The process $X(t)$ is wide sense stationary $(WSS)$ with $R_XX(τ) = 3δτ$. It is applied to a linear, time invariant $(LTI)$ system with the following input output relationship:
$Y'(t)+2Y(t)=X(t),Y(0)=0$ (for any $t$)
(a) find the impulsive response $h(t)$ of the system.
(b) Use $h(t)$ to calculate $R_{XY}(\tau)$,$R_{YY}(\tau).$
MY WORKING:
a. By taking the laplace transform of the given Differential Equation we have:
$SY(s)-Y(0)+2Y(s)=X(s)\Rightarrow SY(s)+2Y(s)=X(s) \Rightarrow Y(s)(S+2)=X(s)$
Which gives, $H(s)=\frac{Y(s)}{X(s)}=\frac{1}{s+2}$
Taking inverse laplace transform of $H(s)$, we get required $h(t)$, which is:
$\frac{Y(t)}{X(t)}=h(t)=e^{-2t}$
(b) Now $R_{XY}(\tau)=E[X(t_1)Y(t_2)]=E[X(t_1)X(t_2)h(t_2)]=E[X(t_1)X(t_2)e^{-2t}].$ I don't know how to evaluate this expectation, because I don't know what X(t) value is. can anyone please help me?