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)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “The question for the country now is how to secure a more equal distribution of property among the people. There can be no republican institutions with vast masses of property permanently in a few hands, and large masses of voters without property.... Let no man get by inheritance, or by will, more than will produce at four per cent interest an income ... of fifteen thousand dollars] per year, or an estate of five hundred thousand dollars.”
    —Rutherford Birchard Hayes (1822–1893)

    “The function of literature, through all its mutations, has been to make us aware of the particularity of selves, and the high authority of the self in its quarrel with its society and its culture. Literature is in that sense subversive.”
    —Lionel Trilling (1905–1975)