Decoupling of Fourier Reconstruction System for Shifts of Several Signals

Abstract

We consider the problem of ``algebraic reconstruction'' of linear combinations of shifts of several signals $f_1,\dots ,f_k$ from the Fourier samples. For each $r=1,\dots ,k$ we choose sampling set $S_r$ to be a subset of the common set of zeroes of the Fourier transforms $\mathcal{F}(f_{ł}),ł\ne r$, on which $\mathcal{F}(f_r)\ne 0$. 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.

References in corpus (3)

• Signal Acquisition from Measurements via Non-Linear Models
• An "algebraic" reconstruction of piecewise-smooth functions from integral measurements
• Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data