Table of Contents

Preface

PART 1. 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. Introduction to Estimation Theory with Application in Spectral Analysis

3.1. Introduction

3.2. Covariance-based estimation

3.3. Performance assessment of some spectral estimators

3.4. Bibliography

Chapter 4. Time-Series Models

4.1. Introduction

4.2. Linear models

4.3. Exponential models

4.4. Nonlinear models

4.5. Bibliography

PART 2. 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 3. PARAMETRIC METHODS

Chapter 6. Spectral Analysis by Parametric Modeling

6.1. Which kind of parametric models?

6.2. AR modeling

6.3. ARMA modeling

6.4. Prony modeling

6.5. Order selection criteria

6.6. Examples of spectral analysis using parametric modeling

6.7. 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 MV and generalized MV

7.6. A new estimator: the CAPNORM estimator

7.7. Bibliography

Chapter 8. Subspace-Based Estimators and Application to Partially Known Signal Subspaces

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 the subspace-based methods performance

8.8. Adaptive research of subspaces

8.9. Integrating a priori known frequencies into the MUSIC criterion

8.10. Bibliography

PART 4. ADVANCED CONCEPTS

Chapter 9. Multidimensional Harmonic Retrieval: Exact, Asymptotic, and Modified Cramér-Rao Bounds

9.1. Introduction

9.2. CanDecomp/Parafac decomposition of the multidimensional harmonic model

9.3. CRB for the multidimensional harmonic model

9.4. Modified CRB for the multidimensional harmonic model

9.5. Conclusion

9.6. Appendices

9.7. Bibliography

Chapter 10. Introduction to Spectral Analysis of Non-Stationary Random Signals

10.1. Evolutive spectra

10.2. Non-parametric spectral estimation

10.3. Parametric spectral estimation

10.4. Bibliography

Chapter 11. Spectral Analysis of Non-uniformly Sampled Signals

11.1. Applicative context

11.2. Theoretical framework

11.3. Generation of a randomly sampled stochastic process

11.4. Spectral analysis using undated samples

11.5. Spectral analysis using dated samples

11.6. Perspectives

11.7. Bibliography

Chapter 12. Space–Time Adaptive Processing

12.1. STAP, spectral analysis, and radar signal processing

12.2. Space–time processing as a spectral estimation problem

12.3. STAP architectures

12.4. Relative advantages of pre-Doppler and post-Doppler STAP

12.5. Conclusion

12.6. Bibliography

12.7. Glossary

Chapter 13. Particle Filtering and Tracking of Varying Sinusoids

13.1. Particle filtering

13.2. Application to spectral analysis

13.3. Bibliography

List of Authors

Index

To my children: Guillaume, Aurélien, Anastasia, Virginia, Enrica

First published 2011 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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© ISTE Ltd 2011

The rights of Francis Castanié to be identified as the authors 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

Digital spectral analysis: parametric, non-parametric, and advanced methods / edited by Francis Castanié.

p. cm.

Includes bibliographical references and index.

Includes bibliographical references and index.

ISBN 978-1-84821-277-0

1.Spectral theory (Mathematics) 2.Signal processing--Digital techniques--Mathematics. 3.Spectrum analysis. I. Castanié, Francis.

QA280.D543 2011

621.382′2--dc23

2011012246

British Library Cataloguing-in-Publication Data

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

ISBN 978-1-84821-277-0

Preface

The oldest concept dealt with in signal processing is the concept of frequency, and its operational consequence, the constant quest of frequency contents of a signal. This concept is obviously linked to the fascination for sine functions, deriving from the 19th Century discovery of the “magic” property of the sine functions, or more precisely complex exponentials: it is the only function set able to cross what we call today “linear invariant systems” without going out of this set, i.e. the output of such a system remains in the same class as the input. The distribution of qualifying quantities, such as power or energy of a signal over frequency, the so-called spectrum, is probably one of the most studied topics in signal processing. The concept of frequency itself is not limited to time-related functions, but is much more general: in particular image processing deals with space frequency concept related to geometrical length units, etc.

We can, with good reason, wonder on the pertinence of the importance given to spectral approaches. From a fundamental viewpoint, they relate to the Fourier transformation, projection of signals, on the set of special periodic functions, which include complex exponential functions. By generalizing this concept, the basis of signal vector space can be made much wider (Hadamard, Walsh, etc.), while maintaining the essential characteristics of the Fourier basis. In fact, the projection operator induced by the spectral representations measures the “similarity” between the projected quantity and a particular basis: they have henceforth no more – and no less – relevance 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 or resolution) and 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 frequency sensitivity).

Whatever the reasons may be, spectral descriptors are the most commonly used in signal processing; realizing this, 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. The term digital refers to today’s most widely spread technical means to implement the proposed methods of the analysis.

It is not essential that we must devote ourselves 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 controlled by the availability of technical tools that allow the analysis to be performed. It must kept in mind that the first spectral analyzers used optical analyzers (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 indeed that the theoretical tools were centered on concepts of selective filtering, and sustained by the theory of time-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 approaches, adapting the time axis to discrete time signals. Second, 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 tools, the existence of such methods freed the imagination, by allowing the use of descriptors derived from the domain of parametric modeling. This has its origin in a field of statistics known as analysis of chronological series (see [BOX 70]), the so-called time series, which was successfully applied by G. Yule (1927) for the determination of periodicities of the number of sun spots; but it is actually the present availability of sufficiently powerful digital methods, from the mid-1970s, which led the community of signal processing 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 world is not an easy task – the mere quantitative assessment of the frequency resolution or spectral variances becomes a complicated problem.

Part 1 brings together the processing tools that contribute to spectral analysis. Chapter 1 lists the basics of the signal theory needed 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 used in industrial spectral analysis.

Part 3 obviously deals with parametric methods, studying first the methods based on models of time series, Capons methods and its variants, and then the estimators based on the concepts of sub-spaces.

The fourth and final part is devoted to advanced concepts. It 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] of space-time processing and proposes an inroad in most recent particle filtering-based methods.

Francis CASTANIÉ

May 2011

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 F., Temps-fréquence: Concepts et Outils, Hermès Science, Paris, 2005.

[HLA 08] HLAWATSCH F., AUGER F. (eds), Time-frequency Analysis: Concepts and Methods, ISTE Ltd, London and John Wiley & Sons, New York, 2008.

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

PART 1

Tools and Spectral Analysis

Chapter 1

Fundamentals 1

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, e.g. see Chapter 4 and [LAC 00]). Here, we discuss 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 x(t) 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 x(t) itself.

The deterministic signals are usually separated into classes, representing integral properties of x(t), strongly linked to some quantities known by physicists.

Finite energy signals verify the integral properties in equations [1.1] and [1.2] with continuous or discrete time:

[1.1] images

[1.2] images

We recognize the membership of x(t) to standard function spaces (noted as L2 or l2), as well as the fact that this integral, to within some dimensional constant (an impedance in general), represents the energy E of the signal.

Signals of finite average power verify:

[1.3] images

[1.4] images

If we accept the idea that the sums of equation [1.1] or [1.2] represent “energies”, those of equation [1.3] or [1.4] then represent powers.

It is clear that these integral properties correspond to mathematical characteristics whose morphological behavior along the time axis is very different: the finite energy signals will be in practice “pulse-shaped”, or “transient” signals such that |x(t)| → 0 for |t| → ∞. This asymptotic behavior is not at all necessary to ensure the convergence of the sums, and yet all practical finite energy signals verify it. For example, the signal below is of finite energy:

(This type of damped exponential oscillatory waveform is a fundamental signal in the analysis of linear systems that are invariant by translation.)

Figure 1.1. Electromagnetic interference signal and its decomposition

On the more complex example of Figure 1.1, we see a finite energy signal of the form:

where the four components xi(t) start at staggered times (ti). Its shape, even though complex, is supposed to reflect a perfectly reproducible physical experiment.

Finite power signals will be, in practice, permanent signals, i.e. not canceling at infinity. For example:

[1.5] images

An example is given in Figure 1.2. It is clear that there is obviously a start and an end, but its mathematical model cannot take this unknown data into account, and it is relevant to represent it by an equation of the type [1.5]. This type of signal, modulated in amplitude and angle, is fundamental in telecommunications.

Figure 1.2. Frequency-modulated sine curve

1.1.2. Random signals

The deterministic models of signals described in the previous section are absolutely unrealistic, in the sense that they do not take into account any inaccuracy, or irreproducibility, even if partial. To model the uncertainty on the signals, several approaches are possible today, but the one that was historically adopted at the origin of the signal theory is a probabilistic approach.

In this approach, the signal is regarded as resulting from a drawing in a probability space. The observation () (th outcome or “realization”) is one of the results of the random event, but is no longer signal: the latter is now constituted by all the possible outcomes (, ). The information of the signal is no longer contained in one of the realizations (), but in the probability laws that govern (, ). Understanding this fact is essential: it is not relevant, in this approach, to look for representations of the signal in the form of transformations on one of the outcomes that could well demonstrate the properties sought; here, we must “go back” as far as possible to the probability laws of (, ), from the observation of one (or a finite number) of the outcomes, generally over a finite time interval. How, from the data { () = 1,…,, [,]}, we can go back to the laws of (, ) is the subject of the estimation theory (see ).

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