Spectral Analysis E-Book

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

Seitenzahl: 288

Veröffentlichungsjahr: 2013

Beschreibung

This book deals with these parametric methods, first discussing those based on time series models, Capon's method and its variants, and then estimators based on the notions of sub-spaces. However, the book also deals with the traditional "analog" methods, now called non-parametric methods, which are still the most widely used in practical spectral analysis.

Leseprobe

Table of Contents

Preface

Specific Notations

PART I. Tools and Spectral Analysis

Chapter 1. Fundamentals

1.1. Classes of signals

1.2. Representations of signals

1.3. Spectral analysis: position of the problem

1.4. Bibliography

Chapter 2. Digital Signal Processing

2.1. Introduction

2.2. Transform properties

2.3. Windows

2.4. Examples of application

2.5. Bibliography

Chapter 3. Estimation in Spectral Analysis

3.1. Introduction to estimation

3.2. Estimation of 1st and 2nd order moments

3.3. Periodogram analysis

3.4. Analysis of estimators based on

3.5. Conclusion

3.6. Bibliography

Chapter 4. Time-Series Models

4.1. Introduction

4.2. Linear models

4.3. Exponential models

4.4. Non-linear models

4.5. Bibliography

PART II. Non-Parametric Methods

Chapter 5. Non-Parametric Methods

5.1. Introduction

5.2. Estimation of the power spectral density

5.3. Generalization to higher order spectra

5.4. Bibliography

PART III. Parametric Methods

Chapter 6. Spectral Analysis by Stationary Time Series Modeling

6.1. Parametric models

6.2. Estimation of model parameters

6.3. Properties of spectral estimators produced

6.4. Bibliography

Chapter 7. Minimum Variance

7.1. Principle of the MV method

7.2. Properties of the MV estimator

7.3. Link with the Fourier estimators

7.4. Link with a maximum likelihood estimator

7.5. Lagunas methods: normalized and generalized MV

7.6. The CAPNORM estimator

7.7. Bibliography

Chapter 8. Subspace-based Estimators

8.1. Model, concept of subspace, definition of high resolution

8.2. MUSIC

8.3. Determination criteria of the number of complex sine waves

8.4. The MinNorm method

8.5. “Linear” subspace methods

8.6. The ESPRIT method

8.7. Illustration of subspace-based methods performance

8.8. Adaptive research of subspaces

8.9. Bibliography

Chapter 9. Introduction to Spectral Analysis of Non-Stationary Random Signals

9.1. Evolutive spectra

9.2. Non-parametric spectral estimation

9.3. Parametric spectral estimation

9.4. Bibliography

List of Authors

Index

First published in France in 2003 by Hermès Science/Lavoisier entitled “Analyse spectrale”

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 5DXUK

ISTE USA4308 Patrice RoadNewport Beach, CA 92663USA

www.iste.co.uk

© LAVOISIER, 2003

© ISTE Ltd, 2006

The rights of Francis Castanié to be identified as the author of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Analyse spectrale. English

Spectral analysis: parametric and non-parametric digital methods / edited by Francis Castanié. -- 1st ed.

p. cm.

Includes bibliographical references and index.

ISBN-13: 978-1-905209-05-7

ISBN-10: 1-905209-05-3

1. Signal processing--Digital techniques. 2. Spectrum analysis--Statistical methods. I. Castanié, Francis. II. Title.

TK5102.9 .A452813

621.382’2--dc22

2006012689

British Library Cataloguing-in-Publication Data

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

ISBN 10: 1-905209-05-3

ISBN 13: 978-1-905209-05-7

Preface

An immediate observation for those who deal with the processing of signals is the omnipresence of the concepts of frequency, which are called “spectra” here. This presence is important in the fields of theoretical research as well as in most applied sectors of engineering. We can, with good reason, wonder on the pertinence of the importance given to spectral approaches. On the fundamental plane, they relate to the Fourier transformation, projection of signals, signal descriptors, on the basis of special periodic functions, which include complex exponential functions. By generalizing this concept, the basis of functions can be wider (Hadamard, Walsh, etc.), while maintaining the essential characteristics of the Fourier basis. By projections, the spectral representations measure the “similarity” between the projected quantity and a particular base: they have henceforth no more — and no less — legitimacy than this.

The predominance of this approach in signal processing is not only based on this reasonable (but dry) mathematical description, but probably has its origins in the fact that the concept of frequency is in fact a perception through various human “sensors”: the system of vision, which perceives two concepts of frequency (time-dependent for colored perception, and spatial via optical concepts of separating power), hearing, which no doubt is at the historical origin of the perceptual concept of frequency — Pythagoras and the “Music of the Spheres” — and probably other proprioceptive sensors (all those who suffer from seasickness have a direct physical experience of the frequential sensitivity).

Whatever the reasons may be, spectral descriptors are the most commonly used in signal processing; starting with this acknowledgment, the measurement of these descriptors is therefore a major issue: this is the reason for the existence of this book, dedicated to this measurement that is classically christened as Spectral Analysis.

It is not essential that one must devote oneself to a tedious hermeneutic to understand spectral analysis through countless books that have dealt with the subject (we will consult with interest the historical analysis of this field given in [MAR 87]). If we devote ourselves to this exercise in erudition concerning the cultural level, we realize that the theoretical approaches of current spectral analysis were structured from the late 1950s; these approaches have the specific nature of being controled by the availability of technical tools that allow the analysis to be performed. One can in this regard remember that the first spectral analyzers used optical techniques (spectrographs), then at the end of this archaeological phase — which is generally associated with the name of Isaac Newton — spectral analysis got organized around analog electronic technologies. We will not be surprised consequently that the theoretical tools were centered on concepts of selective filtering, and sustained by the theory of continuous signals (see [BEN 71]). At this time, the criteria qualifying the spectral analysis methods were formulated: frequency resolution or separating power, variance of estimators, etc. They are still in use today, and easy to assess in a typical linear filtering context.

The change to digital processing tools was first done by transposition of earlier analog transposition approaches, rewriting time first simply in terms of discrete time signals. Secondly, a simultaneous increase in the power of processing tools and algorithms opened the field up to more and more intensive digital methods. But beyond the mere availability of more comfortable digital resolution tools, the existence of such methods freed the imagination, by allowing the use of descriptors from the field of parametric modeling. This has its origin in a field of statistics known as analysis of chronological series (see [BOX 70]), which was successfully created by G. Yule (1927) for the determination of periodicities of the number of sun spots; but it is actually the availability of sufficiently powerful digital methods, in the mid 1970s, which led the community of signal processors to consider parametric modeling as a tool for spectral analysis, with its own characteristics, including the possibilities to obtain “super-resolutions” and/or to process signals of very short duration. We will see that characterizing these estimators with the same criteria as estimators from the analog word is not an easy thing: the mere quantitative assessment of the frequency resolution or spectral variances becomes a complicated problem.

The first part brings together the processing tools that contribute to spectral analysis. Chapter 1 lists the bases of the signal theory needed in order to read the following parts of the book: the informed reader could obviously skip this. Next, digital signal processing, the theory of estimation and parametric modeling of time series are presented.

The “classical” methods, known nowadays as non-parametric methods, form part of the second part. The privative appearing in the qualification of “non-parametric” must not be seen as a sign of belittling these methods: they are the most employed in spectral analysis.

The third and last part obviously deals with parametric methods, studying first the methods based on models of chronological series, Capron’s methods and its variants, then the estimators based on the concepts of sub-spaces.

The last chapter of this part provides an opening to parametric spectral analysis of non-stationary signals, a subject with great potential, which is tackled in greater depth in another book of the IC2 series [HLA 05].

Francis CASTANIÉ

Bibliography

[BEN 71] BENDAT J. S., PIERSOL A. G., Random Data: Analysis and Measurement Procedures, Wiley Intersciences, 1971.

[BOX 70] BOX G., JENKINS G., Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco, 1970.

[HLA 05] HLAWATSCH F., AUGER R., OVARLEZ J.-P., Temps-fréquence: concepts et outils, IC2 series, Hermès Science/Lavoisier, Paris, 2005.

[MAR 87] MARPLE S., Digital Spectral Analysis with Applications, Prentice Hall, Englewood Cliffs (NJ), 1987.

Specific Notations

Dirichlet’s kernel

IFT

()

Inverse Fourier transform

DFT

()

Discrete Fourier transform

DIFT

()

Discrete inverse Fourier transform

Continuous time mean

Discrete time mean

(Auto)spectral Energy or Power density

Interspectral Energy or Power density

S

xx...x

(

f

1

,f

2

,..., f

n

)

Multispectrum

PART I

Tools and Spectral Analysis

Chapter 1

Fundamentals1

1.1. Classes of signals

Every signal-processing tool is designed to be adapted to one or more signal classes and presents a degraded or even deceptive performance if applied outside this group of classes. Spectral analysis too does not escape this problem, and the various tools and methods for spectral analysis will be more or less adapted, depending on the class of signals to which they are applied.

We see that the choice of classifying properties is fundamental, because the definition of classes itself will affect the design of processing tools.

Traditionally, the first classifying property is the deterministic or non-deterministic nature of the signal.

1.1.1. Deterministic signals

The definitions of determinism are varied, but the simplest is the one that consists of calling any signal that is reproducible in the mathematical sense of the term as a deterministic signal, i.e. any new experiment for the generation of a continuous time signal x(t) (or discrete time x(k)) produces a mathematically identical signal. Another subtler definition, resulting from the theory of random signals, is based on the exactly predictive nature of x(t)t > t0 from the moment that it is known for t < t0 (singular term of the Wold decomposition for example; see Chapter 4 and [LAC 00]). We will discuss here only the definition based on the reproducibility of x(t), as it induces a specific strategy on the processing tools: as all information of the signal is contained in the function itself, any bijective transformation of () will also contain all this information. Representations may thus be imagined, which, without loss of information, will demonstrate the characteristics of the signal better than the direct representation of the function () itself.