Wiener-Khinchin theorem ($\Gamma (\tau )=\int S(\Omega )e^{-i\Omega \tau }d\Omega$)

For a complex weakly stationary process with finite second moment and continuous uncentered autocorrelation $\Gamma (\tau )=\mathbb{E}[\overline{X_t}{X}_{t+\tau }]$, there is a unique finite positive spectral measure of a stationary random field. With the optical sign convention, write it asThe sign reverses the frequency coordinate relative to the alternative positive-exponential convention. To see the positivity behind the theorem, expand $\mathbb{E}|{\sum }_j{c}_j{X}_{t_j}{|}^2\ge 0$. This makes the correlation a positive-definite function, so Bochner's theorem supplies the measure.^[ref]

An ordinary power spectral density $S$ exists when $dF=S(\omega )d\omega$. If $\Gamma$ is integrable, $S(\omega )=(2\pi {)}^{-1}\int \Gamma (\tau )e^{i\omega \tau }d\tau$. Stationarity alone does not guarantee such a density. For $X_t=Z$ at every time and $\mathbb{E}|Z{|}^2=1$, the correlation is one and $F={\delta }_0$.

The normalized Lorentzian line shape $S(\omega )=\kappa /[\pi ((\omega -{\omega }_0{)}^2+{\kappa }^2)]$ gives $\Gamma (\tau )=e^{-\kappa |\tau |}{e}^{-i{\omega }_0\tau }$. Its width therefore describes decay of temporal coherence. A persistent spectral line instead contributes an atom, which cannot be treated as an ordinary integrable density.