The _Wiener-Khinchin theorem_ states that the [[Power-spectral density]] that corresponds with a [temporal autocorrelation function,](Autocorrelation%20functions.md) $C(t)$ is its [Fourier transform.](Time-frequency%20Fourier%20transform.md) We may thus write the power-spectral density as $S(\omega)$ as $S(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t} C(t)$ ^dd903c # Derivation of the Wiener-Khinchin theorem One way of arriving at the [Wiener-Khinchin theorem](Wiener-Khinchin%20theorem.md) is by taking the definition of the [temporal autocorrelation function](Autocorrelation%20functions.md#Temporal%20autocorrelation%20function), [$C(t')=\langle x(t+t')x(t')\rangle =\lim_{\tau\rightarrow \infty}\int_{-\tau/2}^{\tau/2}dt'\,x(t+t')x(t')$](autocorrelation%20functions#^467186) and taking the [Fourier transform](Time-frequency%20Fourier%20transform.md) of the term $x(t')$ within the time window $\tau,$ this is expressed as [$x_\tau(\omega)=\mathcal{F}[x(t')]_\omega=\int_{-\tau/2}^{\tau/2}dt'\,x(t')e^{i\omega t'}$](Power-spectral%20density#^8dec19)^a82a49 Using this [Fourier transform](Wiener-Khinchin%20theorem#^a82a49) we may derive the [power-spectral density](Power-spectral%20density.md) by taking its amplitude, squaring it and taking its average such that $\langle|x_\tau(\omega)|^2\rangle = \int_{-\tau/2}^{\tau/2}dt''\,\int_{-\tau/2}^{\tau/2}dt'\, \langle x(t'')x(t')\rangle e^{i\omega (t''-t')}$$=\int_{-\tau/2}^{\tau/2}dt''\,\int_{-\tau/2}^{\tau/2}dt'\,C(t''-t') e^{i\omega (t''-t')}$In order to evaluate the inner integral while preserving the same time dependence for the [autocorrelation function](Autocorrelation%20functions.md) we change its limits such that $\langle|x_\tau(\omega)|^2\rangle=\int_{-\tau/2}^{\tau/2}dt'\,\int_{-\tau/2+t'}^{\tau/2-t'}dt\,C(t) e^{i\omega t}$$=\int_{-\tau}^{\tau}dt\,\int_{-\tau/2+|t|/2}^{\tau/2-|t|/2}dt'\,C(t) e^{i\omega t}$ ^98b1b5 Evaluating the inner integral gives us $\langle|x_\tau(\omega)|^2\rangle=\int_{-\tau}^{\tau}dt\,(\tau-|t|)C(t) e^{i\omega (t)}$ ^f09643 In accordance with the definition of the [power spectral density](Power-spectral%20density#^1cca41) $S(\omega)=\lim_{\tau\rightarrow \infty}\frac{1}{\tau}\int_{-\tau}^{\tau}dt\,(\tau-|t|)C(t) e^{i\omega t}=\lim_{\tau\rightarrow \infty}\cancel{\frac{\tau}{\tau}}\int_{-\tau}^{\tau}dt\,C(t) e^{i\omega t}-\lim_{\tau\rightarrow \infty}\frac{|t|}{\tau}\int_{-\tau}^{\tau}dt\,C(t) e^{i\omega t}$ The second term goes to zero under the limit and in the case where the limit may be absorbed by the limits of the integral we're left with the [Wiener-Khinchin theorem](Wiener-Khinchin%20theorem.md), [$S(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t} C(t).$](Wiener-Khinchin%20theorem#^dd903c) %%The derivation is poorly motivated in the notes by Chase Broedersz. The limits switcheroo is strange in the inner integral. Try to explain it a bit if you can %% ^4ae31f --- # Recommended Reading The derivation and definition given here is based on the following set of lecture notes: * [Broedersz, C. P., T M1, _Advanced Statistical Physics_, Lecture Notes, Summer 2018](Broedersz,%20C.%20P.,%20T%20M1,%20Advanced%20Statistical%20Physics,%20Lecture%20Notes,%20Summer%202018..md) pgs. 161-162. Here the [Wiener-Khinchin theorem](Wiener-Khinchin%20theorem.md) is introduced in the context of dissipation in stochastic systems. #StatisticalPhysics