On Gabor orthonormal bases over finite prime fields

TL;DR

This paper characterizes Gabor orthonormal bases over finite prime fields, linking them to tiling and spectral properties, and explores the structure and existence of Gabor windows with various support conditions.

Contribution

It establishes necessary and sufficient conditions for indicator functions to be Gabor windows and constructs examples of windows with full support and non-indicator magnitude.

Findings

Indicator functions are Gabor windows iff they tile and are spectral.

Positive functions with support size equal to modulation set size are unimodular Gabor windows.

Existence of Gabor windows with full support and non-indicator magnitudes.

Abstract

We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.