Table 2 Existing WVD associated with LCT
From: On a new Wigner-Ville distribution associated with linear canonical transform
Formula | Literature |
|---|---|
\( W_{A}(t,u)=\int _{\mathbb {R}} X_{A}\left (u+\frac {\tau }{2}\right) X_{A}^{\ast }\left (u-\frac {\tau }{2}\right)e^{-jv\tau }d\tau \) | [14] |
\(W_{A}^{x}(t,u)=\int _{\mathbb {R}} x\left (t+\frac {\tau }{2}\right) x^{\ast }\left (t-\frac {\tau }{2}\right)K_{A}(u,\tau)d\tau \) | [15] |
\(W_{A}(t,u)=\int _{\mathbb {R}} X_{A}\left (\frac {u+u^{\prime }}{2}\right) X_{A}^{\ast }\left (\frac {u+u^{\prime }}{2}\right)e^{j2\pi u^{\prime }t}du^{\prime }\) | [16] |
\( W_{x}^{A_{1},A_{2},A_{3}}(t,u)=\frac {1}{\sqrt {j2\pi b_{3}}}\int _{\mathbb {R}} X_{A_{1}}\left (t+\frac {\tau }{2}\right) X_{A_{2}}^{\ast }\left (t-\frac {\tau }{2}\right)K_{A_{3}}(u,\tau)d\tau \) | Â |
where A1,A2,A3 are different parameter matrices of LCT. | [18] |
\( W_{x}^{A_{1},A_{0}}(t,u)=\frac {1}{\sqrt {j2\pi b_{0}}}\int _{\mathbb {R}} X_{A}\left (t+\frac {\tau }{2}\right) x^{\ast }\left (t-\frac {\tau }{2}\right)K_{A_{0}}(u,\tau)d\tau \) | Â |
where A0=[a0 b0, c0, d0] is the parameter matrix of LCT. | [19] |
\( {LWD}_{A}^{x}(t,u)=2\int _{\mathbb {R}} X_{A}\left (\omega +bu\right) X_{\overline {A}}^{\ast }\left (\omega -bu\right)e^{-jbdu^{2}}K_{A}^{\ast }(\omega,2t)d\omega \) | Â |
where \(\overline {A}=[a\ \ -b;\ \ -c \ \ d]\). | [20] |
\( {WDOL}_{x}(t,u)=\int _{\mathbb {R}} x\left (t+\frac {\tau }{2}\right) x^{\ast }\left (t-\frac {\tau }{2}\right)h_{A}(u,t)d\tau \), | Â |
where \(h_{A}=K_{A}(u,\tau)e^{\frac {j}{2b}[at^{2}+2t(u_{0}-u)\-2u({du}_{0}-{bw}_{0})+du^{2}]}\). | [22] |