Parameter estimation for nonincreasing exponential sums by Prony-like methods

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Let zj:=efj with fj∈(-∞,0]+i[-π,π) be distinct nodes for j=1,…,M. With complex coefficients cj≠0, we consider a nonincreasing exponential sum h(x):=c1ef1x+⋯+cMefMx(x⩾0). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if 2,N sampled values h(k) (k=0,…,2N-1; N⩾M) are given. This parameter estimation problem is a nonlinear inverse problem. For noiseless sampled data, we describe the close connections between Prony-like methods, namely the classical Prony method, the matrix pencil method, and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.