Objective:

Gabor multipliers are an efficient time-variant filtering tool 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 (the time-frequency mask or symbol). The result is then synthesized.

Transforms other than the Gabor transform, particularly the wavelet transform, are more suitable for certain applications. The concept of multipliers can easily be extended in this case. This results in the introduction of operators called wavelet multipliers, which will be investigated in detail in this project. The project aims to precisely define wavelet multipliers' mathematical properties and optimize their use in applications.

Method:

The problem will be approached using modern wavelet theory, harmonic analysis tools, and numeric tools. Formulation and demonstration of analytic statements will be conducted jointly with systematic numeric experiments in order to study the properties of wavelet multipliers.

The following topics will be investigated in the project:

  • Eigenvalues and eigenvectors of wavelet multipliers
  • Invertibility and injectivity of wavelet multipliers
  • Reproducing kernel invariance
  • Discretization and implementation of wavelet multiplier
  • Best approximation of operators by wavelet multipliers and identification of wavelet multipliers

Application:

The applications of wavelet multipliers in signal processing are numerous and include any application requiring time-variant filtering. Some applications of wavelet multipliers will be investigated further in the parallel projects:

  • Mathematical Modeling of Auditory Time-Frequency Masking Functions
  • Improvement of Head-Related Transfer Function Measurements
  • Advanced Method of Sound Absorption Measurements

Publications:

  • Anaïk Olivero: "Expérimentation des multiplicateurs temps-échelle" (On the time-scale multipliers) Master thesis under the supervision of R. Kronland-Martinet and B. Torrésani, June 2008