NIST Technical Note 1337: Characterization of Clocks and Oscillators

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DR. BARNES:I think that's true and you may have different reasons to do regressionanalysis, if your purpose is to measure a drift and understand theconfidence intervals, then I think what has been presented is reasonable,if you have as your purpose to look--to see if there are indications offunny behavior in a curve that has such strong curvature or drift thatyou can't get it on graph paper without doing that, I think it is a veryreasonable thing to do. I think looking at the data is one of thehealthiest things any analyst can do.582TN-295
NIST Technical Note 1318, 1990.VARIANCES BASED ON DATA WITH DEAD TIME BETWEEN THE MEASUREMENTSJames A. BarnesAustron, Inc.Boulder, Colorado 80301andDavid W. AllanTime and Frequency DivisionNational Institute of Standards and TechnologyBoulder, Colorado 80303The accepted definition of frequency stability in the time domain isthe two-sample variance (or Allan variance). It is based on themeasurement of average frequencies over adjacent time intervals,with no "dead time" between the intervals. The primary advantagesof the Allan variance are that (1) it is convergent for manyencountered noise models for which the conventional variance isdivergent; (2) it can distinguish between many important anddifferent spectral noise types; (3) the two-sample approach relatesto many practical implementations; for example, the rms change of anoscillator's frequency from one period to the next; and (4) Allanvariances can be easily estimated at integer multiples of the sampleinterval.In 1974 a table of bias functions which related variance estimateswith various configurations of number of samples and dead time tothe Allan variance was published [1]. The tables were based onnoises with pure power-law spectral densities.Often situations occur that unavoidably have dead time betweenmeasurements, but still the conventional variances are notconvergent. Some of these applications are outside of the time-andfrequencyfield. Also, the dead times are often distributedthroughout a given average, and this distributed dead time is nottreated in the 1974 tables.This paper reviews the bias functions Bl(N,r,~), and B2(r,~) andintroduces a new bias function, B3(2,M,r,~), to handle the commonlyoccurring cases of the effect of distributed dead time on thecomputed variances. Some convenient and easy-to-interpretasymptotic limits are reported. A set of tables for the biasfunctions are included at the end of this paper.Key words: Allan variance; bias functions; data sampling and deadtime; dead time between the measurement; definition of frequencystability; distributed dead time; two-sample variance1TN-296
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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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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
Page 253 and 254: and frequency stability of precisio
Page 255 and 256: PHASE P~OT Clook No. ~- e I"d,....
Page 257 and 258: BL-\5ES .-\.\D \·A.Rl.-\.\CES OF S
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Page 265 and 266: N-3n+lEj=1Mod Oy2 (t) =0 2(t) {.! +
Page 267 and 268: (a)1SIMULATED NOISE(b)1-10- 2- --10
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
Page 277 and 278: '" Dr 22 = -7.507E-16 (about -6.5E-
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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
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Page 319 and 320: -2,THE BIAS FUNCTION, 8 2 (r,p.)Fig
Page 321 and 322: AppendixWith reference to figure 1,
Page 323 and 324: 8lIN,r,lll) for r = .011\1\ N= ~ 81
Page 325 and 326: BUN,r,IItJI for r =/tl\ !'I= 8 16 3
Page 327 and 328: 81 (N,r,lIUl for r =ItJ \ IF 4 8 16
Page 329 and 330: __ ....... ~...........__ .........
Page 331 and 332: BllN,r,kl) for r = 1024It! \ N= 4 B
Page 333 and 334: B2(r,llU)It! \ r = .0001 .0003 .001
Page 335 and 336: B3( 2, P1,I", IIU) far r '" .01~\ ~
Page 337 and 338: 83(2,I'f,r,~)for r '"""'III \ 2 4 B
Page 339 and 340: B312.I'I,r,llU) for r " 4I\J \ pt::
Page 341 and 342: B3(2,I'I,r,lIJ) Fer r ~ MI\J\ PI::
Page 343 and 344: B3(2,"',r,ail for r = 1024MIl \ I'l
Page 345 and 346: E. APPENDIX - Notes and ErrataThe n
Page 347 and 348: Influence of Pressure and Humidity
Page 349 and 350: 22. page TN-160In eqs (101), (102),
Page 351 and 352: 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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