my objective is to prove why we can't use fft (which uses DFT) to calculate the PSD on a continuous time process.
The teacher instractures word by word are:
go to Matlab and read help on fft(), copy the DTF formula it is implementing and compare to the formula that should be implemented in our case, since we have a continuous time process and not a discrete time process! Then complete the computation of PSD for continuous time, as you said: from X(f) and X*(f). Then conclude whether it is possible to use fft() function and obtain the same result as you developed for continuous time. If yes, explain how.
my calculation of the PSD is $$S_X(f)=\int_{-\infty}^\infty e^{-2\pi f\tau j}R_X(\tau) d\tau = \frac{\sigma^2}{2\alpha}\int_{-\infty}^\infty e^{-2\pi f\tau j}e^{-\alpha |\tau |}d\tau=\frac{\sigma ^2}{4\pi ^2f^2+\alpha ^2}$$
Matlab's DTF: $$\sum_{j=1}^nX(j)W_n^{(j-1)(k-1)}$$ Where $$W_n=e^{\frac{-2\pi i}{n}}$$
The process is $$X'(t)+\alpha X(t) = Z(t), t \ge 0, X(0) = 0 $$ where Z is zero mean white Gaussian noise.
Any ideas? Thank you!