Parameter estimation for multivariate exponential sums

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 be a multivariate exponential sum, i.e., h is a finite linear combination of complex exponentials with pairwise different frequency vectors. Determine all parameters of h, i.e., all frequency vectors, all coefficients, and the number of exponentials, if finitely many equispaced sampled data of h are given. Using Ingham–type inequalities, the stability of the reconstructed exponential sum h̃ is discussed both in the square and uniform norm. Further 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 by a novel algorithm, the sparse approximate Prony method (SAPM), where we use only some data sampled along few lines. The first part of SAPM estimates the frequency vectors by 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.