NIST Technical Note 1337: Characterization of Clocks and Oscillators

IEEE TRANSACTIONS ON INSTRUMENTATlON AND MEASUREMENT. VOL. 'IM·33. NO. 4. DECEMBER 1984333L INTRODUCTIONMany works [11-[ 5 J have been devoted to the characterizationof the frequency stability of ultrastable frequency sourcesand have shown that the frequency noise of a generator can beeasily characterized by means of the "two-sample variance"[2] of frequency fluctuations, which is also known as the"Allan variance" [2) in the special case where the dead timebetween samples is zero.An algorithm for frequency measurements has been developedby J. J. Snyder [6 J, [7]. It increases the resolution offrequency meters, in the presence of white phase noise. It hasbeen considered in detail by D. W. Allan and 1. A. Barnes [8].They have defined a function called the "modified Allanvariance" and they have analyzed its properties for the commonlyencountered components of phase or frequency fluctuations[3]. For that purpose, the authors of [81 have expressedthe modified Allan variance in terms of the autocorrelation ofthe phase fluctuations. For each noise component, they havecomputed the modified Allan variance and deduced an empiricalexpression for the ratio between the modified Allan varianceand the Allan variance.In this paper, we show that the analytical expression of thisratio can be obtained directly, even for the noise componentsfor which the autocorrelation of phase functions is not definedfrom the mathematical point of view. We give the theoreticalexpressions and compare them with those published in [8).The precision of the estimate of the modified Allan varianceis discussed and results related to white phase and white frequencynoises are presented.II. BACKGROUND AND DEFINITIONSIn the time domain, the characterization of frequency stabilityis currently achieved by means of the two-sample variance(2) (at(2, T, T» of fractional frequency fluctuations. It isdefined as(0;(2, T, T» = i «Ybi - J!I