NIST Technical Note 1337: Characterization of Clocks and Oscillators

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246 A. L. LANCE, W. D. SEAL, AND F. LABAARspectral density. The terminology for single-sideband versus double-sidebandsignals is totally distinct from the one-sided spectral density versus twosidedspectral terminology. They are totally different concepts. The definitionsand concepts of spectral density are set forth in NBS Technical Note632 (Shoaf et al., 1973).C. SPECTRAL DENSITIES OF PHASE FLUCTUATIONS IN THEFREQUENCY DOMAn,; *The spectral density 5 y (/) of the instantaneous fractional frequencyfluctuations y(t) is defined as a measure of frequency stability, as set forthby Barnes et at. (1970). Sif) is the one-sided spectral density ofIrequencyfluctuatiOns on a "per hertz" basis, i.e., the dimensionality is Hz - 1. Therange of Fourier frequency 1 is from ~ero to infinity. S6'(/)' in hertz squaredper hertz, is the spectral density of/requency fluctuations bv. It is calculatedas5 (/) = -. (bvrms)2 (16)6\" bandwidth used in the measurement of bv rmsThe range of the Fourier frequency 1 is from zero to infinity.The spectral density of phaseflucruations is a normalized frequency domainmeasure of phase fluctuation sidebands. S6"'(/)' in radians squared perhertz. is the one-sided spectral density of the phase fluctuations on a "perhertz ,. basis:5 (/) = &Prms (17)6 bandwidth used in the measurement of bt/>rms'The power spectral densities ofphase and frequency fluctuation are related byS6",(f) = (v~!f)S}'(/). (18)The range of the Fourier frequency 1 is from zero to infinity.5 m (f). in radians squared Hertz squared per hertz is the spectral densityof angular frequency fluctuations 150:The defined spectral densities have the following relationships:(19)S6,(f) = v6S/f) = (l/2n)2S 6o (f) = 12S6"'(/); (20)S6"'(/) = (l/W)2S 60 (f) = (vo!f)25y(/) = [5 6 .(/)/1 2 ]. (21)Note that Eq. (20) is hertz squared per hertz, whereas Eq. (21) is in radianssquared per hertz.The term 5.;rfP(V), in watts per hertz, is the spectral density of the (square.. See Appendix Note # 30TN-197
7. PHASE ~OISE AND AM NOISE MEASUREMENTS247root of) radio frequency power P. The power of a signal is dispersed over thefrequency spectrum owing to noise, instability, and modulation. This conceptis similar to the concept of spectral density of voltage fluctuationsSbv(f)· Typically, Sbv(f) is more convenient for characterizing a basebandsignal where voltage, rather than power, is relevant. S.;rfP(v) is typicallymore convenient for characterizing the dispersion of the signal power inthe vicinity of the nominal carrier frequency Va' To relate the two spectraldensities, it is necessary to specify the impedance associated with the signal.A definition of frequency stability that relates the actual sideband powerof phase fluctuations with respect to the carrier power level, discussed byGlaze (1970), is called !I!(f). For a signal with PM and with no AM, !I!(f)is the normalized version of S.;rfP(v), with its frequency parameter f referencedto the signal's average frequency Va as the origin such that f equalsV - Va' If the signal also has AM, !I!(f) is the normalized version of thoseportions of S.;rfP(v) that are phase-modulation sidebands.Because I is the Fourier frequency difference (v - va), the range of risfrom minus Va to plus infinity. Since !I!(f) is a normalized density (phasenoise sideband power),(22) *!I!(f) is defined as the ratio of the power in one sideband, referred to theinput carrier frequency on a per hertz of bandwidth spectral density basis,to the total signal power, at Fourier frequency difference f from the carrier,per one device. It is a normalized frequency domain measure of phase fluctuationsidebands, expressed in decibels relative to the carrier per hertz:power density (one phase modulation sideband)!I!(f) = . . (23)carner powerFor the types of signals under consideration, by definition the two phasenoisesidebands (lower sideband and upper sideband, at - f and f from Va,respectively) of a signal are approximately coherent with each other, andthey are of approximately equal intensity.It was previously show that the measurement of phase fluctuations (phasenoise) required driving a double-balanced mixer with two signals in phasequadrature so the FM-AM conversion resulted in voltage fluctuations atthe mixer output that were analogous to the phase fluctuations. The operationof the mixer when it is driven at quadrature is such that the amplitudesof the two phase sidebands are added linearly in the output of the mixer,resulting in four times as much power in the output as would be present ifonly one of the phase sidebands were allowed to contribute to the output., See Appendix Note # 31TN-198
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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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Page 171 and 172: 142 Rep. 580--2Reprinted, with perm
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Page 175 and 176: 146 Rep. 580-2The values of ha are
Page 177 and 178: 148 Rep. SSO-2TABLE II - Translalio
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Page 181 and 182: 522 LESAGE AND AUDOIN01., 1971J:S,I
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Page 185 and 186: 526LESAGE AND AUDOINQuartz ClJstalF
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Page 201 and 202: 7. PHASE NOISE AND AM NOISE MEASURE
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Page 211 and 212: 7. PHASE :'-iOrSE AND AM NOISE MEAS
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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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