Gabor families in l2(Zd)

This paper addresses Gabor families in l(2)(Z(d)). The discrete Gabor families have interested many researchers due to their good potential for digital signal processing. Gabor analysis in l(2)(Z(d)) is more complicated than that in l(2)(Z) 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)(Z(d)); 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)(Zd); 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.