We do not address discretization issues in this paper. However, it is worth noting, that in practice, we generally wish to compute the chirplet transform of a discrete-time signal, and it is sometimes the case that the mother chirplet is also discrete-time and has no closed-form mathematical description. Thus, dilation would require resampling, and contraction would require antialiasing. In this case, the largest subspace we might obtain would be the subspace that omits both dilation and tiling-size, leaving us with the four-dimensional parameter space:
S_t_c,f_c,0,c,d \!=\!
C_t_c,f_c,0,c,d \: g \: \: s \!