Applications of nonlinear filters with the linear-in-the-parameter structure

The subject of this thesis is the application of nonlinear lters, with the linear-in-the-parameter structure, to time series prediction and channel equalisation problems. In particular, the Volterra and the radial basis function (RBF) expansion techniques are considered to implement the nonlinear lter structures. These approaches, however, will generate lters with very large numbers of parameters. As large lter models require signi cant implementation complexity, they are undesirable for practical implementations. To reduce the size of the lter, the orthogonal least squares (OLS) algorithm is considered to perform model selection. Simulations were conducted to study the e ectiveness of subset models found using this algorithm, and the results indicate that this selection technique is adequate for many practical applications. The other aspect of the OLS algorithm studied is its implementation requirements. Although the OLS algorithm is very e cient, the required computational complexity is still substantial. To reduce the processing requirement, some fast OLS methods are examined. Two major applications of nonlinear lters are considered in this thesis. The rst involves the use of nonlinear lters to predict time series which possess nonlinear dynamics. To study the performance of the nonlinear predictors, simulations were conducted to compare the performance of these predictors with conventional linear predictors. The simulation results con rm that nonlinear predictors normally perform better than linear predictors. Within this study, the application of RBF predictors to time series that exhibit homogeneous nonstationarity is also considered. This type of time series possesses the same characteristic throughout the time sequence apart from local variations of mean and trend. The second application involves the use of lters for symbol-decision channel equalisation. The decision function of the optimal symbol-decision equaliser is rst derived to show that it is nonlinear, and that it may be realised explicitly using a RBF lter. Analysis is then carried out to illustrate the di erence between the optimum equaliser's performance and that of the conventional linear equaliser. In particular, the e ects of delay order on the equaliser's decision boundaries and bit error rate (BER) performance are studied. The minimum mean square error (MMSE) optimisation criterion for training the linear equaliser is also examined to illustrate the sub-optimum nature of such a criterion. To improve the linear equaliser's performance, a method which adapts the equaliser by minimising the BER is proposed. Our results indicate that the linear equalisers performance is normally improved by using the minimum BER criterion. The decision feedback equaliser (DFE) is also examined. We propose a transformation using the feedback inputs to change the DFE problem to a feedforward equaliser problem. This uni es the treatment of the equaliser structures with and without decision feedback. Declaration of originality This thesis was composed entirely by myself. The work reported herein was conducted exclusively by myself, with the exception of the chapters mentioned below, in the Department of Electrical Engineering at the University of Edinburgh. The work presented in Chapter 5 was performed by Dr Sheng Chen (University of Portmouth) and myself under the direction of Dr Bernard Mulgrew. Eng Siong CHNG July 1995 i Acknowledgements I would like to thank the following people for their invaluable assistance during the course of this PhD: Dr Bernard Mulgrew for his patience and willingness to share his insight throughout my stay in Edinburgh. Professor Peter Grant for his support and guidance. Dr Sheng Chen for his help in all the papers we wrote together. Mr Achilleas Stogillou and Mr John Thompson for their many hours speaking about Mathematics, RBF networks, time series prediction and channel equalisation problems. Dr Gavin Gibson for providing guidance to the work performed in Chapters 4 and 5. Dr Iain Scott for proofreading several chapters of the thesis. Dr Martin Reekie for proofreading the entire thesis. Yee Ling for her wonderful company for so many years. And lastly, to my parents and family for their love and understanding in letting me study so far away from home. ii Acknowledgements To my parents, and to Yee Ling. iii