Gabor fusion frames generated by difference sets

TL;DR

This paper explores Gabor fusion frames generated by difference sets, analyzing their coherence, sparse recovery performance, and optimal packing properties, revealing their advantages over traditional generators like Alltop and random vectors.

Contribution

It introduces the use of difference sets for Gabor fusion frames, demonstrating their optimal sparsity and tightness as Grassmannian packings, and compares their effectiveness in sparse recovery.

Findings

Difference set-based Gabor systems form equiangular tight frames.

They are shown to be optimally sparse fusion frames.

Numerical comparisons indicate competitive sparse recovery performance.

Abstract

Collections of time- and frequency-shifts of suitably chosen generators (Alltop or random vectors) proved successful for many applications in sparse recovery and related fields. It was shown in \cite{xia2005achieving} that taking a characteristic function of a difference set as a generator, and considering only the frequency shifts, gives an equaingular tight frame for the subspace they span. In this paper, we investigate the system of all $N^2$ time- and frequency-shifts of a difference set in dimension $N$ via the mutual coherence, and compare numerically its sparse recovery effectiveness with Alltop and random generators. We further view this Gabor system as a fusion frame, show that it is optimally sparse, and moreover an equidistant tight fusion frame, i.e. it is an optimal Grassmannian packing.