Wigner distribution associated with the symplectic coordinates transformation

The celebrated -Wigner distribution (-WD), a generalization of the ordinary Wigner distribution (WD), has attracted much attention in the literature for its wide use in time-frequency analysis. However, it fails to break through the time-frequency resolution limit of the WD. Due to more degrees of freedom, the matrix Wigner distribution (MWD), a generalization of the -WD and a special case of the -Wigner distribution, seems to be an effective tool to tackle this challenge. In this study, we focus on a specific MWD, i.e., the so-called symplectic Wigner distribution (SWD), through the change of coordinates with the general symplectic matrix. We disclose an equivalence relation between the uncertainty product in SWD domains and that in the classical settings, from which we obtain the optimal symplectic matrix constraint condition under which the SWD achieves higher time-frequency resolution than the WD. We also provide the simulation example associated with linear frequency-modulated signals for verifying the established superresolution theory.