We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions.
We derive a method for the reconstruction of non-stationary signals with structured phase functions using only a small number of signal measurements. Our approach employs generalized shift operators as well as the generalized Prony method. Our goal is to reconstruct a variety of sparse signal models using a small number of signal measurements.
In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be recovered from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and show, that this approach can be directly used to reconstruct classical exponential sums from non-equispaced data. Furthermore, we derive a modification of Prony's method for sparse approximation with exponential sums which leads to a non-linear least-squares problem.
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