# what is the co-variance of system output cooresponding to a non-iid random variable

Consider the following system.$$\DeclareMathOperator{\sign}{sign}$$ \begin{aligned} &\dot{e_1}=c_1e_1+c_2e_2+\eta\\ &\dot{e_2}=c_3e_1+c_4e_2-c_5 \sign(e_1)+a \eta \end{aligned} which $$\eta$$ is a gaussian random variable and {$$a,c_1,c_2,c_3,c_4 \in \mathbb{R}$$}, such that: $$$$A=\begin{bmatrix}{c_1\ \ \ c_2\\c_3\ \ \ c_4}\end{bmatrix}$$$$
is an stable matrix, and $$c_5> 0$$. The output of this system is the following discontinuous signal: $$u=c_5 \sign(e_1)$$ In addition as a result of sliding on the sliding surface, we conclude that, this signal is with probability of 0.5, $$c_5$$ and possibility of 0.5, $$-c_5$$.

Unfortunately this random signal is not i.i.d and this brings up this question:

If you consider the following LTI system. \begin{aligned} &\dot{x}=Fx+Gu\\ &y_f=Cx \end{aligned} which $$x \in \mathbb{R}^2$$. and $$F$$ is an stable matrix.

WHAT IS THE CO-VARIANCE OF ----> $$y_f$$? In addition is there any body who know any source that has dealt with the discontinuous signal that I have presented, I even don't know what is its specific name to search for any source on google, thanks, please post me some useful key-words