NIST Technical Note 1337: Characterization of Clocks and Oscillators

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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'
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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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From: Precision FrequenC)' Control,
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12 FREQUENCY AND TIME MEASUREMENT 1
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12 FREQUENCY AND TIME MEASUREMENT19
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12 FREQUENCY AND TIME MEASUREMENT 2
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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.
Page 121 and 122: 408 CHAPTER BIBLIOGRAPHIESJackisch.
Page 123 and 124: 410 CHAPTER BIBLIOGRAPHIESlesage. P
Page 125 and 126: 412 CHAPTER BIBLIOGRAPHIESPaul. F.
Page 127 and 128: 414 CHAPTER BIBLIOGRAPHIESSorden. J
Page 129 and 130: 416 CHAPTER BIBLIOGRAPHIESYoshimura
Page 131: 648IEEE TRANSACTIONS ON ULTRASONICS
Page 135 and 136: 652 IEEE TRANSACTIONS ON ULTRASONIC
Page 137 and 138: IEEE TRANSACTIONS ON ULTRASONICS. F
Page 139 and 140: Powet spectral analysis of the outp
Page 141 and 142: major difficulty is designing a mix
Page 143 and 144: The phase noise in the source can c
Page 145 and 146: detail elsewhere [12]. The low pass
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Page 153 and 154: All.n, D. Y., .nd D...s, H., Picose
Page 155 and 156: 105Characterization of Frequency St
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Page 163 and 164: :l frequcllcy ~eparatioll froll1 th
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Page 167 and 168: B\R~f:S eL al.: CIHRACTERIZ.,TIOI<
Page 169 and 170: 8IR:-IES et al.: CH.'R.'CTER[Z.
Page 171 and 172: 142 Rep. 580--2Reprinted, with perm
Page 173 and 174: i44 Rep. 580-2If the initial sampli
Page 175 and 176: 146 Rep. 580-2The values of ha are
Page 177 and 178: 148 Rep. SSO-2TABLE II - Translalio
Page 179 and 180: ISO Rep. 580-2, 738-2BIBLIOGRAPHYAL
Page 181 and 182: 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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