Frontiers in Applied Mathematics and Statistics (May 2020)
Prony-Type Polynomials and Their Common Zeros
Abstract
The problem of hidden periodicity of a bivariate exponential sum f(n)=∑j=1Najexp(-i〈ωj,n〉), where a1, …, aN ∈ ℂ\{0} and n ∈ ℤ2, is to recover frequency vectors ω1,…,ωN∈[0,2π) 2 using finitely many samples of f. Recently, this problem has received a lot of attention, and different approaches have been proposed to obtain its solution. For example, Kunis et al. [1] relies on the kernel basis analysis of the multilevel Toeplitz matrix of moments of f. In Cuyt et al. [2], the exponential analysis has been considered as a Padé approximation problem. In contrast to the previous method, the algorithms developed in Diederichs and Iske [3] and Cuyt and Wen-Shin [4] use sampling of f along several lines in the hyperplane to obtain the univariate analog of the problem, which can be solved by classical one-dimensional approaches. Nevertheless, the stability of numerical solutions in the case of noise corruption still has a lot of open questions, especially when the number of parameters increases. Inspired by the one-dimensional approach developed in Filbir et al. [5], we propose to use the method of Prony-type polynomials, where the elements ω1, …, ωN can be recovered due to a set of common zeros of the monic bivariate polynomial of an appropriate multi-degree. The use of Cantor pairing functions allows us to express bivariate Prony-type polynomials in terms of determinants and to find their exact algebraic representation. With respect to the number of samples the method of Prony-type polynomials is situated between the methods proposed in Kunis et al. [1] and Cuyt and Wen-Shin [4]. Although the method of Prony-type polynomials requires more samples than Cuyt and Wen-Shin [4], numerical computations show that the algorithm behaves more stable with regard to noisy data. Besides, combining the method of Prony-type polynomials with an autocorrelation sequence allows the improvement of the stability of the method in general.
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