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.