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Daniel Potts, Manfred Tasche: Parameter estimation for nonincreasing exponential sums by Prony-like methods

Daniel Potts, Manfred Tasche: Parameter estimation for nonincreasing exponential sums by Prony-like methods

Author(s):
Daniel Potts
Manfred Tasche
Title:
Daniel Potts, Manfred Tasche: Parameter estimation for nonincreasing exponential sums by Prony-like methods
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 04, 2012
Mathematics Subject Classification:
65D10 [Smoothing, curve fitting]
41A45 [Approximation by arbitrary linear expressions]
65F15 [Eigenvalues, eigenvectors]
65F20 [Overdetermined systems, pseudoinverses]
94A12 [Signal theory (characterization, reconstruction, filtering, etc.)]
Abstract:
Let $z_j:={\mathrm e}^{f_j}$ with $f_j \in {\mathbb C}$ and $0 < |z_j| \le 1$ be distinct nodes for $j=1,\ldots, M$. Let $h(x) := c_1\,{\mathrm e}^{f_1\,x} +\, \ldots\,+ c_M\,{\mathrm e}^{f_M\,x}$ $(x\ge 0)$ be a nonincreasing exponential sum with complex coefficients $c_j \neq 0$. 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 $h(k)$ $(k=0,\ldots,2N-1;\, N\ge M)$ 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.
Keywords:
Parameter estimation, nonincreasing exponential sum, Prony--like method, exponential fitting problem, ESPRIT, matrix pencil factorization, companion matrix, Prony polynomial, eigenvalue problem, rectangular Hankel matrix, nonlinear approximation, parse trigonometric polynomial, sparse Fourier approximation
Language:
English
Publication time:
04/2012