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)=∑j=1Kγjcos(2πajt+bj), where the frequency parameters aj∈ R (or aj∈ i 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 2 K+ 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≥ 2 K+ 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.