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Parameter estimation for nonincreasing exponential sums by Prony-like methods. (English) Zbl 1281.65021
Summary: Let \(z_j := e^{f_j}\) with \(f_j \in (-\infty, 0] + i[-{\pi}, {\pi})\) be distinct nodes for \(j = 1, \dots, M\). With complex coefficients \(c_j \neq 0\), we consider a nonincreasing exponential sum \(h(x) := c_1e^{f_1x} + \cdots + c_Me^{f_Mx}\) \((x \geq 0)\). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of \(h\), if \(2N\) sampled values \(h(k)\) \((k = 0, \dots, 2N - 1;~ N \geq M)\) 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 the QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

MSC:
65D10 Numerical smoothing, curve fitting
11L03 Trigonometric and exponential sums, general
11Y05 Factorization
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A22 Matrix pencils
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