I am trying to understand the definition of wide sense stationary on my own and probably have some silly questions. Wikipedia says, wide sense stationary is a process with constant mean and autocorrelation function over time. Now, lets assume a process,
$$X(n) = 0.9X(n-1) + V(n) + U(n),$$ where $V$ is white noise and $U$ is an external input.
- Lets assume $U(n) = 0\quad \forall n$, i.e. $X_1(n) = 0.9X_1(n-1)+V(n)$.
In this case can $X_1$ be called WSS? What does mean being constant actually implies? Can I assume $0.9X(n-1)$ is mean since it is deterministic and say mean is decreasing over time, and this is not WSS? - Lets assume $U(n) = C \text{ const.}$, i.e. $X_2(n) = 0.9X_2(n-1) + V(n) + C$
If $X_1$ was WSS, I believe this should be WSS as well. Please confirm. - If $U(n)$ is an external signal changing over time, i.e. $X_3=X$
then would still be WSS, if $X_p= 0.9X_p(n-1)+V(n)$ were originally? In this case $X$ is depended on a signal which is not part of the original process.