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Birkhäuser

Gabor Analysis and Algorithms

Theory and Applications

  • Book
  • © 1998

Overview

Editors:
  1. Hans G. Feichtinger

    1. Department of Mathematics, University of Vienna, Vienna, Austria

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    You can also search for this editor in PubMed   Google Scholar

  2. Thomas Strohmer

    1. Department of Mathematics, University of Vienna, Vienna, Austria

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Applied and Numerical Harmonic Analysis (ANHA)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction

    • Hans G. Feichtinger, Thomas Strohmer
    Pages 1-31

  3. The duality condition for Weyl-Heisenberg frames

    • A. J. E. M. Janssen
    Pages 33-84

  4. Gabor systems and the Balian-Low Theorem

    • John J. Benedetto, Christopher Heil, David F. Walnut
    Pages 85-122

  5. A Banach space of test functions for Gabor analysis

    • Hans G. Feichtinger, Georg Zimmermann
    Pages 123-170

  6. Pseudodifferential operators, Gabor frames, and local trigonometric bases

    • Richard Rochberg, Kazuya Tachizawa
    Pages 171-192

  7. Perturbation of frames and applications to Gabor frames

    • Ole Christensen
    Pages 193-209

  8. Aspects of Gabor analysis on locally compact abelian groups

    • Karlheinz Gröchenig
    Pages 211-231

  9. Quantization of TF lattice-invariant operators on elementary LCA groups

    • Hans G. Feichtinger, Werner Kozek
    Pages 233-266

  10. Numerical algorithms for discrete Gabor expansions

    • Thomas Strohmer
    Pages 267-294

  11. Oversampled modulated filter banks

    • Helmut Bölcskei, Franz Hlawatsch
    Pages 295-322

  12. Adaptation of Weyl-Heisenberg frames to underspread environments

    • Werner Kozek
    Pages 323-352

  13. Gabor representation and signal detection

    • Ariela Zeira, Benjamin Friedlander
    Pages 353-380

  14. Multi-window Gabor schemes in signal and image representations

    • Yehoshua Y. Zeevi, Meir Zibulski, Moshe Porat
    Pages 381-407

  15. Gabor kernels for affine—invariant object recognition

    • Jezekiel Ben-Arie, Zhiqian Wang
    Pages 409-426

  16. Gabor’s signal expansion in optics

    • Martin J. Bastiaans
    Pages 427-451

  17. Back Matter

    Pages 453-496
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Keywords

About this book

In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency­ shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi­ cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri­ ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta­ bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.

Reviews

"... a definitive survey of the subject shows connections and interactions between theory, numerical algorithms, and applications." —Mathematical Reviews

Editors and Affiliations

  • Department of Mathematics, University of Vienna, Vienna, Austria

    Hans G. Feichtinger, Thomas Strohmer

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