Some studies on characterization and modelling of stochastic processes in the multiscale framework

This thesis is an attempt to obtain a multiscale representation for stochastic processes. This thesis, besides obtaining the multiscale model, derives an alternative characterization for the filter bank. This alternative characterization is an illustrative representation of the transformations taking place in the signal as it propagates through the filter bank. The derivation of this characterization also leads to certain interesting relationships between the input and output correlations and spectrums of certain multirate filters (for WSCS inputs), which are elements of the filter bank. The thesis builds a framework for the proposed multiscale model, by obtaining an illustrative characterization of the filter bank. For this an alternative representation for wide sense cyclostationary (WSCS) processes is -obtained. This representation is important, as it is well known that the output of a filter bank, for a wide sense stationary (WSS) or a WSCS input, is WSCS. This alternative characterization of WSCS process is a time frequency representation of the WSCS process. The WSCS process, with period M, can be characterized using M distinct correlation sequences. The Fourier Transform of these correlation sequences gives M distinct spectrums, which can be used to represent the WSCS process, with period M. Using this alternative characterization of WSCS processes, we obtain the inputoutput relationships of certain multirate filters (for WSCS inputs). In particular those multirate filters that constitute the filter bank, have been discussed in detail. Once these relationships are available, an alternative characterization of the filter bank is obtained. This characterization explicitly brings out the transfonnations that take place in a signal as it propagates through a filter-bank. Relationships between the input and the output