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Table 4 Other useful properties of new WVDL

From: On a new Wigner-Ville distribution associated with linear canonical transform

Property

Formula

Nonlinearity

\({WVDL}_{x}^{A}(t,u)\neq {WVDL}_{x_{1}}^{A}(t,u)+{WVDL}_{x_{2}}^{A}(t,u)\)

Linear canonical time shift

\( {WVDL}_{\mathbf {T}_{s}^{A}x}^{A}(t,u) ={WVDL}_{x}^{A}(t-s, u)\)

Linear canonical modulation

\({WVDL}_{\mathbf {M}_{u_{0}}^{A}x}^{A}(t,u)={WVDL}_{x}^{A}(t, u-u_{0})\)

Dilation

\({WVDL}_{\mathbf {D}_{t_{0}}x}^{A}(t,u)=\frac {1}{t_{0}}{WVDL}_{x}^{A_{1}}\left (\frac {t}{t_{0}}, t_{0}u\right)\) where \(A_{1}=\left [ t_{0}^{2}a\ \ b; \ \frac {c}{t_{0}^{2}} \ \ d\right ]\)

Reconstruction formula

\(x(t)=\frac {1}{x^{\ast }(0)}e^{-j\frac {a}{2b}t^{2}}\int _{\mathbb {R}} {WVDL}_{x}^{A}\left (\frac {t}{2},u \right)e^{j\frac {t}{b}u}du\)

Convolution

\({WVDL}_{x_{1} \circledast x_{2}}^{A}(t,u)=2\pi |b|\int _{\mathbb {R}}{WVDL}_{x_{1}x_{2}}^{A}(w,u) \)

 

\(\cdot {WVDL}_{x_2x_{1}}^{A}(t-w,u-aw)dw\)