Time-Frequency/Time-Scale Analysis: Flandrin: 9780122598708: Book: Technology: eCampus.com

This highly acclaimed work has so far been available only in French. It is a detailed survey of a variety of techniques for time-frequency/time-scale analysis (the essence of "Wavelet Analysis"). This book has broad and comprehensive coverage of a topic of keen interest to a variety of engineers, especially those concerned with signal and image processing. Flandrin provides a discussion of numerous issues and problems that arise from a mixed description in time and frequency, as well as problems in interpretation inherent in signal theory.

Key Features
* Detailed coverage of both linear and quadratic solutions
* Various techniques for both random and deterministic signals

Preface

Foreword

(8)

Chapter 1. The Time-Frequency Problem

(40)

1.1. The Time-Frequency Duality and Its Bars

(16)

1.1.1. Fourier Analysis

(1)

Limitations

(1)

Citations

(1)

1.1.2. Heisenberg-Gabor Uncertainty Principle

(2)

The time-frequency inequality

(2)

Interpretations

(2)

1.1.3. Slepian-Pollak-Landau Theory

(1)

Concentrations

(1)

Sampling

(2)

The eigenvalue equation

(2)

Approximation of bandlimited signals

(1)

Approximative dimension of a signal

(1)

Inequality of the concentrations

(2)

1.2. Leaving Fourier?

(16)

1.2.1. Local Quantities

(1)

Instantaneous frequency

(4)

Group delay

(1)

Interpretative remarks

(1)

An example

(4)

1.2.2. Nonstationary Signals

(1)

Definition

(2)

Generalizations

(2)

Variations

(2)

1.3. Towards Time-Frequency: Several Approaches

(4)

1.3.1. The Time-Frequency Plane and Its Three Readings

(1)

Frequency (time)

(1)

Time (frequency)

(1)

Time-frequency

(1)

1.3.2. Decompositions, Distributions, Models

(1)

Decompositions

(1)

Distributions

(1)

Models

(1)

1.3.3. Moving and Joint, Adaptive and Evolutionary Methods

(1)

Moving and joint

(1)

Adaptive and evolutionary

(1)

Chapter 1 Notes

(3)

Chapter 2. Classes of Solutions

(134)

2.1. An Introduction with Historical Landmarks

(16)

2.1.1. Short-Time Fourier and Instantaneous Spectrum

(1)

Sonagram and spectrogram

(1)

Restrictions

(2)

2.1.2. Atomic Decompositions

(1)

Gabor

(1)

Variations

(1)

Wavelets

(1)

2.1.3. Pseudo-Densities

(1)

Wigner-Ville

(2)

Page

(1)

Rihaczek

(2)

Extensions

(1)

Unification

(2)

2.1.4. The Parallel to Quantum Mechanics

(1)

Different concerns

(1)

Some intersections

(1)

2.2. Atomic Decompositions

(37)

2.2.1. Projections and Bases -- General Principles

(1)

Discrete bases

(1)

Continuous bases

(2)

Frames

(1)

2.2.2. Time-Frequency Examples

(1)

Short-time Fourier

(3)

Obstruction established by Balian-Low Theorem

(2)

Gabor and variants

(1)

2.2.3. Time-Scale Examples

(1)

Continuous wavelets

(4)

Discrete wavelets

(3)

Multiresolution analyses and orthonormal bases

(6)

Pyramidal algorithms

(3)

Some wavelet bases

(5)

2.2.4. A "Detection-Estimation" Viewpoint

(1)

Ambiguity functions

(4)

Atoms and matched filtering

101

(2)

2.3. The Energy Distributions

103

(47)

General setting

103

(1)

Covariance principles

104

(1)

2.3.1. Construction of the Bilinear Classes

104

(1)

Time-frequency

104

(3)

Time-scale

107

(2)

2.3.2. The Troika of Parameterizations-Definitions-Properties

109

(1)

Definitions

110

(4)

Constraints

114

(2)

Cohen's class

116

(16)

Affine class

132

(10)

2.3.3. Results of Exclusion and Conditional Uniqueness

142

(1)

Wigner's Theorem

143

(2)

Some results of exclusion

145

(2)

Some results on conditional uniqueness

147

(3)

2.4. The Power Distributions

150

(24)

2.4.1. From Deterministic to Random Signals

150

(1)

Decompositions and fluctuations

150

(1)

Distributions and expectation values

151

(1)

Cramer and beyond

152

(1)

2.4.2. The Orthogonal (or Almost Orthogonal) Solutions

153

(1)

Karhunen decompositions

153

(1)

Priestley spectrum

154

(2)

Tjostheim, Melard, Grenier approach

156

(4)

2.4.3. The Frequency Solutions

160

(1)

Harmonizable signals

160

(5)

Wigner-Ville spectrum

165

(5)

2.4.4. Some Links Between the Different Spectra

170

(1)

Continuous time

170

(1)

Discrete time

171

(3)

Chapter 2 Notes

174

(9)

Chapter 3. Issues of Interpretation

183

(126)

3.1. About the Bilinear Classes

185

(28)

3.1.1. The Different Parameterizations

185

(2)

Time-frequency

187

(1)

Time-time

188

(3)

Frequency-frequency

191

(1)

Frequency-time

191

(3)

3.1.2. Parameterizations, Operators and Correspondence Rules

194

(1)

Why operators?

194

(1)

The operator of time-frequency shifts

195

(2)

Correspondence rules

197

(2)

Kernels

199

(1)

Weyl calculus

200

(1)

Moments

201

(4)

Dilations and ambiguities

205

(3)

3.1.3. Time-Frequency or Time-Scale?

208

(1)

Fourier scale

208

(2)

Mellin scale

210

(3)

Analysis and decision statistics

213

(1)

3.2. The Wigner-Ville Distribution and Its Geometry

213

(76)

3.2.1. Wigner-Ville versus Spectrogram

213

(1)

Structure of the distributions

213

(2)

Pseudo-Wigner-Ville

215

(1)

Supports

216

(1)

Localization to chirps

217

(4)

Spectrogram and reassignment

221

(4)

Discretization

225

(1)

3.2.2. The Mechanism of Interferences

226

(2)

Construction principle

228

(3)

A different perspective from the ambiguity plane

231

(1)

Inner and outer interferences

232

(2)

Approximation by the method of stationary phase

234

(3)

Singularities and catastrophes

237

(6)

Interferences, localization, and symmetries

243

(2)

Generalization to the s-Wigner distribution

245

(2)

Generalization to the affine distributions

247

(5)

3.2.3. Reduction of the Interferences

252

(1)

Analytic signal

252

(1)

Wigner-Ville and atomic decompositions

252

(2)

Smoothing

254

(1)

Coupled smoothing

255

(1)

Separable smoothing

256

(5)

Joint smoothing

261

(8)

Variable and/or adapted smoothing

269

(5)

"Image" approaches

274

(1)

3.2.4. Usefulness of the Interferences

274

(1)

Unitarity

274

(1)

Phase information

275

(2)

What is a component?

277

(2)

3.2.5. Statistical Estimation of the Wigner-Ville Spectrum

279

(1)

Assumptions

280

(1)

Classes of estimators

281

(1)

Bias

282

(1)

Variance

283

(1)

Examples

284

(5)

3.3. About the Positivity

289

(12)

3.3.1. Some Problems Caused by the Nonpositivity

289

(4)

3.3.2. Positivity by the Signal

293

(1)

An example

293

(1)

Hudson's theorem

294

(1)

Random signals and positive spectra

295

(1)

3.3.3. Positivity by the Distribution

296

(1)

Positive distributions

296

(1)

Positive smoothing

297

(3)

A stochastic interpretation

300

(1)

Chapter 3 Notes

301

(8)

Chapter 4. Time-Frequency as a Paradigm

309

(50)

4.1. Localization

311

(16)

4.1.1. Heisenberg-Gabor Revisited

311

(1)

Example 1

311

(2)

Example 2

313

(2)

4.1.2. Energy Concentration

315

(1)

Problem formulation

315

(1)

The general eigenvalue equation

316

(1)

Restriction to ellipsoidal domains

316

(7)

Interpretations and conjecture

323

(1)

4.1.3. Other Time-Frequency Inequalities

323

(1)

L(p)-norms

324

(1)

Localization and stationary phase

325

(2)

4.2. Signal Analysis

327

(15)

4.2.1. Time-Frequency, Time-Scale, and Spectral Analysis

327

(1)

Paving and marginal distributions

328

(1)

The example of "1/f-noise"

329

(1)

Analysis of self-similar processes

330

(4)

4.2.2. Nonstationary Characteristics

334

(1)

Distance from the stationary case

334

(2)

Demodulation

336

(2)

Local singularities

338

(2)

Evolutionary singularities

340

(2)

4.3. Decision Statistics

342

(12)

4.3.1. Matched Time-Frequency Filtering

343

(2)

4.3.2. Maximum Likelihood Estimators for Gaussian Processes

345

(1)

Classical solution

345

(1)

Time-frequency formulation

346

(1)

4.3.3. Some Examples

347

(1)

Rayleigh channel

347

(1)

Detection of chirps and Doppler tolerance

347

(2)

Locally optimal detection

349

(1)

Time-frequency jitter

350

(1)

A broader class of time-frequency receptors

351

(3)

Chapter 4 Notes

354

(5)

Bibliography

359

(22)

Index

381


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