## Foundations of Time-Frequency AnalysisTime-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 |

335 | |

355 | |

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### Common terms and phrases

Balian-Low theorem Banach algebra Banach space basis for L2(Rd bounded operator boundedness C S(Rd Chapter coefficients compact support compute condition Consequently converges unconditionally convolution Corollary decay defined Definition dense density equivalent estimate finite Fourier series frame bounds frame for L2(Rd frequency function spaces Gabor expansions Gabor frame operator Gabor system Gaussian Heisenberg group Hilbert space implies inequality inversion formula irreducible kernel theorem Kohn-Nirenberg lattice mathematical matrix modulation spaces multiplication obtain operator norm orthonormal basis Plancherel's theorem pointwise polynomial Proof properties Proposition pseudodifferential operators quantum mechanics Riesz basis satisfies sequence Sg,g short-time Fourier transform signal analysis STFT subspace symplectic tempered distributions tight frame time-frequency analysis time-frequency plane time-frequency representations time-frequency shifts tion uncertainty principle unconditional convergence unitary operator v-moderate Vgf(x,u wavelet theory wavelet transform weight Weyl Wigner distribution Wilson bases window g Zak transform

### Popular passages

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.