Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients
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In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y(t) = ∑_j=1^M( ∑_m=0^n_j γ_j,m t^m) e^2πλ_j t, where the frequency parameters λ_j∈ℂ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of y with regard to a finite interval [0,P] ⊂ℝ with P>0. Our method requires at most 2N+2 Fourier coefficients c_k(y) to recover all parameters of y, where N:=∑_j=1^M (1+n_j) denotes the order of y. The recovery is based on the observation that for λ_j∉i/Pℤ the terms of y possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [12]. If a sufficiently large set of L Fourier coefficients of y is available (i.e., L > 2N+2), then our recovery method automatically detects the number M of terms of y, the multiplicities n_j for j=1, … , M, as well as all parameters λ_j, j=1, … , M and γ_j,m j=1, … , M, m=0, … , n_j, determining y. 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.
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