ウィグナー分布 ウィグナー分布関数の性質

ウィキペディア

ウィグナー分布

出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/02/22 09:04 UTC 版)

ウィグナー分布関数の性質

ウィグナー分布関数には、以下のような特徴的性質がある。

Projection property

${\begin{aligned}|x(t)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\|X(f)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}$

Energy property

$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }W_{x}(t,f)\,df\,dt=\int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df$

Recovery property

${\begin{aligned}\int _{-\infty }^{\infty }W_{x}\left({\frac {t}{2}},f\right)e^{i2\pi ft}\,df&=x(t)x^{*}(0)\\\int _{-\infty }^{\infty }W_{x}\left(t,{\frac {f}{2}}\right)e^{i2\pi ft}\,dt&=X(f)X^{*}(0)\end{aligned}}$

Mean condition frequency and mean condition time

${\begin{aligned}X(f)&=|X(f)|e^{i2\pi \psi (f)},\quad x(t)=|x(t)|e^{i2\pi \phi (t)},\\{\text{if }}\phi '(t)&=|x(t)|^{-2}\int _{-\infty }^{\infty }fW_{x}(t,f)\,df\\{\text{ and }}-\psi '(f)&=|X(f)|^{-2}\int _{-\infty }^{\infty }tW_{x}(t,f)\,dt\end{aligned}}$

Moment properties

${\begin{aligned}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }t^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }t^{n}|x(t)|^{2}\,dt\\\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }f^{n}|X(f)|^{2}\,df\end{aligned}}$

Real properties

$W_{x}^{*}(t,f)=W_{x}(t,f)$

Region properties

${\begin{aligned}{\text{If }}x(t)&=0{\text{ for }}t>t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t>t_{0}\\{\text{If }}x(t)&=0{\text{ for }}t<t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t<t_{0}\end{aligned}}$

Multiplication theorem

${\begin{aligned}{\text{If }}y(t)&=x(t)h(t)\\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(t,\ rho)W_{h}(t,f-\rho )\,d\rho \end{aligned}}$

Convolution theorem

${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t-\tau )h(\tau )\,d\tau \\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(\rho ,f)W_{h}(t-\rho ,f)\,d\rho \end{aligned}}$

Correlation theorem

${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t+\tau )h^{*}(\tau )\,d\tau {\text{ then }}\\W_{y}(t,\omega )&=\int _{-\infty }^{\infty }W_{x}(\rho ,\omega )W_{h}(-t+\rho ,\omega )\,d\rho \end{aligned}}$

Time-shifting covariance

${\begin{aligned}{\text{If }}y(t)&=x(t-t_{0})\\{\text{then }}W_{y}(t,f)&=W_{x}(t-t_{0},f)\end{aligned}}$

Modulation covariance

${\begin{aligned}{\text{If }}y(t)&=e^{i2\pi f_{0}t}x(t)\\{\text{then }}W_{y}(t,f)&=W_{x}(t,f-f_{0})\end{aligned}}$

Scale covariance

${\begin{aligned}{\text{If }}y(t)&={\sqrt {a}}x(at){\text{ for some }}a>0{\text{ then }}\\{\text{then }}W_{y}(t,f)&=W_{x}(at,{\frac {f}{a}})\end{aligned}}$