Constructing super Gabor frames: the rational time-frequency lattice case

For a time-frequency lattice Λ = Aℤd × Bℤd, it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det(AB)| = 1/L. The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles ℝd by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.