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
