Gabor Analysis and Algorithms : Theory and Applications Hardback
Edited by Hans G. Feichtinger, Thomas Strohmer
Part of the Applied and Numerical Harmonic Analysis series
Hardback
Description
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.
Information
-
Item not Available
- Format:Hardback
- Pages:496 pages, XVI, 496 p.
- Publisher:Birkhauser Boston Inc
- Publication Date:18/12/1997
- Category:
- ISBN:9780817639594
Information
-
Item not Available
- Format:Hardback
- Pages:496 pages, XVI, 496 p.
- Publisher:Birkhauser Boston Inc
- Publication Date:18/12/1997
- Category:
- ISBN:9780817639594
