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Gabor Analysis and Algorithms

Theory and Applications

  • Hans G. Feichtinger
  • Thomas Strohmer

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

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Hans G. Feichtinger, Thomas Strohmer
    Pages 1-31
  3. A. J. E. M. Janssen
    Pages 33-84
  4. John J. Benedetto, Christopher Heil, David F. Walnut
    Pages 85-122
  5. Hans G. Feichtinger, Georg Zimmermann
    Pages 123-170
  6. Richard Rochberg, Kazuya Tachizawa
    Pages 171-192
  7. Hans G. Feichtinger, Werner Kozek
    Pages 233-266
  8. Helmut Bölcskei, Franz Hlawatsch
    Pages 295-322
  9. Ariela Zeira, Benjamin Friedlander
    Pages 353-380
  10. Yehoshua Y. Zeevi, Meir Zibulski, Moshe Porat
    Pages 381-407
  11. Jezekiel Ben-Arie, Zhiqian Wang
    Pages 409-426
  12. Martin J. Bastiaans
    Pages 427-451
  13. Back Matter
    Pages 453-496

About this book

Introduction

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.

Keywords

Analysis Signal algorithm algorithms calculus detection harmonic analysis image processing numerical analysis object recognition operator pattern recognition

Editors and affiliations

  • Hans G. Feichtinger
    • 1
  • Thomas Strohmer
    • 1
  1. 1.Department of MathematicsUniversity of ViennaViennaAustria

Bibliographic information