NIST Technical Note 1337: Characterization of Clocks and Oscillators

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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
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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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12 FREOU::NCY AND TIME MEASUREMENT2
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12 FREQUENCY AND TIME MEASUREMENTI~
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12 FREOUENCV AND TIME MEASUREMENT21
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228 SAMUEL R. STEIN9.3 GHzOSCILLATO
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230 SAMUEL R. STEINThe combination
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232 SAMUEl R STEINHowever, the spec
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400 CHAPTER BIBLIOGRAPHIESAllan. D.
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402 CHAPTER BIBLIOGRAPHIESBaugh. R.
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404 CHAPTER BIBLIOGRAPHIESConway. A
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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 and 132: 648IEEE TRANSACTIONS ON ULTRASONICS
Page 133 and 134: 650 IEEE TRANSACTIONS ON ULTRASONIC
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
Page 147 and 148: NJ:",--0~.8~..(/)-110-1:20-130-140m
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Page 151 and 152: 2. Copy.r,ion Beeween Frequency and
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 159 and 160: e4.RNES et al.: CH."R.ACTERrZ.4.TIO
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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: 142 Rep. 580--2Reprinted, with perm
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
Page 183 and 184: 524 LESAGE AND AUDOINKl- 1410- 11 \
Page 185 and 186: 526LESAGE AND AUDOINQuartz ClJstalF
Page 187 and 188: 528 LESAGE AND AUDOINMinrFrequent,r
Page 189 and 190: 530 LESAGE AND AUOOINsuch as c:r;(T
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Page 193 and 194: 534 LESAGE AND AUDOINof measurement
Page 195 and 196: LESAGE AND AUDOIN2 - Vo )]53610- 1
Page 197 and 198: 538 LESAGE AND AUWINstability measu
Page 199 and 200: Copyright Ie) 1984 Academic Press.
Page 201 and 202: 7. PHASE NOISE AND AM NOISE MEASURE
Page 207 and 208: 7. PHASE ~OISE AND AM NOISE MEASURE
Page 211 and 212: 7. PHASE :'-iOrSE AND AM NOISE MEAS
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7. PHASE NOISE AND AM NOISE MEASURE
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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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