A Group-theoretical Approach to Gabor Analysis

We describe new methods to obtain non-orthogonal Gabor expansions of discrete and finite signals and reconstruction of signals from regularly sampled STFT-values by series expansions. By this we understand the expansion of a signal of a given length n, into a (finite) series of coherent building blocks, obtained from a Gabor atom through discrete time- and frequency shift operators. Although bump-type atoms are natural candidates the approach is not restricted to such building blocks. Also the set of time/frequency shift operators does not have to be a (product) lattice, but just an ordinary (additive) subgroup of the time/frequency-plane, which is naturally identified with the two-dimensional n 2 n cyclic group. In contrast, other, non-separable subgroups, turn out to be more interesting for our task: the efficient determination of a suitable set of coefficients for the coherent expansion. It is sufficient to determine the so-called dual Gabor atom. The existence and basic properties of this dual atom are well known in the case of lattice groups. As use here that this is true for general groups. But more importantly, we demonstrate that the conjugate gradient method reduces the computational complexity drastically. Given the dual atom the required Gabor coefficients are obtained as short time Fourier coefficients of the given signal with the dual atom being the moving window