NIST Technical Note 1337: Characterization of Clocks and Oscillators

L + h o 2,ADDITIONAL VARIANCES THAT HAY BE USEP TO PESCRIBEFREOOENcy INSWILITIU IN TIlE TItlE ool1llIN1. Modified Allan or Modified Tw0-$4mple Vari.nce~,2ll.lThis implicitly assumes that the random driving mechanismfor each term is independent of the others. Inaddition, there is the implicit assumption that themechanism is valid over all Fourier frequencies, whichmay not always be true.The values of h$ are characteristic models of oscillatorfrequency noise. For integer values (as Often seems tobe the case for reasonable models), ~ = -a - I,for -3 ~ a ~ I, and ~ ~ -2 for Q ~ 1, where 0,2(,)_r».Table 2 gives the coefficients of the translation amongche frequency stability measures from time domain tofrequency domain and from frequency domain to timedomain.The slope characteristics of the five independent noiseprocesses are plotted in the frequency and time domains1n Fig. 1 (log-log scale).Log Sy (t)f--~-2 I I I f 2/I I I V'\, f- II If l 17I'\.J I IfO I 1/1I I I I I ILog FOUrier Frequency, fInstead of the use of 0 2(,), a "Modified Variance" Mod0,2(,) may be used to characterize frequency iostabilities(Stein, 1985; Allan, 1987). It has the property ofyielding different dependence on r for white phase noiseand flicker phase noise. The dependenc. for Hod 0,(')is ,-1/2 and ,-1 respectively. Mod ~,2(,) is defineda.!:Mod ~2(,) '" 2y 2rwhere N is the original number of time measurementsspaced by,. and, '" m,. the sample time of choice(N=M+l). A device or signal shall be characterized by aplot of a,(') or 0,1(,) or Mod ar(,) or Kod 0,1(,) vs.sampling time '. or by tabulating discrece values or byequivalent means such as a statement of power laws(Appendix I).2. Other VariancesSeveral other variances have been introduced by workersin this field. In pareicular, before the introductionof the ~.o-sample variance, it was standard practice touse che sample variance, SI, defined as2sLog S¢(t)i\ I I I I I I I\f-4\ I I I I\I I I I I I~ I I I Ij ~ fl-31 I II 1\1 I I II I ~ f-~ I I I, I 1\ I I III I I \ f-1 II I~ I I fO I II , I , I I I I I ILog Fourier Frequency, fIn practico it may be obeained from a set of measurementsof the frequency of the oscillator as2 1 Ns = L (y - y)2N i"'l iThe sample variance diverges for some types of noiseand, therefore, is not generally useful.Ocher variances based on the structure function approachcan also be defined (Lindsey and Chi, 1976). Forexample. there are the Hadamard variance. the threesamplevariance and the high pass variance (Rutman1978). They are occasionally used in research andscientific works for specific purposes. such asdifferentiating betveen different t;rpes of noise and fordealing vi,h sys~ematics and sidebands in the speccrum.App;:mrx IIILeg cr y (T),,,I -1IT, I I I I I I r1"\,1 T-;-~ I I I I ITtn I~I 1,.-0 I IIXII , I I I IfLog Sample Time, 1"Slope characteristics of the fiveindependent noise I:rocesses.FrGtll>: 1423Allan. D. ~., Statistics of atomic frequency srandards.Proc. IEEE, Vol. ~.22l-230, (Feb 1966).Allan, D. ~ .. et al., Perfo~ance, modeling. and simula~i~naf ~om. ~.slum be4m clQC~. r~oc. 27~hAnnual Symposium on Frequency Control, 334-346,(1973) .Allan, D. W., The measurement of frequency and frequencystability of precision oscillators. P~~c. 6ehAnnual Precise Time and Time Incerval ••anningMe.:ing. 109-142, (Dec 1974).TN-143