Question

The impulse response of an LTI system can be obtained by

• A. differentiating the unit ramp response
• B. differentiating the unit step response
• C. integrating the unit ramp response
• D. integrating the unit step response

Hint:

Impulse response is the derivative of the step response for any LTI system.

Answer 1

The Correct Option is B

This is a conceptual, direct-recall question about how standard test-signal responses relate to each other in an LTI system, so here is the reasoning behind each choice rather than a numeric calculation.
The unit ramp is the time-integral of the unit step, and the unit step is the time-integral of the unit impulse; by linearity, the same chain of integrals links the ramp response, step response, and impulse response of any LTI system.
Going from the step response back to the impulse response therefore requires one differentiation, which is exactly \( h(t) = \dfrac{d}{dt}s(t) \), so "differentiating the unit step response" is correct.
Differentiating the ramp response only reaches the step response (one step short), while integrating either the ramp or step response moves further away from the impulse response rather than toward it. \[ \boxed{h(t) = \dfrac{d}{dt}s(t)} \]

Answer 2

Another way to see this is by tracking the standard test signals directly: the impulse, step, and ramp inputs are related to each other by successive integrations, so their outputs (for a fixed LTI system) must obey the exact same relationships.

1. Option "differentiating the unit ramp response": The ramp signal itself is two integrations away from the impulse (impulse to step to ramp), so differentiating the ramp response only removes one of those two integrations, landing on the step response rather than the impulse response.
2. Option "differentiating the unit step response": The step signal is exactly one integration away from the impulse signal, so a single differentiation of the step response exactly reverses that one integration and lands precisely back on the impulse response, matching the required relationship.
3. Option "integrating the unit ramp response": Integrating adds yet another layer beyond the ramp, moving progressively further from the impulse response rather than back toward it.
4. Option "integrating the unit step response": Integrating the step response produces the ramp response, which is the opposite direction from what is needed to recover the impulse response.

Tracing the chain of integrations that link impulse, step, and ramp signals shows that only one operation, applied to only one of the four responses, correctly reverses exactly one level of integration.

So the correct answer is differentiating the unit step response.