Suppose we have a random process $ x (t) $ with a given spectrum, so with a given time correlation function $ R_ {xx} (\tau) $. Further suppose that each random variable. for each $ t $ be distributed as a Gaussian $ \mathcal {N} (x; \mu, \sigma ^ 2) $ with the same mean and variance. Having only the possibility to generate $N $ uncorrelated realizations, extracting $N $ times (once every $\Delta t $ seconds) from such distribution $ \mathcal {N} (x; \mu, \sigma ^ 2) $, how can I transform them to obtain equivalent correlated realizations of the process $ x (t) $, according to $R_ { xx} (\tau) $?
Thanks in advance.