NIST Technical Note 1337: Characterization of Clocks and Oscillators

TIME AND FREQUENCY5311974] critically depends on the precision of thep(p - 1)/2 frequency comparisons which can beperformed by arranging oscillators in pairs. Furthermore,the uncertainty in the determination ofO'y(T) translates directly into the uncertainty indetermining the h", coefficients if the frequencygenerator is perturbed by noise processes modeledby (7).The precision in the estimation of time domainmeasurements of frequency stability has been consideredby several authors [Tausworthe, 1972; Lesageand Audoin, 1973; Yoshimura. 1978]. It hasbeen determined for most of the experimentalsituations which can be encountered in the twosamplevariance characterization of frequencystability, with or without dead time (P. Lesage andC. Audoin, private communication, 1978).Calculation of the expectation value of the twosamplevariance according to (25) requires an inImitenumber of data. But, in practice, only m countingresults are available, and one calculates the estimatedaverage of the two-sample variance as follows:1 ... -1 _t).~(2, T, T, m) = L (Yhl - Y1)2 (29)2(m - I) k-IOne can easily show that the expectation valueof ()~(2, T, T, m) equals the averaged two-samplevariance with dead time. Thus the rmite numberof measurements does not introduce bias in theestimation of the two-sample variance.The estimated averaged two-sample variance(EATSV) being a random function of m, we needto characterize the uncertainty~ in the estimation.We thus introduce the variance of the EATSV,according to the common understanding of avariance. We set0'2 [b~(2, T, T. m»)= ([~;(2, T. T, m) - (0:(2, T, T») 2) (30)With the expression (29) of the EATSV we get0'2 [b~(2, T, T, m))withI]2("'.1 ",-I= [ 2: 2: 1l,1l/ )2(m - I) '_I /-1(31)Il, = (f,... - y,)~ - 2(