Block-equivalent finite Gabor frames

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.