My first contact, ca. 1980 with the Communication Theory Group at TU Vienna was Franz Hlawatsch, who explained to me that he was working on the \textcolor{blue}{Wigner distribution}, following his advisor W. Mecklenbr\"auker. I told him about what is now called Feichtinger's algebra $S_0(R^d)$ and he pointed out to me that this was closely related to the claim of D. Gabor concerning the representation of signals as double sums using time-frequency shifted copies of the Gaussian. Already in 1989 H. Reiter published his work on the Metaplectic group using this Segal algebra. In the same year G. Folland's book on Phase Space Analysis appeared.


Meanwhile Gabor Analysis is a well-established field mathematical analysis and in particular the Banach Triple $(S_0,L_2,S_0*)$, consisting of Feichtinger's algebra, the Hilbert space $L_2$ an the dual space, the space of mild distributions (which appears to be the correct ambient vector space of all objects, treated as signals in
\newpage application areas) are established as key objects. Even classical Fourier analysis (DFT versus distributional Fourier transform) or the description of pseudo-differential operators (slowly varying systems in mobile communication) can be treated elegantly in this setting.

Chirp signals are important objects in many areas, including optics. Their Wigner transform is a Dirac measure concentrated along a line. In mathematics these functions of the form $\exp(2 \pi t^2)$ (an their dilated versions) are known as characters of second degree. Together with the Fourier transform and the usual dilation operators the (obviously unitary) pointwise multipliers by such chirps define a group of unitary operators, the so-called \textcolor{red}{metaplectic group} (the group of linear canonical transformations \newline in the engineering literature, or the special affine Fourier \newline transform, with the Fractional Fourier transform as special case).

Starting from the observation that these unitary operators extend naturally to Banach Gelfand Triple automorphism allows to put these operators into a natural context, going beyond the Hilbert space setting. Despite the fact that discrete Wigner transforms, discrete Hermite functions and in particular discrete version of the Fractional Fourier transform (a commutative subgroup of the metaplectic group) have been studied in many engineering papers the results available so far are usually based on heavy computations and do not reveal the structural properties. As time permits such aspects and preliminary answers to pending questions will be offered.