The work presented in this text relates to research work in the general area of adaptive filter theory and practice which has been carried out at the Department of Electrical Engineering, University of Edinburgh since 1977. Much of the earlier work in the department was devoted to looking at the problems associated with the physical implementation of these structures. This text relates to research which has been undertaken since 1984 which is more involved with the theoretical development of adaptive algorithms. The text sets out to provide a coherent framework within which general adaptive algorithms for finite impulse response adaptive filters may be evaluated. It further presents one approach to the problem of finding a stable solution to the infinite impulse response adaptive filter problem. This latter objective being restricted to the communications equaliser application area. The authors are indebted to a great number of people for their help, guidance and encouragement during the course of preparing this text. We should first express our appreciation for the support given by two successive heads of department at Edinburgh, Professor J. H. Collins and Professor J. Mavor. The work reported here could not have taken place without their support and also that of many colleagues, principally Professor P. M. Grant who must share much of the responsibility for instigating this line of research at Edinburgh.

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1 Introduction.- 1.1 Adaptive Signal Processing.- 1.2 The Adaptive Filter.- 1.3 Modes of Operation.- 1.4 Application of Adaptive Filters.- 1.5 Summary.- 2 Adaptive Fir Filter Algorithms.- 2.1 Introduction.- 2.2 Optimum Linear Estimation.- 2.3 Sampled Matrix Inversion.- 2.4 Least Squares Estimation.- 2.5 Stochastic Gradient Methods.- 2.6 Self-Orthogonalising Algorithms.- 2.7 Summary and Complexity Comparison.- 3 Performance Comparisons.- 3.1 Introduction.- 3.2 System Identification.- 3.3 Channel Equalisation.- 3.4 Summary and Conclusions.- 4 A Self-Orthogonalising Block Adaptive Filter.- 4.1 Introduction.- 4.2 Theoretical Development.- 4.3 A Practical Algorithm.- 4.4 Computational Complexity.- 4.5 Simulation Results.- 4.6 Conclusions.- 5 The Infinite Impulse Response Linear Equaliser.- 5.1 Introduction.- 5.2 The Linear Equaliser.- 5.3 FIR and IIR Equaliser Performance.- 5.4 System Identification.- 5.5 Conclusions.- 6 An Adaptive IIR Equaliser.- 6.1 Introduction.- 6.2 The Kalman Filter.- 6.3 The Kalman Filter as an IIR Equaliser.- 6.4 An Adaptive Kalman Equaliser.- 6.5 RLS System Identification.- 6.6 Conclusions.- 7 Conclusions.- 7.1 Summary.- 7.2 Limitations and Further Work.- Appendix A The Fast Kalman Algorithm.- Appendix B The RLS Lattice Algorithm.- Appendix C Circular and Linear Convolution.- References.