequenC)' Control,
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12 FREQUENCY AND TIME MEASUREMENT 1
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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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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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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