How to generate time series with a predefined auto correlation and cross-correlation among the series

I want to generated $N$ time series, each with length of $N_t$, i.e., I am considering $N_t$ time instants. The generated time series must have an auto correlation of $rho_a$, i.e., the correlation between every two successive samples taken at $t$ and $t-1$ must have a correlation of $rho_a$. Also, the cross correlation between every two time series $i$ and $j$ must be $rho_{ij}$. Neglecting the cross correlation between the different time series, this is a straightforward and it can be done by auto regressive model AR(1). For each of the N time series, I do the following (assume 3 time series to be generated). I generate the time samples according to $X_t=rho_a X_{t-1}+u_t$, $Y_t=rho_a Y_{t-1}+v_t$ and $Z_t=rho_a Z_{t-1}+epsilon_t$ where $u_t,v_t,epsilon_t$ are $N(0,1)$. However, considering the cross correlation, this is not applicable as the generated time series does not have the predefined cross correlation. Note that a similar question addresses a similar problem but it is only valid for two time series (not multiple time series). Here is the link :How to simulate two correlated AR(1) time series?