Wiener-Khintchine Theorem

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

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

Also recall that the Fourier Transform of $E(t)$ is defined by

$$E(\tau)=\int_{-\infty}^\infty E_\nu e^{-2\pi i\nu\tau}\,d\nu,$$ (2)

giving a Complex Conjugate of

$$E^*(\tau)=\int_{-\infty}^\infty E_\nu^* e^{2\pi i\nu\tau}\,d\nu.$$ (3)

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

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

$$= \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty E_\nu^*E_{\nu'} e^{-2\pi i\tau(\nu'-\nu)} e^{-2\pi i\nu't}\,d\tau\,d\nu\,d\nu'$$

$$= \int_{-\infty}^\infty \int_{-\infty}^\infty E_\nu^* E_{\nu'} \delta(\nu'-\nu)e^{-2\pi i\nu't}\,d\nu\,d\nu'$$

$$= \int_{-\infty}^\infty E_\nu^* E_\nu e^{-2\pi i\nu t}\, d\nu$$

$$= \int_{-\infty}^\infty \vert E_\nu\vert^2 e^{-2\pi i\nu t}\,d\nu$$

$$= {\mathcal F}[\,\vert E_\nu\vert^2],$$ (4)

so, amazingly, the Autocorrelation is simply given by the Fourier Transform of the Absolute Square of $E(\nu)$,

$$C(t) = {\mathcal F}[\vert E(\nu)\vert^2].$$ (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