Optimal solution of redundant measurements in frequency domain

Collini Francesca, Bureau international des poids et mesures

The current definition of the second as unit of time in the Système International (SI) relies on cesium frequency standards. However, in the last decades, several other atomic transitions demonstrated superior performance with respect to the cesium and prompted the community to reconsider the definition of the second. To this end, a class of standards νk (k = 1, 2, . . . , NS), with NS indicating the number of different reference transitions, denoted as Secondary Representations of the Second (SRS), has been identified as particularly promising to be used for a new definition of the second. From the mathematical point of view, the problem to be solved is to find an optimal solution for νk, with input data given by a redundant and correlated data set expressed in form of frequency ratios νi/νj. To optimize the frequency values νk, two different methods are usually used: a direct least squares approach and a method based on the closed loops in a graph theory framework. Given the non-linear nature of the problem, the least squares method applies a linear approximation through Taylor expansion, while the closed loops algorithm works with logarithms of the input data. We focused on the latter algorithm, proving that the two different approaches converge to the same solution for the set of the considered input values. Theoretically, the logarithms of all frequency ratios in each closed loop should add up to zero. By imposing this condition to experimental data, we obtain a set of equations to be used in the Lagrange multiplier method and optimized in a least squares sense.

Area: CS51 - Optimality: theoretical and applied results (Cristina Zucca)

Keywords: Optimization; redundant dataset; graph theory

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