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

Step 1: Define impulse and step responses.
The impulse response $h(t)$ of an LTI system is the output when the input is a unit impulse $\delta(t)$.
Step 2: Relationship between unit step and impulse.
The unit impulse is the derivative of the unit step function:
\[ \delta(t) = \frac{d}{dt}u(t) \]
Step 3: System response relation.
If $s(t)$ is the step response of the system, then:
\[ h(t) = \frac{d}{dt}s(t) \]
Step 4: Eliminate incorrect options.
Integrating responses does not yield the impulse response directly. Differentiating the ramp response gives the step response, not impulse.
Step 5: Final conclusion.
Hence, the impulse response can be obtained by differentiating the unit step response.

Answer 2

This question asks how the impulse response \( h(t) \) of an LTI system relates to its response to other standard test inputs (ramp and step). The key relationship is that each standard input is the time-integral of the previous one: the ramp is the integral of the step, and the step is the integral of the impulse. Because the system is linear and time-invariant, the same relationship holds between the corresponding output responses.

1. Option "differentiating the unit ramp response": Differentiating the ramp response undoes one level of integration and produces the step response, not the impulse response; this option describes how to get the step response from the ramp response, one level short of impulse response.
2. Option "differentiating the unit step response": Since the unit step \( u(t) \) is the integral of the unit impulse \( \delta(t) \), and the system is linear, the step response \( s(t) \) is also the integral of the impulse response \( h(t) \), i.e. \( s(t) = \int_{-\infty}^{t} h(\tau)\,d\tau \). Differentiating both sides gives \( h(t) = \dfrac{d}{dt}s(t) \), which directly recovers the impulse response.
3. Option "integrating the unit ramp response": Integrating the ramp response moves further away from the impulse response, not closer; integration is the operation that goes from impulse toward step toward ramp, not backward, so this cannot produce \( h(t) \).
4. Option "integrating the unit step response": This also moves in the wrong direction along the impulse-step-ramp chain, producing the ramp response instead of undoing back to the impulse response.

Only differentiation applied to the step response correctly reverses the one integration step that separates it from the impulse response.

So the correct answer is differentiating the unit step response.