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Book synopsis

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.

Book details

Book category Books in English Mathematics & science Mathematics Calculus & mathematical analysis

• Full title: Gabor Analysis and Algorithms
• Subtitle: Theory and Applications
• Author: Hans G. Feichtinger, Thomas Strohmer
• Language: English
• Binding: Paperback
• Number of pages: 496
• EAN: 9781461273820
• ISBN: 146127382X
• ID: 06795211
• Publisher: Springer-Verlag New York Inc.
• Weight: 783 g
• Dimensions: 235 × 155 × 29 mm
• Date of publishing: 11. October 2012

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