Which statement is true about the relationship between impulse response and step response?

• A. Impulse response is derivative to step response
• B. Step response is derivative to impulse response
• C. They are unrelated
• D. They are identical

Answer: (A)

In linear time-invariant systems, the impulse response and step response are intimately connected because a unit step is the integral of a Dirac delta. The impulse response h(t) is the output to a delta input, while the step response s(t) is the output to a unit step input. For an LTI system, the step response is the integral of the impulse response: s(t) = ∫0^t h(τ) dτ. Differentiating both sides with respect to time gives ds/dt = h(t). So the impulse response is the derivative of the step response. The step response and impulse response are thus related by differentiation and integration, not identical, and they are not unrelated.