Time Frequency Signal Analysis and Processing

A Comprehensive Reference

Edited by

Boualem Boashash Director, Signal Processing Research Queensland University of Technology

Brisbane, Australia

2003

ELSEVIER

Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

Contents

Preface vii

List of Contributors xxiii

Part I: Introduction to the Concepts of TFSAP 1

Chapter 1: Time-Frequency Concepts (B. Boashash) 3

Overview 3 1.1 The Need for a Time-Frequency Distribution (TFD) 4

1.1.1 Representation of Three Real-Life Signals 4 1.1.2 Time-Domain Representation 5 1.1.3 Frequency-Domain Representation 7 1.1.4 Joint Time-Frequency Representation 9 1.1.5 Desirable Characteristics of a TFD 11

1.2 Signal Formulations and Characteristics in the (t, f) Domain . . . . 12 1.2.1 Signal Models used in (t,f) Methods 12 1.2.2 Analytic Signals 13 1.2.3 Hubert Transform; Analytic Associate 14 1.2.4 Duration, Bandwidth, BT Product 15 1.2.5 Asymptotic Signals 18 1.2.6 Monocomponent vs. Multicomponent Signals 19

1.3 Instantaneous Frequency and Time-Delay 19 1.3.1 Instantaneous Frequency (IF) 19 1.3.2 IF and Time Delay (TD) 21 1.3.3 Mean IF and Group Delay (GD) 23 1.3.4 Relaxation Time, Dynamic Bandwidth 26

1.4 Summary and Discussion 26

Chapter 2: Heuristic Formulation of Time-Frequency Distributions (B. Boashash) 29

Overview 29 2.1 Method 1: The Wigner-Ville Distribution 30

2.1.1 Knife-Edge IF Indication 30 2.1.2 Formulation of the Signal Kernel 30 2.1.3 The Wigner Distribution 31 2.1.4 The Wigner-Ville Distribution 33

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2.2 Method 2: Time-Varying Power Spectral Density 36 2.2.1 Spectra of Non-Stationary Random Processes 36 2.2.2 Estimating the Wigner-Ville Spectrum 37

2.3 Method 3: Windowed FT (STFT, Spectrogram, Gabor Transform) . 38 2.3.1 STFT and Spectrogram 38 2.3.2 Optimal Window Length of the Spectrogram 39 2.3.3 STFT vs. Gabor Transform 41

2.4 Method 4: Filtered Function of Time 42 2.4.1 Filter Banks and the Sonograph 42 2.4.2 Equivalence to Spectrogram 43

2.5 Method 5: Instantaneous Power Spectra 43 2.5.1 Page Distribution 43

2.6 Method 6: Energy Density 45 2.6.1 Rihaczek's Complex Energy Density 45 2.6.2 Levin's Real Energy Density 46 2.6.3 Windowed Rihaczek and Levin Distributions 46

2.7 Relationship between TFDs 47 2.7.1 Spectrogram 47 2.7.2 Wigner-Ville Distribution 48 2.7.3 Rihaczek Distribution 48 2.7.4 Levin Distribution 49 2.7.5 Windowed Rihaczek Distribution 49 2.7.6 Windowed Levin Distribution 50 2.7.7 Page Distribution 50 2.7.8 Relationship between the WVD and Other TFDs 51 2.7.9 Other Populär TFDs 51

2.8 Summary and Discussion 52

Chapter 3: Theory of Quadratic TFDs (B. Boashash) 59

Overview 59 3.1 The WVD 60

3.1.1 Properties of the WVD 60 3.1.2 Limitations of the WVD 62

3.2 Formulations of Quadratic TFDs 66 3.2.1 Time-Lag Formulations and Other Domain Definitions . . . . 66 3.2.2 Time-Frequency Formulation 67 3.2.3 Doppler-Lag Formulation and TFD Design 69 3.2.4 Doppler-Frequency Formulation 70 3.2.5 Examples of Simple TFD Formulations 71

3.3 Properties of Quadratic TFDs 72 3.3.1 Desirable Properties 72 3.3.2 TFD Properties & Equivalent Kernel Constraints 74

Contents xi

3.3.3 Examples of TFDs with Specific Properties 74 3.4 Summary, Discussion and Conclusions 76

Part II: Fundamental Principles of TFSAP 83

Chapter 4: Time-Frequency Signal and System Analysis 85

Overview 85 4.1 Analytic Signal & Instantaneous Frequency

(B. Picinbono) 86 4.1.1 The Problem 86 4.1.2 Analytic Signal and Canonical Pair 86 4.1.3 Phase Signals, Regulär Case 89 4.1.4 Singular and Asymptotic Phase Signals 92 4.1.5 Summary and Conclusions 93

4.2 Cross-terms & Localization in Quadratic Time-frequency Distributions (P. Flandrin) 94 4.2.1 Identifying Cross-Terms 94 4.2.2 Reducing Cross-Terms 96 4.2.3 Cross-Terms and Localization 98 4.2.4 Summary and Conclusions 100

4.3 The Covariance Theory of Time-Frequency Analysis (F. Hlawatsch and G. Tauböck) 102 4.3.1 The Covariance Principle 102 4.3.2 Time-Frequency Displacement Operators 103 4.3.3 Covariant Signal Representations: Group Domain 104 4.3.4 The Displacement Function 106 4.3.5 Covariant Signal Representations: Time-Frequency Domain . 109 4.3.6 Example: Hyperbolic Wavelet Transform & Hyperbolic Class 110 4.3.7 Summary and Conclusions 112

4.4 Uncertainty in Time-Frequency Analysis (Paulo M. Oliveira and Victor Barroso) 114 4.4.1 The Time-Frequency Plane 115 4.4.2 Information and Spectral Estimation 116 4.4.3 Summary and Conclusions 121

4.5 Generalized TFRs via Unitary Transforms (R. G. Baraniuk) 122 4.5.1 Three Approaches to Joint Distributions 123 4.5.2 Linking Signal and Axis Transformations 124 4.5.3 Examples of Linked Signal/Axis Transformations 125 4.5.4 Summary and Conclusions 127

4.6 Signal Measures in the Time-Frequency Plane (G. Jones) 128

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4.6.1 Time-Frequency Analysis 128 4.6.2 Density Distributions and Energy Distributions 128 4.6.3 Signal Measures in Time-Frequency 129 4.6.4 Properties & Interpretation of Local Measurements in TF . . 131 4.6.5 Application of Local TF Measures to Energy Distributions . 132 4.6.6 Example Result for an Adaptive Energy Distribution 133 4.6.7 Summary and Conclusions 134

4.7 Time-Frequency Transfer Function Calculus of Linear Time-Varying Systems (G. Matz and F. Hlawatsch) 135 4.7.1 Linear Time-Varying Systems 135 4.7.2 The Generalized Weyl Symbol 135 4.7.3 The Generalized Spreading Function 137 4.7.4 Underspread LTV Systems 138 4.7.5 Time-Frequency Transfer Function Calculus 140 4.7.6 Summary and Conclusions 143

4.8 Wigner Distribution and Fractional Fourier Transform (T. Alieva and M. J. Bastiaans) 145 4.8.1 Time-Frequency Representations 145 4.8.2 Wigner Distribution and Ambiguity Function 145 4.8.3 Fractional Fourier Transform 146 4.8.4 Fractional Power Spectrum and Radon-Wigner Transform . . 147 4.8.5 Fractional Fourier Transform Moments 148 4.8.6 Applications 151 4.8.7 Summary and Conclusions 152

4.9 Gabor Spectrogram (S. Qian) 153 4.9.1 Power Spectrum 153 4.9.2 Gabor Spectrogram 153 4.9.3 Numerical Simulations 156 4.9.4 Summary and Conclusions 158

Chapter 5: Design of Time-Frequency Distributions 159

Overview 159 5.1 Ambiguity Functions

(P. Flandrin) 160 5.1.1 The Radar/Sonar Problem 160 5.1.2 Definitions of Ambiguity Functions 160 5.1.3 Properties of Narrowband Ambiguity Functions 162 5.1.4 Remarks on Wideband Ambiguity Functions 166 5.1.5 Summary and Conclusions 166

5.2 Reduced Interference Time-Frequency Distributions (William J. Williams) 168 5.2.1 Nonstationarity, Resolution and Interference 168

Contents XÜi

5.2.2 The Reduced Interference Distribution 169 5.2.3 Kernel Selection for RID 170 5.2.4 Comparisons of TFD Results 175 5.2.5 Summary and Conclusions 175

5.3 Adaptive Time-Frequency Analysis (R. G. Baraniuk and D. L. Jones) 178 5.3.1 Adaptive Short-Time Fourier Transforms 178 5.3.2 Adaptive Quadratic Representations 180 5.3.3 Summary and Conclusions 183

5.4 Polynomial Wigner-Ville Distributions (B. Boashash and G. R. Putland) 185 5.4.1 Polynomial FM Signals 185 5.4.2 Principles of Formulation of Polynomial WVDs 185 5.4.3 IF Estimates with Zero Deterministic Bias 187 5.4.4 Calculation of Coefficients 188 5.4.5 Examples 189 5.4.6 Multicomponent Signals and Polynomial TFDs 191 5.4.7 Summary and Conclusions 191

5.5 Design of Polynomial TFDs, with Applications (M. Benidir) 193 5.5.1 Decompositions of Polynomial Derivatives 193 5.5.2 Design of Time-Frequency Distributions 194 5.5.3 Estimation of the Phase of a PPS 198 5.5.4 Summary and Conclusions 201 5.5.5 Appendix 201

5.6 Time-Frequency Representations Covariant to Group Delay Shifts (A. Papandreou-Suppappola) 203 5.6.1 Group Delay Shift Covariance Property 203 5.6.2 Classes of GDS Covariant QTFRs 206 5.6.3 Simulation Example 210 5.6.4 Summary and Conclusions 211

5.7 Design of High-Resolution Quadratic TFDs with Separable Kernels (B. Boashash and G. R. Putland) 213 5.7.1 RIDs and Quadratic TFDs 213 5.7.2 Separable Kernel Formulations 213 5.7.3 Properties 216 5.7.4 Design Examples of Separable-Kernel TFDs 217 5.7.5 Results and Discussion 218 5.7.6 Summary and Conclusions 222

5.8 Fractional Fourier Transform and Generalized-Marginal TFDs (X.-G. Xia) 223 5.8.1 Fractional Fourier Transform 223 5.8.2 Generalized-Marginal Time-Frequency Distribution 224

Contents

5.8.3 Summary and Conclusions 228

Part III: Time-Frequency Methods 229

Chapter 6: Implementation and Realization of TFDs 231

Overview 231 6.1 Discrete Time-Frequency Distributions

(B. Boashash and G. R. Putland) 232 6.1.1 The Discrete Wigner-Ville Distribution (DWVD) 232 6.1.2 The Windowed DWVD 234 6.1.3 The Discrete Quadratic TFD 235 6.1.4 Desirable Properties; Kernel Constraints 239 6.1.5 Examples 241 6.1.6 Summary and Conclusions 241

6.2 Quadratic and Higher Order Time-Frequency Analysis Based on the STFT (LJ. Stankovic) 242 6.2.1 STFT Based Realization of the Quadratic Representations . 242 6.2.2 Discrete Realization of the Basic S-Method Form 245 6.2.3 STFT Based Realization of Higher Order Representations . . 248 6.2.4 Summary and Conclusions 250

6.3 Gabor's Signal Expansion for a Non-Orthogonal Sampling Geometry (M. J. Bastiaans and A. J. van Leest) 252 6.3.1 Historical Perspective 252 6.3.2 Gabor's Signal Expansion on a Rectangular Lattice 252 6.3.3 Fourier Transform and Zak Transform 253 6.3.4 Rational Oversampling 254 6.3.5 Non-Orthogonal Sampling 256 6.3.6 Gabor's Signal Expansion on a Non-Orthogonal Lattice . . . 257 6.3.7 Summary and Conclusions 259

6.4 Spectrogram Decompositions of Time-Frequency Distributions (W. J. Williams and S. Aviyente) 260 6.4.1 Decomposition Based Approaches 260 6.4.2 Decomposition of Time-Frequency Kernels 261 6.4.3 Development of the Method 261 6.4.4 Wigner Example 262 6.4.5 Optimum Orthogonal Windows 263 6.4.6 Kernel Decomposition Results 266 6.4.7 Summary and Conclusions 266

6.5 Computation of Discrete Quadratic TFDs (B. Boashash and G. R. Putland) 268 6.5.1 General Computational Procedure 268 6.5.2 Computation of the Analytic Signal 268

Contents xv

6.5.3 Real-Time Computation of TFDs 269 6.5.4 Computational Approximations for Discrete-Time Kernels . . 270 6.5.5 Special Case: Direct Form of the Discrete Spectrogram . . . 272 6.5.6 Sample Code Fragments 274 6.5.7 The TFSA package 278 6.5.8 Summary and Conclusions 278

Chapter 7: Measures, Performance Assessment and Enhancement 279

Overview 279 7.1 Time-Frequency Analysis Based on the Affine Group

(J. Bertrand and P. Bertrand) 280 7.1.1 Scale Transformations in TF Analysis of Real Signals . . . . 280 7.1.2 Tomographie Derivation of the Affine Wigner Function . . . . 282 7.1.3 Discussion in terms of Corrections for Wigner Function . . . 284 7.1.4 Hyperbolic Chirps and Affine Group Extension 286 7.1.5 Unitarity Property and Some of its Consequences 287 7.1.6 Summary and Conclusions 288

7.2 Time-Frequency Reassignment (F. Auger, P. Flandrin and E. Chassande-Mottin) 290 7.2.1 Basic Principle 290 7.2.2 Variations and Related Approaches 294 7.2.3 Summary and Conclusions 295

7.3 Measuring Time-Frequency Distributions Concentration (LJ. Stankovic) 297 7.3.1 Concentration Measurement 297 7.3.2 Numerical Examples 301 7.3.3 Parameter Optimization 302 7.3.4 Summary and Conclusions 304

7.4 Resolution Performance Assessment for Quadratic TFDs (B. Boashash and V. Sucic) 305 7.4.1 Selecting and Comparing TFDs 305 7.4.2 Performance Criteria for TFDs 306 7.4.3 Resolution Performance Measure for TFDs 308 7.4.4 Application to TFD Selection for a Multicomponent Signal . 309 7.4.5 Use of the Performance Measure in Real-Life Situations . . . 310 7.4.6 Summary and Conclusions 313

7.5 Joint-Domain Representations via Discrete-Domain Frames (J. M. Morris and S. M. Joshi) 315 7.5.1 Frames and Reconstruction Collections 315 7.5.2 Product-Function Frames 316 7.5.3 Cascaded Frames 319 7.5.4 Summary and Conclusions 321

XVI Contents

Chapter 8: Multi-Sensor and Time-Space Processing 323

Overview 323 8.1 Blind Source Separation Using Time-Frequency Distributions

(K. Abed-Meraim, A. Belouchrani and A. R. Leyman) 324 8.1.1 Separation of Instantaneous Mixtures 325 8.1.2 Separation of Convolutive Mixtures 328 8.1.3 Illustrative Examples 331 8.1.4 Summary and Conclusions 332

8.2 Spatial Time-Frequency Distributions and Their Applications (M. G. Amin and Y. Zhang) 334 8.2.1 Spatial Time-Frequency Distributions 334 8.2.2 Fundamental Properties 335 8.2.3 Examples 337 8.2.4 Crossterm Issues in STFD 341 8.2.5 Summary and Conclusions 342

8.3 Quadratic Detection in Arrays using TFDs (A. M. Rao and D. L. Jones) 344 8.3.1 The Detection Problem 344 8.3.2 Quadratic Detection in a Linear Array 345 8.3.3 TFD Based Array Detection 346 8.3.4 Summary and Conclusions 347

8.4 Implementation of STFDs-Based Source Separation Algorithms (A. Belouchrani) 349 8.4.1 The Spatial TFD (STFD) 349 8.4.2 STFDs-Based Source Separation 351 8.4.3 Implementation of the Whitening 352 8.4.4 Selection of Auto-Terms and Cross-Terms 353 8.4.5 Implementation of JD and JOD 354 8.4.6 Summary and Conclusions 355

8.5 Underdetermined Blind Source Separation for FM-like Signals (K. Abed-Meraim, L-T. Nguyen and A. Belouchrani) 357 8.5.1 Data Model and Assumptions 357 8.5.2 Separation using Vector Clustering 358 8.5.3 Separation using Monocomponent Extraction 361 8.5.4 Summary and Conclusions 366

Part IV: Statistical Techniques 369

Chapter 9: Random Processes and Noise Analysis 371

Overview 371

Contents xvii

9.1 Analysis of Noise in Time-Frequency Distributions (LJ. Stankovic) 372 9.1.1 Wigner Distribution 372 9.1.2 Noise in Quadratic Time-Frequency Distributions 374 9.1.3 Noisy Signals 376 9.1.4 Numerical Example 380 9.1.5 Summary and Conclusions 380

9.2 Statistical Processing of Dispersive Systems and Signals (A. Papandreou-Suppappola, B.-G.Iem, G. F. Boudreaux-Bartels) . . 382 9.2.1 Processing Tools For Time-Varying Systems and Signals . . . 382 9.2.2 Dispersive Time-Frequency Symbols 384 9.2.3 Special Cases of Dispersive Time-Frequency Symbols 386 9.2.4 Analysis Application Examples 388 9.2.5 Summary and Conclusions 390

9.3 Robust Time-Frequency Distributions (V. Katkovnik, I. Djurovic and LJ. Stankovic) 392 9.3.1 Robust Spectrogram 392 9.3.2 Realization of the Robust STFT 394 9.3.3 Robust Wigner Distribution 396 9.3.4 Example 398 9.3.5 Summary and Conclusions 399

9.4 Time-Varying Power Spectra of Nonstationary Random Processes (G. Matz and F. Hlawatsch) 400 9.4.1 Nonstationary Random Processes 400 9.4.2 The Generalized Wigner-Ville Spectrum 400 9.4.3 The Generalized Evolutionary Spectrum 401 9.4.4 The Generalized Expected Ambiguity Function 403 9.4.5 Underspread Processes 404 9.4.6 Time-Varying Spectral Analysis of Underspread Processes . . 405 9.4.7 Summary and Conclusions 408

9.5 Time-Frequency Characterization of Random Time-Varying Channels (G. Matz and F. Hlawatsch) 410 9.5.1 Time-Varying Channels 410 9.5.2 WSSUS Channels 411 9.5.3 Underspread WSSUS Channels 414 9.5.4 Summary and Conclusions 418

Chapter 10: Instantaneous Frequency Estimation and Localization 421

Overview 421 10.1 Iterative Instantaneous Frequency Estimation for Random Signals

(A. El-Jaroudi and M. K. Emresoy) 422 10.1.1 IF Estimation: Introduction and Background 422 10.1.2 Iterative Algorithm for IF Estimation 423

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10.1.3 Convergence of the Estimation Algorithm 424 10.1.4 Summary and Conclusions 427

10.2 Adaptive Instantaneous Frequency Estimation Using TFDs (LJ. Stankovic) 429 10.2.1 Optimal Window Width 429 10.2.2 Adaptive Algorithm 430 10.2.3 Numerical Example 434 10.2.4 Summary and Conclusions 436

10.3 IF Estimation for Multicomponent Signals (Z. M. Hussain and B. Boashash) 437 10.3.1 Time-Frequency Peak IF Estimation 437 10.3.2 Properties of IF Estimates Based on Quadratic TFDs . . . . 439 10.3.3 Design of Quadratic TFDs for Multicomponent IF Estimation 441 10.3.4 An Adaptive Algorithm for Multicomponent IF Estimation . 443 10.3.5 Summary and Conclusions 445

10.4 Analysis of Polynomial FM Signals in Additive Noise (P. O'Shea and B. Barkat) 447 10.4.1 The Polynomial Wigner-Ville Distributions 447 10.4.2 Higher Order Ambiguity Functions 452 10.4.3 Comparison of PWVDs & Higher Order Ambiguity Functions 453 10.4.4 Appendix: Asymptotic MSE of a PWVD-Based IF Estimate 453 10.4.5 Summary and Conclusions 455

10.5 IF Estimation of FM Signals in Multiplicative Noise (B. Barkat and B. Boashash) 457 10.5.1 Random Amplitude Modulation 457 10.5.2 Linear FM Signal 457 10.5.3 Polynomial FM Signals 460 10.5.4 Time-Varying Higher-Order Spectra 462 10.5.5 Summary and Conclusions 463

Chapter 11: Time-Frequency Synthesis and Filtering 465

Overview 465 11.1 Linear Time-Frequency Filters

(F. Hlawatsch and G. Matz) 466 11.1.1 Time-Frequency Design of Linear, Time-Varying Filters . . . 466 11.1.2 Explicit Design— The Generalized Weyl Filter 467 11.1.3 Implicit Design I—The STFT Filter 468 11.1.4 Implicit Design II —The Gabor Filter 470 11.1.5 The Discrete-Time Case 471 11.1.6 Simulation Results 473 11.1.7 Summary and Conclusions 474

11.2 Time-Varying Filter via Gabor Expansion (S. Qian) 476

Contents xix

11.2.1 Filtering a Six-Cylinder Engine Sound 476 11.2.2 Discrete Gabor Expansion 476 11.2.3 Time-Varying Filtering 479 11.2.4 Numerical Simulation 480 11.2.5 Summary and Conclusions 480

11.3 Time-Frequency Filtering of Speech Signals in Hands-Free Telephone Systems (S. Stankovic) 481 11.3.1 Time-Variant Filtering of Speech Signals 482 11.3.2 Summary and Conclusions 486

11.4 Signal Enhancement by Time-Frequency Peak Filtering (B. Boashash and M. Mesbah) 489 11.4.1 Signal Enhancement and Filtering 489 11.4.2 Time-Frequency Peak Filtering 489 11.4.3 Accurate TFPF 492 11.4.4 Discrete-Time Algorithm for TFPF 493 11.4.5 Examples and Results 496 11.4.6 Summary and Conclusions 496

Chapter 12: Detection, Classification and Estimation 499

Overview 499 12.1 Optimal Time-Frequency Detectors

(A. M. Sayeed) 500 12.1.1 Time-Frequency Detection 500 12.1.2 Time-Frequency Representations 500 12.1.3 Time-Frequency Detection Framework 502 12.1.4 Extensions 507 12.1.5 Summary and Conclusions 508

12.2 Time-Frequency Signal Analysis and Classification Using Matching Pursuits (A. Papandreou-Suppappola and S. B. Suppappola) . . . . 510 12.2.1 Signal Time-Frequency Structures 510 12.2.2 Matching Pursuits for Analysis and Classification 510 12.2.3 Simulation Example 515 12.2.4 Summary and Conclusions 517

12.3 System Identification using Time-Frequency Filtering (X.-G. Xia) 519 12.3.1 Problem Description 519 12.3.2 Time-Frequency Filtering 520 12.3.3 Denoising for Received Signals through a Noisy Channel . . . 522 12.3.4 System Identification 524 12.3.5 Summary and Conclusions 527

12.4 Time-Frequency Methods for Signal Estimation and Detection (F. Hlawatsch and G. Matz) 528 12.4.1 Nonstationary Signal Estimation 528

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12.4.2 Nonstationary Signal Detection 531 12.4.3 Summary and Conclusions 535

Part V: Engineering Applications 539

Chap te r 13: Time-Prequency Methods in Communications 541

Overview 541 13.1 Time-Frequency Interference Mitigation in Spread Spectrum

Communication Systems (M. G. Amin and A. R. Lindsey) 542 13.1.1 Spread-Spectrum Systems and Interference 542 13.1.2 Typical Signal Model 544 13.1.3 A Time-Frequency Distribution Perspective 544 13.1.4 Example 546 13.1.5 Summary and Conclusions 547

13.2 Communication Over Linear Dispersive Channels: A Time-Frequency Perspective (A. M. Sayeed) 549 13.2.1 Linear Dispersive Channels 549 13.2.2 Time-Frequency Model for Dispersive Channels 549 13.2.3 Communication over Dispersive Channels 551 13.2.4 Summary and Conclusions 557

13.3 Eigenfunctions of Underspread Linear Communication Systems (S. Barbarossa) 558 13.3.1 Eigenfunctions of Time-Varying Systems 558 13.3.2 Systems with Spread Function Confmed to a Straight Line . 559 13.3.3 Analytic Models for Eigenfunctions of Underspread Channels 560 13.3.4 Optimal Waveforms for LTV Digital Communications . . . . 566 13.3.5 Summary and Conclusions 567

13.4 Fractional Autocorrelation for Detection in Communications (O. Akay and G. F. Boudreaux-Bartels) 568 13.4.1 Fractional Fourier Transform 568 13.4.2 Fractional Convolution and Correlation 569 13.4.3 Fractional Autocorrelation and the Ambiguity Function . . . 572 13.4.4 Detection and Chirp Rate Parameter Estimation of Chirps . 573 13.4.5 Summary and Conclusions 574

Chap te r 14: Time-Frequency Methods in Radar, Sonar & Acoustics 577

Overview 577 14.1 Special Time-Frequency Analysis of Helicopter Doppler Radar Data

(S. L. Marple Jr.) 578 14.1.1 Dynamic Range Considerations in TF Analysis 578 14.1.2 Classical Linear and Quadratic TFDs 578 14.1.3 Alternative High-Resolution Linear TFD 581

Contents xxi

14.1.4 Application to Simulated and Actual Data 583 14.1.5 Summary and Conclusions 589

14.2 Time-Frequency Motion Compensation Algorithms for ISAR Imaging (S. Barbarossa) 590 14.2.1 Echo from a Rotating Rigid Body 591 14.2.2 Signal Analysis based on Time-Frequency Representations . . 593 14.2.3 Parametric Estimation of Instantaneous Phases 593 14.2.4 Summary and Conclusions 596

14.3 Flight Parameter Estimation using Doppler and Lloyd's Mirror Effects (B. G. Ferguson and K. W. Lo) 597 14.3.1 Acoustical Doppler Effect 597 14.3.2 Acoustical Lloyd's Mirror Effect 599 14.3.3 Time-Frequency Signal Analysis 601 14.3.4 Source Parameter Estimation: An Inverse TF Problem . . . . 601 14.3.5 Summary and Conclusions 604

14.4 Wigner-Ville Analysis of High Frequency Radar Measurements of a Surrogate Theater Ballistic Missile (G. J. Frazer) 605 14.4.1 Experiment Description 605 14.4.2 Signal Description 605 14.4.3 Signal Model 607 14.4.4 Instantaneous Doppler Estimation 608 14.4.5 Results 608 14.4.6 Summary and Conclusions 612

14.5 Time-Frequency Methods in Sonar (V. Chandran) 615 14.5.1 Principles of Sonar 615 14.5.2 Classical Methods used in Sonar 618 14.5.3 Time-Frequency Approach to Sonar 619 14.5.4 Prony and Higher-Order Spectral Methods in Sonar 622 14.5.5 Dispersion and Angle Frequency Representation 623 14.5.6 Summary and Conclusions 624

Chapter 15: Time-Frequency Diagnosis and Monitoring 627

Overview 627 15.1 Time-Frequency Analysis of Electric Power Disturbances

(E. J. Powers, Y. Shin and W. M. Grady) 628 15.1.1 Time-Frequency Analysis: Reduced Interference Distribution 628 15.1.2 Power Quality Assessment via Time-Frequency Analysis . . . 629 15.1.3 Application of IF for Disturbance Propagation 631 15.1.4 Summary and Conclusions 634

15.2 Combustion Diagnosis by TF Analysis of Car Engine Signals (J. F. Böhme and S. Carstens-Behrens) 635 15.2.1 Knocking Combustions 635

xxii Contents

15.2.2 Signal Models 635 15.2.3 Signal Analysis using Wigner-Ville Spectrum 637 15.2.4 Signal Analysis using S-Method 639 15.2.5 Summary and Conclusions 641

15.3 Power Class Time-Frequency Representations and their Applications (A. Papandreou-Suppappola, F. Hlawatsch, G. F. Boudreaux-Bartels) 643 15.3.1 Power Class Quadratic Time-Frequency Representations . . . 643 15.3.2 Power Class Applications 647 15.3.3 Summary and Conclusions 649

15.4 Image Distortion Analysis using the Wigner-Ville Distribution (A. Beghdadi and R. lordache) 651 15.4.1 Image Quality & Joint Spatial/Spatial-Freq. Representations 651 15.4.2 Continuous 2D Wigner-Ville Distribution 652 15.4.3 Discrete 2D Wigner-Ville Distribution 655 15.4.4 An Image Dissimilarity Measure based on the 2D WVD . . . 659 15.4.5 Summary and Conclusions 660

15.5 Time-Frequency Detection of EEG Abnormalities (B. Boashash, M. Mesbah and P. Colditz) 663 15.5.1 EEG Abnormalities and Time-Frequency Processing 663 15.5.2 EEG Seizures in Newborns 663 15.5.3 Data Acquisition 664 15.5.4 Selection of a Time-Frequency Distribution 664 15.5.5 EEG Pattern Analysis 665 15.5.6 Analysis of Time-Frequency Seizure Patterns 665 15.5.7 Time-Frequency Matched Detector 666 15.5.8 Summary and Conclusions 669

15.6 Time-Frequency Based Machine Condition Monitoring and Fault Diagnosis (M. Mesbah, B. Boashash and J. Mathew) 671 15.6.1 Machine Condition Monitoring and Fault Diagnosis 671 15.6.2 Time-Frequency Analysis Methods 675 15.6.3 Examples of Condition Monitoring Using TFA 677 15.6.4 Summary and Conclusions 681

Chapter 16: Other Applications (B. Boashash) 683

Time-Frequency Bibliography 685

Time-Frequency Index 719