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 $ae^{-at}$.

Answer 2

Given the unit step response \( s(t) = 1 - e^{-at} \), the impulse response \( h(t) \) can be found either by direct differentiation or, as a check, by verifying the reverse relationship: integrating each candidate impulse response should reproduce the given step response.

1. Option \( ae^{-at} \): Integrating this candidate from \( 0 \) to \( t \) gives \( \displaystyle\int_0^t ae^{-a\tau}\,d\tau = \left[-e^{-a\tau}\right]_0^t = 1 - e^{-at} \), which exactly reproduces the given step response \( s(t) \). This confirms the candidate is consistent with the data.
2. Option \( a^{-1}e^{-at} \): Integrating this from \( 0 \) to \( t \) gives \( \dfrac{1}{a}\int_0^t e^{-a\tau}\,d\tau = \dfrac{1}{a^2}\left(1 - e^{-at}\right) \), which carries an extra factor of \( 1/a^2 \) and does not match \( s(t) = 1 - e^{-at} \) unless \( a = 1 \); it fails in general.
3. Option \( (1-a^{-1})e^{-at} \): This expression is a constant multiple of \( e^{-at} \) with a coefficient that depends on \( a \) in an unnatural way; integrating it does not simplify to \( 1 - e^{-at} \) for general \( a \), so it does not satisfy the step-impulse relationship.
4. Option \( (1-a)e^{-at} \): Similarly, integrating this gives a result scaled by \( (1-a)/a \), which again does not reduce to the required \( 1 - e^{-at} \) form except for a coincidental specific value of \( a \), so it is not correct in general.

Only the candidate whose integral reproduces the exact given step response is the true impulse response.

So the correct answer is \( ae^{-at} \).