Oversampled finite-discrete Gabor transforms

This paper considers the construction and fast computation of non-periodic discretizations of the Gabor transform. We first derive a reasonable finite version of the continuous transform by discretizing the integrals over regular in the time-frequency domain. We next apply the novel decomposition of the Gabor transform presented by Redding and Newsam (see Proceedings of the International Conference on Acoustics, Speech and Signal Processing III, p.5-8, 1994) to reduce the discrete system to a collection of independent overdetermined Toeplitz mosaic systems. For the special case of an integer oversampling ratio, appealing to new results on the generalized inverses of Toeplitz mosaic systems then establishes that the discrete oversampled transforms and their generalized inverses can be repeatedly computed in O(J log J) operations, after an initial setup cost of O(Jlog² J) operations to appropriately factorize the inverse (J is the number of samples in the time-frequency domain). The paper thus generalizes the results of Zibulski and Zeevi (see IEEE Transactions on Signal Processing vol.42, p.942-945, 1994) on the fast computation of oversampled Gabor transforms and their inverses by removing the requirement for a periodic window function