Weyl-Heisenberg frames and Balian-Low theorem in ℓ2(Z)

Abstract

This paper focuses on the reconstruction of digital signals in multirate digital signal processing. We study the discrete time Weyl-Heisenberg (DTWH) system for oversampling schemes and describe the frame operator of DTWH frames as the composition of the sampling operator and interpolation operator. Using discrete time Zak transform, we characterize the dual discrete time Weyl-Heisenberg tight frames, DTWH frames, and tight DTWH frames based on oversampling schemes. In addition, we discuss the applications of DTWH frames to the reconstruction of Weyl-Heisenberg systems in periodic space L2([0, 1)). Finally, we give Balian-Low theorem for an orthonormal basis formed by DTWH systems and weak Balian-Low theorem for exact DTWH frames in sequence space ℓ2(Z).