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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The field of adaptive systems design is one which may be regarded as mature, having been the subject of considerable research effort in the areas of control and signal processing for more than 30 years. Indeed a number of books [1,2,3,4,5] on the subject have appeared in recent years which serve to illustrate the range and diversity of adaptive systems. However none of these texts has provided a coherent framework within which it is possible to evaluate the range of adaptive algorithms available, and their comparative merits in terms of performance, complexity and stability. It is also clear from the published literature that algorithms relating to filters having finite impulse response (FIR) have been much more successful than those relating to infinite impulse response (IIR) filters.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 2. Adaptive FIR Filter Algorithms

Abstract
The aims of this chapter are threefold; (i) to describe and define a broad selection of adaptive FIR filter algorithms, (ii) to give an indication of the convergence performance that currently available theoretical results would predict for these algorithms, and (iii) to provide a comparison of the computational requirements of the algorithms. The function of an adaptive FIR filter algorithm was identified in the seminal work of Widrow [29,30,31], and that is to find the optimum FIR filter from available data rather than from the second order statistics of the data. Widrow used the Wiener minimum mean-square error (MMSE) definition of optimum [6]. Thus in section 2.2 the MMSE cost function is defined and an expression for the optimum MMSE FIR filter is given in terms of autocorrelation and cross-correlation functions [32]. To illustrate the role of the Wiener FIR filter in the design of adaptive filter systems, the important problem of system identification is examined.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 3. Performance Comparisons

Abstract
While logistics preclude a comparison of the complete set of algorithms that have been mentioned in chapter 2, it is possible to examine the performance of a subset whose elements are representative of the three classes into which adaptive filters may be divided. These three classes are: (i) stochastic gradient search algorithms such as the LMS algorithm of subsection 2.5.1 and the BLMS algorithm of subsection 2.5.2, (ii) self-orthogonalising or transform domain algorithms such as the sliding DFT structure of subsection 2.6.1, and (iii) least squares techniques such as the simple RLS algorithm of subsection 2.4.1. The convergence performance of one or two algorithms from each class will be studied by computer simulation.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 4. A Self-Orthogonalising Block Adaptive Filter

Abstract
Traditionally the recursive least squares (RLS) and the least mean (LMS) algorithms have been considered to be the two major alternatives in adaptive finite impulse response (FIR) filtering. They represent the two extremes in a trade off of convergence performance against computational complexity. The conventional RLS algorithm of section 2.4.1 requires a number of computations per new data point that is a function of the square of the number of coefficients, N, in the FIR filter i.e. order N2 or O (N2). This contrasts sharply with the LMS algorithm which requires O (N) computations. However it is evident from the simulation results of chapter 3 that the RLS algorithm offers consistent rapid mean square error (MSE) convergence properties whereas the convergence properties of the LMS algorithm are generally poorer and dependent upon the input signal conditioning [36]. Fortunately the computational complexity of the RLS algorithm can be reduced by exploiting the shifting property of the input vector to yield the fast algorithms [35,65,68,60,61], such as those which are developed in appendix A and appendix B. Although the fast algorithms offer RLS convergence properties at O (N) computations they still represents a computational load which is significantly higher than the LMS algorithm.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 5. The Infinite Impulse Response Linear Equaliser

Abstract
In many adaptive filtering problems, solutions that use purely FIR filters can provide acceptable performance [87,11,100]. Indeed FIR filters are generally to be preferred as they are unconditionally stable and because of the wide selection of well understood adaptive FIR filter algorithms that are available, cf. chapter 2. However these FIR realisations suffer from problems of indeterminate order when it is necessary to model transfer function poles. In particular, when the poles of transfer function are close to the unit circle in the z-plane, a FIR filter of high order may be required to meet a particular performance goal [101]. The obvious alternative has been the adoption of adaptive IIR filters.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 6. An Adaptive IIR Equaliser

Abstract
While the IIR Wiener filter exhibits a distinct performance advantage over a FIR filter of the same order when used to equalise a known channel, significant problems are encountered with the former when the channel is unknown or time-varying and an adaptive filter structure is required. An initial approach to the problem might be to postulate an adaptive algorithm such as those suggested in [101] that would recursively estimate the coefficients of the IIR Wiener filter in the same manner as the LMS algorithm [31] is used to estimate the coefficients of the FIR Wiener filter. However, in the process of adaptation, there is a finite probability that the poles of the filter will move outside the unit circle in the z-plane. This can lead to instability if the poles remain outside the unit circle for an extended period [101]. As discussed in chapter 5, adaptive IIR filter algorithms do exist whose convergence in a mean sense is assured but few theoretical results are available with which to predict the MSE convergence properties ot these algorithms.
Bernard Mulgrew, Colin F. N. Cowan

Chapter 7. Conclusions

Summary
A broad selection of adaptive finite impulse response (FIR) filter algorithms was examined to assess their theoretical convergence performance and computational requirements. From this examination a classification system has been specified in which the available algorithms are grouped into three classes according to convergence performance and computational complexity. These three classes are: (i) stochastic gradient (SG) algorithms, (ii) self-orthogonalising (SO) algorithms and (iii) recursive least squares (RLS) algorithms. Formerly classes (ii) and (iii) had been grouped together. Movement from class (i) through (ii) to (iii) improves convergence performance at the expense of increasing computational complexity.
Bernard Mulgrew, Colin F. N. Cowan

Backmatter

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