Finite Gabor systems and uncertainty principle for block sliding discrete Fourier transform

In this paper, we study the finite Gabor system for oversampling schemes. A characterization of dual finite Gabor tight frame using discrete time Zak transform is given. Also, a method to calculate the coefficients of the finite Gabor system expansion in the case of oversampling and a necessary and sufficient condition for the existence of biorthogonal pair of Riesz basis in l² (Z_L) is given. Further, we introduce the notion of block sliding discrete Fourier transform (BSDFT) which reduces the computational complexity and give uncertainty principle for BSDFT. An uncertainty principle for two finite Parseval Gabor frames in terms of sparse representations is given. Finally, using the notion of numerical sparsity, an uncertainty principle for finite Gabor frames is given.