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.
