Symplectic Wigner Distribution in the Linear Canonical Transform Domain: Theory and Application

This paper devotes to combine the chirp basis function transform and symplectic coordinates transform to yield a novel Wigner distribution (WD) associated with the linear canonical transform (LCT), named as the symplectic WD in the LCT domain (SWDL). It incorporates the merits of the symplectic WD (SWD) and the WD in the LCT domain (WDL), achieving stronger capability in the non-stationary signal time-frequency feature extraction while maintaining the same level of computational complexity. Some essential properties of the SWDL are derived, including marginal distributions, energy conservations, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time translation property, frequency modulation property, and time translation and frequency modulation property. Heisenberg’s uncertainty principles of the SWDL are formulated, giving rise to three kinds of lower bounds attainable respectively by Gaussian enveloped complex exponential (GECE) signal, Gaussian (G) signal and Gaussian enveloped chirp (GEC) signal. The optimal symplectic matrices corresponding to the highest time-frequency resolution are generated by solving these attainable lower bounds optimization (minimization) problems. The time-frequency resolution of the SWDL is compared with those of the SWD and WDL to demonstrate its superiority in GECE, G and GEC signals time-frequency energy concentration. Three synthesis examples are carried out to verify the feasibility and reliability of the theoretical analysis. Practical radar scenarios are also performed to illustrate that the SWDL outperforms some existing methods in linear frequency-modulated signals frequency rate feature extraction.