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}

Read more about this topic:  Wigner Distribution Function

Famous quotes containing the words properties of the, properties of, properties, distribution and/or function:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
    —Ralph Waldo Emerson (1803–1882)

    “Advocating the mere tolerance of difference between women is the grossest reformism. It is a total denial of the creative function of difference in our lives. Difference must be not merely tolerated, but seen as a fund of necessary polarities between which our creativity can spark like a dialectic.”
    —Audre Lorde (1934–1992)