Contents

Preface

FIRST PART. FUNDAMENTAL CONCEPTS AND METHODS

Chapter 1. Time-Frequency Energy Distributions: An Introduction

1.1. Introduction

1.2. Atoms

1.3. Energy

1.4. Correlations

1.5. Probabilities

1.6. Mechanics

1.7. Measurements

1.8. Geometries

1.9. Conclusion

1.10. Bibliography

Chapter 2. Instantaneous Frequency of a Signal

2.1. Introduction

2.2. Intuitive approaches

2.3. Mathematical definitions

2.4. Critical comparison of the different definitions

2.5. Canonical pairs

2.6. Phase signals

2.7. Asymptotic phase signals

2.8. Conclusions

2.9. Bibliography

Chapter 3. Linear Time-Frequency Analysis I: Fourier-Type Representations

3.1. Introduction

3.2. Short-time Fourier analysis

3.3. Gabor transform; Weyl-Heisenberg and Wilson frames

3.4. Dictionaries of time-frequency atoms; adaptive representations

3.5. Applications to audio signals

3.6. Discrete algorithms

3.7. Conclusion

3.8. Acknowledgements

3.9. Bibliography

Chapter 4. Linear Time-Frequency Analysis II: Wavelet-Type Representations

4.1. Introduction: scale and frequency

4.2. Continuous wavelet transform

4.3. Discrete wavelet transform

4.4. Filter banks and wavelets

4.5. Generalization: multi-wavelets

4.6. Other extensions

4.7. Applications

4.8. Conclusion

4.9. Acknowledgments

4.10. Bibliography

Chapter 5. Quadratic Time-Frequency Analysis I: Cohen’s Class

5.1. Introduction

5.2. Signal representation in time or in frequency

5.3. Representations in time and frequency

5.4. Cohen’s class

5.5. Main elements

5.6. Conclusion

5.7. Bibliography

Chapter 6. Quadratic Time-Frequency Analysis II: Discretization of Cohen’s Class

6.1. Quadratic TFRs of discrete signals

6.2. Temporal support of TFRs

6.3. Discretization of the TFR

6.4. Properties of discrete-time TFRs

6.5. Relevance of the discretization to spectral analysis

6.6. Conclusion

6.7. Bibliography

Chapter 7. Quadratic Time-Frequency Analysis III: The Affine Class and Other Covariant Classes

7.1. Introduction

7.2. General construction of the affine class

7.3. Properties of the affine class

7.4. Affine Wigner distributions

7.5. Advanced considerations

7.6. Conclusions

7.7. Bibliography

SECOND PART. ADVANCED CONCEPTS AND METHODS

Chapter 8. Higher-Order Time-Frequency Representations

8.1. Motivations

8.2. Construction of time-multifrequency representations

8.3. Multilinear time-frequency representations

8.4. Towards affine multilinear representations

8.5. Conclusion

8.6. Bibliography

Chapter 9. Reassignment

9.1. Introduction

9.2. The reassignment principle

9.3. Reassignment at work

9.4. Characterization of the reassignment vector fields

9.5. Two variations

9.6. An application: partitioning the time-frequency plane

9.7. Conclusion

9.8. Bibliography

Chapter 10. Time-Frequency Methods for Non-stationary Statistical Signal Processing

10.1. Introduction

10.2. Time-varying systems

10.3. Non-stationary processes

10.4. TF analysis of non-stationary processes – type I spectra

10.5. TF analysis of non-stationary processes – type II spectra

10.6. Properties of the spectra of underspread processes

10.7. Estimation of time-varying spectra

10.8. Estimation of non-stationary processes

10.9. Detection of non-stationary processes

10.10. Conclusion

10.11. Acknowledgements

10.12. Bibliography

Chapter 11. Non-stationary Parametric Modeling

11.1. Introduction

11.2. Evolutionary spectra

11.3. Postulate of local stationarity

11.4. Suppression of a stationarity condition

11.5. Conclusion

11.6. Bibliography

Chapter 12. Time-Frequency Representations in Biomedical Signal Processing

12.1. Introduction

12.2. Physiological signals linked to cerebral activity

12.3. Physiological signals related to the cardiac system

12.4. Other physiological signals

12.5. Conclusion

12.6. Bibliography

Chapter 13. Application of Time-Frequency Techniques to Sound Signals: Recognition and Diagnosis

13.1. Introduction

13.2. Loudspeaker fault detection

13.3. Speaker verification

13.4. Conclusion

13.5. Bibliography

List of Authors

Index

First published in France in 2005 by Hermes Science/Lavoisier entitled “Temps-fréquence: concepts et outils”

First published in Great Britain and the United States in 2008 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, 2008

© LAVOISIER, 2005

The rights of Franz Hlawatsch and François Auger 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

Temps-fréquence. English
Time-frequency analysis: concepts and methods / edited by Franz Hlawatsch and François Auger.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-1-84821-033-2
1.  Signal processing--Mathematics. 2.  Time-series analysis. 3.  Frequency spectra. I.  Hlawatsch, F. (Franz) II.  Auger, François. III.  Title.
TK5102.9.T435 2008
621.382′2--dc22

2006015556

British Library Cataloguing-in-Publication Data

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

ISBN: 978-1-84821-033-2

Preface

Is time-frequency a mathematical utopia or, on the contrary, a concept imposed by the observation of physical phenomena? Various “archetypal” situations demonstrate the validity of this concept: musical notes, a linear chirp, a frequency shift keying signal, or the signal analysis performed by our auditory system. These examples show that “frequencies” can have a temporal localization, even though this is not immediately suggested by the Fourier transform. In fact, very often the analyzed phenomena manifest themselves by oscillating signals evolving with time: to the examples mentioned above, we may add physiological signals, radar or sonar signals, acoustic signals, astrophysical signals, etc. In such cases, the time-domain representation of the signal does not provide a good view of multiple oscillating components, whereas the frequency-domain representation (Fourier transform) does not clearly show the temporal localization of these components. We may conjecture that these limitations can be overcome by a time-frequency analysis where the signal is represented as a joint function of time and frequency – i.e., over a “time-frequency plane” – rather than as a function of time or frequency. Such an analysis should constitute an important tool for the understanding of many processes and phenomena within problems of estimation, detection or classification.

We thus have to find the mathematical transformation that allows us to map the analyzed signal into its time-frequency representation. Which “generalized Fourier transform” establishes this mapping? At this point, we find ourselves confronted with a fundamental limitation, known as the uncertainty principle, that excludes any precise temporal localization of a frequency. This negative result introduces some degree of uncertainty, or even of arbitrariness, into time-frequency analysis. One of its consequences is that we can never consider a transformation as the only correct time-frequency transformation, since time-frequency localization cannot be verified in an exact manner.

Is time-frequency an ill-posed problem then? Maybe, since it does not have a unique solution. However, this ambiguity and mathematical freedom have led to the definition of a great diversity of time-frequency transformations. Today, the chimeric concept of time-frequency analysis is materialized by a multitude of different transformations (or representations) that are based on principles even more diverse than the domains from which they originated (signal processing, mathematics, quantum mechanics, etc.). These principles and signal analysis or processing methods are just as useful in real-life applications as they are interesting theoretically.

Thus, is time-frequency a reality today? This is what we attempt to demonstrate in this book, in which we describe the principles and methods that make this field an everyday fact in industry and research. Written at the end of a period of approximately 25 years in which the discipline of time-frequency analysis witnessed an intensive development, this tutorial-style presentation is addressed mainly to researchers and engineers interested in the analysis and processing of non-stationary signals. The book is organized into two parts and consists of 13 chapters written by recognized experts in the field of time-frequency analysis. The first part describes the fundamental notions and methods, whereas the second part deals with more recent extensions and applications.

The diversity of viewpoints from which time-frequencyanalysis can be approached is demonstrated in Chapter 1, “Time-Frequency Energy Distributions: An Introduction”. Several of these approaches – originating from quantum mechanics, pseudodifferential operator theory or statistics – lead to the same set of fundamental solutions, for which they provide complementary interpretations. Most of the concepts and methods discussed in this introductory chapter will be developed in the following chapters.

Chapter 2, entitled “Instantaneous Frequency of a Signal”, studies the concept of a “time-dependent frequency”, which corresponds to a simplified and restricted form of time-frequency analysis. Several definitions of an instantaneous frequency are compared, and the one appearing most rigorous and coherent is discussed in detail. Finally, an in-depth study is dedicated to the special case of phase signals.

The two following chapters deal with linear time-frequency methods. Chapter 3, “Linear Time-Frequency Analysis I: Fourier-Type Representations”, presents methods that are centered about the short-time Fourier transform. This chapter also describes signal decompositions into time-frequency “atoms” constructed through time and frequency translations of an elementary atom, such as the Gabor and Wilson decompositions. Subsequently, adaptive decompositions using redundant dictionaries of multi-scale time-frequency atoms are discussed.

Chapter 4, “Linear Time-Frequency Analysis II: Wavelet-Type Representations”, discusses “multi-resolution” or “multi-scale” methods that are based on the notion of scale rather than frequency. Starting with the continuous wavelet transform, the chapter presents orthogonal wavelet decompositions and multi-resolution analyses. It also studies generalizations such as multi-wavelets and wavelet packets, and presents some applications (compression and noise reduction, image alignment).

Quadratic (or bilinear) time-frequency methods are the subject of the three following chapters. Chapter 5, “Quadratic Time-Frequency Analysis I: Cohen’s Class”, provides a unified treatment of the principal elements of Cohen’s class and their main characteristics. This discussion is helpful for selecting the Cohen’s class time-frequency representation best suited for a given application. The characteristics studied concern theoretical properties as well as interference terms that may cause practical problems. This chapter constitutes an important basis for several of the methods described in subsequent chapters.

Chapter 6, “Quadratic Time-Frequency Analysis II: Discretization of Cohen’s Class”, considers the time-frequency analysis of sampled signals and presents algorithms allowing a discrete-time implementation of Cohen’s class representations. An approach based on the signal’s sampling equation is developed and compared to other discretization methods. Subsequently, some properties of the discrete-time version of Cohen’s class are studied.

The first part of this book ends with Chapter 7, “Quadratic Time-Frequency Analysis III: The Affine Class and Other Covariant Classes”. This chapter studies quadratic time-frequency representations with covariance properties different from those of Cohen’s class. Its emphasis is placed on the affine class, which is covariant to time translations and contractions-dilations, similarly to the wavelet transform in the linear domain. Other covariant classes (hyperbolic class, power classes) are then considered, and the role of certain mathematical concepts (groups, operators, unitary equivalence) is highlighted.

The second part of the book begins with Chapter 8, “Higher-Order Time-Frequency Representations”, which explores multilinear time-frequency analysis. The class of time-multifrequency representations that are covariant to time and frequency translations is presented. Time-(mono)frequency representations ideally concentrated on polynomial modulation laws are studied, and the corresponding covariant class is presented. Finally, an opening towards multilinear affine representations is proposed.

Chapter 9, “Reassignment”, describes a technique that is aimed at improving the localization of time-frequency representations, in order to enable a better interpretation by a human operator or a better use in an automated processing scheme. The reassignment technique is formulated for Cohen’s class and for the affine class, and its properties and results are studied. Two recent extensions – supervised reassignment and differential reassignment – are then presented and applied to noise reduction and component extraction problems.

The two following chapters adopt a statistial approach to non-stationarity and time-frequency analysis. Various definitions of a non-parametric “time-frequency spectrum” for non-stationary random processes are presented in Chapter 10, “Time-Frequency Methods for Non-stationary Statistical Signal Processing”. It is demonstrated that these different spectra are effectively equivalent for a subclass of processes referred to as “underspread”. Subsequently, a method for the estimation of time-frequency spectra is proposed, and finally the use of these spectra for the estimation and detection of underspread processes is discussed.

Chapter 11, “Non-stationary Parametric Modeling”, considers non-stationary random processes within a parametric framework. Several different methods for non-stationary parametric modeling are presented, and a classification of these methods is proposed. The development of such a method is usually based on a parametric model for stationary processes, whose extension to the non-stationary case is obtained by means of a sliding window, adaptivity, parameter evolution or non-stationarity of a filter input.

The two chapters concluding this book are dedicated to the application of time-frequency analysis to measurement, detection, and classification tasks. Chapter 12, “Time-Frequency Representations in Biomedical Signal Processing”, provides a well-documented review of the contribution of time-frequency methods to the analysis of neurological, cardiovascular and muscular signals. This review demonstrates the high potential of time-frequency analysis in the biomedical domain. This potential can be explained by the fact that diagnostically relevant information is often carried by the non-stationarities of biomedical signals.

Finally, Chapter 13, “Application of Time-Frequency Techniques to Sound Signals: Recognition and Diagnosis”, proposes a time-frequency technique for supervised non-parametric decision. Two different applications are considered, i.e., the classification of loudspeakers and speaker verification. The decision is obtained by minimizing a distance between a time-frequency representation of the observed signal and a reference time-frequency function. The kernel of the time-frequency representation and the distance are optimized during the training phase.

As the above outline shows, this book provides a fairly extensive survey of the theoretical and practical aspects of time-frequency analysis. We hope that it will contribute to a deepened understanding and appreciation of this fascinating subject, which is still witnessing considerable developments.

We would like to thank J.-P. Ovarlez for important contributions during the initial phase of this work, G. Matz for helpful assistance and advice, and, above all, P. Flandrin for his eminent role in animating research on time-frequency analysis.

Vienna and Saint Nazaire, June 2008.

FIRST PART

Fundamental Concepts and Methods

Chapter 1

Time-Frequency Energy Distributions: An Introduction 1

Abstract: The basic tools for an “energetic” time-frequency analysis may be introduced in various ways, which find their theoretical roots in quantum mechanics or the theory of pseudo-differential operators as well as in signal theory. Each of these points of view casts a specific light on the same mathematical objects, with complementary interpretations (in terms of atoms, devices, covariances, correlations, probabilities, measurements, symmetries, etc.), some of which are briefly discussed here.

Keywords: energy distribution, general classes, covariance principles, measurement devices, operators.

1.1. Introduction

Until quite recently, “classical” signal processing was confronted with a paradoxical situation. On the one hand, it was clearly recognized that most of the signals emitted by natural and/or artificial systems had different forms of time dependence of their structural properties (spectral content, statistical laws, transfer function, etc.). On the other hand, however, the standard tools used to analyze and process such signals were generally based on assumptions of a steady state or “stationarity”. Insofar as “non-stationarities” are not merely in no way exceptional, but very often carry the most important information about a signal, it proved necessary to develop general approaches capable of, for example, going beyond Fourier-type methods. In this light, the concept of “time-frequency” has progressively emerged as a natural (and increasingly widely accepted) paradigm. One of its main characteristics is the non-uniqueness of its tools, which reveals the diversity of the possible forms of non-stationarity and, at the same time, is a consequence of the intrinsic limitations existing between canonically conjugate variables (i.e., variables related by the Fourier transform).

As is proved by their historical development (see for example [FLA 99a, Chapter 2] and [JAF 01]), it appears that the basic distributions for time-frequency analysis can be introduced in a large number of ways that have their roots not only in signal theory, but also in quantum mechanics, in statistics, or in the theory of pseudo-differential operators. Indeed, each of these points of view casts a specific light on the same mathematical objects, and provides complementary interpretations in terms of atoms, devices, covariances, correlations, probabilities, measurements, mechanical or optical analogies, symmetries, etc.

The purpose of this chapter is to organize the web of multiple paths leading to time-frequency distributions. We shall limit ourselves to the case of energy (and therefore mainly quadratic) distributions, and insist on the utility and the justifications of a small number of key distributions.

We note that most of the quadratic distributions mentioned here (spectrogram, scalogram, Wigner-Ville, Bertrand, etc.) play a central role in time-frequency analysis and deserve to be discussed more specifically. However, the objective of this chapter is not to provide the reader with an exhaustive presentation of these methods, or to compare them (for this, see for example [BOU 96, COH 95, FLA 99a, HLA 92, HLA 95, MEC 97] or [AUG 97]), but rather to emphasize the various motivations that have led to their introduction.

1.2. Atoms

Before considering a time-frequency analysis in terms of energy, an intuitive approach would be to linearly decompose a signal into a set of “building blocks” on which we can impose “good” localization properties in time as well as in frequency. More specifically, the value taken by a signal x(t) at a time t0 can be expressed in an equivalent manner by

where δt(τ): = δ(t τ) is the Dirac distribution in t, or by

where we use the notation images. If the first decomposition favors the temporal description, the second (in which images is the Fourier transform of the signal x(t)) is based on a dual interpretation in terms of waves. It is these two antinomical points of view that are to be reconciled by a joint description in terms of time and frequency. To this end, the two previous decompositions may be replaced by a third, intermediate, which can be written as

(1.1) images

The functions images thus involved enable a transition between the previous extreme situations (perfect localization in time and no localization in frequency when images, perfect localization in frequency and no localization in time when images. In fact, they play a role of time-frequency atoms in that they are supposed to be constituents of any signal and to possess joint localization properties that are as ideal as possible (“elementarity”, within the limits of the time-frequency inequalities of Heisenberg-Gabor type [FOL 97, GAB 46]). For such a decomposition to be completely meaningful, it must naturally be invertible, so that we have1

(1.2) images

which makes λ x(t, f) a linear time-frequency representation of x(t).

There is obviously a great arbitrariness in the choice of such a representation. A simple way to proceed consists of generating the family of atoms images by the action of a group of transformations acting on a single primordial element g(·). It is in this way that the choice images images leads to the family of short-time Fourier transforms with window g(·), while setting images (with images and g(·) with zero mean) yields the family of .

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