Power spectral density and the Wiener-Khintchine theorem
For a stationary process X, the Power Spectral Density (PSD) \(S_x\) is equivalent to the Fourier transform of the autocorrelation function \(R_x\).
\(S_x\left(f\right)=\int_{-\infty}^{+\infty}{R_x(t)e^{-j2\pi ft}dt}\) (26)
Overview of Provided Definitions:
\(S_x\left(f\right)=E\left[\left|\hat{X}(f)\right|^2\right] and R_x\left(t\right)=E\left[x\left(\tau\right)x\left(\tau+t\right)\right]\) (27)
A process is considered stationary if its statistics (mean, variance, etc.) remain invariant under translation.
The Wiener-Khintchine theorem allows the interpretation of Digital Signal Processing (DSP) as a power distribution.
\(S_x\left(f\right)=\int_{-\infty}^{+\infty}{\left|S_{xx}(f)\right|^2df=p\left(x\right):\ \ \ }\ p\left(x\right)= \lim\limits_{T \to \infty} \frac {1} {N}\int_{0}^{T}{\left|x(t)\right|^2dt}\) (28)