Discrete Stochastic Processes and Optimal Filtering E-Book

Jean-Claude Bertein

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 172

Veröffentlichungsjahr: 2012

Beschreibung

Optimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using MATLAB.

Leseprobe

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

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

7.5. Appendices

7.6. Examples treated using Matlab software

Table of Symbols and Notations

Bibliography

Index

To our families.

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

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

First edition published 2007 by ISTE Ltd

Second edition published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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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John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd 2007, 2010

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.

Discrete stochastic processes and optimal filtering / Jean-Claude Bertein, Roger Ceschi. -- 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-181-0

1. Signal processing--Mathematics. 2. Digital filters (Mathematics) 3. Stochastic processes. I. Ceschi, Roger. II.Title.

TK5102.9.B465 2009

621.382′2--dc22

2009038813

British Library Cataloguing-in-Publication Data

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

ISBN 978-1-84821-181-0

Preface

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

It makes up a necessary stage in the most diverse domains: calculation of the orbits or guidance of aircraft in the aerospace or aeronautic domain, calculation of filters in the telecommunications or command systems domain, or even in the processing of seismic signals domain — the list is endless.

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

In their work, the authors have taken pains to stress the 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 turn to other works to acquire a sound knowledge of the subjects studied.

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

Introduction

The objective of this book is the progressive and rigorous presentation of the bases of discrete optimal filtering. 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 discusses 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, “Introduction to Discrete Time Processes”, is by its very nature more “physics-based” than the preceding chapters and can be considered as an introduction to numerical filtering. Results that are essential for what follows will be given in this chapter.

Chapter 4, “Estimation”, brings us the pre-requisites essential for the construction of optimal filters. The results obtained on projections in Hilbert spaces make up the cornerstone of future demonstrations.

Chapter 5 discusses the Wiener filter, an electronic device well adapted to processing second-order stationary signals. 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 adaptation or convergence process, we again encounter the Wiener filter.

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

Each chapter is enhanced by a series of exercises with answers, and solutions to 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

Recalling that the set of real n -tuples, can be fitted into two laws:

making it a vector space of dimension n .

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