Time-Frequency Analysis

Franz Hlawatsch is an Associate Professor at the Vienna University of Technology, Austria. His research interests are in the areas of time-frequency signal processing, non-stationary statistical signal processing and wireless communications. Francois Auger is Head of the Physical Measurements Department of the Technology Institute of the University of Saint Nazaire, France. His current research interests include motor control, embedded control and signal processing with FPGAs, spectral estimation and time-frequency representations.

• Preface 13FIRST PART. FUNDAMENTAL CONCEPTS AND METHODS 17Chapter 1. Time-Frequency Energy Distributions: An Introduction 19Patrick FLANDRIN1.1. Introduction 191.2. Atoms 201.3. Energy 211.3.1. Distributions 221.3.2. Devices 221.3.3. Classes 231.4. Correlations 261.5. Probabilities 271.6. Mechanics 291.7. Measurements 291.8. Geometries 321.9. Conclusion 331.10.Bibliography 34Chapter 2. Instantaneous Frequency of a Signal 37Bernard PICINBONO2.1. Introduction 372.2. Intuitive approaches 382.3. Mathematical definitions 402.3.1. Ambiguity of the problem 402.3.2. Analytic signal and Hilbert transform 402.3.3. Application to the definition of instantaneous frequency 422.3.4. Instantaneous methods 452.4. Critical comparison of the different definitions 462.4.1. Interest of linear filtering 462.4.2. Bounds of the quantities introduced 462.4.3. Instantaneous nature 472.4.4. Interpretation by the average 482.5. Canonical pairs 492.6. Phase signals 502.6.1. Blaschke factors 502.6.2. Oscillatory singularities 542.7. Asymptotic phase signals 572.7.1. Parabolic chirp 572.7.2. Cubic chirp 592.8. Conclusions 592.9. Bibliography 60Chapter 3. Linear Time-Frequency Analysis I: Fourier-Type Representations 61Remi GRIBONVAL3.1. Introduction 613.2. Short-time Fourier analysis 623.2.1. Short-time Fourier transform 633.2.2. Time-frequency energy maps 643.2.3. Role of the window 663.2.4. Reconstruction/synthesis 713.2.5. Redundancy 713.3. Gabor transform; Weyl-Heisenberg and Wilson frames 713.3.1. Sampling of the short-time Fourier transform 713.3.2. Weyl-Heisenberg frames 723.3.3. Zak transform and “critical” Weyl-Heisenberg frames 743.3.4. Balian-Low theorem 753.3.5. Wilson bases and frames, local cosine bases 753.4. Dictionaries of time-frequency atoms; adaptive representations 773.4.1. Multi-scale dictionaries of time-frequency atoms 773.4.2. Pursuit algorithm 783.4.3. Time-frequency representation 793.5. Applications to audio signals 803.5.1. Analysis of superimposed structures 803.5.2. Analysis of instantaneous frequency variations 803.5.3. Transposition of an audio signal 823.6. Discrete algorithms 823.6.1. Fast Fourier transform 833.6.2. Filter banks: fast convolution 833.6.3. Discrete short-time Fourier transform 853.6.4. Discrete Gabor transform 863.7. Conclusion 863.8. Acknowledgements 873.9. Bibliography 87Chapter 4. Linear Time-Frequency Analysis II: Wavelet-Type Representations 93Thierry BLU and Jerome LEBRUN4.1. Introduction: scale and frequency 944.2. Continuous wavelet transform 954.2.1. Analysis and synthesis 954.2.2. Multiscale properties 974.3. Discrete wavelet transform 984.3.1. Multi-resolution analysis 984.3.2. Mallat algorithm 1044.3.3. Graphical representation 1064.4. Filter banks and wavelets 1074.4.1. Generation of regular scaling functions 1084.4.2. Links with approximation theory 1114.4.3. Orthonormality and bi-orthonormality/perfect reconstruction 1124.4.4. Polyphase matrices and implementation 1144.4.5. Design of wavelet filters with finite impulse response 1144.5. Generalization: multi-wavelets 1164.5.1. Multi-filter banks 1164.5.2. Balancing and design of multi-filters 1184.6. Other extensions 1214.6.1. Wavelet packets 1214.6.2. Redundant transformations: pyramids and frames 1224.6.3. Multi-dimensional wavelets 1234.7. Applications 1244.7.1. Signal compression and denoising 1244.7.2. Image alignment 1254.8. Conclusion 1254.9. Acknowledgments 1264.10. Bibliography 126Chapter 5. Quadratic Time-Frequency Analysis I: Cohen’s Class 131Francois AUGER and Eric CHASSANDE-MOTTIN5.1. Introduction 1315.2. Signal representation in time or in frequency 1325.2.1. Notion of signal representation 1325.2.2. Temporal representations 1335.2.3. Frequency representations 1345.2.4. Notion of stationarity 1355.2.5. Inadequacy of monodimensional representations 1365.3. Representations in time and frequency 1375.3.1. “Ideal” time-frequency representations 1375.3.2. Inadequacy of the spectrogram 1405.3.3. Drawbacks and benefits of the Rihaczek distribution 1425.4. Cohen’s class 1425.4.1. Quadratic representations covariant under translation 1425.4.2. Definition of Cohen’s class 1435.4.3. Equivalent parametrizations 1445.4.4. Additional properties 1455.4.5. Existence and localization of interference terms 1485.5. Main elements 1555.5.1. Wigner-Ville and its smoothed versions 1555.5.2. Rihaczek and its smoothed versions 1575.5.3. Spectrogram and S transform 1585.5.4. Choi-Williams and reduced interference distributions 1585.6. Conclusion 1595.7. Bibliography 159Chapter 6. Quadratic Time-Frequency Analysis II: Discretization of Cohen’s Class 165Stephane GRASSIN6.1. Quadratic TFRs of discrete signals 1656.1.1. TFRs of continuous-time deterministic signals 1676.1.2. Sampling equation 1676.1.3. The autocorrelation functions of the discrete signal 1686.1.4. TFR of a discrete signal as a function of its generalized ACF 1696.1.5. Discussion 1716.1.6. Corollary: ambiguity function of a discrete signal 1726.2. Temporal support of TFRs 1736.2.1. The characteristic temporal supports 1736.2.2. Observations 1756.3. Discretization of the TFR 1766.3.1. Meaning of the frequency discretization of the TFR 1766.3.2. Meaning of the temporal discretization of the TFR 1766.3.3. Aliased discretization 1776.3.4. “Non-aliased”discretization 1796.4. Properties of discrete-time TFRs 1806.4.1. Discrete-time TFRs 1816.4.2. Effect of the discretization of the kernel 1826.4.3. Temporal inversion 1826.4.4. Complexcon jugation 1836.4.5. Real-valued TFR 1836.4.6. Temporal moment 1836.4.7. Frequency moment 1846.5. Relevance of the discretization to spectral analysis 1856.5.1. Formulation of the problem 1856.5.2. Trivial case of a sinusoid 1876.5.3. Signal with linear frequency modulation 1876.5.4. Spectral analysis with discretized TFRs 1886.6. Conclusion 1896.7. Bibliography 189Chapter 7. Quadratic Time-Frequency Analysis III: The Affine Class and Other Covariant Classes 193Paulo GONCALVES, Jean-Philippe OVARLEZ and Richard BARANIUK7.1. Introduction 1937.2. General construction of the affine class 1947.2.1. Bilinearity of distributions 1947.2.2. Covariance principle 1957.2.3. Affine class of time-frequency representations 1987.3. Properties of the affine class 2017.3.1. Energy 2017.3.2. Marginals 2027.3.3. Unitarity 2027.3.4. Localization 2037.4. Affine Wigner distributions 2067.4.1. Diagonal form of kernels 2067.4.2. Covariance to the three-parameter affine group 2097.4.3. Smoothed affine pseudo-Wigner distributions 2117.5. Advanced considerations 2167.5.1. Principle of tomography 2167.5.2. Operators and groups 2177.6. Conclusions 2227.7. Bibliography 223SECOND PART. ADVANCED CONCEPTS AND METHODS 227Chapter 8. Higher-Order Time-Frequency Representations 229Pierre-Olivier AMBLARD8.1. Motivations 2298.2. Construction of time-multifrequency representations 2308.2.1. General form and desirable properties 2308.2.2. General classes in the symmetric even case 2318.2.3. Examples and interpretation 2368.2.4. Desired properties and constraints on the kernel 2378.2.5. Discussion 2398.3. Multilinear time-frequency representations 2408.3.1. Polynomial phase and perfect concentration 2408.3.2. Multilinear time-frequency representations: general class 2428.4. Towards affine multilinear representations 2438.5. Conclusion 2468.6. Bibliography 247Chapter 9. Reassignment 249Eric CHASSANDE-MOTTIN, Francois AUGER, and Patrick FLANDRIN9.1. Introduction 2499.2. The reassignment principle 2509.2.1. Classical tradeoff in time-frequency and time-scale analysis 2509.2.2. Spectrograms and scalograms re-examined and corrected by mechanics 2529.2.3. Generalization to other representations 2549.2.4. Link to similar approaches 2579.3. Reassignment at work 2579.3.1. Fast algorithms 2589.3.2. Analysis of a few simple examples 2599.4. Characterization of the reassignment vector fields 2659.4.1. Statistics of the reassignment vectors of the spectrogram 2659.4.2. Geometrical phase and gradient field 2679.5. Two variations 2699.5.1. Supervised reassignment 2699.5.2. Differential reassignment 2709.6. An application: partitioning the time-frequency plane 2719.7. Conclusion 2749.8. Bibliography 274Chapter 10. Time-Frequency Methods for Non-stationary Statistical Signal Processing 279Franz HLAWATSCH and Gerald MATZ10.1. Introduction 27910.2. Time-varying systems 28110.3. Non-stationary processes 28310.4. TF analysis of non-stationary processes – type I spectra 28510.4.1. GeneralizedWigner-Ville spectrum 28510.4.2. TF correlations and statistical cross-terms 28610.4.3. TF smoothing and type I spectra 28710.4.4. Properties of type I spectra 28910.5. TF analysis of non-stationary processes – type II spectra 28910.5.1. Generalized evolutionary spectrum 28910.5.2. TF smoothing and type II spectra 29110.6. Properties of the spectra of underspread processes 29110.6.1. Approximate equivalences 29210.6.2. Approximate properties 29510.7. Estimation of time-varying spectra 29610.7.1. A class of estimators 29610.7.2. Bias-variance analysis 29710.7.3. Designing an estimator 29910.7.4. Numerical results 30010.8. Estimation of non-stationary processes 30210.8.1. TF formulation of the optimum filter 30310.8.2. TF design of a quasi-optimum filter 30410.8.3. Numerical results 30510.9. Detection of non-stationary processes 30610.9.1. TF formulation of the optimum detector 30910.9.2. TF design of a quasi-optimum detector 31010.9.3. Numerical results 31110.10. Conclusion 31310.11. Acknowledgements 31510.12. Bibliography 315Chapter 11. Non-stationary Parametric Modeling 321Corinne MAILHES and Francis CASTANIE11.1. Introduction 32111.2. Evolutionary spectra 32211.2.1. Definition of the “evolutionary spectrum”32211.2.2. Properties of the evolutionary spectrum 32411.3. Postulate of local stationarity 32511.3.1. Sliding methods 32511.3.2. Adaptive and recursive methods 32611.3.3. Application to time-frequency analysis 32811.4. Suppression of a stationarity condition 32911.4.1. Unstable models 32911.4.2. Models with time-varying parameters 33211.4.3. Models with non-stationary input 34011.4.4. Application to time-frequency analysis 34611.5. Conclusion 34811.6. Bibliography 349Chapter 12. Time-Frequency Representations in Biomedical Signal Processing 353Lotfi SENHADJI and Mohammad Bagher SHAMSOLLAHI12.1. Introduction 35312.2. Physiological signals linked to cerebral activity 35612.2.1. Electroencephalographic (EEG) signals 35612.2.2. Electrocorticographic (ECoG) signals 35912.2.3. Stereoelectroencephalographic (SEEG) signals 35912.2.4. Evoked potentials (EP) 36212.3. Physiological signals related to the cardiac system 36312.3.1. Electrocardiographic (ECG) signals 36312.3.2. R-R sequences 36512.3.3. Late ventricular potentials (LVP) 36712.3.4. Phonocardiographic (PCG) signals 36912.3.5. Doppler signals 37212.4. Other physiological signals 37212.4.1. Electrogastrographic (EGG) signals 37212.4.2. Electromyographic (EMG) signals 37312.4.3. Signals related to respiratory sounds (RS) 37412.4.4. Signals related to muscle vibrations 37412.5. Conclusion 37512.6. Bibliography 376Chapter 13. Application of Time-Frequency Techniques to Sound Signals: Recognition and Diagnosis 383Manuel DAVY13.1. Introduction 38313.1.1. 38413.1.2. Sound signals 38413.1.3. Time-frequency analysis as a privileged decision-making tool 38413.2. Loudspeaker fault detection 38613.2.1. Existing tests 38613.2.2. A test signal 38813.2.3. A processing procedure 38913.2.4. Application and results 39113.2.5. Use of optimized kernels 39513.2.6. Conclusion 39913.3. Speaker verification 39913.3.1. Speaker identification: the standard approach 39913.3.2. Speaker verification: a time-frequency approach 40313.4. Conclusion 40513.5. Bibliography 406List of Authors 409Index 413