NIST Technical Note 1337: Characterization of Clocks and Oscillators

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12 FREQUENCY AND TIME MEASUREMENT19512.1.1 Relationship between the Power Spectrum and thePhase SpectrumThe power spectrum differs from a delta function b(j) due to the presenceof the amplitude- and phase-noise terms, c(c) and ¢(t), respectively, includedin Eq. (12-1). Usually the noise modulation separates into two distinctcomponents, so that one observes a very narrow feature called the carrierabove the level of a relatively broad pedestal. The frequency that separatesthe carrier and pedestal is denoted fe' Below this frequency the spectraldensity of the carrier exceeds that of the pedestal. Assuming that the amplitudenoise is negligible compared to the phase noise and that the phasemodulation is small, the relationship between the power and phase spectrais given by (Walls and DeMarchi, 1975)STS(j) ~ VJ e-1(fcl x {J(f), if -fc
196 SAMUEL R STEINperforming measurements or the description of measurement results. ThisIsituation was complicated by the difficulty of comparing the various:~eSCriPtions. A measure of stability is often used to summarize someImportant feature of the performance of the standard. It may therefore not befpossible to translate from one measure to another even though the respective!measurement processes are fully described and all relevant parameters aregiven. This situation resulted in a strong pressure to achieve a higher degreeof uniformity.In order to reduce the difficulty of comparing devices measured in separatelaboratories, the IEEE convened a committee to recommend uniformmeasures of frequency stability. The recommendations made by the committeeare based on the rigorous statistical treatment of ideal oscillators thatobey a certain model (Barnes ee ai., 1971). Most importantly, these oscillatorsare assumed to be elements of a stationary ensemble. A random process isstationary if no translation of the time coordinate changes the probabilitydistribution of the process. That is, ifone looks at the ensemble at one instantof time, then the distribution in values for a process within the ensemble isexactly the same as the distribution at any other instant of time. The elementsof the ensemble are not constant in time, but as one element changes valueother elements of the ensemble assume previous values. Thus, it is notpossible to determine the particular time when the measurement was made.The stationary noise model has been adopted because many theoreticalresults, particularly those related to spectral densities, are valid only for thiscase. It is important for the statistician to exercise considerable care sinceexperimentally one may measure quantities approximately equal to either theinstantaneous frequency of the oscillator or the instantaneous phase. But theideal quantities approximated by these measurements may not both bestationary. The instantaneous angular frequency is conventionally defined asthe time derivative of the total oscillator phase. Thus,and the instantaneous frequency is writtendw(r) = dr [2rrvoe + et>(e)], (12-7)1 det>v(e) = Vo +--. (12-8)2n dtFor precision oscillators. the second term on the right-hand side is quitesmall, and it is useful to define the fractional frequency\'le) - Vo 1 det> dxy(t)= =--=- (12-9)• Vo 2n\'0 de de 'TN-66
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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
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
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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 70 and 71: From: Precision FrequenC)' Control,
Page 72 and 73: 12 FREQUENCY AND TIME MEASUREMENT 1
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Page 80 and 81: 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
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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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