Discrete Gabor structures and optimal representations

The idea of Gabor´s (1946) signal expansion is to represent a signal in terms of a discrete set of time-shifted and frequency modulated signals that are localized in the time-frequency (or phase) space. We present detailed descriptions of the block and banded structures for the Gabor matrices. Based on the explicit descriptions of the sparsity of such matrices, we can establish the sparse form of the Gabor matrix and obtain the dual Gabor atom (mother wavelet), the inverse of the Gabor frame operator, and carry out the discrete finite Gabor transform in a very efficient way. Some explicit sufficient and also necessary conditions are derived for a Gabor atom g to generate a Gabor frame with respect to a TF-lattice (a, b)