info prev up next book cdrom email home

Wiener-Khintchine Theorem

Recall the definition of the Autocorrelation function $C(t)$ of a function $E(t)$,

C(t) \equiv \int_{-\infty}^\infty E^*(\tau)E(t+\tau)\,d\tau.
\end{displaymath} (1)

Also recall that the Fourier Transform of $E(t)$ is defined by
E(\tau)=\int_{-\infty}^\infty E_\nu e^{-2\pi i\nu\tau}\,d\nu,
\end{displaymath} (2)

giving a Complex Conjugate of
E^*(\tau)=\int_{-\infty}^\infty E_\nu^* e^{2\pi i\nu\tau}\,d\nu.
\end{displaymath} (3)

Plugging $E^*(\tau)$ and $E(t+\tau)$ into the Autocorrelation function therefore gives

$\displaystyle C(t)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \left[{\int_{-\infty}^\infty E_\nu^* e^{2\p...
...[{\int_{-\infty}^\infty E_{\nu'} e^{-2\pi i\nu'(t+\tau)}\, d\nu'}\right]\,d\tau$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty E_\nu^*E_{\nu'} e^{-2\pi i\tau(\nu'-\nu)} e^{-2\pi i\nu't}\,d\tau\,d\nu\,d\nu'$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty E_\nu^* E_{\nu'} \delta(\nu'-\nu)e^{-2\pi i\nu't}\,d\nu\,d\nu'$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty E_\nu^* E_\nu e^{-2\pi i\nu t}\, d\nu$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \vert E_\nu\vert^2 e^{-2\pi i\nu t}\,d\nu$  
  $\textstyle =$ $\displaystyle {\mathcal F}[\,\vert E_\nu\vert^2],$ (4)

so, amazingly, the Autocorrelation is simply given by the Fourier Transform of the Absolute Square of $E(\nu)$,
C(t) = {\mathcal F}[\vert E(\nu)\vert^2].
\end{displaymath} (5)

The Wiener-Khintchine theorem is a special case of the Cross-Correlation Theorem with $f=g$.

See also Autocorrelation, Cross-Correlation Theorem, Fourier Transform

© 1996-9 Eric W. Weisstein