Estimating the instabilities of N clocks by means of comparison measurements
The estimation of individual instabilities of N clocks, compared by measuring the differences of their readings, is considered without assuming a priori any hypotheses on their uncorrelation. Instabilities of the N clocks are described by a complete (non-diagonal) N x N covariance matrix R. Only differences of clock readings are available in order to estimate R. Statistical processing of these data allows one to calculate the (N-1)x(N-l) covariance matrix S of the differences relative to the N-th(reference) clock. By analyzing the relationships tying R and S, several pieces of information can be inferred and, in particular, the conditions for the validity of the uncorrelation hypothesis are established. The estimation of R from S is not unique: in any case R must be positive definite. A theorem states that R is positive definite if and only if its determinant is positive. Nevertheless infinitely many acceptable choices of R still fulfill the condition of positive definiteness. This paper shows that, by increasing the number N of compared clocks, the amount of arbitrariness in estimating R is reduced. The analysis of some experimental data illustrates the capability of the method.