Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier Coefficients
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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=1^Kγ_j cos(2π a_j t + b_j), where the frequency parameters a_j∈ℝ (or a_j∈iℝ) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in <cit.>. 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>2K+2 Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients
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