Parameter estimation for nonincreasing exponential sums by Prony–like methods

Let zj := e fj with fj ∈ (−∞, 0] + i [−π, π) be distinct nodes for j = 1, . . . ,M . With complex coefficients cj 6= 0, we consider a nonincreasing exponential sum h(x) := c1 e f1 x + . . . + cM e fM x (x ≥ 0). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if 2N 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.