1.1 First Integral Transforms and Function Spaces 25
1.1.1 Fourier Transforms 26
1.1.2 Hilbert Transform, Analytic Signal 33
1.2 Sampling and Aliasing 34
1.3 Wiener's Deterministic Spectral Theory 39
1.4 Deterministic Spectral Theory for Time Series 42
1.4.1 Sample Autocovariance and Autocorrelation Functions 43
1.4.2 Spectral Representation and Periodogram 46
1.4.3 The Correlogram - Periodogram Duality 49
1.5 Time-Frequency Representations 50
1.5.1 The Concept of Time-Frequency 50
1.5.2 Instantaneous Frequency & Group Delay 53
1.5.3 Non-Stationary and Locally Stationary Processes 54
1.5.4 Wigner-Ville and Related Representations 54
1.5.5 The Ambiguity Function 56
1.5.6 Linear Time-Frequency Representations 58
1.5.7 Representing a Time-Frequency Transform 60
1.6 Examples and S-Commands 61
1.7 Notes & Complements 62
Chapter 2: Stationary Processes 65
2.1 Stationary Processes 65
2.2 Spectral Representations 72
2.3 Nonparametric Spectral Estimation 77
2.4 Spectral Estimation in Practice 81
2.5 Examples and S-Commands 88
2.5.1 White Noise Spectrum 88
2.5.2 Auto Regressive Models 89
2.5.3 Monthly CO2 Concentrations 90
2.6 Notes & Complements 94
Chapter 3: Gabor Transform 99
3.1 Definitions and First Properties 99
3.1.1 Basic Definitions 99
3.1.2 Redundancy and its Consequences 101
3.1.3 Invariance Properties 102
3.2 Commonly Used Windows 104
3.3 Examples 107
3.3.1 Academic Signals 107
3.3.2 Discussion of some ``Real Life'' Examples 113
3.4 Examples and S-Commands 120
3.4.1 Gabor Functions 122
3.4.2 CGT of Simple (Deterministic) Signals 122
3.4.3 ``Real Life'' Examples 124
3.5 Notes & Complements 125
Chapter 4: Wavelet Transform 127
4.1 Definitions and Basic Properties 127
4.1.1 Basic Definitions 127
4.1.2 Redundancy 131
4.1.3 Invariance 132
4.1.4 A Simple Reconstruction Formula 133
4.2 Continuous Multiresolutions 134
4.3 Commonly Used Analyzing Wavelets 135
4.3.1 Complex-valued Progressive Wavelets 135
4.3.2 Real-valued Wavelets 137
4.4 Wavelet Singularity Analysis 139
4.4.1 Hölder Regularity 139
4.4.2 Oscillating Singularities and Trigonometric Chirps 143
4.5 First Examples of Wavelet Analyses 144
4.5.1 Academic Examples 144
4.5.2 Examples of Time-Scale Analysis 148
4.5.3 Non-Academic Signals 150
4.6 Examples and S-Commands 155
4.6.1 Morlet wavelets 155
4.6.2 Wavelet Transforms 155
4.6.3 Real Signals 157
4.7 Notes & Complements 158
Chapter 5: Discrete Transforms, Algorithms 161
5.1 Frames 162
5.1.1 Gabor Frames 164
5.1.2 Critical Density: the Balian-Low Phenomenon 167
5.1.3 Wavelet Frames 168
5.2 Dyadic Wavelet Transform 171
5.2.1 Taking Large Scales into Account 173
5.2.2 The Discrete Dyadic Wavelet Transform 174
5.2.3 Local Extrema and Zero Crossings Representations 177
5.3 Matching Pursuit 178
5.3.1 The Regression Pursuit Method 180
5.3.2 Time-Frequency Atoms 181
5.4 Wavelet Orthonormal Bases 182
5.4.1 Multiresolution Analysis and Orthonormal Bases 183
5.4.2 Simple Examples 185
5.4.3 Computations of the Wavelet Coefficients 191
5.5 Playing with Time-Frequency Localization 195
5.5.1 Wavelet Packets 195
5.5.2 Local Trigonometric Bases 198
5.5.3 The ``Best Basis" Strategy 200
5.6 Algorithms, Implementation 202
5.6.1 Direct Quadratures 202
5.6.2 FFT-Based Algorithms 203
5.6.3 Filter Bank Approaches to the Wavelet Transform 205
5.6.4 Approximate Algorithms 209
5.7 Examples and S-Commands 212
5.7.1 Localization of the Wavelets and Gabor Functions 212
5.7.2 Dyadic Wavelet Transform and Local Extrema 213
5.8 Notes & Complements 214
Chapter 6: Stochastic Processes 221
6.1 Second Order Processes 221
6.1.1 Introduction 222
6.1.2 The Karhunen-Loeve Transform 223
6.1.3 Approximation of Processes on an Interval 226
6.1.4 Time-Varying Spectra 229
6.2 Time-Frequency Analysis of Stationary Processes 233
6.2.1 Gabor Analysis 233
6.2.2 Wavelet Analysis 240
6.2.3 Self-Similarity of WAN Traffic 244
6.3 First Steps Towards Non-Stationarity 246
6.3.1 Locally Stationary Processes 246
Least Square Optimization 251
Minimum bias optimization 252
Families of windows 252
6.3.2 Processes with Stationary Increments and Fractional Brownian Motion 254
6.4 Examples and S-Commands 262
6.5 Notes & Complements 268
Chapter 7: Frequency Modulated Signals 271
7.1 Asymptotic Signals 272
7.1.1 The Canonical Representation of a Real Signal 272
7.1.2 Asymptotic Signals and the Exponential Model 273
7.2 Ridge and Local Extrema 276
7.2.1 ``Differential" Methods 276
7.2.2 ``Integral'' Methods: Spaces of Ridges 279
7.2.3 Ridges as Graphs of Functions 281
7.2.4 Ridges as ``Snakes'' 283
7.2.5 Bayesian Interpretation 284
7.2.6 Variations on the Same Theme 285
7.3 Algorithms for Ridge Estimation 286
7.3.1 The ``Corona" Method 286
7.3.2 The ``ICM" Method 288
7.3.3 The Choice of the Parameters 288
7.3.4 Examples 289
7.4 The ``Crazy Climbers'' Algorithm 291
7.4.1 Crazy Climbers 293
7.4.2 Chaining 295
7.4.3 Examples 296
7.5 Reassignment Methods 298
7.5.1 Overview 298
The Crazy Climber Algorithm as Reassignment 299
7.5.2 The Synchrosqueezed Wavelet Transform 299
Synchrosqueezing 299
Chaining 301
An example 301
7.6 Examples and S-Commands 303
7.6.1 Transform Modulus Maxima 303
7.6.2 Ridge Detection Methods 303
7.6.3 The Multiridge Problem 306
7.6.4 Reassignment 307
7.7 Notes & Complements 307
Chapter 8: Statistical Reconstructions 309
8.1 Nonparametric Regression 309
8.2 Regression by Thresholding 311
8.2.1 A First Example 311
8.2.2 Thresholding Wavelet Coefficients 312
8.3 The Smoothing Spline Approach 314
8.4 Reconstruction from Extrema 318
8.5 Reconstruction from Ridge Skeletons 328
8.5.1 The Case of the Wavelet Transform 330
8.5.2 The Case of the Gabor Transform 337
8.5.3 Examples 342
8.6 Examples and S-Commands 346
8.6.1 Transient Detection from Dyadic Wavelet Transform Extrema 346
8.6.2 Reconstructions of Frequency Modulated Signals 348
8.7 Notes & Complements 352
Chapter 9: Downloading and Installing 357
9.1 Downloading Swave 357 9.2 Installing Swave on a Unix Platform 358 9.3 Troubleshooting 358
Chapter 10: The Swave S functions 361
Chapter 11: The Swave S utilities 409
