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)

    “In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
    —Ralph Waldo Emerson (1803–1882)

    “Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the implications and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.”
    —Susanne K. Langer (1895–1985)