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 h0 be a multivariate exponential sum, i.e., h0 is a finite linear combination of complex exponentials with distinct frequency vectors. Determine all parameters of h0, i.e., all frequency vectors, all coefficients, and the number of exponentials, if finitely many sampled data of h0 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. 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 h0 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 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.