Effective support, Dirac combs, and signal recovery
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 $γ$, meaning they have the form $f(x)=\sum_{i=1}^γ a_i 1_{A_i}(x)$, where the sets $A_i \subset {\mathbb Z}_N^d$ are disjoint, $a_i$ are complex numbers, and $γ\leq N^d$. We will define the concept of effective support of these signals and show that if $γ$ 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.