Representations of Gabor frame operators

Abstract

Gabor theory is concerned with expanding signals as linear combinations of elementary signals that are obtained from a single function (the window) by shifting it in time and frequency over integer multiples of a time shift parameter and a frequency shift parameter. In these expansion problems a key role is played by the Gabor frame operator associated with the set of elementary signals used in the expansions. The Gabor frame operator determines whether stable expansions exist for any finite-energy signal (that is, whether we have indeed a frame), and, if so, gives a recipe for computing the expansion coefficients by using the canonically associated dual frame. In this contribution we consider the Gabor frame operator and associated dual frames in the time domain, the frequency domain, the time-frequency domain, and, for rational values of the sampling factor´µfactor´µ ½ , the Zak transform domain. We thus have the opportunity to address the basic problems-whether we have a Gabor frame and how we can compute a dual frame-in any of these domains we find, depending on and , , convenient. The representations in the time domain and the frequency domain are conveniently discussed in the more general context of shift-invariant systems, and for this we present certain parts of what is known as Ron-Shen theory, adapted to our needs with emphasis on computational aspects. This contribution contains many examples, counter-intuitive and confusing results , statements that one would like to be true but that are not and vice versa, etc., to show that Gabor theory, despite its rapid development in the last ten years, is still far from being completed.