5 How to find Power Spectrum (PSD) of a random signal \(x\left ( t\right ) \)

input: random signal \(x\left ( t\right ) \)

output: PSD of \(x\left ( t\right ) \)

Algorithm:

1. Find autocorrelation \(R_{x}\left ( \tau \right ) \) of \(x\left ( t\right ) \)
2. Find the Fourier Transform of \(R_{x}\left ( \tau \right ) \). The result is the PSD of \(x\left ( t\right ) \) called \(S_{x}\left ( f\right ) \)

Another method (this below works if not random \(x\left ( t\right ) \)) , why? can’t find FT for random process?

1. Find Fourier Transform \(X\left ( f\right ) \) of \(x\left ( t\right ) \)
2. Find the \(\left \vert X\left ( f\right ) \right \vert ^{2}=X\left ( f\right ) X^{\ast }\left ( f\right ) \)