Abstract
Let g be a Gabor window of length N and ( a, b) be a pair of lattice constants, a and b are considered as time and frequency gaps for a TF-lattice with N 2 ab elements. The corresponding time-frequency shifts of g form a so-called Gabor family. In this paper, we investigate the structural properties of the discrete Gabor transforms. We address the problem of finding the best approximation of a signal x ϵ C n by linear combinations of a Gabor family. We consider critical sampling, oversampling and undersampling, and do not assume that the Gabor family is a frame. For the task we determine the (generalized) dual Gabor atom (GDGA). This amounts to determining the pseudoinverse of the Gabor matrix and can be solved by the conjugate-gradient (CG) algorithm with O( N) complexity for fixed lattice constants ( a, b). We provide an easy practical criterion for checking whether a Gabor triple ( g, a, b) generates a Gabor frame or not. We propose an efficient algorithm for estimating the Gabor frame bounds and an algorithm for determining tight Gabor atoms.
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