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Wiley - Time-frequency analysis concepts and methods

'Wiley - Time-frequency analysis concepts and methods'
This page intentionally left blank Time-Frequency Analysis This page intentionally left blank Time-Frequency Analysis Concepts and Methods Edited by Franz Hlawatsch François Auger First published in France in 2005 by Hermes Science/Lavoisier entitled “Temps-fréquence: concepts et outils” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley Weyl-Heisenberg and Wilson frames . . . . . . . . .71 3.3.1. Sampling of the short-time Fourier transform . . . . . . . . . . .71 3.3.2. Weyl-Heisenberg frames . . . . . . . . . . . . . . . . . . . . . . .72 3.3.3. Zak transform and “critical” Weyl-Heisenberg frames . . . . . .74 3.3.4. Balian-Low theorem . . . . . . . . . . . . . . . . . . . . . . . . .75 3.3.5. Wilson bases and frames, local cosine bases . . . . . . . . . . . .75 3.4. Dictionaries of time-frequency atoms; adaptive representations . . . .77 3.4.1. Multi-scale dictionaries of time-frequency atoms . . . . . . . . .77 3.4.2. Pursuit algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .78 3.4.3. Time-frequencyrepresentation. . . . . . . . . . . . . . . . . . .79 3.5. Applications to audio signals . . . . . . . . . . . . . . . . . . . . . . . .80 3.5.1. Analysis of superimposed structures . . . . . . . . . . . . . . . .80 3.5.2. Analysis of instantaneous frequency variations . . . . . . . . . .80 3.5.3. Transposition of an audio signal . . . . . . . . . . . . . . . . . . .82 3.6. Discrete algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 3.6.1. Fast Fourier transform . . . . . . . . . . . . . . . . . . . . . . . .83 3.6.2. Filter banks: fast convolution . . . . . . . . . . . . . . . . . . . .83 3.6.3. Discrete short-time Fourier transform . . . . . . . . . . . . . . . .85 3.6.4. Discrete Gabor transform. . . . . . . . . . . . . . . . . . . . . .86 3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 Contents7 3.8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 3.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 Chapter 4. Linear Time-Frequency Analysis II: Wavelet-Type Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 Thierry BLUand Jérôme LEBRUN 4.1. Introduction: scale and frequency . . . . . . . . . . . . . . . . . . . . .94 4.2. Continuous wavelet transform . . . . . . . . . . . . . . . . . . . . . . .95 4.2.1. Analysis and synthesis . . . . . . . . . . . . . . . . . . . . . . . .95 4.2.2. Multiscale properties . . . . . . . . . . . . . . . . . . . . . . . . .97 4.3. Discrete wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . .98 4.3.1. Multi-resolution analysis . . . . . . . . . . . . . . . . . . . . . . .98 4.3.2. Mallat algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . .104 4.3.3. Graphical representation . . . . . . . . . . . . . . . . . . . . . . .106 4.4. Filter banks and wavelets . . . . . . . . . . . . . . . . . . . . . . .
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