Explorations in Time-Frequency Analysis

9.21 MB·English

by  Patrick Flandrin

Half-title Page: i
Title page Page: iii
Copyright information Page: iv
Dedication Page: v
Table of contents Page: vii
Acknowledgments Page: x
Preface Page: xiii
Notation Page: xv
1 Introduction Page: 1
Part I Basics and Constraints Page: 7
2 Small Data Are Beautiful Page: 9
2.1 Gravitational Waves Page: 9
2.2 Bats Page: 11
2.3 Riemann-Like Special Functions Page: 14
2.4 Chirps (Everywhere) Page: 16
3 Of Signals and Noise Page: 21
3.1 Order versus Disorder Page: 21
3.2 Signals Page: 22
3.3 Noise Page: 24
4 On Time, Frequency, and Gauss Page: 29
4.1 Gauss Page: 29
4.2 From Gauss to Fourier Page: 31
4.3 From Gauss to Shannon-Nyquist Page: 31
4.4 From Gauss to Gabor Page: 32
5 Uncertainty Page: 35
5.1 Variance Page: 35
5.2 Entropy Page: 38
5.3 Ubiquity and Interpretation Page: 39
6 From Time and Frequency to Time-Frequency Page: 40
6.1 Correlation and Ambiguity Page: 40
6.2 Distribution and Wigner Page: 43
6.3 Spectrograms, Cohen, and the Like Page: 46
7 Uncertainty Revisited Page: 50
7.1 L[sub(2)]-Norm Page: 50
7.2 L[sub(p)]-Norms and Entropy Page: 51
7.3 Concentration and Support Page: 51
7.4 Variance Page: 53
7.5 Uncertainty and Time-Frequency Localization Page: 54
8 On Stationarity Page: 56
8.1 Relative Stationarity Page: 57
8.2 Testing Stationarity Page: 60
Part II Geometry and Statistics Page: 67
9 Spectrogram Geometry 1 Page: 69
9.1 One Logon Page: 69
9.2 Two Logons Page: 70
9.3 Many Logons and Voronoi Page: 73
10 Sharpening Spectrograms Page: 77
10.1 Reassignment Page: 78
10.2 Multitaper Reassignment Page: 83
10.3 Synchrosqueezing Page: 88
10.4 Sparsity Page: 90
10.5 Wedding Sharpening and Reconstruction Page: 96
11 A Digression on the Hilbert–Huang Transform Page: 98
11.1 Empirical Mode Decomposition Page: 98
11.2 Huang's Algorithm Page: 100
11.3 The Hilbert–Huang Transform Page: 100
11.4 Pros, Cons, and Variations Page: 101
12 Spectrogram Geometry 2 Page: 106
12.1 Spectrogram, STFT, and Bargmann Page: 106
12.2 Reassignment Variations Page: 107
12.3 Attractors, Basins, Repellers, and Contours Page: 111
13 The Noise Case Page: 116
13.1 Time-Frequency Patches Page: 116
13.2 Correlation Structure Page: 118
13.3 Logon Packing Page: 121
14 More on Maxima Page: 124
14.1 A Randomized Lattice Model Page: 124
14.2 Ordinates and Maxima Distributions Page: 129
14.3 Voronoi Page: 134
15 More on Zeros Page: 139
15.1 Factorizations Page: 139
15.2 Density Page: 143
15.3 Pair Correlation Function Page: 144
15.4 Voronoi Page: 145
15.5 Delaunay Page: 149
15.6 Signal Extraction from “Silent” Points Page: 153
15.7 Universality Page: 161
15.8 Singularities and Phase Dislocations Page: 164
16 Back to Examples Page: 168
16.1 Gravitational Waves Page: 168
16.2 Bats Page: 175
16.3 Riemann-Like Special Functions Page: 188
17 Conclusion Page: 197
18 Annex: Software Tools Page: 199
References Page: 201
Index Page: 210