In linear time-invariant systems, which statement correctly describes the relationship between step response and impulse response?

• A. The impulse response is the derivative of the step response
• B. The impulse response equals the step response
• C. The step response is the derivative of the impulse response
• D. The impulse response is unrelated to the step response

Answer: (A)

In a linear time-invariant system, the output for any input is the convolution of that input with the impulse response. A unit step input can be thought of as the integral of a delta impulse, so feeding a unit step into the system gives a output that is the integral of the impulse response: y_step(t) = ∫0^t h(τ) dτ. That means the step response is the accumulated effect of the impulse response over time.

Because differentiation and integration are inverse operations, differentiating the step response yields the impulse response: h(t) = d/dt [s(t)] for t > 0 (and similarly h(t) = 0 for t < 0 if the system is causal). Put differently, the impulse response is the derivative of the step response, and the step response is the integral of the impulse response.