Diagonalizing the Gabor frame operator


Helmut Bölcskei, Hans G. Feichtinger, and Franz Hlawatsch


Proc. IEEE UK Sympos. Applications of Time-Frequency and Time-Scale Methods, Univ. of Warwick, Coventry, UK, pp. 249-255a, Aug. 1995.

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The 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.


Gabor expansion, DFT filter banks, Weyl-Heisenberg frames, Zak transform, oversampling

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