WAVELET ANALYSIS AND FREQUENCY BAND DECOMPOSITIONS

Abstract. In many applications such as parameter identification of oscillating systems in civil enginee-ring, speech processing, image processing and others we are interested in the frequency con-tent of a signal locally in time. As a start wavelet analysis provides a time-scale decomposi-tion of signals, but this wavelet transform can be connected with an appropriate time-frequency decomposition. For instance in Matlab are defined pseudo-frequencies of wavelet scales as frequency centers of the corresponding bands. This frequency bands overlap more or less which depends on the choice of the biorthogonal wavelet system. Such a definition of frequency center is possible and useful, because different frequencies predominate at different dyadic scales of a wavelet decomposition or rather at different nodes of a wavelet packet decomposition tree. The goal of this work is to offer better algorithms for characterising frequency band behaviour and for calculating frequency centers of orthogonal and biorthogonal wavelet systems. This will be done with some product formulas in frequency domain. Now the connecting procedu-res are more analytical based, better connected with wavelet theory and more assessable. This procedures doesn’t need any time approximation of the wavelet and scaling functions. The method only works in the case of biorthogonal wavelet systems, where scaling functions and wavelets are defined over discrete filters. But this is the practically essential case, because it is connected with fast algorithms (FWT, Mallat Algorithm). At the end corresponding to the wavelet transform some closed formulas of pure oscillations are given. They can generally used to compare the application of different wavelets in the FWT regarding it’s frequency behaviour. 1