Estimating the Instabilities of N Clocks by Measuring Differences of their Readings

The estimation of individual instabilities of N clocks, when only differences of clock readings are available, is examined without any a priori assumption about the lack of correlation between clocks. The instability of the N clocks is described by the N × N covariance matrix R but only the (N - 1) × (N - 1) covariance matrix S of the differences referred to the N-th clock may be estimated directly from measurements. The estimation of R from S is not unique but the solution domain is limited by the positive definiteness of R. The features of this domain are analysed and the conditions for the validity of the uncorrelation hypothesis are established. The modifications of the solution domain, when the number N of compared clocks increases, is examined showing that each added clock can reduce the domain and therefore the arbitrariness in estimating R. Examples with some experimental data illustrate the capabilities of this approach.