Time-Frequency Modelling

The overriding aim of this thesis is to investigate the bene ts of focusing time-frequency analysis on particular regions of the time-frequency plane. The thesis examines aspects of such a regionalisation in the analysis of both deterministic signals and stochastic processes. The majority of deterministic energetic time-frequency representations are nonparametric, indicating the distribution of the energy of a signal in the time-frequency plane, but providing no further information about the time-frequency structure. This thesis develops a semi-parametric time-frequency model to simultaneously describe the time-frequency energetic structure of a signal and provide an indication of its time-frequency complexity. The model aims to identify `time-frequency components' within the signal, to indicate how their energy is distributed in the time-frequency plane, and thereby to probabilistically associate every location in the plane with each identi ed component. The thesis investigates a number of applications of the model including time-frequency decomposition, noise reduction, and the design of perfect reconstruction lter banks. The model is also used to generate regionally optimised time-frequency distributions. The thesis develops a framework for regional time-frequency analysis of nonstationary random processes and LTV systems, de ning the class of `regionally underspread' processes and systems and developing approximate regional transfer function relationships. The framework is used to de ne a measure of the strength of the linear relationship between two regionally underspread processes in a speci c region of the time-frequency plane. The measure, termed `regional coherence', is useful when the linear relationship remains consistent over a time-frequency region. The time-frequency model and the regional coherence measure are both used in the analysis of slow wave potentials recorded from the motor-cortex of a monkey during the execution of a motor task. iv