Diagonalizing the Gabor frame operator
AuthorsHelmut Bölcskei, Hans G. Feichtinger, and Franz Hlawatsch
ReferenceProc. IEEE UK Sympos. Applications of Time-Frequency and Time-Scale Methods, Univ. of Warwick, Coventry, UK, pp. 249-255a, Aug. 1995.
AbstractThe Gabor expansion is a signal decomposition into time-frequency shifted versions of a window function. Computation of the expansion coefficients requires a "dual'' window. This paper discusses fast algorithms for calculating the dual window. We consider situations where the Gabor frame operator can be expressed---either directly or in a transform domain---as a multiplication operator, and hence the dual window can be calculated by pointwise division. In the cases of critical sampling and integer oversampling, the Zak transform allows to do this independently of the original window. In the general case (including the case of rational oversampling), one has to make restrictions about the window's temporal or spectral support. We furthermore obtain expressions for the eigenfunctions, eigenvalues, and frame bounds of the Gabor frame operator, and we derive an efficient algorithm for the construction of tight Gabor-type (Weyl-Heisenberg) frames.
KeywordsGabor expansion, DFT filter banks, Weyl-Heisenberg frames, Zak transform, oversampling
Download this document:
Copyright Notice: © 1995 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.