Time-variant filters are gaining more importance in today's signal processing applications. Also, there wavelet analysis has numerous applications. The goal of this project is to investigate time-variant systems based on wavelet analysis.
The concept of multipliers can be easily extended to wavelet frames. This means the coefficients of a wavelet analysis are multiplied by a fixed symbol and then resynthesized. The influence of the special structures of these sequences on the resulting operators will be investigated.
The theory of Pseudo-Differential Operators (PDO) can be translated to the wavelet case. How operators of interest in the investigation of multipliers, like the Kohn-Nirenberg correspondence, are translated to this case is of particular interest. Natural starting points for the research are:
Use dilations in the definition of the spreading function instead of modulation.
Define a special wavelet kernel function by using a weak formulation:
< K f , g > = < k , Wg f >
A very useful application for this project is an analysis-modification-synthesis system based on the wavelet analysis. With some language manipulation, this could be called a "Wavelet Phase Vocoder".
The application investigated in this project is the measurement of reflection coefficients. The wavelet analysis is preferable for signals containing transient parts. It is essential to separate the impulse responses of different reflections in order to calculate the absorption coefficient of a sound-proof wall. The impulse responses can be easily separated in a scalogram, and they can be extracted by using a wavelet multiplier.