NIST Technical Note 1337: Characterization of Clocks and Oscillators

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SCM)4-M-R(O) - 4-M-R(r) - 2 oMoG(MT)M-l+ 2 I (M-n)[2 o G(nT) - G«M+n)T) - G«M-n)T)] 0n=1Following the work of Barnes and Allan [2,3], we can define the function U(r)by the relation,U(r) = 2 oR(O) - 2-R(r),and also defineF(nr) = G(nT)/U(r),where r = T/r_the form,The function U(r) for power-law power spectral densities hasU(r)which yieldsF(nr)Finally, the working relation can be written asB 3 (2,M,r,p)20M + M-F(Mr)M-l-I (M-n) [2 oF(nr) - F«M+n)r)n=1[2 + F(r)] 0MJL+2- F«M-n)r)]18TN-313
8lIN,r,lll) for r = .011\1\ N= ~ 816 32 M 128 2S6 512 102~ INF....--..................... _...------_..... - ..---- --- _... -_ ... -_ .......--..--..------_ ... - ..._-- ----------..--------------- - ......... ------ _..--------- ---_ .....---------2 1.000f+OO 1.000E+OO 1.000E+OO 1.000£+00 I.OOOE+OO I.OOOE+00 \.OOOE+OO I.OOOE+OO I. OOOE+OO I. 000f+00-1.8 1.001E+OO 1.210E+OO I. 3bOf+00 1.S45£+OO I. m£+OO 2. 06'9E+OO 2.349£+00 2. ~ S6E+00 2.~9~+OO 2.512£+00-1.6 I. 199£+00 I. ~87E+OO I. 893E+00 2.~SSE+OO 3. ;mE+00 ~. 440E+00 5.505E+OO 6.002£+00 6.1~+OO 6. 309E+00-I.~ 1.328£+00 1.8S6E+OO 2.688E+OO 3. 991E+OO 6.OS4E+OO 9. 500E+00 1.282E+01 1.~S6E+01 1.532£+01 1.S8SE+01-1.2 I. 482E+OO 2.W>E+OO 3.BBOE+OO 6.599£+00 1.1~5£+O1 2.026E+OI 2.9~7E+01 3.4$bE+01 3.7S4f:+01 3. 980E+01-I I.667E+OO 3.000E+OO 5.667E+OO I. IOOE+OI 2. 167E+OI ~.25SE+OI 6. 628E+OI 8. 19OE+OI 9.064E+O\ 1.000E+02-.8 I. IJ:8.4E+00 3.86OE+OO 8.303E+OO 1.828E+01 ~. ~1E+OI 8. 691E+OI I. 443E+02 1.86BE+02 2. I37E+02 2.519E+02-.6 2. 134E+{I() 4.959£+00 1.205£+01 2. 977E+01 7. 296E+01 I.697E+02 2.99SE+02 ~.076E+02 ~.8SIE+02 6.~23E+02-.4 2.0\07E+OO 6. 279E+OO I. 703E+Ol 4.655E+01 1.2~7E+02 3.106E+02 5.814£+02 8. 357E+02 1.~3E+03 1.715£+03-.2 2.677E+OO 7.714£+00 2.296£+01 6.836E+01 1.976E+02 5.22OE+02 1.03bE+03 1.58OE+03 2. 08-4E+03 5. 58IE+030 2.912E+OO 9.075E+OO 2.909E+01 9.282E+01 2.S57E+02 7.95IE+02 I.672E+03 2.719E+03 3. 826E+03.2 3.089E+OO 1.01BE+01 3.44S£+01 1.162£+02 3. 763E+02 1.097E+03 2.446£+03 4. 263E+03 6. 453E+03.4 3.203E+OO 1.095f+01 3. 857E+01 1. 354E+02 4. 572E+02 1.39OE+03 3. 287E+03 6. I72E+03 1.014£+04.6 3.268E+OO I. 143E+01 4. 133E+01 I. 495f+02 5.22IE+02 1. M9E+03 ~.I~2E+03 8.~I9E+03 1.5J4£+04.8 3.302£+00 1.170E+OI 4.303E+01 1.59OE+02 5. 708E+02 I.869E+03 ~.991E+03 1.104E+04 2.189£+041 3.319£+00 1.185E+O\ 4. 4Q3E+0 I 1.6S3E+02 6.066E+02 2.05SE+03 5. 842E+03 I. ~12E+04 3.109£+041.2 3. 327E+00 1.192£+01 ~. 46IE+OI 1.693E+02 6. 33OE+02 2.215£+03 6.717E+03 I.782E+04 4.3B3E+041.4 3.33Of+OO 1. I96E+OI ~.~94E+01 I. 72OE+02 6.53OE+02 2. 359E+03 7.MOE+03 2.234E+04 6.169£+041.6 3.332E+00 1.198£+01 4.514E+01 I.738E+02 6.688£+02 2. 493E+03 8.b37E+03 2. 793E+04 8. 697E+041.8 3. 333E+1)1) 1.199£+01 4.526E+OI I. 750E+02 6.819£+02 2. 623E+03 9.737E+03 3.~93E+04 1.23IE+052 3. 333E+OO 1.2OOE+OI 4.533E+01 1. 76OE+02 6.933£+02 2.752£+03 1.097E+04 4. 37BE+04 1.749£+05BICN,r,lIIU) for r = .03Mu \ N= 4 8 16 32 64 128 2S6 512 1024 INF------------------------------------------------------.--------------------------------------------------------------------2 1.000E+OO 1.000E+OO 1.000E+00 I.OOOE+OO 1.000E+OO I.OOOE+OO I.OOOE+OO I.OOOE+OO 1.000E+OO 1.000E+OO-I. 8 I. 09IE+OO 1.211E+OO 1.366E+OO 1.573£+00 I. 827E+00 I. 9SOE+OO 1.995£+00 2.010E+OO 2.015E+OO 2. 01 6E+OO-1.6 1.199£+00 I. 49\E+00 1.910E+OO 2. 532E+0
Page 1 and 2:
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
Page 54 and 55:
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
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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
Page 271 and 272: IEEE TRANSACTIONS ON INSTRUYlENTATI
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-
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
Page 289 and 290: andG == N (N - 1) (N - 2)Also, the
Page 291 and 292: APPENDIX BREGRESSIONS ON LINEAR AND
Page 293 and 294: REFERENCES1. D.W. Allan, "The Measu
Page 295 and 296: 5.12 )( 10 - 7 SEC
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
Page 309 and 310: where aZ(N,T,r) is the expected sam
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
Page 317 and 318: I!II.....I~I.....cIQ)IEQ) I~I~(J)
Page 319 and 320: -2,THE BIAS FUNCTION, 8 2 (r,p.)Fig
Page 321: AppendixWith reference to figure 1,
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
Page 354: 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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