Gabor Analysis and Algorithms: Theory and Applications / Edition 1

by Hans G. Feichtinger, Thomas Strohmer

Overview

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 coefficients 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.

Product Details

ISBN-13:	9780817639594
Publisher:	Birkhäuser Boston
Publication date:	12/18/1997
Series:	Applied and Numerical Harmonic Analysis
Edition description:	1998
Pages:	496
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

1 The duality condition for Weyl-Heisenberg frames.- 2 Gabor systems and the Balian-Low Theorem.- 3 A Banach space of test functions for Gabor analysis.- 4 Pseudodifferential operators, Gabor frames, and local trigonometric bases.- 5 Perturbation of frames and applications to Gabor frames.- 6 Aspects of Gabor analysis on locally compact abelian groups.- 7 Quantization of TF lattice-invariant operators on elementary LCA groups.- 8 Numerical algorithms for discrete Gabor expansions.- 9 Oversampled modulated filter banks.- 10 Adaptation of Weyl-Heisenberg frames to underspread environments.- 11 Gabor representation and signal detection.- 12 Multi-window Gabor schemes in signal and image representations.- 13 Gabor kernels for affine-invariant object recognition.- 14 Gabor’s signal expansion in optics.

Editorial Reviews

Surveys the subject of Gabor analysis, showing the connections and interactions between theory, numerical algorithms, and applications. The first part covers the mathematical foundations of Gabor analysis, including Weyl-Heisenberg frames, duality conditions, the uncertainty principle, and the Balian-Low Theorem, as well as a group theoretical approach. The second part presents numerical algorithms and selected applications in signal and image processing. Annotation c. by Book News, Inc., Portland, Or.

Booknews

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

From the Publisher