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Tables of Contents for Time-Frequency/Time-Scale Analysis
Chapter/Section Title
Page #
Page Count
Preface
xi
Foreword
1
8
Chapter 1. The Time-Frequency Problem
9
40
1.1. The Time-Frequency Duality and Its Bars
10
16
1.1.1. Fourier Analysis
10
1
Limitations
10
1
Citations
11
1
1.1.2. Heisenberg-Gabor Uncertainty Principle
12
2
The time-frequency inequality
14
2
Interpretations
16
2
1.1.3. Slepian-Pollak-Landau Theory
18
1
Concentrations
18
1
Sampling
19
2
The eigenvalue equation
21
2
Approximation of bandlimited signals
23
1
Approximative dimension of a signal
24
1
Inequality of the concentrations
24
2
1.2. Leaving Fourier?
26
16
1.2.1. Local Quantities
26
1
Instantaneous frequency
27
4
Group delay
31
1
Interpretative remarks
31
1
An example
32
4
1.2.2. Nonstationary Signals
36
1
Definition
36
2
Generalizations
38
2
Variations
40
2
1.3. Towards Time-Frequency: Several Approaches
42
4
1.3.1. The Time-Frequency Plane and Its Three Readings
43
1
Frequency (time)
43
1
Time (frequency)
43
1
Time-frequency
43
1
1.3.2. Decompositions, Distributions, Models
43
1
Decompositions
44
1
Distributions
44
1
Models
44
1
1.3.3. Moving and Joint, Adaptive and Evolutionary Methods
45
1
Moving and joint
45
1
Adaptive and evolutionary
45
1
Chapter 1 Notes
46
3
Chapter 2. Classes of Solutions
49
134
2.1. An Introduction with Historical Landmarks
50
16
2.1.1. Short-Time Fourier and Instantaneous Spectrum
50
1
Sonagram and spectrogram
51
1
Restrictions
52
2
2.1.2. Atomic Decompositions
54
1
Gabor
54
1
Variations
55
1
Wavelets
56
1
2.1.3. Pseudo-Densities
57
1
Wigner-Ville
58
2
Page
60
1
Rihaczek
60
2
Extensions
62
1
Unification
62
2
2.1.4. The Parallel to Quantum Mechanics
64
1
Different concerns
64
1
Some intersections
65
1
2.2. Atomic Decompositions
66
37
2.2.1. Projections and Bases -- General Principles
66
1
Discrete bases
66
1
Continuous bases
67
2
Frames
69
1
2.2.2. Time-Frequency Examples
70
1
Short-time Fourier
70
3
Obstruction established by Balian-Low Theorem
73
2
Gabor and variants
75
1
2.2.3. Time-Scale Examples
76
1
Continuous wavelets
76
4
Discrete wavelets
80
3
Multiresolution analyses and orthonormal bases
83
6
Pyramidal algorithms
89
3
Some wavelet bases
92
5
2.2.4. A "Detection-Estimation" Viewpoint
97
1
Ambiguity functions
97
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