Local Aspects of Time-Frequency-Analysis. Variation on a Theme.

Time-frequency analysis aims at providing simultaneous information on a signal`s time- and frequency content. In order to clarify this idea, time-frequency representations are often compared to a music-score, which, in fact, very efficiently conveys the information which frequency, or rather pitch, should sound at which instant. For the mathematical understanding, however, this comparison may be misleading, as the uncertainty principle, due to Heisenberg, does not permit an exact separation of signal components in the time-frequency domain. In the analysis of signals with Gabor frames, this fact leads to a trade-off between good time- and good frequency resolution. The first main topic of this project is the investigation of a new class of frames, denoted by "Quilted Gabor Frames (QGF)". By their construction, QGF allow for the adaptation of both the analysis window and the time- and frequency shift parameters to the properties of a signal or class of signals under consideration. One typical example would be given by the analysis of music signals, which often requires wide windows, implying good frequency resolution, in low frequency bands. On the other hand, in areas of the time-frequency domain, where the signal is dominated by rather percussive elements, which determine the rhythmical structure, short windows and dense time-sampling will lead to optimal results. By allowing for different Gabor frames locally in the time-frequency sense, QGF allow for analysis methods adapted to this kind of situation. As the structure of the Heisenberg group may no more be used in a global sense, completely new methods are necessary and will be developed in the course of the project, in order to achieve appropriate results. The second main topic of the project, indeed closely connected with the first one, is the investigation of families of time-frequency localization operators derived form the continuous time-frequency representation short-time Fourier transform. By means of these families of localization operators, a characterisation of certain Banach spaces, in particular the so called modulation spaces, will be given. This characterization, being an important result in itself, will also yield technical tools for the investigation of properties of QGF. Furthermore, a generalization of classical Gabor frames, which allows for more than just one analysis windows, will be studied (Multi-window Gabor frames). Here, the desired results aim for qualitative and quantitative statements on the replaceability of a given set of generating windows by a different, in its time-frequency characteristics similar one. Gabor-Multipliers are the discrete, i.e. sampled versions of the above-mentioned time-frequency localization operators. The analysis of two new models for generalization of classical Gabor-multipliers forms the fourth and last main topic of the proposed project. In generalization of the classical approach of modification of time- frequency coefficients by multiplication with a so called weight function, interaction between coefficients in different but close time-frequency points is included in the approximation of more general operators. An application example for these generalizations is the modelling of channel operators in wireless communication. The four main topics are connected by common ideas and technical tools as well as the motivation by concrete applications. The achieved results will therefore - partly with partners from the applied sciences - be realized in the context of the project and with a focus on real-life applications.