GROUP THEORETICAL APPROACH TO GABOR ANALYSIS

We describe new methods to obtain nonorthogonal Gabor expansions of di screte and finite signals and reconstruction of signals from regularly sampled short time fourier transform (STFT) values by series expansio ns. 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. Altho ugh bump-type atoms are natural candidates, the approach is not restri cted to such building blocks. Also the set of time- and frequency-shif t operators does not have to be a (product) lattice, but just an ordin ary (additive) subgroup of the time/frequency plane, which is naturall y identified with the 2-D n x n cyclic group. In contrast, other nonse parable subgroups turn out to be more interesting for our task: the ef ficient determination of a suitable set of coefficients for the cohere nt expansion. It is sufficient to determine the so-called dual Gabor a tom. The existence and basic properties of this dual atom are well kno wn in the case of lattice groups. It is shown that this is true for ge neral groups. But more importantly, we demonstrate that the conjugate gradient method reduces the computational complexity drastically.