Gabor expansion and frame theory

Authors

Helmut Bölcskei

Reference

Diploma thesis, Vienna University of Technology, Sept. 1994.

Abstract

This thesis treats the Gabor expansion, one of the major linear time-frequency signal representations. The Gabor expansion, proposed by D. Gabor in 1946, is the decomposition of a signal into a set of time-shifted and modulated versions of an elementary window function. A significant part of this work is devoted to the theory of frames, which constitutes the mathematical background for the Gabor expansion. We provide a general treatment of the Gabor expansion for arbitrary windows and arbitrary sampling densities in the time-frequency plane. First of all, a detailed presentation of the Zak transform and the theory of frames, mathematical tools necessary for a discussion of the Gabor expansion, is given. The Zak transform is a linear signal representation that is of fundamental theoretical and practical importance in the context of the Gabor expansion. The theory of frames in general, and Weyl-Heisenberg frames in particular, yields important information about the mathematical properties of the Gabor expansion (existence of the expansion, uniqeness of the expansion coefficients, numerical properties). Using the Zak transform and results from frame theory, we finally consider methods for the calculation of the Gabor coefficients.

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