NIST Technical Note 1337: Characterization of Clocks and Oscillators

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where four quantities must be calculated from the data:Sx ...NLn=1x nNLn=lSnxNIn=1X n nSxxN= L Xn Zn=1DefineaB = bcWith these definitions, Eq. A3 can be rewritten in the matrix form:---X N T B + E: (A4)and the coefficients, B, which minimize the squared errors are given by:a '"'" B = = T • ( N'N )-1 • ( N'X ) (A5)'" cThe advantage of evenly spaced data for these regressions is that, with a bit ofalgebra, the matrix, ( N'N )-1, can be written down in closed form:A B C( N'N )-1 B D E 1 / G (A6)whereC E F* A :=: 3 [3 (N + 1) + 2JB = -18 (2N + 1)C = 30D 12 (2N + 1) (BN + 11) / [(N + 1) (N + 2)]E -180 / (N + 2)F = 180 / {(N + 1) (N + 2)J* See Appendix Note #40566TN-279
andG == N (N - 1) (N - 2)Also, the inverse of T is just:=The complete solution for the regression parameters can be summarized as follows:There are four quantities which must calculate from the data:NSx I X n Snnx = I X n n2n=1 n=1NNSnx I X n n Sxx = I X 2n=l n=1Nfor n == 1, 2, 3, "" N. Based on these four quantities, the regression parametersare calculated from the seven following equations:n'" a (A Sx + 'a B Snx + '0 2 C Snnx) / G'"2b (B Sx + '0 D Sox + '0 E Snnx) / (G '0)'" 2 2c '" (C Sx + '0 E Sox + '0 F Snnx) / (G '0 )"'2°2 '"'" - To c Snnx) / (N - 3)= (Sxx a Sx'" 2 = 0 2 A / G°a;b 2 0 2 D / (G '0 2 )'" 2°c 8 2 F / (G T 04)561TN-280
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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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Page 237 and 238: 7. PHASE NOISE AND AM NOISE MEASURE
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Page 251 and 252: average frequency over the interval
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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
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Page 279 and 280: Coefficient of simple determination
Page 281 and 282: 100%CUMULATIVE PERIODOGRAM ~50%(,
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Page 285 and 286: TABLE 6.STANDARD DEVIATIONSPROCEDUR
Page 287: APPENDIX AREGRESSION ANALYSIS(Equal
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
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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
Page 313 and 314: Since the time-domain definition fo
Page 315 and 316: Stability Using Non-zero Dead-Time
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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
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Page 331 and 332: BllN,r,kl) for r = 1024It! \ N= 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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