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., 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=1Kγjcos(2πajt+bj) , where frequency parameters $$a_{j} \in {\mathbb {R}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">aj∈R (or {\mathrm i} xmlns:mml="http://www.w3.org/1998/Math/MathML">aj∈iR ) are pairwise different. Our is based on recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal use set classical coefficients f with regard to fixed interval (0, P $$P>0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">P>0 . Even though all terms may non- -periodic, our requires at most $$2K+2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">2K+2 $$c_{n}(f)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">cn(f) recover We show that case exact data, terminates after $$K+1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">K+1 steps. The also detect number K if priori unknown and $$L \ge 2K+2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">L≥2K+2 available. Therefore provides alternative known numerical approaches recovery Prony’s method.