Abstract
The problem of hidden periodicity of a bivariate exponential sum $ {f({\bf n})=\sum_{j=1}^{N}a_j\exp{(-\mathrm{i}\langle{\bm \omega}_j, {\bf n}\rangle)}}, $ where $a_1,\dots,a_N \in \mathbb{C}\backslash\{0\}$ and ${\bf n}\in \mathbb{Z}^2$, is to recover frequency vectors ${\bm \omega}_1, \dots, {\bm \omega}_N \in [0,2\pi)^2$ 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, \cite{KunisProny} relies on the kernel basis analysis of the multilevel Toeplitz matrix of moments of $f$. In \cite{CuytProny2}, the exponential analysis has been considered as a Pad\'e approximation problem. In contrast to the previous method, the algorithms developed in \cite{BDAR,Cuyt} 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 \cite{FMPrestin}, we propose to use the method of Prony-type polynomials, where the elements ${\bm \omega}_1, \dots, {\bm \omega}_N $ 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 \cite{KunisProny, Cuyt}. Although the method of Prony-type polynomials requires more samples than \cite{Cuyt}, 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.
Highlights
Let N ∈ N be an integer, a1, a2, . . . , aN ∈ C\{0} and ωj = ∈ [0, 2π )2 with ωj = ωk for j = k, j, k = 1, . . . , N
Motivated by Pan and Saff [18] and Filbir et al [5], we propose the method of Prony-type polynomials in the two-dimensional case, where the parameters z1, . . . , zN can be recovered as a set of common zeros of the monic bivariate polynomial of an appropriate multi-degree
Let α = (α1, α2) and β = (β1, β2) be elements of Z2+; we say that α is greater than β with respect to the Graded Lexicographic Order (Grlex) α >grlex β, if |α| > |β| or |α| = |β| and α1 − β1 is positive
Summary
ZN using finitely many samples f is called the problem of parameter estimation of an exponential sum f. In contrast to the previous one, the algorithms developed in Potts and Tasche [15], 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. Motivated by Pan and Saff [18] and Filbir et al [5], we propose the method of Prony-type polynomials in the two-dimensional case, where the parameters z1, . First one needs to study in detail properties of such multivariate polynomials, and to analyze a structure of ideals and varieties they build, which causes certain technical challenges which we hope to overcome in future
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