@dpiponiDan Piponi I was reading about this today:
https://en.wikipedia.org/wiki/Linear_canonical_transformation
This helps generalise some of the fractional Fourier transform ideas by associating them with a 2x2 matrix in time-frequency space.
What I find interesting here is that the Wigner function is trying to approximately expand in two (very different) bases at once, which is impossible but it's doing the best it can. Then these transformations are allowing manipulation of the direct product space of the two bases.