Time-Frequency Signal Analysis and Processing

Transcription

1 Time-Frequency Signal Analysis and Processing A Comprehensive Reference Editor: Boualem Boashash

2 The front cover illustration is a time frequency representation of a Noisy Miner (Manorina melanocephala) bird song. The signal came from the John Gould s Birds of Australia CD ROM, published by Protoavis Productions, RMB 4375, Seymour, Vic 3660, Australia.

3 Time-Frequency Signal Analysis and Processing A Comprehensive Reference Edited by Boualem Boashash Director, Signal Processing Research Queensland University of Technology Brisbane, Australia

4 Information on the Editor (for Library of Congress) Name: Boualem Boashash Date of birth: June 21, 1953 Work address: Director, Signal Processing Research Queensland University of Technology 2 George Street / GPO Box 2434 Brisbane, Q 4001, Australia Tel Fax b.boashash@qut.edu.au ISBN:

5 Preface Time-Frequency Signal Analysis and Processing (TFSAP) is a collection of theory and algorithms used for analysis and processing of non-stationary signals, as found in a wide range of applications including telecommunications, radar, and biomedical engineering. This book brings together the main knowledge of TFSAP, from theory to applications, in a user-friendly reference suitable for both expert and non-expert readers. The contents of the book include: 1. a comprehensive tutorial introduction to TFSAP, accessible to anyone who has taken a first course in signals and systems; 2. more specialized theory and algorithms, concisely presented by some of the leading authorities on the respective topics; and 3. studies of key applications, written by leading researchers, showing how to use TFSAP methods to solve practical problems. The motivation for producing this book was twofold: My original and widely used decade-old tutorial on TFSAP [1] needed updating in two respects. First, some of the advances of the last decade are sufficiently fundamental to warrant inclusion in an introductory treatment, while others are sufficiently important to demand coverage in any comprehensive review of TFSAP. Second, new applications have widened the range of disciplines interested in TFSAP, and thus reduced the common background knowledge that may be expected of readers. Part I of this book addresses these needs. The need for a standard language of discourse became apparent in 1990 while I was editing the 23 contributions to the first comprehensive book in the field [2]. These seminal contributions to TFSAP led to further developments throughout the 1990s, including some significant advances in practical methods suitable for non-stationary signals. These efforts continued apace as this book was being written. Such rapid progress produced a variety of new terminologies and notations that were in need of standardization and inclusion in an updated reference book. The organization of this book uses five Parts, each Part including several Chapters, and each Chapter comprising several Articles. Part I introduces the basic concepts while Parts II to V cover more advanced or specialized areas. vii

6 viii Preface Part I defines and explains the basic concepts of TFSAP, intuitively derives a variety of well-known time-frequency distributions (TFDs), and then reduces them to a common form. This leads to the general treatment of quadratic TFDs in Chapter 3, which should be regarded as the core of the book and as a prerequisite for the later chapters. Part II gives more details on some fundamental topics of TFSAP, such as TFD design and signal analysis in the (t, f) plane. Part III describes specialized techniques used in implementation, measurement and enhancement of TFDs. Part IV presents the key statistical techniques for TFSAP of noisy signals, including a full treatment of detection and classification methods. Part V describes a representative selection of TFSAP applications, encompassing telecommunications, radar, sonar, power generation, image quality, automotive applications, machine condition monitoring, and biomedical engineering. Usability is enhanced by an updated consolidated bibliography (alphabetical by author) and a two-level index (which also serves as a dictionary of abbreviations). Under the standard review procedure used for this book, each Article had two (usually external) reviewers concentrating on scientific rigor and accuracy, plus two anonymous internal reviewers concentrating on clarity and consistency. Acknowledgments are due to a number of people who made possible the completion of this book. Foremost among them are my two sons, who aided me to continue this work during and after my wife s final illness, thus contributing to my sense of balance and purpose during this difficult period. I thank all authors and reviewers, and the organizers of the Special Sessions on TFSAP at ISSPA conferences, for their expertise, timely effort and professionalism, and for facilitating the exchange of ideas between contributors to this book. I thank my research students and the SPRC staff for valuable assistance. In particular, Gavin Putland assisted with the technical editing of portions of Part I and was responsible for the final mix-down of the authors L A TEX TM and PostScript TM files. References Boualem Boashash, Editor. [1] B. Boashash, Time-frequency signal analysis, in Advances in Spectrum Analysis and Array Processing (S. Haykin, ed.), vol. 1, ch. 9, pp , Englewood Cliffs, NJ: Prentice-Hall, [2] B. Boashash, ed., Time-Frequency Signal Analysis: Methods and Applications. Melbourne/N.Y.: Longman-Cheshire/Wiley, 1992.

7 Contents Preface List of Contributors vii xxiii Part I: Introduction to the Concepts of TFSAP 1 Chapter 1: Time-Frequency Concepts (B. Boashash) 3 Overview The Need for a Time-Frequency Distribution (TFD) Representation of Three Real-Life Signals Time-Domain Representation Frequency-Domain Representation Joint Time-Frequency Representation Desirable Characteristics of a TFD Signal Formulations and Characteristics in the (t, f) Domain Signal Models used in (t, f) Methods Analytic Signals Hilbert Transform; Analytic Associate Duration, Bandwidth, BT Product Asymptotic Signals Monocomponent vs. Multicomponent Signals Instantaneous Frequency and Time-Delay Instantaneous Frequency (IF) IF and Time Delay (TD) Mean IF and Group Delay (GD) Relaxation Time, Dynamic Bandwidth Summary and Discussion Chapter 2: Heuristic Formulation of Time-Frequency Distributions (B. Boashash) 29 Overview Method 1: The Wigner-Ville Distribution Knife-Edge IF Indication Formulation of the Signal Kernel The Wigner Distribution The Wigner-Ville Distribution ix

8 x Contents 2.2 Method 2: Time-Varying Power Spectral Density Spectra of Non-Stationary Random Processes Estimating the Wigner-Ville Spectrum Method 3: Windowed FT (STFT, Spectrogram, Gabor Transform) STFT and Spectrogram Optimal Window Length of the Spectrogram STFT vs. Gabor Transform Method 4: Filtered Function of Time Filter Banks and the Sonograph Equivalence to Spectrogram Method 5: Instantaneous Power Spectra Page Distribution Method 6: Energy Density Rihaczek s Complex Energy Density Levin s Real Energy Density Windowed Rihaczek and Levin Distributions Relationship between TFDs Spectrogram Wigner-Ville Distribution Rihaczek Distribution Levin Distribution Windowed Rihaczek Distribution Windowed Levin Distribution Page Distribution Relationship between the WVD and Other TFDs Other Popular TFDs Summary and Discussion Chapter 3: Theory of Quadratic TFDs (B. Boashash) 59 Overview The WVD Properties of the WVD Limitations of the WVD Formulations of Quadratic TFDs Time-Lag Formulations and Other Domain Definitions Time-Frequency Formulation Doppler-Lag Formulation and TFD Design Doppler-Frequency Formulation Examples of Simple TFD Formulations Properties of Quadratic TFDs Desirable Properties TFD Properties & Equivalent Kernel Constraints

9 Contents xi Examples of TFDs with Specific Properties Summary, Discussion and Conclusions Part II: Fundamental Principles of TFSAP 83 Chapter 4: Time-Frequency Signal and System Analysis 85 Overview Analytic Signal & Instantaneous Frequency (B. Picinbono) The Problem Analytic Signal and Canonical Pair Phase Signals, Regular Case Singular and Asymptotic Phase Signals Summary and Conclusions Cross-terms & Localization in Quadratic Time-frequency Distributions (P. Flandrin) Identifying Cross-Terms Reducing Cross-Terms Cross-Terms and Localization Summary and Conclusions The Covariance Theory of Time-Frequency Analysis (F. Hlawatsch and G. Tauböck) The Covariance Principle Time-Frequency Displacement Operators Covariant Signal Representations: Group Domain The Displacement Function Covariant Signal Representations: Time-Frequency Domain Example: Hyperbolic Wavelet Transform & Hyperbolic Class Summary and Conclusions Uncertainty in Time-Frequency Analysis (Paulo M. Oliveira and Victor Barroso) The Time-Frequency Plane Information and Spectral Estimation Summary and Conclusions Generalized TFRs via Unitary Transforms (R. G. Baraniuk) Three Approaches to Joint Distributions Linking Signal and Axis Transformations Examples of Linked Signal/Axis Transformations Summary and Conclusions Signal Measures in the Time-Frequency Plane (G. Jones)

10 xii Contents Time-Frequency Analysis Density Distributions and Energy Distributions Signal Measures in Time-Frequency Properties & Interpretation of Local Measurements in TF Application of Local TF Measures to Energy Distributions Example Result for an Adaptive Energy Distribution Summary and Conclusions Time-Frequency Transfer Function Calculus of Linear Time-Varying Systems (G. Matz and F. Hlawatsch) Linear Time-Varying Systems The Generalized Weyl Symbol The Generalized Spreading Function Underspread LTV Systems Time-Frequency Transfer Function Calculus Summary and Conclusions Wigner Distribution and Fractional Fourier Transform (T. Alieva and M. J. Bastiaans) Time-Frequency Representations Wigner Distribution and Ambiguity Function Fractional Fourier Transform Fractional Power Spectrum and Radon-Wigner Transform Fractional Fourier Transform Moments Applications Summary and Conclusions Gabor Spectrogram (S. Qian) Power Spectrum Gabor Spectrogram Numerical Simulations Summary and Conclusions Chapter 5: Design of Time-Frequency Distributions 159 Overview Ambiguity Functions (P. Flandrin) The Radar/Sonar Problem Definitions of Ambiguity Functions Properties of Narrowband Ambiguity Functions Remarks on Wideband Ambiguity Functions Summary and Conclusions Reduced Interference Time-Frequency Distributions (William J. Williams) Nonstationarity, Resolution and Interference

11 Contents xiii The Reduced Interference Distribution Kernel Selection for RID Comparisons of TFD Results Summary and Conclusions Adaptive Time-Frequency Analysis (R. G. Baraniuk and D. L. Jones) Adaptive Short-Time Fourier Transforms Adaptive Quadratic Representations Summary and Conclusions Polynomial Wigner-Ville Distributions (B. Boashash and G. R. Putland) Polynomial FM Signals Principles of Formulation of Polynomial WVDs IF Estimates with Zero Deterministic Bias Calculation of Coefficients Examples Multicomponent Signals and Polynomial TFDs Summary and Conclusions Design of Polynomial TFDs, with Applications (M. Benidir) Decompositions of Polynomial Derivatives Design of Time-Frequency Distributions Estimation of the Phase of a PPS Summary and Conclusions Appendix Time-Frequency Representations Covariant to Group Delay Shifts (A. Papandreou-Suppappola) Group Delay Shift Covariance Property Classes of GDS Covariant QTFRs Simulation Example Summary and Conclusions Design of High-Resolution Quadratic TFDs with Separable Kernels (B. Boashash and G. R. Putland) RIDs and Quadratic TFDs Separable Kernel Formulations Properties Design Examples of Separable-Kernel TFDs Results and Discussion Summary and Conclusions Fractional Fourier Transform and Generalized-Marginal TFDs (X.-G. Xia) Fractional Fourier Transform Generalized-Marginal Time-Frequency Distribution

12 xiv Contents Summary and Conclusions Part III: Time-Frequency Methods 229 Chapter 6: Implementation and Realization of TFDs 231 Overview Discrete Time-Frequency Distributions (B. Boashash and G. R. Putland) The Discrete Wigner-Ville Distribution (DWVD) The Windowed DWVD The Discrete Quadratic TFD Desirable Properties; Kernel Constraints Examples Summary and Conclusions Quadratic and Higher Order Time-Frequency Analysis Based on the STFT (LJ. Stanković) STFT Based Realization of the Quadratic Representations Discrete Realization of the Basic S-Method Form STFT Based Realization of Higher Order Representations Summary and Conclusions Gabor s Signal Expansion for a Non-Orthogonal Sampling Geometry (M. J. Bastiaans and A. J. van Leest) Historical Perspective Gabor s Signal Expansion on a Rectangular Lattice Fourier Transform and Zak Transform Rational Oversampling Non-Orthogonal Sampling Gabor s Signal Expansion on a Non-Orthogonal Lattice Summary and Conclusions Spectrogram Decompositions of Time-Frequency Distributions (W. J. Williams and S. Aviyente) Decomposition Based Approaches Decomposition of Time-Frequency Kernels Development of the Method Wigner Example Optimum Orthogonal Windows Kernel Decomposition Results Summary and Conclusions Computation of Discrete Quadratic TFDs (B. Boashash and G. R. Putland) General Computational Procedure Computation of the Analytic Signal

13 Contents xv Real-Time Computation of TFDs Computational Approximations for Discrete-Time Kernels Special Case: Direct Form of the Discrete Spectrogram Sample Code Fragments The TFSA package Summary and Conclusions Chapter 7: Measures, Performance Assessment and Enhancement 279 Overview Time-Frequency Analysis Based on the Affine Group (J. Bertrand and P. Bertrand) Scale Transformations in TF Analysis of Real Signals Tomographic Derivation of the Affine Wigner Function Discussion in terms of Corrections for Wigner Function Hyperbolic Chirps and Affine Group Extension Unitarity Property and Some of its Consequences Summary and Conclusions Time-Frequency Reassignment (F. Auger, P. Flandrin and E. Chassande-Mottin) Basic Principle Variations and Related Approaches Summary and Conclusions Measuring Time-Frequency Distributions Concentration (LJ. Stanković) Concentration Measurement Numerical Examples Parameter Optimization Summary and Conclusions Resolution Performance Assessment for Quadratic TFDs (B. Boashash and V. Sucic) Selecting and Comparing TFDs Performance Criteria for TFDs Resolution Performance Measure for TFDs Application to TFD Selection for a Multicomponent Signal Use of the Performance Measure in Real-Life Situations Summary and Conclusions Joint-Domain Representations via Discrete-Domain Frames (J. M. Morris and S. M. Joshi) Frames and Reconstruction Collections Product-Function Frames Cascaded Frames Summary and Conclusions

14 xvi Contents Chapter 8: Multi-Sensor and Time-Space Processing 323 Overview Blind Source Separation Using Time-Frequency Distributions (K. Abed-Meraim, A. Belouchrani and A. R. Leyman) Separation of Instantaneous Mixtures Separation of Convolutive Mixtures Illustrative Examples Summary and Conclusions Spatial Time-Frequency Distributions and Their Applications (M. G. Amin and Y. Zhang) Spatial Time-Frequency Distributions Fundamental Properties Examples Crossterm Issues in STFD Summary and Conclusions Quadratic Detection in Arrays using TFDs (A. M. Rao and D. L. Jones) The Detection Problem Quadratic Detection in a Linear Array TFD Based Array Detection Summary and Conclusions Implementation of STFDs-Based Source Separation Algorithms (A. Belouchrani) The Spatial TFD (STFD) STFDs-Based Source Separation Implementation of the Whitening Selection of Auto-Terms and Cross-Terms Implementation of JD and JOD Summary and Conclusions Underdetermined Blind Source Separation for FM-like Signals (K. Abed-Meraim, L-T. Nguyen and A. Belouchrani) Data Model and Assumptions Separation using Vector Clustering Separation using Monocomponent Extraction Summary and Conclusions Part IV: Statistical Techniques 369 Chapter 9: Random Processes and Noise Analysis 371 Overview

15 Contents xvii 9.1 Analysis of Noise in Time-Frequency Distributions (LJ. Stanković) Wigner Distribution Noise in Quadratic Time-Frequency Distributions Noisy Signals Numerical Example Summary and Conclusions Statistical Processing of Dispersive Systems and Signals (A. Papandreou-Suppappola, B.-G. Iem, G. F. Boudreaux-Bartels) Processing Tools For Time-Varying Systems and Signals Dispersive Time-Frequency Symbols Special Cases of Dispersive Time-Frequency Symbols Analysis Application Examples Summary and Conclusions Robust Time-Frequency Distributions (V. Katkovnik, I. Djurović and LJ. Stanković) Robust Spectrogram Realization of the Robust STFT Robust Wigner Distribution Example Summary and Conclusions Time-Varying Power Spectra of Nonstationary Random Processes (G. Matz and F. Hlawatsch) Nonstationary Random Processes The Generalized Wigner-Ville Spectrum The Generalized Evolutionary Spectrum The Generalized Expected Ambiguity Function Underspread Processes Time-Varying Spectral Analysis of Underspread Processes Summary and Conclusions Time-Frequency Characterization of RandomTime-Varying Channels (G. Matz and F. Hlawatsch) Time-Varying Channels WSSUS Channels Underspread WSSUS Channels Summary and Conclusions Chapter 10: Instantaneous Frequency Estimation and Localization 421 Overview Iterative Instantaneous Frequency Estimation for Random Signals (A. El-Jaroudi and M. K. Emresoy) IF Estimation: Introduction and Background Iterative Algorithm for IF Estimation

16 xviii Contents Convergence of the Estimation Algorithm Summary and Conclusions Adaptive Instantaneous Frequency Estimation Using TFDs (LJ. Stanković) Optimal Window Width Adaptive Algorithm Numerical Example Summary and Conclusions IF Estimation for Multicomponent Signals (Z. M. Hussain and B. Boashash) Time-Frequency Peak IF Estimation Properties of IF Estimates Based on Quadratic TFDs Design of Quadratic TFDs for Multicomponent IF Estimation An Adaptive Algorithm for Multicomponent IF Estimation Summary and Conclusions Analysis of Polynomial FM Signals in Additive Noise (P. O Shea and B. Barkat) The Polynomial Wigner-Ville Distributions Higher Order Ambiguity Functions Comparison of PWVDs & Higher Order Ambiguity Functions Appendix: Asymptotic MSE of a PWVD-Based IF Estimate Summary and Conclusions IF Estimation of FM Signals in Multiplicative Noise (B. Barkat and B. Boashash) Random Amplitude Modulation Linear FM Signal Polynomial FM Signals Time-Varying Higher-Order Spectra Summary and Conclusions Chapter 11: Time-Frequency Synthesis and Filtering 465 Overview Linear Time-Frequency Filters (F. Hlawatsch and G. Matz) Time-Frequency Design of Linear, Time-Varying Filters Explicit Design The Generalized Weyl Filter Implicit Design I The STFT Filter Implicit Design II The Gabor Filter The Discrete-Time Case Simulation Results Summary and Conclusions Time-Varying Filter via Gabor Expansion (S. Qian)

17 Contents xix Filtering a Six-Cylinder Engine Sound Discrete Gabor Expansion Time-Varying Filtering Numerical Simulation Summary and Conclusions Time-Frequency Filtering of Speech Signals in Hands-Free Telephone Systems (S. Stanković) Time-Variant Filtering of Speech Signals Summary and Conclusions Signal Enhancement by Time-Frequency Peak Filtering (B. Boashash and M. Mesbah) Signal Enhancement and Filtering Time-Frequency Peak Filtering Accurate TFPF Discrete-Time Algorithm for TFPF Examples and Results Summary and Conclusions Chapter 12: Detection, Classification and Estimation 499 Overview Optimal Time-Frequency Detectors (A. M. Sayeed) Time-Frequency Detection Time-Frequency Representations Time-Frequency Detection Framework Extensions Summary and Conclusions Time-Frequency Signal Analysis and Classification Using Matching Pursuits (A. Papandreou-Suppappola and S. B. Suppappola) Signal Time-Frequency Structures Matching Pursuits for Analysis and Classification Simulation Example Summary and Conclusions System Identification using Time-Frequency Filtering (X.-G. Xia) Problem Description Time-Frequency Filtering Denoising for Received Signals through a Noisy Channel System Identification Summary and Conclusions Time-Frequency Methods for Signal Estimation and Detection (F. Hlawatsch and G. Matz) Nonstationary Signal Estimation

18 xx Contents Nonstationary Signal Detection Summary and Conclusions Part V: Engineering Applications 539 Chapter 13: Time-Frequency Methods in Communications 541 Overview Time-Frequency Interference Mitigation in Spread Spectrum Communication Systems (M. G. Amin and A. R. Lindsey) Spread-Spectrum Systems and Interference Typical Signal Model A Time-Frequency Distribution Perspective Example Summary and Conclusions Communication Over Linear Dispersive Channels: A Time-Frequency Perspective (A. M. Sayeed) Linear Dispersive Channels Time-Frequency Model for Dispersive Channels Communication over Dispersive Channels Summary and Conclusions Eigenfunctions of Underspread Linear Communication Systems (S. Barbarossa) Eigenfunctions of Time-Varying Systems Systems with Spread Function Confined to a Straight Line Analytic Models for Eigenfunctions of Underspread Channels Optimal Waveforms for LTV Digital Communications Summary and Conclusions Fractional Autocorrelation for Detection in Communications (O. Akay and G. F. Boudreaux-Bartels) Fractional Fourier Transform Fractional Convolution and Correlation Fractional Autocorrelation and the Ambiguity Function Detection and Chirp Rate Parameter Estimation of Chirps Summary and Conclusions Chapter 14: Time-Frequency Methods in Radar, Sonar & Acoustics 577 Overview Special Time-Frequency Analysis of Helicopter Doppler Radar Data (S. L. Marple Jr.) Dynamic Range Considerations in TF Analysis Classical Linear and Quadratic TFDs Alternative High-Resolution Linear TFD

19 Contents xxi Application to Simulated and Actual Data Summary and Conclusions Time-Frequency Motion Compensation Algorithms for ISAR Imaging (S. Barbarossa) Echo from a Rotating Rigid Body Signal Analysis based on Time-Frequency Representations Parametric Estimation of Instantaneous Phases Summary and Conclusions Flight Parameter Estimation using Doppler and Lloyd s Mirror Effects (B. G. Ferguson and K. W. Lo) Acoustical Doppler Effect Acoustical Lloyd s Mirror Effect Time-Frequency Signal Analysis Source Parameter Estimation: An Inverse TF Problem Summary and Conclusions Wigner-Ville Analysis of High Frequency Radar Measurements of a Surrogate Theater Ballistic Missile (G. J. Frazer) Experiment Description Signal Description Signal Model Instantaneous Doppler Estimation Results Summary and Conclusions Time-Frequency Methods in Sonar (V. Chandran) Principles of Sonar Classical Methods used in Sonar Time-Frequency Approach to Sonar Prony and Higher-Order Spectral Methods in Sonar Dispersion and Angle Frequency Representation Summary and Conclusions Chapter 15: Time-Frequency Diagnosis and Monitoring 627 Overview Time-Frequency Analysis of Electric Power Disturbances (E. J. Powers, Y. Shin and W. M. Grady) Time-Frequency Analysis: Reduced Interference Distribution Power Quality Assessment via Time-Frequency Analysis Application of IF for Disturbance Propagation Summary and Conclusions Combustion Diagnosis by TF Analysis of Car Engine Signals (J. F. Böhme and S. Carstens-Behrens) Knocking Combustions

20 xxii Contents Signal Models Signal Analysis using Wigner-Ville Spectrum Signal Analysis using S-Method Summary and Conclusions Power Class Time-Frequency Representations and their Applications (A. Papandreou-Suppappola, F. Hlawatsch, G. F. Boudreaux-Bartels) Power Class Quadratic Time-Frequency Representations Power Class Applications Summary and Conclusions Image Distortion Analysis using the Wigner-Ville Distribution (A. Beghdadi and R. Iordache) Image Quality & Joint Spatial/Spatial-Freq. Representations Continuous 2D Wigner-Ville Distribution Discrete 2D Wigner-Ville Distribution An Image Dissimilarity Measure based on the 2D WVD Summary and Conclusions Time-Frequency Detection of EEG Abnormalities (B. Boashash, M. Mesbah and P. Colditz) EEG Abnormalities and Time-Frequency Processing EEG Seizures in Newborns Data Acquisition Selection of a Time-Frequency Distribution EEG Pattern Analysis Analysis of Time-Frequency Seizure Patterns Time-Frequency Matched Detector Summary and Conclusions Time-Frequency Based Machine Condition Monitoring and Fault Diagnosis (M. Mesbah, B. Boashash and J. Mathew) Machine Condition Monitoring and Fault Diagnosis Time-Frequency Analysis Methods Examples of Condition Monitoring Using TFA Summary and Conclusions Chapter 16: Other Applications (B. Boashash) 683 Time-Frequency Bibliography 685 Time-Frequency Index 719

21 List of Contributors K. Abed-Meraim, Ecole Nationale Superieure des Telecommunication (Telecom Paris), Dept. TSI (Signal & Image Processing), 46 rue Barrault, Paris, France. O. Akay, Dokuz Eylül University, Department of Electrical and Electronics Engineering, Izmir, Turkey. T. Alieva, Technische Universiteit Eindhoven, Faculteit Elektrotechniek, 5600 MB Eindhoven, Netherlands. M. G. Amin, Villanova University, Department of Electrical and Computer Engineering, Villanova, PA 19085, USA. F. Auger, Centre for Research and Technology Transfer (CRTT), Boulevard de l Universite, Saint Nazaire, Cedex 44602, France. S. Aviyente, The University of Michigan, Department of Electrical Engineering and Computer Science, Ann Arbor, MI 48109, USA. R. Baraniuk, Rice University, Department of Electrical & Computer Engineering, Houston, TX , USA. S. Barbarossa, University of Rome La Sapienza, INFOCOM Department, Rome, Italy. B. Barkat, Nanyang Technological University, School of Electrical and Electronic Engineering, Singapore V. Barroso, Instituto Superior Tecnico, Instituto de Sistemas e Robotica/DEEC, Lisboa, Portugal. M. J. Bastiaans, Technische Universiteit Eindhoven, Faculteit Elektrotechniek, 5600 MB Eindhoven, Netherlands. A. Beghdadi, L2TI-Institut Galilee, Universite Paris 13, Villetaneuse, France. A. Belouchrani, Ecole Nationale Polytechnique, EE Department, Algiers, Algeria. M. Benidir, LSS-Supelec, Universite de Paris-Sud, Gif-sur-Yvette, France. J. Bertrand, LPTMC, University Paris VII, Paris, Cedex 05, France. P. Bertrand, LPTMC, University Paris VII, Paris, Cedex 05, France. B. Boashash, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia. J. F. Böhme, Ruhr-Universitaet Bochum, Department of Electrical Engineering and Information Science, Bochum, Germany. G. F. Boudreaux-Bartels, University of Rhode Island, Department of Electrical and Computer Engineering, Kingston RI 02881, USA. S. Carstens-Behrens, Robert Bosch GmbH, Stuttgart, Germany. xxiii

22 xxiv List of Contributors V. Chandran, EESE, Queensland University of Technology, Brisbane, Qld 4001, Australia. E. Chassande-Mottin, Max Planck Institut fur Gravitationphysik, Albert Einstein Institut, D Golm, Germany. P. Colditz, Perinatal Research Centre, Royal Women s Hospital, Brisbane, 4029, Australia. I. Djurovic, Kyoto Institute of Technology, Department of Mechanical and Systems Engineering, Matsugasaki, Sakyo Kyoto , Japan. A. El-Jaroudi, University of Pittsburgh, Department of Electrical Engineering, Pittsburgh, PA 15261, USA. M. K. Emresoy, University of Pittsburgh, Department of Electrical Engineering, Pittsburgh, PA 15261, USA. B. G. Ferguson, Defence Science and Technology Organisation, Pyrmont, NSW 2009, Australia. P. Flandrin, Laboratoire de Physique, (UMR 5672 CNRS), Ecole Normale Superieure de Lyon, Lyon Cedex 07, France. G. J. Frazer, Defence Science and Technology Organisation, Surveillance Systems Division, Edinburgh, SA 5111, Australia. W. M. Grady, The University of Texas at Austin, Department of Electrical and Computer Engineering, Austin, TX , USA. F. Hlawatsch, Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology, A-1040 Vienna, Austria. Z. M. Hussain, RMIT, School of Electrical and Computer Systems Engineering, Melbourne, Victoria 3001, Australia. B.-G. Iem, Kangnung National University, Electronics Department, Kangwon-do , Republic of Korea. R. Iordache, Signal Processing Laboratory, Tampere University of Technology, FIN Tampere, Finland. D. L. Jones, University of Illinois at Urbana-Champaign, Department of Electrical and Computer Engineering, Urbana IL 61801, USA. G. Jones, Sicom Systems Ltd., Fonthill, Ontario L0S 1E0, Canada. S. M. Joshi, Lucent Technologies, Alameda CA 94501, USA. V. Katkovnik, Kwangju Institute of Science and Technology, Department of Mechatronics, Puk-gu, Kwangju , Republic of Korea. A. R. Leyman, Center for Wireless Communication, National University of Singapore, 10 Kent Ridge Crescent, Singapore, A. R. Lindsey, U.S. Air Force Research Laboratory, Rome, NY 13441, USA. Nguyen Linh-Trung, Ecole Nationale Superieure des Telecommunication (Telecom Paris), Dept. TSI (Signal & Image Processing), 46 rue Barrault, Paris, France.

23 List of Contributors xxv K. W. Lo, Defence Science and Technology Organisation, Pyrmont, NSW 2009, Australia. L. Marple Jr., ORINCON Corporation, San Diego, CA 92121, USA. J. Mathew, CIEAM, Queensland University of Technology, Brisbane, Qld 4001, Australia. G. Matz, Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology, A-1040 Vienna, Austria. M. Mesbah, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia. J. M. Morris, University of Maryland Baltimore County, Computer Science and Electrical Engineering Department, Catonsville, MD 21250, USA. P. M. Oliveira, Escola Naval, DAEI, Base Naval de Lisboa, Alfeite, 2800 Almada, Portugal. P. O Shea, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia. A. Papandreou-Suppappola, Telecommunications Research Center, Arizona State University, Department of Electrical Engineering, Tempe AZ , USA. B. Picinbono, Laboratoire des Signaux et Systemes, Supelec, Ecole Superieure d Electricite, Gif-sur-Yvette, France. E. J. Powers, The University of Texas at Austin, Department of Electrical and Computer Engineering, Austin, TX , USA. G. R. Putland, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia. S. Qian, National Instruments Corp., Austin TX 78759, USA. A. M. Rao, University of Illinois at Urbana-Champaign, Digital Signal Processing Group, Urbana IL 61801, USA. A. M. Sayeed, University of Wisconsin-Madison, Department of Electrical and Computer Engineering, Madison, WI 53706, USA. Y. J. Shin, The University of Texas at Austin, Department of Electrical and Computer Engineering, Austin, TX , USA. LJ. Stanković, University of Montenegro, Elektrotehnicki Fakultet, Podgorica, Montenegro, Yugoslavia. S. Stanković, University of Montenegro, Elektrotehnicki Fakultet, Podgorica, Montenegro, Yugoslavia. V. Sucic, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia. S. B. Suppappola, Acoustic Technologies, Mesa, AZ 85204, USA. G. Tauboeck, Telecommunications Research Center Vienna (FTW), A-1040 Vienna, Austria.

24 xxvi List of Contributors A. J. van Leest, Technische Universiteit Eindhoven, Faculteit Elektrotechniek, 5600 MB Eindhoven, Netherlands. W. J. Williams, The University of Michigan, Department of Electrical Engineering and Computer Science, Ann Arbor, MI 48109, USA. X.-G. Xia, University of Delaware, Department of Electrical and Computer Engineering, Newark DE 19716, USA. Y. Zhang, Villanova University, Department of Electrical and Computer Engineering, Villanova, PA 19085, USA.

25 Part I Introduction to the Concepts of TFSAP

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27 Chapter 1 Time-Frequency Concepts 0 Time-frequency signal analysis and processing (TFSAP) concerns the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution (TFD), which is intended to show how the energy of the signal is distributed over the two-dimensional time-frequency space. Processing of the signal may then exploit the features produced by the concentration of signal energy in two dimensions (time and frequency) instead of only one (time or frequency). The first chapter begins the introductory tutorial which constitutes Part I of the book. This tutorial updates the one given in [1] by including recent advances, refining terminology, and simplifying both the presentations of concepts and formulations of methodologies. Reading the three chapters of Part I will facilitate the understanding of the later chapters. The three sections of Chapter 1 present the key concepts needed to formulate time-frequency methods. The first (Section 1.1) explains why time-frequency methods are preferred for a wide range of applications in which the signals have time-varying characteristics or multiple components. Section 1.2 provides the signal models and formulations needed to describe temporal and spectral characteristics of signals in the time-frequency domain. It defines such basic concepts as analytic signals, the Hilbert transform, the bandwidth-duration product and asymptotic signals. Section 1.3 defines the key quantities related to time-frequency methods, including the instantaneous frequency (IF), time-delay (TD) and group delay (GD). 0 Author: Boualem Boashash, Signal Processing Research Centre, Queensland University of Technology, Brisbane, Australia. Reviewers: K. Abed-Meraim, A. Beghdadi, M. Mesbah, G. Putland and V. Sucic. 3

28 4 Chapter 1: Time-Frequency Concepts 1.1 The Need for a Time-Frequency Distribution (TFD) The two classical representations of a signal are the time-domain representation s(t) and the frequency-domain representation S(f). In both forms, the variables t and f are treated as mutually exclusive: to obtain a representation in terms of one variable, the other variable is integrated out. Consequently, each classical representation of the signal is non-localized w.r.t. the excluded variable; that is, the frequency representation is essentially averaged over the values of the time representation at all times, and the time representation is essentially averaged over the values of the frequency representation at all frequencies. In the time-frequency distribution, denoted by ρ(t, f), the variables t and f are not mutually exclusive, but are present together. The TFD representation is localized in t and f Representation of Three Real-Life Signals The usefulness of representing a signal as a function of both time and frequency is illustrated by considering three signals of practical importance: 1. Sinusoidal FM signal: Monophonic television sound, like monophonic FM radio, is transmitted on a frequency-modulated carrier. If the audio signal is a pure tone of frequency f m (the modulating frequency), then the frequency of the carrier is of the form f i (t) = f c + f d cos[2πf m t + φ] (1.1.1) where t is time, f i (t) is the frequency modulation law (FM law), f c is the mean (or center ) carrier frequency, f d is the peak frequency deviation, and φ allows for the phase of the modulating signal. The amplitude of the carrier is constant. 2. Linear FM signal: Consider a sinusoidal signal of total duration T, with constant amplitude, whose frequency increases from f 0 to f 0 + B at a constant rate α = B/T. If the origin of time is chosen so that the signal begins at t = 0, the FM law may be written f i (t) = f 0 + αt ; 0 t T. (1.1.2) In an electronics laboratory, such a signal is called a linear frequency sweep, and might be used in an automated experiment to measure the frequency response of an amplifier or filter. In mineral exploration, a linear FM signal is called a chirp or Vibroseis signal, and is used as an acoustic ping for detecting underground formations [2, 3]. 3. Musical performance: A musical note consists of a number of components of different frequencies, of which the lowest frequency is called the fundamental and the remainder are called overtones [4, p. 270]. These components are

29 The Need for a Time-Frequency Distribution (TFD) 5 Fig : An example of musical notation [5]. Roughly speaking, the horizontal dimension is time and the vertical dimension is frequency. present during a specific time interval and may vary in amplitude during that interval. In modern musical notation, each note is represented by a head. The vertical position of the head (together with other information such as the clef and key signature) indicates the pitch, i.e. the frequency of the most prominent component (usually the fundamental). The horizontal position of the head in relation to other symbols indicates the starting time, and the duration is specified by the shading of the head, attached bars, dots, stem and flags, and tempo markings such as Allegro. The choice of instrument each instrument being characterized by its overtones and their relationships with the fundamental is indicated by using a separate stave for each instrument or group of instruments, or a pair of staves for a keyboard instrument. Variations in amplitude are indicated by dynamic markings such as mp and crescendo. Fig illustrates the system. By scanning a set of staves vertically, one can see which fundamentals are present on which instruments at any given time. By scanning the staves horizontally, one can see the times at which a given fundamental is present on a given instrument. Each of the three above signals has a time-varying frequency or time-varying frequency content. Such signals are referred to as non-stationary. The three examples described above are comprehensible partly because our sense of hearing readily interprets sounds in terms of variations of frequency or frequency content with time. However, conventional representations of a signal in the time domain or frequency domain do not facilitate such interpretation, as shown below Time-Domain Representation Any signal can be described naturally as a function of time, which we may write s(t). This representation leads immediately to the instantaneous power, given by s(t) 2, which shows how the energy of the signal is distributed over time; the total signal energy is E = s(t) 2 dt. (1.1.3)