Decoupling of Fourier Reconstruction System for Shifts of Several Signals

TL;DR

This paper presents a method for algebraic reconstruction of linear combinations of shifted signals from Fourier samples, decoupling the system into separate generalized Prony systems for each signal.

Contribution

It introduces a decoupling approach using specific sampling sets to reduce the reconstruction system into independent generalized Prony systems for each signal.

Findings

Reconstruction system can be decoupled into separate systems for each signal.

Conditions for unique solvability of the decoupled systems are discussed.

Examples illustrate the decoupling and solvability conditions.

Abstract

We consider the problem of ``algebraic reconstruction'' of linear combinations of shifts of several signals $f_1,\ldots,f_k$ from the Fourier samples. For each $r=1,\ldots,k$ we choose sampling set $S_r$ to be a subset of the common set of zeroes of the Fourier transforms ${\cal F}(f_\l), \ \l \ne r$, on which ${\cal F}(f_r)\ne 0$. We show that in this way the reconstruction system is reduced to $k$ separate systems, each including only one of the signals $f_r$. Each of the resulting systems is of a ``generalized Prony'' form. We discuss the problem of unique solvability of such systems, and provide some examples.