NIST Technical Note 1337: Characterization of Clocks and Oscillators

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jo0N:; IR(n) = mod a~(f)a~(r)VS. n r = nroI I I1'I I ,,'.2 I 11111' 1 1 I ""~ 0825(i -I 0.6755~ IX 0 0.5022I>-'>-'----c(( 0.15Ir2" S (f) =h f exy ex'"0.01! I I ! I I ! I I I ! I ! I ! ! ! N ! I ! ! I I ! ! ,1 2 5 1a 2 5 100 2 5 1000n, Number of Samples AveragedFigure 2. Ratio of mod a~( r) to a~( r) as a function of the number of time samples averagedtogether to calculate either variance.
of the frequency measured in this manner over the interval r. The estimate of frequency,obtained by averaging the phase or time data, is degr~ded by about 10 percent from that obtainedby using just the end points [3]. This is analog~us to the algorithm for calculating modair) which, in this case, underestimates the uncerta~ty in measuring frequency by J2. Forwhite phase modulation (PM) noise, a = 2, the optimu~ estimator for frequency is obtained byaveraging the time or phase data over the interval r. i This is analogous to the algorithm forcalculating mod ay( r) which yields the one-sigma unbertainty in the estimate for frequencymeasured in this manner. This estimate for frequency ~s In better than that provided byay(r).Based on these considerations, it is our opinion that mod a y(r) can be profitably usedImuch more often than it is now. The presence of significant high-frequency FM or PM noise inthe measurement system, in an oscillator, or in an oscillator slaved to a frequency reference, isvery common. The use of mod CTy(r) in such circums~ances allows one to more quickly assesssystematic errors and long-term frequency stability. In lother words, a much more precise valuefor the frequency or the time of a signal (for a given m~surement interval) can be derived usingmod air) when n is large.iThe primary reasons for using ay( r) are that it ~s well known, it is simple to calculate, itis the most efficient estimator for FM noise (a ~ 0), :tnd it has a unique value for all r. Theadvantages of mod aye r) are cited in the above paragrkph. There are some situations where astudy of both air) and mod air) can be even more Gevealing than either one. The disadvantagesof using mod av(r) are that it is more complex tp calculate and thus requires more computertime and it has 'not been commonly used in the lit~rature, so interpretation of the results ismore difficult to reconcile with published information. Another concern sometimes raised is thatmod a/ r) does not have a unique value in regions ~ominated by FM noise (a ~ 0). Withrapidly increasing computer speeds, the computational d~sadvantage is disappearing. The correctedand expanded information presented in figure 2 ahd table 2 addresses the concern aboutuniqueness. :The primary disadvantage of using a/ r) is that ~he results can be too conservative. Thatis, if the level of high-frequency FM noise is high, the~ the results are biased high, and it cantake much longer (often orders of magnitude longer) to! characterize the underlying low-frequencyperformance of the signal under test. :References to Section A.6I[1] Walls, F. L, John Gary, Abbie O'Gallagher, [Roland Sweet and Linda Sweet, "TimeDomain Frquency Stability Calculated from the! Frequency Domain Description: Use ofthe SIGINT Software Package to Calculate T~e Domain Frequency Stability from theFrequency Domain," NISTIR 89-3916, 1989.I[2] Bernier, LG., Theoretical Analysis of the Modified Allan Variance, Proceedings of the41st Frequency Control Symposium, IEEE Cataldgue No. 87CH2427-3, 116, 1987.I[3] Allan, D.W. and J.E. Gray, Comments on the dctober 1970 Metrologia Paper "The U.S.Naval Observatory Time Reference and Perfoimance of a Sample of Atomic Clocks,"International Journal of ScientifiC Metrology, 7, 7~, 1971.!IN-12
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 13 and 14: The first of these (D.l) by Lesage
Page 15 and 16: Table 1. Guide to Selection of Meas
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Page 19: Table 2. Ratio of mod a~('r) to a~(
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
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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
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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
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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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