Gabor families in l2(Zd)

Abstract

This paper addresses Gabor families in $l^2\left({\mathbb{Z}}^d\right)$. The discrete Gabor families have interested many researchers due to their good potential for digital signal processing. Gabor analysis in $l^2\left({\mathbb{Z}}^d\right)$ is more complicated than that in $l^2\left(\mathbb{Z}\right)$ since the geometry of the lattices generated by time-frequency translation matrices can be quite complex in this case. In this paper, we characterize window functions such that they correspond to complete Gabor families (Gabor frames) in $l^2\left({\mathbb{Z}}^d\right)$; obtain a necessary and sufficient condition on time-frequency translation for the existence of complete Gabor families (Gabor frames, Gabor Riesz bases) in $l^2\left({\mathbb{Z}}^d\right)$; characterize duals with Gabor structure for Gabor frames; derive an explicit expression of the canonical dual for a Gabor frame; and prove its norm minimality among all Gabor duals.