It is well known, that the effect of the Fourier transform on a
spectrogram (the absolute value squared of a STFT) is just
rotation by 90 degrees. Hence one may ask whether there are
operators which correspond to a rotation by an arbitrary angle.
This is in fact possible and well known for decades, in many
different appearances, under the name of a Fractional Fourier
transform. There are many different approaches, starting in
different communities, which have lead to studies in the last 50
years (engineers, physicists, mathematicians). The most powerful
(to our mind) is the view-point of Andre Weil, who introduced the
metaplectic group (containing the group of Fractional FTs) in
his famous paper in Acta Mathematica ([7])