Table of Contents

Preface

Introduction

Chapter 1: Random Vectors

1.1. Definitions and general properties

1.2. Spaces L¹ (dP) and L²(dP)

1.3. Mathematical expectation and applications

1.4. Second order random variables and vectors

1.5. Linear independence of vectors of L² (dP)

1.6. Conditional expectation (concerning random vectors with density function)

1.7. Exercises for Chapter 1

Chapter 2: Gaussian Vectors

2.1. Some reminders regarding random Gaussian vectors

2.2. Definition and characterization of Gaussian vectors

2.3. Results relative to independence

2.4. Affine transformation of a Gaussian vector

2.5. The existence of Gaussian vectors

2.6. Exercises for Chapter 2

Chapter 3: Introduction to Discrete Time Processes

3.1. Definition

3.2. WSS processes and spectral measure

3.3. Spectral representation of a WSS process

3.4. Introduction to digital filtering

3.5. Important example: autoregressive process

3.6. Exercises for Chapter 3

Chapter 4: Estimation

4.1. Position of the problem

4.2. Linear estimation

4.3. Best estimate — conditional expectation

4.4. Example: prediction of an autoregressive process AR (1)

4.5. Multivariate processes

4.6. Exercises for Chapter 4

Chapter 5: The Wiener Filter

5.1. Introduction

5.2. Resolution and calculation of the FIR filter

5.3. Evaluation of the least error

5.4. Resolution and calculation of the IIR filter

5.5. Evaluation of least mean square error

5.6. Exercises for Chapter 5

Chapter 6: Adaptive Filtering: Algorithm of the Gradient and the LMS

6.1. Introduction

6.2. Position of problem [WID 85]

6.3. Data representation

6.4. Minimization of the cost function

6.5. Gradient algorithm

6.6. Geometric interpretation

6.7. Stability and convergence

6.8. Estimation of gradient and LMS algorithm

6.9. Example of the application of the LMS algorithm

6.10. Exercises for Chapter 6

Chapter 7: The Kalman Filter

7.1. Position of problem

7.2. Approach to estimation

7.3. Kalman filtering

7.4. Exercises for Chapter 7

Appendix A

Appendix B

Table of Symbols and Notations

Bibliography

Index

First published in France in 2005 by Hermes Science/Lavoisier entitled “Processus stochastiques discrets et filtrages optimaux”

First published in Great Britain and the United States in 2007 by ISTE Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd, 2007

© LAVOISIER, 2005

The rights of Jean-Claude Bertein and Roger Ceschi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Bertein, Jean-Claude.

[Processus stochastiques discrets et filtrages optimaux. English]

Discrete stochastic processes and optimal filtering/Jean-Claude Bertein, Roger Ceschi.

p. cm.

Includes index.

“First published in France in 2005 by Hermes Science/Lavoisier entitled “Processus stochastiques discrets et filtrages optimaux”.”

ISBN 978-1-905209-74-3

1. Signal processing--Mathematics. 2. Digital filters (Mathematics) 3. Stochastic processes.

I. Ceschi, Roger. II. Title.

TK5102.9.B465 2007

621.382'2--dc22

2007009433

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 13: 978-1-905209-74-3

To our families

We wish to thank Mme Florence François for having typed the manuscript, and M. Stephen Hazlewood who assured the translation of the book

Preface

Discrete optimal filtering applied to stationary and non-stationary signals allows us to process in the most efficient manner possible, according to chosen criteria, all of the problems that we might meet in situations of extraction of noisy signals.

This constitutes the necessary stage in the most diverse domains: the calculation of the orbits or guidance of aircraft in the aerospace or aeronautic domain, the calculation of filters in the telecommunications domain, or in the domain of command systems, or again in that of the processing of seismic signal — the list is non-exhaustive.

Furthermore, the study and the results obtained from discrete signals lend themselves easily to the calculator.

In their book, the authors have taken pains to stress educational aspects, preferring this to displays of erudition; all of the preliminary mathematics and probability theories necessary for a sound understanding of optimal filtering have been treated in the most rigorous fashion. It should not be necessary to have to turn to other works to acquire a sound knowledge of the subjects studied.

Thanks to this work, the reader will be able not only to understand discrete optimal filtering but also will be able easily to go deeper into the different aspects of this wide field of study.

Introduction

The object of this book is to present the bases of discrete optimal filtering in a progressive and rigorous manner. The optimal character can be understood in the sense that we always choose that criterion at the minimum of the norm -L² of error.

Chapter 1 tackles random vectors, their principal definitions and properties.

Chapter 2 covers the subject of Gaussian vectors. Given the practical importance of this notion, the definitions and results are accompanied by numerous commentaries and explanatory diagrams.

Chapter 3 is by its very nature more “physics” heavy than the preceding ones and can be considered as an introduction to digital filtering. Results that will be essential for what follows will be given.

Chapter 4 provides the pre-requisites essential for the construction of optimal filters. The results obtained on projections in Hilbert spaces constitute the cornerstone of future demonstrations.

Chapter 5 covers the Wiener filter, an electronic device that is well adapted to processing stationary signals of second order. Practical calculations of such filters, as an answer to finite or infinite pulses, will be developed.

Adaptive filtering, which is the subject of Chapter 6, can be considered as a relatively direct application of the determinist or stochastic gradient method. At the end of the process of adaptation or convergence, the Wiener filter is again encountered.

The book is completed with a study of Kalman filtering which allows stationary or non-stationary signal processing; from this point of view we can say that it generalizes Wiener’s optimal filter.

Each chapter is accentuated by a series of exercises with answers, and resolved examples are also supplied using Matlab software which is well adapted to signal processing problems.

Chapter 1

Random Vectors

1.1. Definitions and general properties

If we remember that n = {x = (x₁,…,xn)| xj ;j=1 to n}, the set of real n -tuples can be fitted to two laws: and making it a vector space of dimension n.

The basis implicitly considered on n will be the canonical base ℓ₁ = (1, 0,…, 0),…, ℓn = (0,…, 0, 1) and x n expressed in this base will be denoted:

Definition of a real random vector

Beginning with a basic definition, without concerning ourselves at the moment with its rigor: we can say simply that a real vector linked to a physical or biological phenomenon is random if the value taken by this vector is unknown and the phenomenon is not completed.

For typographical reasons, the vector will instead be written XT = (X₁,…, Xn) or even X = (X₁,…, Xn) when there is no risk of confusion.

In other words, given a random vector X and Β ⊂ n we do not know if the assertion (also called the event) (X Β) is true or false:

However, we do usually know the “chance” that X Β; this is denoted P (X B) and is called the probability of the event (X B).

After completion of the phenomenon, the result (also called the realization) will be denoted

when there is no risk of confusion.

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Lesen Sie weiter in der vollständigen Ausgabe!