Some aspects of the time-frequency analysis of random processes

Abstract

This thesis is concerned with the analysis of non-stationary random processes by use of time-frequency techniques. The thesis concentrates on several problems, as we detail below. After a brief introduction in chapter I, chapter two gives a basic introduction to the method of time-frequency analysis of deterministic signals. In particular, we show the effect of windowing of the signal on the shape of the WVD and we derive a formula for the optimal window length. Efficient algorithms for the computation of the Wigner-Ville Distribution (WVD) are described. Chapter three is concerned with the analysis of random signals. Some introductory theory on classes of non-stationary processes is given, and the properties possessed by the WVD and Evolutive Spectrum (ES) are examined. Particular attention is given to the properties of the instantaneous frequency (IF), which is shown to be an inherently important parameter of a non-stationary process. The problem of estimating the ES and the mean IF are considered, and a class of estimators based on Cohen's class of time-frequency distributions (TFD) is proposed. A Karhunen-Loeve expansion for the process is used to deduce expansions for the WVD, IF and estimators of mean of these quantities. These expansions allow greater insight into the structure of these quantities, in particular, showing that the ES and IF may be decomposed linearly into principal time-frequency components. A similar expression is derived for the bias, allowing isolation of those principal components which contribute significantly to the bias. This allows the simplified design of estimators tailored to a priori known process structure. A design procedure which allows the properties of the estimators to be optimised to the desired trade-offs between bias, variance and resolution in various regions of the time-frequency plane is described. Simplified structures for the basis elements of the expansions in the case where the non-stationary process is uncorrelated are deduced. In particular, we show that the IF of a white process possesses X2 statistics with constant mean. We conclude that nonzero probability of negative IF is due only to correlation in the process. These expansions are used to determine the. probability distribution functions for the WVD, IF and the estimators of the mean of these quantities in the Gaussian case, by numerical inversion of the characteristic functions. The method described is numerically stable. The chapter concludes by examining the problem of sampling a non-stationary process. A definition of a band-limited harmonisable process is given, and sampling expansions are derived which are analogous to the stationary case. The problem of defining the instantaneous frequency of a discrete time process is considered.Chapter four considers the problem of cross analysis of non-stationary processes. A new quantity called the time-frequency coherence (TFC) is defined and we show that its properties and interpretation are analogous to the stationary coherence. A class of estimators for the TFC is proposed, and it is shown that in order to be meaningful, these estimators must be based on non-negative TFDs. A functional analysis setting for these TFDs is proposed. A theorem shows that any attempt to remove interference terms in the TFD of a multicomponent signal, is equivalent to the production of a non-negative TFD. The method is applied to the analysis of temperature gradient microstructure signals derived from turbulent fluids. We demonstrate that the method is capable of extracting new information, but we do not attempt to give a detailed physical interpretation. In the fifth chapter, the problem of pattern recognition and classification in the time-frequency domain is addressed. Pattern descriptors based on the eigenvalues and singular values of the ES are examined, and separation properties are considered. A clustering algorithm is used to determine optimal grouping of a number of random processes according to their time-frequency characteristics. An enhanced resolution method based on the generalised singular value decomposition is proposed. Examples illustrate the effectiveness of the method.