Gabor frames, unimodularity, and window decay

Authors

Helmut Bölcskei and Augustus J. E. M. Janssen

Reference

The Journal of Fourier Analysis and Applications, Vol. 6, No. 3, pp. 255-276, May 2000.

Abstract

We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties of g are inherited by its minimal dual and by generalized duals. We consider compactly supported, exponentially decaying, and faster than exponentially decaying window functions. Particularly, we find that g and the minimal dual have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. In the case of integer oversampling unimodularity of the Zibulski-Zeevi matrix is equivalent to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Gabor frame operator, on a result by S. Jaffard, on a functional calculus for Gabor frame operators, on results from the theory of entire functions, and on the theory of polynomial matrices.

Keywords

Gabor expansions, Weyl-Heisenberg frames, tight frames, Zak transform, window decay, unimodularity, polynomial matrices, entire functions

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