Block-equivalent finite Gabor frames

TL;DR

This paper introduces the concept of block-equivalent finite Gabor frames, characterizes when Gabor systems are block-equivalent, and explores conditions leading to diagonal or sparse frame operator matrices.

Contribution

It defines block-equivalence for finite Gabor frames, provides explicit transformations, and characterizes conditions for block-diagonal and diagonal frame operators.

Findings

Gabor systems are block-equivalent when modulation or translation sets are subgroups of 1_N.

Conditions are identified under which the frame operator matrix becomes diagonal.

Geometric conditions on subsets of 1_N induce sparsity and block structures in the frame operator.

Abstract

We study finite systems of vectors whose frame operator matrices are unitarily equivalent, via explicit and computationally efficient unitary transformations, to block-diagonal matrices. We call such systems block-equivalent. We show that a Gabor system $\mathcal{G}=\mathcal{G}(g,L\times K)\subset \mathbb C^N$ is block-equivalent when either the modulation set $L$ or the translation set $K$ is a subgroup of $\mathbb Z_N$. We also characterize situations in which the frame operator matrix becomes diagonal. Finally, we show that geometric conditions on subsets of $\mathbb Z_N$ force certain diagonals of the frame operator matrix of $\mathcal{G}$ to vanish, yielding additional sparsity and block structures.