EXPONENTIAL ANALYSIS TASK – APPLICATIONS, METHODS, PROBLEMS

Abstract

This review article presents the problem of exponential analysis, its applications, limitations and problems that are relevant despite its more than two hundred years of history. During this time, many methods for solving the problem have been developed, but the greatest popularity and efficiency are demonstrated by Prony-like methods: the Padé–Laplace method, Prony method and the matrix pencil method, which are the subject of research in this work. The purpose of the study: to present the current state of the problem of exponential analysis, its main problems and methods for solving them. The main problems of these methods are: the issue of determining the number of exponents, choosing the optimal signal sampling frequency and reducing computational costs. The article presents solutions to these problems. Materials and methods. To solve problems, an analysis of scientific literature was carried out and linear algebra methods were used. Results. Modifications of the methods to solve their problems are described. The algorithm for calculating the Padé approximation with a minimum degree denominator solves the problem of Froissart doublets in the Padé–Laplace method. To increase the accuracy of calculations of Taylor coef-ficients, splines are used in the same method. To select the optimal signal sampling frequency, use the estimation of the matrix condition number in the Prony method. For the matrix pencil method, the problem of computational costs is solved due to its recurrent and multichannel versions. Conclusion. Various formulations of this problem are presented, which find their numerous applications. The limitations, problems of the exponential analysis problem, and their solutions are described. Three parametric methods are considered: Prony, Padé–Laplace and the matrix pencil method.