 Wiener–Khinchin theorem

The Wiener-Khinchin theorem relates the autocorrelation of a weakly stationary process to its power spectral density by a Fourier transform. The displayed convention is appropriate when the optical correlation is $\Gamma (\tau )=⟨f^{\ast }(t)f(t+\tau )⟩$ and the analytic field contains $e^{-i\omega t}$. A Lorentzian line shape of half-width $\kappa$ gives exponential correlation decay $e^{-\kappa |\tau |}$, including the carrier factor $e^{-i\omega \tau }$ when the full field is correlated.

The Wiener-Khinchin theorem, also known as the Wiener-Khinchin theorem for the autocorrelation function, is a fundamental result in the field of signal processing and stochastic processes. It establishes a relationship between the autocorrelation function of a stationary random process and its power spectral density.