Comparison of Extensions of the Three-Cornered Hat and Groslambert Covariance Algorithms for Estimating N-Oscillator Ensemble-Relative Time Stability

摘要

There are multiple options for determination of ensemble-relative Allan Deviation. We focus on the Three-Cornered Hat (TCH) algorithm as extended to N oscillators where N>=3, and the Groslambert Covariance (GCOV) or Allan Covariance algorithm recently reprised by Vernotte, Calosso, and Rubiola. The TCH algorithm uses as inputs pairwise measurements of Allan Deviation, whereas the GCOV algorithm requires time series of pairwise relative phase or frequency measurements. One observation we make is that while Vernotte et al report that the pairwise measurements must be made synchronously between all pairs of oscillators, we found that precise synchronization is not required for the algorithm to work. As long as the measurements are approximately coincident in time, they can be synchronized well enough retroactively. In the case of the extended TCH algorithm, the measurements for each pair of oscillators in the measurement suite are first used to compute pairwise Allan Deviation estimates. Those estimates are then combined in the extended TCH algorithm to produce ensemble-relative ADEV estimates. In the case of the GCOV algorithm, the original pairwise phase differences are first averaged to the desired time scale are then converted by the GCOV algorithm to ensemble-relative ADEV estimates. Both algorithms can be extended to N oscillators. Furthermore, both algorithms can be extended to estimate uncertainties of the ensemble-relative ADEV estimates. A benefit of the GCOV algorithm as reported by Vernotte et al is that it produces physically reasonable ADEV estimates whereas the TCH algorithm can produce negative (and unphysical) values. However, that apparent weakness of the TCH algorithm can be repaired by reformulating TCH as a maximum likelihood problem. The GCOV/Allan Covariance method also produces negative intermediary results which require special treatment, so this is not a significant difference between the two approaches. Overall, in our tests we found that the extended TCH algorithm gave results that were superior to the GCOV/Allan Covariance algorithm, especially at large time scales. TCH also may be preferable to GCOV because the TCH measurements do not require measurements that are either synchronized or approximately time-coincident.