Digital Filters Design for Signal and Image Processing E-Book

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

Seitenzahl: 316

Veröffentlichungsjahr: 2013

Beschreibung

Dealing with digital filtering methods for 1-D and 2-D signals, this book provides the theoretical background in signal processing, covering topics such as the z-transform, Shannon sampling theorem and fast Fourier transform. An entire chapter is devoted to the design of time-continuous filters which provides a useful preliminary step for analog-to-digital filter conversion. Attention is also given to the main methods of designing finite impulse response (FIR) and infinite impulse response (IIR) filters. Bi-dimensional digital filtering (image filtering) is investigated and a study on stability analysis, a very useful tool when implementing IIR filters, is also carried out. As such, it will provide a practical and useful guide to those engaged in signal processing.

Leseprobe

Table of Contents

Introduction

Chapter 1: Introduction to Signals and Systems

1.1. Introduction

1.2. Signals: categories, representations and characterizations

1.3. Systems

1.4. Properties of discrete-time systems

1.5. Bibliography

Chapter 2: Discrete System Analysis

2.1. Introduction

2.2. The z-transform

2.3. The inverse z-transform

2.4. Transfer functions and difference equations

2.5. Z-transforms of the autocorrelation and intercorrelation functions

2.6. Stability

Chapter 3: Frequential Characterization of Signals and Filters

3.1. Introduction

3.2. The Fourier transform of continuous signals

3.3. The discrete Fourier transform (DFT)

3.4. The fast Fourier transform (FFT)

3.5. The fast Fourier transform for a time/frequency/energy representation of a non-stationary signal

3.6. Frequential characterization of a continuous-time system

3.7. Frequential characterization of discrete-time system

Chapter 4: Continuous-Time and Analog Filters

4.1. Introduction

4.2. Different types of filters and filter specifications

4.3. Butterworth filters and the maximally flat approximation

4.4. Equiripple filters and the Chebyshev approximation

4.5. Elliptic filters: the Cauer approximation

4.6. Summary of four types of low-pass filter: Butterworth, Chebyshev type I, Chebyshev type II and Cauer

4.7. Linear phase filters (maximally flat delay or MFD): Bessel and Thomson filters

4.8. Papoulis filters (optimum (On))

4.9. Bibliography

Chapter 5: Finite Impulse Response Filters

5.1. Introduction to finite impulse response filters

5.2. Synthesizing FIR filters using frequential specifications

5.3. Optimal approach of equal ripple in the stop-band and passband

5.4. Bibliography

Chapter 6: Infinite Impulse Response Filters

6.1. Introduction to infinite impulse response filters

6.2. Synthesizing IIR filters

6.3. Bibliography

Chapter 7: Structures of FIR and IIR Filters

7.1. Introduction

7.2. Structure of FIR filters

7.3. Structure of IIR filters

7.4. Realizing finite precision filters

7.5. Bibliography

Chapter 8: Two-Dimensional Linear Filtering

8.1. Introduction

8.2. Continuous models

8.3. Discrete models

8.4. Filtering in the spatial domain

8.5. Filtering in the frequency domain

8.6. Bibliography

Chapter 9: Two-Dimensional Finite Impulse Response Filter Design

9.1. Introduction

9.2. Introduction to 2-D FIR filters

9.3. Synthesizing with the two-dimensional windowing method

9.4. Appendix: spatial window functions and their implementation

9.5. Bibliography

Chapter 10: Filter Stability

10.1. Introduction

10.2. The Schur-Cohn criterion

10.3. Appendix: resultant of two polynomials

10.4. Bibliography

Chapter 11: The Two-Dimensional Domain

11.1. Recursive filters

11.2. Stability criteria

11.3. Algorithms used in stability tests

11.4. Linear predictive coding

11.5. Appendix A: demonstration of the Schur-Cohn criterion

11.6. Appendix B: optimum 2-D stability criteria

11.7. Bibliography

List of Authors

Index

First published in France in 2004 by Hermès Science/Lavoisier entitled “Synthèse de filtres numériques en traitement du signal et des images”

First published in Great Britain and the United States in 2006 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:

ISTE Ltd6 Fitzroy SquareLondon W1T 5DXUKISTE USA4308 Patrice RoadNewport Beach, CA 92663USAwww.iste.co.uk

© ISTE Ltd, 2006

© LAVOISIER, 2004

The rights of Mohamed Najim to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Synthèse de filtres numériques en traitement du signal et des images.

English

Digital filters design for signal and image processing/edited by Mohamed Najim.

p. cm.

Includes index.

ISBN-13: 978-1-905209-45-3

ISBN-10: 1-905209-45-2

1. Electric filters, Digital. 2. Signal processing--Digital techniques.

3. Image processing--Digital techniques. I. Najim, Mohamed. II. Title.

TK7872.F5S915 2006

621.382’2--dc22

2006021429

British Library Cataloguing-in-Publication Data

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

ISBN 10: 1-905209-45-2

ISBN 13: 978-1-905209-45-3

Introduction

Over the last decade, digital signal processing has matured; thus, digital signal processing techniques have played a key role in the expansion of electronic products for everyday use, especially in the field of audio, image and video processing. Nowadays, digital signal is used in MP3 and DVD players, digital cameras, mobile phones, and also in radar processing, biomedical applications, seismic data processing, etc.

This book aims to be a text book which presents a thorough introduction to digital signal processing featuring the design of digital filters. The purpose of the first part (Chapters 1 to 9) is to initiate the newcomer to digital signal and image processing whereas the second part (Chapters 10 and 11) covers some advanced topics on stability for 2-D filter design. These chapters are written at a level that is suitable for students or for individual study by practicing engineers.

When talking about filtering methods, we refer to techniques to design and synthesize filters with constant filter coefficients. By way of contrast, when dealing with adaptive filters, the filter taps change with time to adjust to the underlying system. These types of filters will not be addressed here, but are presented in various books such as [HAY 96], [SAY 03], [NAJ 06].

Chapter 1 provides an overview of various classes of signals and systems. It discusses the time-domain representations and characterizations of the continuoustime and discrete-time signals.

Chapter 2 details the background for the analysis of discrete-time signals. It mainly deals with the z-transform, its properties and its use for the analysis of linear systems, represented by difference equations.

Chapter 3 is dedicated to the analysis of the frequency properties of signals and systems. The Fourier transform, the discrete Fourier transform (DFT) and the fast Fourier transform (FFT) are introduced along with their properties. In addition, the well-known Shannon sampling theorem is recalled.

As we will see, some of the most popular techniques for digital infinite impulse response (IIR) filter design benefit from results initially developed for analog signals. In order to make the reader’s task easy, Chapter 4 is devoted to continuoustime filter design. More particularly, we recall several approximation techniques developed by mathematicians such as Chebyshev or Legendre, who have thus seen their names associated with techniques of filter design.

The following chapters form the core of the book. Chapter 5 deals with the techniques to synthesize finite impulse response (FIR) filters. Unlike IIR filters, these have no equivalent in the continuous-time domain. The so-called windowing method, as a FIR filter design method, is first presented. This also enables us to emphasize the key role played by the windowing in digital signal processing, e.g., for frequency analysis. The Remez algorithm is then detailed.

Chapter 6 concerns IIR filters. The most popular techniques for analog to digital filter conversion, such as the bilinear transform and the impulse invariance method, are presented. As the frequency response of these filters is represented by rational functions, we must tackle the problems of stability induced by the existence of poles of these rational functions.

In Chapter 7, we address the selection of the filter structure and point out its importance for filter implementation. Some problems due to the finite-precision implementation are listed and we provide rules to choose an appropriate structure while implementing filter on fixed point operating devices.

In comparison with many available books dedicated to digital filtering, this title features both 1-D and 2-D systems, and as such covers both signal and image processing. Thus, in Chapters 8 and 9, 2-D filtering is investigated.

Moreover, it is not easy to establish the necessary and sufficient conditions to test the stability of 2-D signals. Therefore, Chapters 10 and 11 are dedicated to the difficult problem of the stability of 2-D digital system, a topic which is still the subject of many works such as [ALA 2003] [SER 06]. Even if these two chapters are not a prerequisite for filter design, they can provide the reader who would like to study the problems of stability in the multi-dimensional case with valuable clarifications. This contribution is another element that makes this book stand out.

The field of digital filtering is often perceived by students as a “patchwork” of formulae and recipes. Indeed, the methods and concepts are based on several specific optimization techniques and mathematical results which are difficult to grasp.

For instance, we have to remember that the so-called Parks-McClellan algorithm proposed in 1972 was first rejected by the reviewers [PAR 72]. This was probably due to the fact that the size of the submitted paper, i.e., 5 pages, did not enable the reviewers to understand every step of the approach [McC 05].

In this book we have tried, at every stage, to justify the necessity of these approaches without recalling all the steps of the derivation of the algorithm. They are described in many articles published during the 1970s in the IEEE periodicals i.e., Transactions on Acoustics Speech and Signal Processing, which has since become Transactions on Signal Processing and Transactions on Circuits and Systems.

Mohamed NAJIMBordeaux

[ALA 2003] ALATA O., NAJIM M., RAMANANJARASOA C. and TURCU F., “Extension of the Schur-Cohn Stability Test for 2-D AR Quarter-Plane Model”, IEEE Trans. on Information Theory, vol. 49, no. 11, November 2003.

[HAY 96] HAYKIN S., Adaptive Filter Theory, 3rd edition, Prentice Hall, 1996.

[McC 05] McCLELLAN J.H. and PARKS W. Th., “A Personal History of the Parks-McClellan Algorithm” IEEE Signal Processing Magazine, pp 82–86, March 2005.

[NAJ 06] NAJIM M., Modélisation, estimation et filtrage optimale en traitement du signal, forthcoming, 2006, Hermes, Paris.

[PAR 72] PARKS W. Th. and McCLELLAN J.H., “Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase,” IEEE Trans. Circuit Theory, vol. CT-19, no. 2, pp 189–194, 1972.

[SAY 03] SAYED A., Fundamentals of Adaptive Filtering, Wiley IEEE Press, 2003.

[SER 06] SERBAN I., TURCU F., NAJIM M., “Schur Coefficients in Several Variables”, Journal of Mathematical Analysis and Applications, vol. 320, issue no. 1, August 2006, pp 293–302.

Chapter 1

Introduction to Signals and Systems1

1.1. Introduction

Throughout a range of fields as varied as multimedia, telecommunications, geophysics, astrophysics, acoustics and biomedicine, signals and systems play a major role. Their frequential and temporal characteristics are used to extract and analyze the information they contain. However, what importance do signals and systems really hold for these disciplines? In this chapter we will look at some of the answers to this question.

First we will discuss different types of continuous and discrete-time signals, which can be termed random or deterministic according to their nature. We will also introduce several mathematical tools to help characterize these signals. In addition, we will describe the acquisition chain and processing of signals.

Later we will define the concept of a system, emphasizing invariant discrete-time linear systems.

1.2. Signals: categories, representations and characterizations

1.2.1. Definition of continuous-time and discrete-time signals

The function of a signal is to serve as a medium for information. It is a representation of the variations of a physical variable.

A signal can be measured by a sensor, then analyzed to describe a physical phenomenon. This is the situation of a tension taken to the limits of a resistance in order to verify the correct functioning of an electronic board, as well as, to cite one example, speech signals that describe air pressure fluctuations perceived by the human ear.

Generally, a signal is a function of time. There are two kinds of signals: continuous and discrete-time.

A continuous-time or analog signal can be measured at certain instants. This means physical phenomena create, for the most part, continuous-time signals.

Figure 1.1.Example of the sleep spindles of an electroencephalogram (EEG) signal

The advancement of computer-based techniques at the end of the 20th century led to the development of digital methods for information processing. The capacity to change analog signals to digital signals has meant a continual improvement in processing devices in many application fields. The most significant example of this is in the field of telecommunications, especially in cell phones and digital televisions. The digital representation of signals has led to an explosion of new techniques in other fields as varied as speech processing, audiofrequency signal analysis, biomedical disciplines, seismic measurements, multimedia, radar and measurement instrumentation, among others.