In linear time-invariant systems, the impulse response is the derivative of which response?

• A. Step Response
• B. Impulse Response
• C. Frequency Response
• D. Input Response

Answer: (A)

In linear time-invariant systems, the step response is the integral of the impulse response. If you feed a unit step into the system, the output is the convolution of the impulse response with the step: s(t) = ∫ h(τ) dτ (over the appropriate range). Differentiating this step response with respect to time recovers the impulse response: h(t) = ds/dt. So the impulse response is the derivative of the step response.

The other options don’t fit this relationship. The impulse response is not the derivative of the impulse response itself, the frequency response is the Fourier transform of the impulse response (not a time-derivative relationship with the step response), and “input response” isn’t the standard term that ties to this derivative connection.