NIST Technical Note 1337: Characterization of Clocks and Oscillators

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andConsider figure 1.2.V(t) = V sin (Lnvt).pLet's assume tnat the maximumvalue of "V" equals 1, hence "V " = 1 We sayp .that the voltage "\I(t)" is normalized to unity.If weknow the freguency of a signal and if thesignal is a sine ~, then we can determine theincremental change in the period "Til (denoted byAt) at a particular angle of phase."Frequency stabil ity is the degree to whichan oscillating signal produces the same valueof frequency for any interval. At, throughouta speci fi ed peri od of time".Let's examine the t ....o waveforms shown infigure 1.3. Frequency stability depends on theamount of time involved in a measurement. Of thetwo oscillating signals, it is evident that "2" ismore stable than "1" from time t 1 to t:) assumingthe horizontal scales are linear in time.+1, I ,v_l~i '~--1II ill "2. '
where V0 ;: nomi na I peak vo Itage amp I i tude,e(t) ;: deviation of amplitude from nominal,"0 ;: nominal fundamental frequency,~(t) ;: deviation of phase from nominal.Ideally "e" and "dl" should equal zero for alltime. However, in the real world there are noperfect oscillators. To determine the extent ofthe noise components "e" and "ill", we shall turnour attention to measurement techniques.The typical precision oscillator, of course,has a very stable sinusoidal voltage output with afrl!quency v and a period of oscillation T, whichis the reciprocal of the frequency (v =1/T). Onegoa1 is to measure the frequency and/or the frequencystability of the sinusoid. Instability isactually measured, but with little confusion it isoften called stability in the literature. Naturally,fluctuations in frequency correspond tofluctuations in the period. Almost all frequencymeasurements, with very few exceptions, are measurementsof phase or of the period fluctuationsin an osci 11 ator, not of frequency, even thoughthe frequency may be the readout. As an example,most frequency counters sense the zero (or nearzero) crossing of the sinusoidal voltage, which isthe point at which the voltage is the most sensitiveto phase fluctuations.One must also realize that any frequencymeasurement involves two oscillators. In someinstances, one oscillator is in the counter. Itis impossible to purely measure only one osc11, ator. In some instances one osci 11 ator may beenough better than the other that the fluctuationsmeasured may be considered essentially those ofthe latter. However, in general because frequencymeasurements are always dual, it is useful todefine:v - vY (t)·C" 1 a (1.2)vaas the fractional frequency difference or deviationof 05cillator one, v l ' with respect to a referenceoscillator V o divided by the nominal frequency v o 'Mow, Yet) is a dimensionless quantity and uSl!fu1in describing oscillator and clock performance;e.g., the time deviation, x(t), of an oscillatorover a period of time t, is simply given by:tx(t) = f y(t')dt' (1. 3)oSince it is impossible to measure instantaneousfrequency, any frequency or fracti ona1 frequl!ncymeasurement always involves sOflle saJllPle time, Ator "t"--some time window through which the oscillatorsare observed; whether it's a picosecond, asecond, or a day, there is always some sampletime. So when determining a fractional frequency,y(t). in fact what is happening is that the timedeviation is being measured say starting at sometime t and again at a later time, t + t. Thedifference in these two time deviations, dividedby t gi ves the average fracti ana1 frequency overthat period t:yet) = x(t + t) - x(t) (1.4)tTau, t, may be called thl! sample time or averagingtime; e.g., it may be determined by the gate timeof a counter.What happens in many cases is that one samplesa number of eyc1 es of an osci 11 ation duri n9 thepreset gate time of a counter; after the gate timehas elapsed, the counter latches the value of thenumber of cyc 1es so that it can be read out,printed, or stored in some othl!r way. Then thereis a delay time for such processing of the databefore the counter anns and starts again on thenext cycle of the oscillation. During the delaytime (or process time), infonnation is lost. Wehave chosen to call it dead time and in someinstances it becomes a problem. Unfortunately fordata processing in typical oscillators the effectsof dead time often hurt most when it is the hardestto avoi d. In other words, for times that areshort comparl!d to a second when it;s very difficultto avoid dead time, that is usually wOeredead time can make a significant difference in thedata analysis. Typically for many oscillators, if3TN-16
Page 1 and 2: Characterization ofClocks and Oscil
Page 3 and 4: PREFACEFor many years following its
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Page 9 and 10: TOPICAL INDEX TO ALL PAPERSThe foll
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Page 15 and 16: Table 1. Guide to Selection of Meas
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Page 19 and 20: Table 2. Ratio of mod a~('r) to a~(
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Page 23: From: Proceedings of the 35th Annua
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
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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
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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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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.
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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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