NIST Technical Note 1337: Characterization of Clocks and Oscillators

336IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-B, NO.4, DECEMBER 1984The EMA V is a random function of m. Its calculation requiresan observatlOn tIme of duration 3m1.We consider f, the fractional deviation of the EMAV relativeto the modified Allan variance defined as follows:f= Mod &;(1) - Mod a~(1)Moda~(r) (16)The standard deviation a(f) off dermes the relative uncertaintyon the measurement of the modified Allan variance due to thefinite number of averaging cycles. We have 'a(e) = M d 2 ( ) {a 2 [Mod a;(1)1p/2 (I 7)o 0y 12 A2where a [Mod ay(r)J denotes the true variance of the EMAVsuch asa 2 [Mod &;(1)J = {(Mod a;(r)J 2 )- [Mod 0;. (r») 2. (18)We assume that the fluctuations y(t) are normally distributed[1O]. One can therefore express ([Mod a; (1')] 2) asm 2 ([Mod 0;(1')]2)= (m 2 + 2m) [Mod a~(r)J2+ 4 mil (m - P){2 1: (n - i)In o}2 +nI (19)pc I1= Iwhere In are integrals which depend on n and on the noise process.We have281l'21'2 n In = I" ~ S (f)cos 61l'np[roo .fX {6 cos 21f[TO i - 4 cos 21l'[ToU + n)- 4 cos 21l'[roU - n) + cos 21l'.f100 + 2n)+ cos 21l'[TO(i - 2n)} df. (20)For each noise component, the expression for a(e) can bededuced from the calculation of integrals involved in (20).These expressions are generally lengthy and complicated exceptfor white phase and white frequency noise modulations, whereintegrals In equal zero. We have limited the present analysisto these two noise components. We get for a(l:)2a(e)=-, for 0: = 2 and O. (21)mWe now compare (21) with previously published results relatedto the estimate of the Allan variance [5 J. For a given timeobservation of duration 3m1', it can be easily deduced from(5) that the relative uncertainty on the estimate of the Allanvariance varies asymptotically as 1.14 m- 1/2 and 1.0 m -112 for0: = 2 and 0, respectively. For these two noise components, theuncertainty on the EMA V is larger than the uncertainty on theestimated Allan variance, but of the same order of magnitude.V. CONCLUSIONWe have calculated the analytical expression for the modifiedAllan variance for each component of the model usuallyconsidered to characterize random frequency fluctuations inprecision oscillators. These expressions have been comparedwith previously published results and the link between theAllan variance and the modified Allan variance has beenspecified.The uncertainty on the estimate of the modified Allan vari­0018-9456/84/0900-0336$01.00 © 1984 IEEEance h~s been studied and numerical values have been reportedfor white phase and white frequency noise modulations.In conclusion, the modified Allan variance appears to be wellsuited for removing the ambiguity between white and flickerphase noise modulation. Nevertheless, the calculation of themodified Allan variance requires signal processing which iscomplicated, compared to the Allan variance. In the presenceof white or flicker phase noise, the Allan variance cannot beeasily deduced from the modified Allan variance. Furthermore,for a given source exhibiting differen t noise components,the determination of the Allan variance from the modifiedone is difficult to perform. For most of time-domain measurements,the use of the Allan variance is preferred.ACKNOWLEDGMENTThe authors would like to express their thanks to Dr. ClaudeAudoin for constructive discussions and valuable comments onthe manuscript.REFERENCES(1) D. W. Allan, "Statistics of atomic frequency standards," Proc.IEEE, voL 54, pp. 221-230, Feb. 1966.[2J J. A. Barnes et 01., "Characterization of frequency stability,"IEEE Trans. Instrum. Meat., vol. 1M-20, pp. 105-120, May 1971.[3] L. S. Cutler and C. L. Searle, "Some aspects of the theory andmeasurement of frequency fluctuations in frequency standards,"Proc. IEEE. vol. 54, pp. 136-154, Feb. 1966.[4) J. Rutman, "Characterization of phase and frequency instabilitiesin precision frequency sources: Fifteen years of progress," Proc.IEEE, vol. 66, pp. 1048-1075, Sept 1978.[5 J P. Lesage and C.· Audoin, "Effect of dead-time on the estimationof the two-sample variance," IEEE Trans. Insrrum. Meas .. volIM-28, pp. 6-10, Mar. 1979.16) J. J. Snyder, "Algorithm for fast digital analysis of interferencefringes," AppL Opt., vol 19, pp. 1223-1225, Apr. 1980.(7] -, "An ultra-high resolution frequency meter," in Proc. 35thAnnu. Symp. Frequency ConTrol (Fort Monmouth, NJ), 1981,pp.464-469.[8] D. W. Allan and J. A. Barnes, "A modified Allan variance withincreased oscillator characterization ability," in Proc. 35th Annu.Symp. Frequency ConTrol (Fort Monmouth, NJ), 1981, pp. 470­474.(9) B. Picinbono, "Processus de diffusion et stationnarite," C.R.Acad. Sci., vol 271, pp. 661-664, Oct 1970.(10) P. Lesage and C. Audoin, "CharacteriZation of frequency stability:Uncertainty due to the finite number of measurements,"IEEE Trans. Instrum. Meas., vol IM·22, pp. 157-161. June 1973.