Nonlinear approximation by sums of nonincreasing exponentials

Many applications in electrical engineering, signal processing, and mathematical physics lead to following approximation problem: Let h be a short linear combination of nonincreasing exponentials with complex exponents. Determine all exponents, all coefficients, and the number of summands from finitely many equispaced sampled data of h. This is a nonlinear inverse problem. This paper is an extension of [11], where only the case of sums of complex exponentials with real frequencies is considered. In the following, we present new results on an approximate Prony method (APM) which is based on [2]. In contrast to [2], we apply perturbation theory for a singular value decomposition of a rectangular Hankel matrix such that we can describe the properties and the numerical behavior of APM in detail. For this inverse problem, the number of sampled data acts as regularization parameter. The first part of APM recovers the exponents. The second part computes the coefficients by an overdetermined linear Vandermonde–type system. Numerical experiments show the performance of our method.