@article {IOPORT.00984099,
    author       = {Qiu, Sigang},
    title        = {Generalized dual Gabor atoms and best approximations by Gabor family.},
    year         = {1996},
    journal      = {Signal Processing},
    volume       = {49},
    number       = {3},
    issn         = {0165-1684},
    pages        = {167-186},
    publisher    = {Elsevier Science, Amsterdam},
    doi          = {10.1016/0165-1684(96)00015-1},
    abstract     = {Summary: 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. We investigate the structural properties of the discrete Gabor transforms. We address the problem of finding the best approximation of a signal $x\in {\bbfC}^{N}$ 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.},
    identifier   = {00984099},
}