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- W3136644154 abstract "In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = sum_{j=1}^{M} left( sum_{m=0}^{n_j} , gamma_{j,m} , t^{m} right) {mathrm e}^{2pi lambda_j t}$, where the frequency parameters $lambda_{j} in {mathbb C}$ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of $y$ with regard to a finite interval $[0,P] subset {mathbb R}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients $c_{k}(y)$ to recover all parameters of $y$, where $N:=sum_{j=1}^{M} (1+n_{j})$ denotes the order of $y$. The recovery is based on the observation that for $lambda_{j} notin frac{mathrm i}{P} {mathbb Z}$ 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 $n_{j}$ for $j=1, ldots , M$, as well as all parameters $lambda_{j}$, $j=1, ldots , M$ and $ gamma_{j,m}$ $j=1, ldots , M$, $m=0, ldots , n_{j}$, 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." @default.
- W3136644154 created "2021-03-29" @default.
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- W3136644154 date "2021-03-13" @default.
- W3136644154 modified "2023-09-27" @default.
- W3136644154 title "Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients" @default.
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