Table 3 Basic properties of new WVDL
From: On a new Wigner-Ville distribution associated with linear canonical transform
Property | Formula |
|---|---|
Conjugation symmetry | \({WVDL}_{x_{1}}^{A}(t,u)=\left [ {WVDL}_{x_{1}}^{A}(t,u)\right ]^{\ast }\), |
\({WVDL}_{x_{1}, x_{2}}^{A}(t,u)=[ {WVDL}_{x_{1}, x_{2}}^{A}(t,u)]^{\ast }\) | Â |
Time shift | \({WVDL}_{\mathbf {T}_{s}x}^{A}(t,u)={WVDL}_{x}^{A}(t-s, u-as)\) |
Modulation | \({WVDL}_{\mathbf {M}_{u_{0}}x}^{A}(t,u)=\frac {1}{2\pi }{WVDL}_{x}^{A}(t, u-{bu}_{0})\) |
Marginal | \(\int _{\mathbb {R}}{WVDL}_{x}^{A}(t,u)du=|x(t)|^{2}\) |
Power | \(\int _{\mathbb {R}} \int _{\mathbb {R}}{WVDL}_{x}^{A}(t,u)dudt=\int _{\mathbb {R}}|x(t)|^{2} dt\) |
Moyal’s formula | \(\int _{\mathbb {R}}\int _{\mathbb {R}}{WVDL}_{x_{1}}^{A}(t,u)[{WVDL}_{x_{2}}^{A}(t,u)]^{\ast }dudt\)\(=\frac {1}{2 \pi |b|}|\langle x_{1}(t), x_{2}(t)\rangle |^{2}\) |