A weakly stationary, continuous-time process has covariance function [C(tau)=sigma^{2} e^{-alpha|tau|}left(cos beta tau-frac{alpha}{beta} sin beta|tau| ight)] Prove that

A weakly stationary, continuous-time process has covariance function

\[C(\tau)=\sigma^{2} e^{-\alpha|\tau|}\left(\cos \beta \tau-\frac{\alpha}{\beta} \sin \beta|\tau|\right)\]

Prove that its spectral density is given by

\[s(\omega)=\frac{2 \sigma^{2} \alpha \omega^{2}}{\pi\left(\omega^{2}+\alpha^{2}+\beta^{2}-4 \beta^{2} \omega^{2}\right)}\]