Wigner Distribution Function - Properties of The Wigner Distribution Function

Properties of The Wigner Distribution Function

The Wigner distribution function has several evident properties listed in the following table.

Remarks
1 Projection property
2 Energy property
3 Recovery property \int_{-\infty}^\infty W_x(t/2,f) e^{i2\pi ft}\,df =x(t)x^*(0) \ \ \ \int_{-\infty}^\infty W_x(t,f/2) e^{i2\pi ft}\,dt =X(f)X^*(0)
4 Mean condition frequency and mean condition time \begin{matrix}X(f)=|X(f)|e^{i2\pi\psi(f)}\ \ \ x(t)=|x(t)|e^{i2\pi\phi(t)} \\ \text{If } \phi'(t)=|x(t)|^{-2}\int_{-\infty}^\infty fW_x(t,f)\,df \\ \text{ and } -\psi'(f)=|X(f)|^{-2}\int_{-\infty}^\infty tW_x(t,f)\,dt \end{matrix}
5 Moment properties \begin{matrix} \int_{-\infty}^\infty \int_{-\infty}^\infty t^nW_x(t,f)\,dt\,df=\int_{-\infty}^\infty t^n|x(t)|^2\,dt \\ \int_{-\infty}^\infty \int_{-\infty}^\infty f^nW_x(t,f)\,dt\,df=\int_{-\infty}^\infty f^n|X(f)|^2\,df \end{matrix}
6 Real properties
7 Region properties \begin{matrix}\text{If } x(t)=0\text{ for }t>t_0\text{ then } W_x(t,f)=0\text{ for }t>t_0 \\ \text{If } x(t)=0\text{ for }t<t_0\text{ then }W_x(t,f)=0\text{ for }t<t_0 \end{matrix}
8 Multiplication theorem
9 Convolution theorem \begin{matrix}\text{If } y(t)=\int_{-\infty}^\infty x(t-\tau)h(\tau)\,d\tau\text{ then } \\ W_y(t,f)=\int_{-\infty}^\infty W_x(\rho,f)W_h(t-\rho,f)\,d\rho \end{matrix}
10 Correlation theorem \begin{matrix}\text{If } y(t)=\int_{-\infty}^\infty x(t+\tau)h^*(\tau)\,d\tau\text{ then } \\ W_y(t,\omega)=\int_{-\infty}^\infty W_x(\rho,\omega)W_h(-t+\rho,\omega)\,d\rho \end{matrix}
11 Time-shifting property \begin{matrix}\text{If } y(t)=x(t-t_0)\text{ then } \\ W_y(t,f)=W_x(t-t_0,f) \end{matrix}
12 Modulation property \begin{matrix}\text{If } y(t)=e^{i2\pi f_0t}x(t)\text{ then } \\ W_y(t,f)=W_x(t,f-f_0) \end{matrix}

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