Rational time–frequency super Gabor frames and their duals

This paper addresses general super Gabor systems with rational time–frequency product lattices. A Zak transform matrix method is developed for such super Gabor systems. We characterize complete super Gabor systems and super Gabor frames. Given a super Gabor frame, we obtain an explicit expression of its canonical dual and a parametrization of all its super Gabor duals and prove that the canonical dual is the norm-minimal one among all super Gabor duals. We also prove that a super Gabor frame has a unique super Gabor dual if and only if it is a Riesz basis.