Very long-term frequency stability: estimation using a special-purpose statistic

We introduce a statistic that can be used for a particularly difficult measurement problem, namely, determining frequency stability for frequency standards and oscillators for averaging times longer than those the traditional Allan deviation can estimate. Theoretical variance #1 ("Theol") has statistical properties that are like "Avar" (Allan variance), with two significant enhancements: (1) it can evaluate frequency stability at longer averaging times than given by the definition of Avar, and (2) it has the highest number of equivalent degrees of freedom (edf) of any estimator of frequency stability. Theol is unbiased relative to Avar for white FM noise, and only moderately biased for the other noises. Given measurements of the time-error function x(t) between two clocks, we have a sequence of time-error samples {x/sub n/ : n=1,..., N/sub x/] with a sampling period between adjacent observations given by /spl tau//sub 0/. In integer multiples of /spl tau//sub 0/, we can obtain an average of fractional-frequency deviates over time /spl tau/=m/spl tau/0, 1 /spl les/m/spl les/N/sub x/-1. Theol is given by Theol(m,T0, N/sub x/)=1/0.75(N/sub x/-m)(mT0)/sup 2/ N/sub x-m///spl Sigma//i=1/2-1//spl Sigma///spl delta/=0 1/(m/-/spl delta/)[x/sub i/-x/sub i/-/spl delta/+m/2)+(x/sub i/+m/sup -/x/sub i/=/spl delta/=m/2)]/sup 2/, for m even, 10 /spl les/m/spl les/ N/sub x/-1, where frequency stability is evaluated at span or stride T/sub s/=0.75mT/sub 0/. This means that the last T/sub s/=0.75(N/sub x/-1)T/sub 0/, or an averaging time corresponding to 3/4 of the duration of a data run, or 50 % longer than the longest T- value of Avar.