Uncertainty principles for the short-time Fourier transform on the lattice

TL;DR

This paper explores various forms of the uncertainty principle for the short-time Fourier transform on the lattice, establishing several inequalities and principles that extend classical results to this discrete-continuous setting.

Contribution

It introduces multiple uncertainty principles for the short-time Fourier transform on the lattice, including orthonormal sequences, Donoho--Stark, Benedicks, and Heisenberg types, along with entropy-based inequalities.

Findings

Established uncertainty principles for orthonormal sequences.

Proved Donoho--Stark and Benedicks-type uncertainty principles.

Derived Heisenberg-type uncertainty inequalities using $k$-entropy.

Abstract

In this paper, we study a few versions of the uncertainty principle for the short-time Fourier transform on the lattice $\mathbb Z^n \times \mathbb T^n$. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho--Stark's uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for this transform on $\mathbb Z^n \times \mathbb T^n$. Also, we obtain the Heisenberg-type uncertainty inequality using the $k$-entropy of the short-time Fourier transform on $\mathbb Z^n \times \mathbb T^n$.