Question

The unit step response of a linear time invariant system is $s\left(t\right)=(2-e^{-2t})u\left(t\right)$. The unit impulse reponse of the system is

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

Answer 1

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

Ans : (b)

For an LTI system,
Impulse response,
$h\left(t\right)=\frac{ds\left(t\right)}{dt}=\frac{d}{dt}\left[\right(2-e^{-2t}\left)u\right(t\left)\right]$
$=2e^{-2t}u\left(t\right)+(2-e^{-2t})\frac{du\left(t\right)}{dt}$
$=2e^{-2t}u\left(t\right)+(2-e^{-2t})\delta \left(t\right)$
$=2e^{-2t}u\left(t\right)+(2-e^0)\delta \left(t\right)\because x\left(t\right)\delta \left(t\right)=x\left(0\right)\delta \left(t\right)$
$=2e^{-2t}u\left(t\right)+\delta \left(t\right)$