Question

Which of the following is the impulse response of the LTI system represented by $\frac{d^{2}y\left(t\right)}{d{t}^2}+4\frac{dy\left(t\right)}{dt}+4y\left(t\right)=x\left(t\right)$

• A. $\delta \left(t\right)+4(t-1)e^{-2t}u\left(t\right)$
• B. $t{e}^{2t}u\left(t\right)$
• C. $t{e}^{-2t}u\left(t\right)$
• D. $(1-t)e^{-2t}u\left(t\right)$

Answer 1

The correct option is C $t{e}^{-2t}u\left(t\right)$

$\frac{d^{2}y\left(t\right)}{d{t}^2}+4\frac{dy\left(t\right)}{dt}+4y\left(t\right)=x\left(t\right)$

Apply Laplace transform on both side, we get
$s^2+4s+4\left)Y\right(s)=X(s)$

$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}=\frac{1}{(s+2{)}^2}$

Apply inverse laplace transform, we get
$h\left(t\right)=t\cdot e^{-2t}u\left(t\right)$