Gabor Analysis and Algorithms : Theory and Applications

Thomas Strohmer

ISBN 10: 0817639594 / ISBN 13: 9780817639594

Editore: Birkhäuser Boston, Birkhäuser Boston, 1997

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Druck auf Anfrage Neuware - Printed after ordering - 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.

Codice articolo 9780817639594

Dati bibliografici

• Titolo: Gabor Analysis and Algorithms : Theory and Applications
• Editore: Birkhäuser Boston, Birkhäuser Boston
• Data di pubblicazione: 1997
• Lingua: Inglese
• ISBN 10: 0817639594
• ISBN 13: 9780817639594
• Rilegatura: Buch
• Condizione: Neu
• Dimensioni: 241 L x 160 A x 34 P (millimetri) 241 millimetri di larghezza per 160 millimetri di altezza per 34 millimetri di profondità
• Peso: 934 grammi

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Riassunto

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.

Recensione

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