L + h o 2,ADDITIONAL VARIANCES THAT HAY BE USEP TO PESCRIBEFREOOENcy INSWILITIU IN TIlE TItlE ool1llIN1. Modified Allan or Modified Tw0-$4mple Vari.nce~,2ll.lThis implicitly assumes that the r<strong>and</strong>om driving mechanismfor each term is independent <strong>of</strong> the others. Inaddition, there is the implicit assumption that themechanism is valid over all Fourier frequencies, whichmay not always be true.The values <strong>of</strong> h$ are characteristic models <strong>of</strong> oscillatorfrequency noise. For integer values (as Often seems tobe the case for reasonable models), ~ = -a - I,for -3 ~ a ~ I, <strong>and</strong> ~ ~ -2 for Q ~ 1, where 0,2(,)_r».Table 2 gives the coefficients <strong>of</strong> the translation amongche frequency stability measures from time domain t<strong>of</strong>requency domain <strong>and</strong> from frequency domain to timedomain.The slope characteristics <strong>of</strong> the five independent noiseprocesses are plotted in the frequency <strong>and</strong> 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 <strong>of</strong> the use <strong>of</strong> 0 2(,), a "Modified Variance" Mod0,2(,) may be used to characterize frequency iostabilities(Stein, 1985; Allan, 1987). It has the property <strong>of</strong>yielding different dependence on r for white phase noise<strong>and</strong> flicker phase noise. The dependenc. for Hod 0,(')is ,-1/2 <strong>and</strong> ,-1 respectively. Mod ~,2(,) is defineda.!:Mod ~2(,) '" 2y 2rwhere N is the original number <strong>of</strong> time measurementsspaced by,. <strong>and</strong>, '" m,. the sample time <strong>of</strong> choice(N=M+l). A device or signal shall be characterized by aplot <strong>of</strong> 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 <strong>of</strong> power laws(Appendix I).2. Other VariancesSeveral other variances have been introduced by workersin this field. In pareicular, before the introduction<strong>of</strong> the ~.o-sample variance, it was st<strong>and</strong>ard 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 <strong>of</strong> measurements<strong>of</strong> the frequency <strong>of</strong> the oscillator as2 1 Ns = L (y - y)2N i"'l iThe sample variance diverges for some types <strong>of</strong> noise<strong>and</strong>, therefore, is not generally useful.Ocher variances based on the structure function approachcan also be defined (Lindsey <strong>and</strong> Chi, 1976). Forexample. there are the Hadamard variance. the threesamplevariance <strong>and</strong> the high pass variance (Rutman1978). They are occasionally used in research <strong>and</strong>scientific works for specific purposes. such asdifferentiating betveen different t;rpes <strong>of</strong> noise <strong>and</strong> fordealing vi,h sys~ematics <strong>and</strong> sideb<strong>and</strong>s 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 <strong>of</strong> the fiveindependent noise I:rocesses.FrGtll>: 1423Allan. D. ~., Statistics <strong>of</strong> atomic frequency sr<strong>and</strong>ards.Proc. IEEE, Vol. ~.22l-230, (Feb 1966).Allan, D. ~ .. et al., Perfo~ance, modeling. <strong>and</strong> simula~i~naf ~om. ~.slum be4m clQC~. r~oc. 27~hAnnual Symposium on Frequency Control, 334-346,(1973) .Allan, D. W., The measurement <strong>of</strong> frequency <strong>and</strong> frequencystability <strong>of</strong> precision oscillators. P~~c. 6ehAnnual Precise Time <strong>and</strong> 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. <strong>of</strong> the 29th AnnualSympo.iua on Frequency Control. 404-411,(Kay 1975).Allan. D. W.• <strong>and</strong> Hellwig, H. W.• Tise Devi.tion <strong>and</strong>Tim. Prediction Error for Clock Specific.tion.Ch.r.ct.rization, .nd Application, Proc. IEEE Position Location <strong>and</strong> 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. <strong>and</strong> Frequ.ncy (Time Domain) Charact.rization, Estimation <strong>and</strong> Prediction <strong>of</strong> Precision <strong>Clocks</strong> .nd <strong>Oscillators</strong>. IEEE Trans. UFFC Vol.~, 647-654 (Nov 1987).Atkinson. Y. K.• Fey, L.• <strong>and</strong> Newman, J .• Spectruatanalysis <strong>of</strong> extremely lov-frequency variations <strong>of</strong>quartz oscillators. Proc. IEEE. Vol. ~, 379-380,(Feb 1963).S.bitch, D., <strong>and</strong> Oliverio, J., Phase noise <strong>of</strong> 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 <strong>of</strong>frequency stability measurements. NSS Tech. <strong>Note</strong>683, (Aug 1976).Sames, J. A., Jones, R. R., Tryon, P. V., <strong>and</strong> Allan,O.Y .• Noise models for atomic clocks. Proc. 14thAnnual Precise Time <strong>and</strong> 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 <strong>of</strong> the Non-St.tionaryClock Error Function, Proc. 2nd European Frequency<strong>and</strong> Ti~e Forum, NeuchAtel (1988).Bl.ckman, R. B., <strong>and</strong> Tukey, J. M., The measurement <strong>of</strong>pow.r spectra, Dover Publication, Inc., New York,NY, (1959).8lair. B. E., Time <strong>and</strong> frequency: Th.ory <strong>and</strong> fundam.nt.ls.NBS Monograph No. 140, US GovernmentPrinting Offic., Washington, DC 20402. (May 1974).Boileau, E., <strong>and</strong> Picinbono, 8., Statistical study <strong>of</strong>phase fluctuations <strong>and</strong> oscillator stability. IEEETrans. Instr. <strong>and</strong> Meas .• Vol. Z2. 66-75,(liar 1976).Br<strong>and</strong>enberger, R., Hadom, F., Halford, 0., <strong>and</strong> Shoaf,J. H., High quality quart& crystal oscillators:frequency-domain <strong>and</strong> time-domain stability. Proc.25th Annual Symposium on Frequency Control. 226230, (1971).Chi, A. R., Th. mechanics <strong>of</strong> translation <strong>of</strong> frequencystability measures becveen frequency <strong>and</strong> timedomainmeasurements. Proc. 9th Annual Precis. Time<strong>and</strong> Ti2e Interval Planning Meeting, (Dec 1977).Cutler, L. S., <strong>and</strong> Searle, C. L.• Some aspects <strong>of</strong> thetheor/ <strong>and</strong> measure=.nt <strong>of</strong> frequency fluctuations infr.q~ney st<strong>and</strong>ards, Pro~. IEEE, Vol. ~, 136-154,(F.b 1966).424De Prins. J., <strong>and</strong> Corn.lissen, G., Analyse spectral•discrete. Eurocon (Lausanne, Swit%erl<strong>and</strong>), (Oct1971) .De Prins, J., De.comet, G.• Gord:i, M., <strong>and</strong> Tuine, J.,Frequency-domain int.rpretation <strong>of</strong> 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 <strong>and</strong> TimeInterval Planning K.eting, 575-618. (Dec 1976).Greetiball. C.A., Initiali%ing a Flicker Nois. Generator.IEEE Trans. Instr. <strong>and</strong> Meas .• Vol. ~, 222-224(Jun. 1986).Groslambert, S., Oliv.r, M.• <strong>and</strong> Uebersfeld, S.,Spectral <strong>and</strong> 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 <strong>and</strong> frequency.Vol. iQ. (May 1972).IEEE-NASA, Proceedings <strong>of</strong> Symposium on short-termfrequency stability. NASA Publication SP 80.(1964) .Proc. IEEE,Jones, R. M., <strong>and</strong> Tryon, P. V., Estimating time fromatomic clocks. NBS Journal <strong>of</strong> Research, Vol. ~,November 17-24. (Jan-Feb 1983).Kartasch<strong>of</strong>f, P., Computer Simulation <strong>of</strong> the ConventionalClock Model. IEEE Trans. Instr. <strong>and</strong> Heas., Vol. ~21, 193-197 (Sept 1979).Kroupa, V. F., Frequency stability: Fundamentals <strong>and</strong>Measurement. IEEE Press. IEEE Selected ReprintSeries. Prepared under the sponsorship <strong>of</strong> the IEEEInstrumentation <strong>and</strong> Heasurement Society, 77-80,(1984).Lesage P., <strong>and</strong> AUdoin, C., Correction to: <strong>Characterization</strong><strong>of</strong> frequency stability: Uncertainty due t<strong>of</strong>inite number <strong>of</strong> measurements. IEEE Trans. Instr.<strong>and</strong> Heas .• Vol. ~, 103. (Mar 1974).Lesage. P.. <strong>and</strong> Audoin, F., A time domain method formeasurement <strong>of</strong> the spectral density <strong>of</strong> fraquencyfluctuations at low Fourier frequencies. Proc.29th Annual Symposium on Frequency Control, ,94403, (May 1975).Lesage, P.. <strong>and</strong> Audoin, C., Correction to: Charactertzation<strong>of</strong> frequency stability: Uncertainty due tothe finite number <strong>of</strong> measurements. I~EE Trans.Instr. <strong>and</strong> Heas., Vol. ~, 270, (Sept 1976).Lesage, P.. <strong>and</strong> Ayi, T., Characteri%ation <strong>of</strong> frequencystability: Analysis <strong>of</strong> modified Allan variance <strong>and</strong>properties <strong>of</strong> its estimate. IEEE Trans. on Instr.<strong>and</strong> Meas., Vol. ~, 332-336. (Dec 1934).Lindsey, W. C., <strong>and</strong> Chie, C. M., Theory <strong>of</strong> oscillatorinstability based upon structure functions. Proc.IEEE, Vol. ~, 1662-1666, (Dec 1976).M<strong>and</strong>etbrot, 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
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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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0 •• 0 •• 0 •• 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
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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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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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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