We can further extend the multidimensional parameter space. Suppose we also allow the chirprate, c, in (3) to be one of the coordinates of the parameter space. The resulting transform is given by:
S_t_c,f_c,(_t),c =
C_t_c,f_c,_t,c g(t) \: \: s(t)
We refer to (10) as the ``Gaussian chirplet transform'' (GCT).
One characteristic of the one-dimensional Gaussian window is that its TF energy distribution is a bi-variate Gaussian function. Therefore its TF energy countours are elliptical, so shearing the TF distribution along the time axis provides no new degrees of freedom that cannot be obtained by combinations of shearing along the frequency axis together with dilation. If we consider other windows, however, we do not, in general, have this degenerate property.