Thomas Peter, Daniel Potts, Manfred Tasche : Nonlinear approximation by sums of exponentials and translates

Author(s):

Thomas Peter
Daniel Potts
Manfred Tasche

Title:

Nonlinear approximation by sums of exponentials and translates

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 5, 2010

Mathematics Subject Classification:

41A30	[Approximation by other special function classes]
65F15	[Eigenvalues, eigenvectors]
65F20	[Overdetermined systems, pseudoinverses]
94A12	[Signal theory (characterization, reconstruction, filtering, etc.)]

Abstract:

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let $h$ be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands, if finitely many perturbed, uniformly sampled data of $h$ are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too.
The second approximation problem is related to the first one and reads as follows: Let $f$ be a linear combination of translates of a 1--periodic window function. Determine all shift parameters, all coefficients, and the number of translates, if finitely many perturbed, uniformly sampled data of $f$ are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

Keywords:

Nonlinear approximation, exponentialsum, exponential fitting, harmonic retrieval, sum of translates, approximate Prony method, nonuniform sampling, parameter estimation, least squares method, signal processing, signal recovery, singular value decomposition, matrix perturbation theory, perturbed rectangular Hankel matrix.

Language:

English

Publication time:

03/2010