Daniel Potts, Manfred Tasche: Parameter estimation for nonincreasing exponential sums by Prony-like methods
- Author(s):
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Daniel Potts
Manfred Tasche
- Title:
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Daniel Potts, Manfred Tasche: Parameter estimation for nonincreasing exponential sums by Prony-like methods
- Electronic source:
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application/pdf
- Preprint series:
- Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 04, 2012
- Mathematics Subject Classification:
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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:
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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
