Discrete-time Wilson System With General Time-frequency Lattices

Due to their potential applications on digital signal processing and wireless com-munications, there have been more and more interests on Wilson systems in l2(Z). Re-cently, Kutyniok and Strohmer have constructed discerte-time Wilson bases for l2(Z) with general time-frequency lattices. On this basis, we focus on Wilson frames for l2(Z) and their duals with general time-frequency lattices whose density are 2K(K is odd). The main work of this thesis is as follows:1.We provide some necessary and sufficient conditions for two Wilson systems in l2(Z) to be dual frames for l2(Z).2.When the window function satisfies some symmetry, we prove that if the Gabor system forms a frame for l2(Z), then the corresponding Wilson system also forms a frame for l2(Z). Moreover, we show that the dual frame having Wilson structure of a Wilson frame can be derived from the canonical Gabor dual of the underlying Gabor frame with A being a rectangular time-frequency lattices.3.When the window function satisfies some symmetry, we obtain characterization of a Wilson system to be a tight frame and an orthonormal basis for l2(Z). We also show that if a Wilson system forms an orthonormal basis for l2(Z), then K=1. Our work covers the known results in references.