**ABSTRACT**
### A Group-theoretical Approach to Gabor Analysis

#### H.G. Feichtinger, T. Strohmer, O. Christensen

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*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.

**Keywords:** algorithms, Gabor analysis, numerical work

Download the paper as a GNU-zipped postscript file (88806 bytes).