A Fast Algorithm for Calculation of Thêo1

Abstract

Thêo1 is a frequency stability statistic which is similar to the Allan variance but can provide stability estimates at longer averaging factors and with higher confidence. However, the calculation of Thêo1 is significantly slower than the Allan variance, particularly for large data sets, due to a worse computational complexity. A faster algorithm for calculating the 'all-τ ' version of Thêo1 is developed by identifying certain repeated sums and removing them with a recurrence relation. The new algorithm has a reduced computational complexity, equal to that of the Allan variance. Computation time is reduced by orders of magnitude for many datasets. The new, faster algorithm does introduce an error due to accumulated floating point errors in very large datasets. The error can be compensated for by increasing the numerical precision used at critical steps. The new algorithm can also be used to increase the speed of ThêoBr and ThêoH which are more sophisticated statistics derived from Thêo1.

Characterizing Frequency Stability Measurements Having Multiple Data Gaps

“…Theo1 is added to ADEV calculations at τ-values beyond ADEV's longest-possible τ-value at T/2 to create a combination plot called THEO-H [4]. Computations are as fast as standard ADEV [12]. ADEV fundamentally measures "selfsimilarity" or underlying long-term autocorrelations.…”

Time-series Imputation Algorithm

“…3. A computation of THEO, which is the best long-term estimator of ADEV, is added as the longest two τ-values beyond 3 x 10 4 s to see the flicker noise, i.e., zero-slope flicker FM [20][21][22]. Note that even with massive gaps filled by the imputation algorithm, white and flicker frequency models are still characterized with the correct, unbiased levels and slopes.…”