Characterization of Frequency Standard Instability by Estimation of Their Covariance Matrix

Patrizia Tavella, Amedeo Prenioli

Abstract:	The popular 3-cornered hat method used for evaluating the noise contributions of individual frequency standards is revisited. This method is used in several cases, but sometimes the results are not consistent because one or more estimated clock variances turn out to be negative. Different causes of this unacceptable result have been conjectured: among them one regards the hypothesis of uncorrelated clocks, essential in this method. Since recently realistic cases of correlation between clocks, mainly due to the environmental conditions, have been observed, this paper proposes an entirety revisited version of the 3-cornered hat method which permits to evaluate the individual variances and also the possible co-variances between clocks, by relaxing the hypothesis of uncorrelation. The uncertainty and the luck of contemporaneity of the measurement series are assumed to be negligible. The lack of the uncorrelation hypothesis calls for a more general mathematical model leading to an underdetermined linear system. The estimates of the (co)variances of the measurement series as well as those of the individual clocks are introduced by means of the scalar product of the related time series and arranged in the respective co-variance matrices S and R. Since co-variance matrix is positive definite by definition, the problem consists in estimating the unknown R, subject to the constraint of positive definiteness, from the known S. Unfortunately, this constraint is not sufficient to estimate R. Therefore a suitable optimization criterion is proposed, which assures the positive definiteness of R and, at the same time, minimizes the global correlation among clocks. Examples of frequency instability measurements processed by the "classical" 3- cornered hat method and the here-revisited method are presented showing that the solution are identical only when the uncorrelation hypothesis doesn't violate the positive definiteness of R
Published in:	Proceedings of the 23th Annual Precise Time and Time Interval Systems and Applications Meeting December 3 - 5, 1991 Pasadena, California
Pages:	265 - 276
Cite this article:	Tavella, Patrizia, Prenioli, Amedeo, "Characterization of Frequency Standard Instability by Estimation of Their Covariance Matrix," Proceedings of the 23th Annual Precise Time and Time Interval Systems and Applications Meeting, Pasadena, California, December 1991, pp. 265-276.
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