Foundations of Time-Frequency Analysis

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Springer Science & Business Media, 2001 - Business & Economics - 359 pages
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Time-frequency analysis is a modern branch of harmonic analysis. It com prises all those parts of mathematics and its applications that use the struc ture of translations and modulations (or time-frequency shifts) for the anal ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.
 

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Contents

Basic Fourier Analysis
3
11 Definition of the Fourier Transform
4
12 The Fundamental Operations
6
13 Fourier Series
12
14 The Poisson Summation Formula
14
15 Gaussians and Plancherels Theorem
16
TimeFrequency Analysis and the Uncertainty Principle
21
22 Uncertainty Principles
26
82 Properties of the Zak Transform
148
83 Gabor Frames and the Zak Transform
156
84 The BalianLow Theorem
162
85 Wilson Bases
167
The Heisenberg Group A Different Point of View
175
92 Representation Theory
181
93 The Stonevon Neumann Theorem
189
94 The Metaplectic Representation and Gabor Frames on General TimeFrequency Lattices
195

23 The Uncertainty Principle of Donoho and Stark
30
24 Quantum Mechanics and the Uncertainty Principle
33
The ShortTime Fourier Transform
37
32 Orthogonality Relations and Inversion Formula
42
33 Liebs Uncertainty Principle
49
34 The Bargmann Transform
53
Quadratic TimeFrequency Representations
59
41 The Spectrogram
60
42 The Ambiguity Function
61
43 The Wigner Distribution
63
44 Positivity of the Wigner Distribution
69
45 Cohens Class
79
Discrete TimeFrequency Representations Gabor Frames
83
51 Frame Theory
85
52 Gabor Frames
93
53 Unconditional Convergence
96
Existence of Gabor Frames
103
62 Boundedness of the Gabor Frame Operator
105
63 Walnuts Representation of the Gabor Frame Operator
111
64 Painless NonOrthogonal Expansions
118
65 Existence of Gabor Frames
120
The Structure of Gabor Systems
127
72 Janssens Representation
130
73 The WexlerRaz Biorthogonality Relations
133
74 The RonShen Duality Principle
135
75 Density of Gabor Frames
138
76 The Variety of Dual Windows
142
Zak Transform Methods
147
Wavelet Transforms
203
Modulation Spaces
215
111 Weights and MixedNorm Spaces
216
112 TimeFrequency Analysis of Distributions
225
113 Theory of Function Spaces
230
114 Generalizations and Variations
239
Gabor Analysis of Modulation Spaces
245
122 Boundedness of Gabor Frame Operators on Modulation Spaces
256
123 Wilson Bases in Modulation Spaces
264
124 Data Compression
272
Window Design and Wieners Lemma
277
132 The Rational Case
279
133 Proof of the Main Theorem
281
134 Operator Algebras
288
135 The Irrational Case
293
136 Banach Frames
297
Pseudodifferential Operators
301
141 Partial Differential Equations
302
142 TimeVarying Systems
305
143 Quantization and the Weyl Calculus
307
144 Kernel Theorems
314
145 Boundedness of Pseudodifferential Operators
317
146 Miscellaneous
324
Appendix
329
References
335
Index
355
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Page 345 - F. Hlawatsch and GF Boudreaux-Bartels, "Linear and quadratic time-frequency signal representations," IEEE Signal Processing Magazine, pp.
Page 353 - YY Zeevi, M. Zibulski. and M. Porat, Multi-window Gabor schemes in signal and image representations, in Gabor Analysis and Algorithms: Theory and Applications, HG Feichtinger and T.
Page 342 - HG Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis, in Gabor Analysis and Algorithms: Theory and Applications.
Page 341 - H. and K. Grochenig, A unified approach to atomic decompositions via integrable group representations.
Page 351 - R. Tolimieri and R. Orr, Poisson summation, the ambiguity function, and the theory of Weyl-Heisenberg frames, J.
Page 346 - S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Rev.
Page 348 - A characterization of the minimal strongly character invariant Segal algebra.
Page 343 - K. Grochenig and C. Heil Modulation spaces and pseudo-differential operators, Integral Equations Operator Theory 34 (1999), 439-457.
Page 341 - Grochenig, Banach spaces related to integrable group representations and their atomic decompositions II, Monatsh. Math., 108(2-3):129-148, 1989. [FG97] HG Feichtinger and K. Grochenig, Gabor frames and time-frequency analysis of distributions, J.

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