Gabor multipliers are an efficient tool for time variant filtering used implicitly in many engineering applications in signal processing. For these operators, the result of a Gabor transform (the sampled version of the Short Time Fourier Transform) is multiplied by a fixed function, called the time-frequency mask or symbol. The result is then synthesized.
While Gabor multipliers are widely and practically used, some of their theoretical properties are not well known. The goal of this project is to improve the mathematical knowledge about Gabor multipliers, in order to optimize their use in applications.
The problem will be approached using modern Gabor theory, harmonic analysis tools, and numeric tools. Formulation and demonstration of analytical statements will be conducted jointly with systematic numeric experiments to study the properties of Gabor multipliers.
The following topics will be investigated in the project:
- Eigenvalues and eigenvectors of Gabor multipliers and their localization
- Invertibility and injectivity of Gabor multipliers
- Reproducing kernel invariance
- Connection of irregular Gabor multipliers and irregular frames of translates
- Discretization and implementation of Gabor multipliers
- Best approximation of operators by Gabor multipliers and identification of Gabor multipliers.
The applications of Gabor multipliers in signal processing are numerous, and include any application requiring time-variant filtering. Some applications of Gabor multipliers will be investigated further in the following parallel projects:
- Mathematical Modeling of Auditory Time-Frequency Masking Functions
- Improvement of Head-Related Transfer Function Measurements
- Advanced Method of Sound Absorption Measurements
The implementation of a Gabor multiplier in the software system STx has already proceeded quite far, see Stx-Mulac.
- Monika Dörfler and Bruno Torrésani, “Representation of operators in the time-frequency domain and generalized Gabor multipliers”, J. Fourier Anal. Appl., 2009 (in press)
- Yohan Frutiger: "Multiplicateurs de Gabor pour les transformations sonores" (Gabor Multipliers for sound transformations) Master thesis under the supervision of R. Kronland-Martinet, June 2008
- F. Jaillet, P. Balazs, M. Dörfler and N. Engelputzeder, “On the Structure of the Phase around the Zeros of the Short-Time Fourier Transform”, NAG/DAGA 2009, International Conference on Acoustics, March 2009, Rotterdam, Nederland
- F. Jaillet, P. Balazs and M. Dörfler, “Nonstationary Gabor Frames”, SampTA'09, 8th International Conference on Sampling and Applications, May 2009, Marseille, France