Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients

Abstract

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) = sum _{j=1}^{K} gamma _{j} , cos (2pi a_{j} t + b_{j}), where the frequency parameters a_{j} in {mathbb {R}} (or a_{j} in {mathrm i} {mathbb {R}}) 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 c_{n}(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 ge 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.