Question

A random process is defined s $X\left(t\right)=6e^{At}$. where ${}^{\prime }{A}^{\prime }$ is random varibale unifrmly distributed in the interval $[0,2]$. Then the autocorrelation function $R_X(t_1,t_2)$ is given as

• A. $\frac{18}{t_1+t_2}[e^{2(t_1+t_2)}-1]$
• B. $18[e^{2(t_1-t_2)}-1]$
• C. $\frac{18}{t_1-t_2}[e^{2(t_1-t_2)}-1]$
• D. $18[e^{2(t_1+t_2)}-1]$

Answer 1

The correct option is A $\frac{18}{t_1+t_2}[e^{2(t_1+t_2)}-1]$

$X\left(t\right)=6e^{At}$
$R_X(t_1,t_2)=E\left[X\right(t_1\left)X\right(t_2\left)\right]=E\left[6e^{A{t}_1}6e^{A{t}_2}\right]$
$=36\left[\frac{1}{2}{\int }_0^2{e}^{A(t_1+t_2)}dA\right]=18{\left[\frac{e^{A(t_1+t_2)}}{t_1+t_2}\right]}_0^2=\frac{18}{t_1+t_2}[e^{2(t_1+t_2)}-1]$