I am trying to understand how I can relate a discrete-time random process to a continuous-time random process sampled at discrete times.
Suppose I have a noise source $N_\tau(t)$ which is derived from unit-amplitude additive white Gaussian noise N(t) that is fed through a single-pole low pass filter $H(s) = \frac{1}{\tau s +1}$.
I understand that the power spectral density of $N_\tau(t)$ is $S_{N_\tau}(\omega) = |H(\omega)|^2 = \frac{1}{(\omega\tau)^2 +1}$. So far so good.
Now I want to sample $N_\tau(t)$ at regular intervals $T$ to obtain a signal $n_\tau[k] = N_\tau(kT)$ -- how would I figure out the standard deviation and spectral density of the discrete samples $n_\tau[k]$?