#### Volume 40, pp. 204-224, 2013.

## Parameter estimation for multivariate exponential sums

Daniel Potts and Manfred Tasche

### Abstract

The recovery of signal parameters from noisy sampled data is an essential problem in digital signal processing. In this paper, we discuss the numerical solution of the following parameter estimation problem. Let $h_0$ be a multivariate exponential sum, i.e., $h_0$ is a finite linear combination of complex exponentials with distinct frequency vectors. Determine all parameters of $h_0$, i.e., all frequency vectors, all coefficients, and the number of exponentials, if finitely many sampled data of $h_0$ are given. Using Ingham-type inequalities, the Riesz stability of finitely many multivariate exponentials with well-separated frequency vectors is discussed in continuous as well as discrete norms. Furthermore, we show that a rectangular Fourier-type matrix has a bounded condition number, if the frequency vectors are well-separated and if the number of samples is sufficiently large. Then we reconstruct the parameters of an exponential sum $h_0$ by a novel algorithm, the so-called sparse approximate Prony method (SAPM), where we use only some data sampled along few straight lines. The first part of SAPM estimates the frequency vectors using the approximate Prony method in the univariate case. The second part of SAPM computes all coefficients by solving an overdetermined linear Vandermonde-type system. Numerical experiments show the performance of our method.

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### Key words

parameter estimation, multivariate exponential sum, multivariate exponential fitting problem, harmonic retrieval, sparse approximate Prony method, sparse approximate representation of signals

### AMS subject classifications

65D10, 65T40, 41A45, 41A63, 65F20, 94A12

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