Adaptive processing. The least mean squares approach with applications in transmission.

*(English)*Zbl 0854.94001
Chichester: Wiley. xix, 456 p. (1995).

This book is concerned with a general, unified approach to adaptive filtering and its applications in transmission. More precisely, it deals with the basic theory of adaptive algorithms when a desired output reference is given (supervised case).

Adaptive filtering is studied in its simplest form, namely using the least mean squares (LMS) approach, that is the one requiring the lowest computational complexity and the most suitable for real-time computation.

Even if large portions of the book are devoted to mathematical developments, the concrete engineering applications are also discussed; in particular, three basic adaptive transmission systems, i.e. the channel equalizer, the echo canceller and the message predictor are repeatedly considered to exemplify the theoretical results.

The book is divided into four parts.

Part 1 deals with the theory of adaptive, finite impulse response (FIR) filters. First, the case of independent successive input vectors is considered and then the case of correlated input vectors is studied under the assumption of a finite memory in the observed signals. The latter situation is a more realistic one and, besides covering practical cases, provides the required rigorous tools to prove exponential convergence to the optimal filter, at least in the absence of measurement noise. Finally, the asymptotic properties of the LMS algorithm are analyzed in the case of measurement noise and its fluctuations around optimality are studied. A few exercises with solutions are also provided in this part of the book.

Part 2 is concerned with practical implementation considerations. In particular, aspects connected with technology, number of components and bandwidth (summarized by the word “complexity”) are studied. Problems connected with precision, describing the difference between the theoretical adaptive filter and its implementation, which in turn involves inaccuracies and noises, are also examined. Furthermore, simplified sign algorithms are considered, where sign functions are used in their implementation. In particular, the clipped algorithm, using the sign of the input data, and the pilot algorithm, using the sign of the error, are studied in detail under the assumption that input and reference signals are independent of past filter parameters.

Part 3 is concerned with the case when the optimal filter is time varying, so that questions connected with the steady state behaviour of the adaptive filter are examined. The corresponding problem is called tracking. The main types of time variations considered in this part are bounded time variations, including deterministic ones, and zero-mean random time variations. The example of digital (nonlinear) phase tracking loop is finally considered.

Part 4 deals with adaptive digital recursive filters, which are infinite impulse response (IIR) filters. They prove particularly useful in transmission systems since they allow a long time impulse response and/or a sharp frequency cut-off with a reasonable computational complexity.

The case of ARMA prediction is examined in detail. It is shown that the problem of optimization of recursive filters presents an intrinsic difficulty due to the fact that the mean square error can admit several minima and there is no explicit formula for the optimal parameters and the associated mean square error. Still, the LMS approach proves useful for the solution of this problem. However, due to the recursive nature of the filter, exact calculations require infinite memory, so that various approximate algorithms are presented with memory size truncated at a finite value.

Finally, the problem of stability is examined for adaptive recursive algorithms showing that, due to the nonlinear nature of the recursive system, including the adaptation loop, the identified parameters exhibit oscillations followed by smooth periods. However, global stability is guaranteed by a self-stabilization property which is enjoyed by most LMS updating algorithms.

Adaptive filtering is studied in its simplest form, namely using the least mean squares (LMS) approach, that is the one requiring the lowest computational complexity and the most suitable for real-time computation.

Even if large portions of the book are devoted to mathematical developments, the concrete engineering applications are also discussed; in particular, three basic adaptive transmission systems, i.e. the channel equalizer, the echo canceller and the message predictor are repeatedly considered to exemplify the theoretical results.

The book is divided into four parts.

Part 1 deals with the theory of adaptive, finite impulse response (FIR) filters. First, the case of independent successive input vectors is considered and then the case of correlated input vectors is studied under the assumption of a finite memory in the observed signals. The latter situation is a more realistic one and, besides covering practical cases, provides the required rigorous tools to prove exponential convergence to the optimal filter, at least in the absence of measurement noise. Finally, the asymptotic properties of the LMS algorithm are analyzed in the case of measurement noise and its fluctuations around optimality are studied. A few exercises with solutions are also provided in this part of the book.

Part 2 is concerned with practical implementation considerations. In particular, aspects connected with technology, number of components and bandwidth (summarized by the word “complexity”) are studied. Problems connected with precision, describing the difference between the theoretical adaptive filter and its implementation, which in turn involves inaccuracies and noises, are also examined. Furthermore, simplified sign algorithms are considered, where sign functions are used in their implementation. In particular, the clipped algorithm, using the sign of the input data, and the pilot algorithm, using the sign of the error, are studied in detail under the assumption that input and reference signals are independent of past filter parameters.

Part 3 is concerned with the case when the optimal filter is time varying, so that questions connected with the steady state behaviour of the adaptive filter are examined. The corresponding problem is called tracking. The main types of time variations considered in this part are bounded time variations, including deterministic ones, and zero-mean random time variations. The example of digital (nonlinear) phase tracking loop is finally considered.

Part 4 deals with adaptive digital recursive filters, which are infinite impulse response (IIR) filters. They prove particularly useful in transmission systems since they allow a long time impulse response and/or a sharp frequency cut-off with a reasonable computational complexity.

The case of ARMA prediction is examined in detail. It is shown that the problem of optimization of recursive filters presents an intrinsic difficulty due to the fact that the mean square error can admit several minima and there is no explicit formula for the optimal parameters and the associated mean square error. Still, the LMS approach proves useful for the solution of this problem. However, due to the recursive nature of the filter, exact calculations require infinite memory, so that various approximate algorithms are presented with memory size truncated at a finite value.

Finally, the problem of stability is examined for adaptive recursive algorithms showing that, due to the nonlinear nature of the recursive system, including the adaptation loop, the identified parameters exhibit oscillations followed by smooth periods. However, global stability is guaranteed by a self-stabilization property which is enjoyed by most LMS updating algorithms.

Reviewer: G.Di Masi (Padova)

##### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93E11 | Filtering in stochastic control theory |

93D21 | Adaptive or robust stabilization |

93E24 | Least squares and related methods for stochastic control systems |