Question

If the unit step response of a network is $(1-e^{-at})$, the unit impulse response is

• A. $ae^{-at}$
• B. $a^{-1}e^{-at}$
• C. $(1-a^{-1})e^{-at}$
• D. $(1-a)e^{-at}$

Hint:

Impulse response is always the time derivative of the step response.

Answer 1

The Correct Option is A

Step 1: Relation between step and impulse response.
For an LTI system, the impulse response $h(t)$ is the derivative of the step response $s(t)$.
Step 2: Given step response.
\[ s(t)=1-e^{-at} \]
Step 3: Differentiate the step response.
\[ h(t)=\frac{d}{dt}(1-e^{-at})=ae^{-at} \]
Step 4: Final conclusion.
Thus, the unit impulse response is $ae^{-at}$.