Time-Frequency Analysis is concerned with the representation of signals as super-position of well localized bump-functions with well-defined frequency content. In the 1D-case one has to find coefficients which allow to write a sound or musical signal as a superposition (double series) of so-called Gabor atoms, which are well localized in time and frequency, comparable to the elementary sounds generated by a kind of micro-tonal synthesizer operating at discrete time-steps. The 2D analogue are local patches of plane waves, i.e. kind of zebra-like building blocks which are added up to represent arbitrary images.

In contrast to wavelets such systems can never have all these good properties and in addition form an orthonormal system. Therefore it is necessary to work with (at least slightly) redundant systems, so called frames (resp. Gabor frames). Nevertheless it is possible to exploit the structure of such systems (in particular invariance properties of the Gabor frame operator) in order to come up with efficient algorithms for (non-orthogonal) signal expansion, with the advantage that the label of a coefficient also provides relevant information (e.g. to which time and at to which frequency it is assigned).

We will discuss various numerical issues arising in this context, mentioning also the problem of connecting continuous models with discrete computations. Finally we indicate how slowly variant channels (in mobile communication) or space-variant blurring operators can be described as Gabor multipliers (operators which perform simple multiplication of the Gabor coefficients) and which ideas can be used to find approximate inverse operators.