Browsing by Author "Petz, Markus"

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form f(t)=∑Kj=1γjcos(2πajt+bj), where the frequency parameters aj∈R (or aj∈iR) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients of f with regard to a fixed interval (0, P) with P>0. Even though all terms of f may be non-P-periodic, our reconstruction method requires at most 2K+2 Fourier coefficients cn(f) to recover all parameters of f. We show that in the case of exact data, the proposed iterative algorithm terminates after at most K+1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and L≥2K+2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

Abstract We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums.