Question

A zero mean white Gaussian random process X(t) is passed through an LTI system whose impluse response h(t) is shown below.

If the two -sided PSD of the input process x(t) is 3 W/Hz, then the variance of the output process of the system will be_____.

• A. 2

Answer 1

The correct option is A 2
$\text{Since X(t) has zero mean, variance of}X\left(t\right)=R_x\left(0\right)$
$\text{So,}{\sigma }_x^2=R_x\left(0\right)$
${\sigma }_x^2=R_y\left(0\right)\because \overline{Y}=\overline{X}H\left(0\right)=\left(0\right)H\left(0\right)=0$
$R_Y\left(\tau \right)=R_X\left(\tau \right)\ast h\left(\tau \right)\ast h(-\tau )=3\delta \left(\tau \right)\ast h\left(\tau \right)\ast h(-\tau )=3\left[h\right(\tau )\ast h(-\tau \left)\right]$
$R_Y\left(0\right)=3\left[{\int }_{-\infty }^{\infty }{h}^2\right(u\left)du\right]$
${\int }_{-\infty }^{\infty }{h}^2\left(u\right)du={\int }_0^2{\left(\frac{1}{2}u\right)}^{2}du=\frac{1}{4}{\left(\frac{u^3}{3}\right)}_0^2=\frac{2}{3}$
$\text{So,}{\sigma }_Y^2=\frac{2}{3}\times 3=2$