Editing Autocorrelation (section)

====Wiener–Khinchin theorem====
The [[Wiener–Khinchin theorem]] relates the autocorrelation function <math>\operatorname{R}_{XX}</math> to the [[spectral density|power spectral density]] <math>S_{XX}</math> via the [[Fourier transform]]:

<math display=block>\operatorname{R}_{XX}(\tau) = \int_{-\infty}^\infty S_{XX}(f) e^{i 2 \pi f \tau} \, {\rm d}f</math>

<math display=block>S_{XX}(f) = \int_{-\infty}^\infty \operatorname{R}_{XX}(\tau) e^{- i 2 \pi f \tau} \, {\rm d}\tau .</math>

For real-valued functions, the symmetric autocorrelation function has a real symmetric transform, so the [[Wiener–Khinchin theorem]] can be re-expressed in terms of real cosines only:

<math display=block>\operatorname{R}_{XX}(\tau) = \int_{-\infty}^\infty S_{XX}(f) \cos(2 \pi f \tau) \, {\rm d}f</math>

<math display=block>S_{XX}(f) = \int_{-\infty}^\infty \operatorname{R}_{XX}(\tau) \cos(2 \pi f \tau) \, {\rm d}\tau .</math>