Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier Coefficients

TL;DR

This paper introduces a new stable method for exactly reconstructing sparse non-harmonic Fourier sums from a minimal set of Fourier coefficients, capable of detecting the number of terms and applicable to non-periodic signals.

Contribution

The paper presents a novel reconstruction algorithm based on rational approximation that requires fewer Fourier coefficients and can determine the number of terms in the sum, improving stability over existing methods.

Findings

Reconstruction requires at most 2K+2 Fourier coefficients.

The iterative algorithm terminates after at most K+1 steps for exact data.

The method can detect the number of terms K if unknown, with L > 2K+2 coefficients.

Abstract

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 $f(t) = \sum_{j=1}^{K} \gamma_{j} \, \cos(2\pi a_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. 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