EAN: 9781461273820

Bilder-Quelle: discount24.de - Sport-Freizeit

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.

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