2021/03/13 by Nadiia Derevianko, Derevianko, Nadiia, Gerlind Plonka +1 · 1 citation
Computer Science · Mathematics · #41A20 #42A16 #42C15 #65D15 #94A12 #Digital Filter Design and Implementation #FOS: Mathematics #Image and Signal Denoising Methods #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #cs.NA #math.NA #msc:41A20 #msc:42A16 #msc:42C15 #msc:65D15 #msc:94A12
paper · pdf · doi:10.48550/arxiv.2103.07743
29 pages, 7 figures
arxiv created 2021/03/13 · openalex publication_date 2021/03/13 · arxiv updated 2021/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y(t) = ∑j=1M ( ∑m=0nj γj,m tm ) \mathrm e2πλj t, where the frequency parameters λj ∈ \mathbb 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] ⊂ \mathbb R with P>0. Our method requires at most 2N+2 Fourier coefficients ck(y) to recover all parameters of y, where N:=∑j=1M (1+nj) denotes the order of y. The recovery is based on the observation that for λj \not∈ (\mathrm i)/(P) \mathbb Z 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 nj for j=1, … , M, as well as all parameters λj, j=1, … , M and γj,m j=1, … , M, m=0, … , nj, 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.