Effective support, Dirac combs, and signal recovery

TL;DR

This paper extends classical Fourier support recovery results to Dirac comb signals with multiple disjoint components, introducing effective support and new recovery conditions under certain distribution assumptions.

Contribution

It introduces the concept of effective support for Dirac combs and derives new recovery conditions that work even when classical support size limits are exceeded.

Findings

Recovery conditions depend on the complexity parameter b3.

New uncertainty principles are established for Dirac comb signals.

Effective support allows successful recovery beyond classical support size limits.

Abstract

Let $f: {\mathbb Z}_N^d \to {\mathbb C}$ be a signal with the Fourier transform $\widehat{f}: \Bbb Z_N^d\to \Bbb C$. A classical result due to Matolcsi and Szucs (\cite{MS73}), and, independently, to Donoho and Stark (\cite{DS89}) states if a subset of frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ of $f$ are unobserved due to noise or other interference, then $f$ can be recovered exactly and uniquely provided that $$ |E| \cdot |S|<\frac{N^d}{2},$$ where $E$ is the support of $f$, i.e., $E=\{x \in {\mathbb Z}_N^d: f(x) \not=0\}$. In this paper, we consider signals that are Dirac combs of complexity $\gamma$, meaning they have the form $f(x)=\sum_{i=1}^{\gamma} a_i 1_{A_i}(x)$, where the sets $A_i \subset {\mathbb Z}_N^d$ are disjoint, $a_i$ are complex numbers, and $\gamma \leq N^d$. We will define the concept of effective support of these signals and show that if $\gamma$ is not too large, a good recovery condition can be obtained by pigeonholing under additional reasonable assumptions on the distribution of values. Our approach produces a non-trivial uncertainty principle and a signal recovery condition in many situations when the support of the function is too large to apply the classical theory.