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

Abstract

AbstractIn 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 $$f(t) = \sum _{j=1}^{K} \gamma _{j} \, \cos (2\pi a_{j} t + b_{j})$$ f ( t ) = ∑ j = 1 K γ j cos ( 2 π a j t + b j ) , where the frequency parameters $$a_{j} \in {\mathbb {R}}$$ a j ∈ R (or $$a_{j} \in {\mathrm i} {\mathbb {R}}$$ a j ∈ 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 $$P>0$$ P > 0 . Even though all terms of f may be non-P-periodic, our reconstruction method requires at most $$2K+2$$ 2 K + 2 Fourier coefficients $$c_{n}(f)$$ 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 $$K+1$$ K + 1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and $$L \ge 2K+2$$ 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.