NIST Technical Note 1337: Characterization of Clocks and Oscillators

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10.7 Time Domain-Frequency Domain TransformsA. Integral transformFigure 10.8 shows the well-Known integraltransform, which transforms a continuous timedomainsignal extending over all time into a.. rlf ~:ca 1 .". :" f" :i :J~ ;4; ,.. 1\ 1'. f'. r~:' ;: H:: ::, I , , , " .: ,t " .1 .. ,f I' U• :: i: H \: :: :1 :: :: = :' 14il ._ " .. ,. I ., ,.. '"FIGURE 10.7nents and the artificial sideband frequency componentsdisappear. In practice, when looking atcomplex input signals, the blOCk time, T, is notsynchronous wi til any component of the transformedpart of the signal. As a resul t, di screte frequencycomponents in the frequency domain haveassociated with them sidebands which come and godependi ng on the phase of the time wi ndow, T,re 1ati ve to the sine components of the i ncomi ngsignal. The effect is much like looking through apicket fence at the sidebands. 12An analogy to the sidelooe leakage and picketfenceeffect is to record the incoming time-varyingsignal on a tape loop, which has a length of time,T. The loop of tape then repeats itself with aperiod of T. This repeating signal is then coupledto a scanni ng or fil ter-type spectrum ana lyzer.The phase d,i scanti nui ty between the end of onepassage of the loop and the beginning of the loopon itself again represents a phase-modulationcomponent, which gives rise to artificial sidebandsin the spectrum analysis. A word of caution this is not what actually happens in a FFT analyzer(i.e., there is no recirculating memory). However,the Fourier transform treats the incoming block asif this were happening.FIGURE 10.8continuous frequency-domain signal extending overall frequency. This is the ideal transform. Inpractice, however, one deals with finite times andbandwidths. The integral transform then, at best,is an estimate of the transform and is so for onlyshort, well-behaved signals. That is, the signalgoes to zero at infinite time and at infinitelyhigh frequency.B. Fouri er seri esThe Foud er-seri es transform assumes peri 0dicity in the time-domain signal for all time.Only one period of the signal (for time T) isrequired for this kind of transform. The Fourierseries treats the incoming signal as periodic withperiod, T, and continuous. The transformed spec~trum is then discrete with infinite harmoniccomponents with frequency spacing Of~. This isshown in figure 10.9.~~Y{\). fl.:.;.i)o.~ JI:~~ .r ""~J)c i~\~.ll~.-._.~.. , ,'.FIGURE 10.9-"/I ,, .34TN--47
XI.TRANSLATION FROM FREQUENCY DOMAIN STABILITYMEASUREMENT TO TIME DOMAIN STABILITY MEASUREMENT ANO VICE-VERSA.ll.l ProcedureKnowing how t.o measure SolI(f) or Sy{f) for apair of asci 11 at01's, 1et us see how to trans latethe ~ower-law noise process to a plot of a 2 (t).yFirst, convert the spectrum data to Sy(f) , thespectral density Df frequency fluctuations (seesections III and VIII).There are two quantities..midl t:DlllPlrtely Sl)ecify S y(f) for a particularpower-law noise precess: (1) the s)ope on alog-log plot for a given range of f and (2) theFIGURE 10.10 amplitUde. Tile sl~ we shall denote by "or";Figure 10.10 shows the transfoTlll from a Si!lll"" therefore ~ is the straight line (on log-logpled t i me- domainsi gna1 to the f1'til1M!ocy domain. scale) which relat.es S (f) to f. The amplitudeyNote that in the frequency domain, t1le result is will be denoted "h "; it is simply the coefficientarepetitive in frequency. This effect, CtmIIIonly of f fDr a range at f. When we examine a plot ofcalled aliasing, is discussed ..arlil~T in sect.iDn spectral density of frequency fluctuations, we are10.3. Figure 10.9 and figure 10.10 show the sym looking at .a representation of the addition of allmetry between the time- and frequency-dmIIain trans t.ile power-law pT'l)a5ses (see sec. IX). We haveforms of discrete lines.CItC. Discrete fourier transform S (f) =YI:(U.l)Fi na11 y , we have the saJlll'l ed, period; c a=(assumed) time-domain signal which ;s t.ransformeddomain representation.10.nThis;s shown in figureIn section IX, five power-law noise processesto a discrete and repetitive (aliased) frequencyweTeoutlined with respect to SolI(f).ar!! the cDllllllonoscillators.Thes!! fiv!!ones encountered wi th preci s i onEquation (8.1) relates these noiseprocesses to Sy(f). One obtainsFIGURE 10. Uwith respect toSif)1. Random walk FM (f-2. ) II =-2(f_l )2. Flicker fM a =-I3. White fH (1) a = 04. Flicker ~ (f) a = 1S. White~ (fZ) a = 2slop!! onlog-logpaperTable ll.l is a list of coefficients fortranslation from a 2(t) to S (f) and from S~(f) toy y '"3STN-48
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Characterization ofClocks and Oscil
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PREFACEFor many years following its
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Page 9 and 10: TOPICAL INDEX TO ALL PAPERSThe foll
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Page 13 and 14: The first of these (D.l) by Lesage
Page 15 and 16: Table 1. Guide to Selection of Meas
Page 17 and 18: 10- 1410- 15c-een10- 1610- 1710- 18
Page 19 and 20: Table 2. Ratio of mod a~('r) to a~(
Page 21 and 22: 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
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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
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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
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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, "
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Page 70 and 71: From: Precision FrequenC)' Control,
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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.
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408 CHAPTER BIBLIOGRAPHIESJackisch.
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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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