Linear invariant estimation of clock phase and trend from noisy phase data

Abstract

We show how to solve two problems of optimal linear estimation from a set of noisy measurements of clock phase. The phase is modelled as a process with stationary dth increments, e.g. a sum of processes with power-law spectra. The additive measurement noise can be any mean-zero process with a known autocovariance function. The estimation targets are the phase at a given time and the coefficient of the overall trend, e.g. the frequency drift. A feasibility condition called ‘invariance’ is imposed according to the target and the degree of non-stationarity of the phase model. The solution of a set of linear equations gives the regression coefficients and the mean squared error of the best linear invariant estimate. Proofs of these results are available from the online version of this journal.