NIST Technical Note 1337: Characterization of Clocks and Oscillators

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The "Allan Variance" 1s defined as:U (1) - a(a) X[12]'{~hl-a+l- It\ -a+l] (10)(5)where the brackets "< >" denote infinite timeaverage. Using equation (4), one may write:It has been shown that typically 0 2(1)Y 1 2varies as 1 1l , and that 1.1 = -a-1 for -3 ~ a ~ +1. 'Hence, we see one of the dimensions of usefulnessof 0y2(t); 1.e., ascertaini ng the dependence on tallows an estimate of a (the power law spectraltype of noise). However, if a ~ +1, then 1.1 =-2,and the 1 dependence becomes somewhat ambiguous asto the type of noise in this region.It is interestingto note that in the region a ~ +1, cry2(1)is bandwi dth (fh) dependent; 1. e., the bandwi dthof the measurement system will affect the value ofo (1), and furthermore, one may use the bandwidthy 3dependence to determine the value of a (see alsoAppendix Ref. Z).Development of the Modified Allan VarianceOne may also write Oy2(1) in terms of ageneralized autocovariance function:whereand wherethe classical autocovariance function of x(t).(7)(8)(9)Using the Fourier transforms of general ized functions,one may determine the coefficients relatingthe power spectral density to o/(t). Ref. 1gives these relationships.It is of interest tonote that U (1) has the following approximate formxin the region a ~ + 1 (see Appendi.: R.. f. 2):Hence, one notes that by changing the reciprocalbandwidth as well as 1, one affects o/(t) insimilar ways, depending on the value of (Y, Fromthis, one should be able to deduce the value of a,since the bandwith dependence becomes stronger fora moving positive from +1, and the 1 dependencebecomes stronger as a moves negative from +1.Onecan change the bandwidth in the hardware or in thesoftware. In the past, it has typically been donein the hardware. 3 James Snyder 4 has shown that itis relatively easy to change the bandwidth in thedata processing by a clever technique and we havefollowed his lead.a newIn particular, we have chosenvariance analysis scheme which coincideswith the Allan variance at the minimum sampletime, 1 , (i.e., minimum data spacing), but whichochanges the bandwidth in the software as thesample time, 1, is changed.Each reading of the time deviation, xi' hasassociated with it an intrinsic nominal (hardware)measurement system bandwi dth, f h' Defi net = _1 . and similarly we may define a softwareh 2nf'bandwidt~, f = fhln, which is lin times narrowersthan the hardware bandwidth. This software bandwidth can be real i zed by averagi ng n adjacentx.'s·, 's1 = n1 where 1 =l/f. We have definedh' s sa modified Allan variance which allows the reciprocalsoftware bandwidth to change linearly with theJsamp 1e time, 1:where 1 = nl 'oG for n = 1.have formed2x . + x.) ] ~ (11)1+n 1 JEq. 11 clearly coincides with Eq.One can see that, in general, wea second difference of three timereadi ngs with each of the three be i ng an average,of n of the x.'s (with non-overlapping averages).As n increases, the (software) bandwidth decreasesand this bandwidth varies just as f s: fh/n.For a fi ni te data set of N I readi ngs of xi(i ~ 1 to N), we may write an estimate:471IN-255
N-3n+lEj=1Mod Oy2 (t) =0 2(t) {.! + .,.:-=1=-- .,,.Y n n2 4n-a+1 _ (2n)-a+1(12) n-1L (n-i)' [_6(a+l + 4(n+i)-«+1 (15)i=1-(2n+i)-a+l + 4(n-i)-a+l - (2n-i)-a+~}Eq. 12 is easy to program, but takes more time toSince we know that a 2(t) is well behaved in thiscompute than for ay
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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.
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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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Page 251 and 252: average frequency over the interval
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Page 263: Proc. 35th Ann. Freq. Control Sympo
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Page 269 and 270: IEEE TRANSACTIONS ON INSTRUMENTATlO
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Page 273 and 274: From: Proceedings of the 15th Annua
Page 275 and 276: where c = Dr/2. In regression analy
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Page 279 and 280: Coefficient of simple determination
Page 281 and 282: 100%CUMULATIVE PERIODOGRAM ~50%(,
Page 283 and 284: 100%CUMULATIVE PER I ODOGRAM50%U'10
Page 285 and 286: TABLE 6.STANDARD DEVIATIONSPROCEDUR
Page 287 and 288: APPENDIX AREGRESSION ANALYSIS(Equal
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Page 291 and 292: APPENDIX BREGRESSIONS ON LINEAR AND
Page 293 and 294: REFERENCES1. D.W. Allan, "The Measu
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Page 297 and 298: 100%CUMULATIVE PERIODOGRAM -50%U'
Page 299 and 300: 100%CUMULATIVE PERIODOGRAM50%(J'l..
Page 301 and 302: FREQUENCY DRIFT AND WHITESTANDARD D
Page 303 and 304: MR. McCASKILL:Well, let me go furth
Page 305 and 306: NIST Technical Note 1318, 1990.VARI
Page 307 and 308: By definition, white noise has a po
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Page 311 and 312: Some useful asymptotic forms of B 3
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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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