Total Variance as an Exact Analysis of the Sample Variance

Given a sequence of fractional frequency deviates, we investigated the relationship between the sample variance of these deviates and the total variance Totvar estimator of the Allan variance. We demonstrated that we can recover exactly twice the sample variance by renormalizing the Totvar estimator and then summing it over dyadic averaging times 1, 2, 4, . . . , 2J along with one additional term that represents variations at all dyadic averaging times greater than 2J. This decomposition of the sample variance mimics a similar theoretical decomposition in which summing the true Allan variance over all possible dyadic averaging times yields twice the process variance. We also establish a relationship between the Totvar estimator of the Allan variance and a biased maximal overlap estimator that uses a circularized version of the original fractional frequency deviates.