Question

The step response of a linear time invariant system is $y\left(t\right)=5e^{-10t}u\left(t\right)$, where u(t) is the unit step function. If the output of the system corresponding to an impulse input $\delta \left(t\right)$ is h(t), then h(t) is

• A. $5u\left(t\right)-50e^{-10t}u\left(t\right)$
• B. $-50e^{-10t}u\left(t\right)$
• C. $5e^{-10t}\delta \left(t\right)$
• D. $5\delta \left(t\right)-50e^{-10t}u\left(t\right)$

Answer 1

The correct option is D $5\delta \left(t\right)-50e^{-10t}u\left(t\right)$

$\text{Step response, s(t) = 5 u(t)}{e}^{-10t}$

$\text{Impulse response,}h\left(t\right)=\frac{ds\left(t\right)}{dt}$

$\therefore h\left(t\right)=\frac{ds\left(t\right)}{dt}=\frac{d\left\{5e^{-10t}u\right(t\left)\right\}}{dt}$

$=-50e^{-10t}u\left(t\right)+5e^{-10t}du\left(t\right)/dt$

$\because \left[\delta \right(t)\text{exists only at t = 0. So put t= 0 in second part]}$

$=-50e^{-10t}u\left(t\right)+5e^{-10\left(0\right)}\delta \left(t\right)dt$

$=-50e^{-10t}u\left(t\right)+5\delta \left(t\right)dt$