In a system where the impulse response is the derivative of the step response, what type of system is described?

• A. Linear Time-Invariant (LTI) System
• B. Nonlinear System
• C. Discrete-Time System
• D. Continuous-Time System

Answer: (A)

In a linear time-invariant system, the impulse response fully characterizes how the system responds to any input due to superposition and time-invariance. The step input is essentially the integral of the impulse response over time, so the system’s step response is the integral of h(t). Differentiating that step response with respect to time brings you back to h(t). In other words, for a causal LTI system, s(t) = ∫0^t h(τ) dτ, and therefore h(t) = ds(t)/dt. This relationship hinges on linearity and time-invariance; nonlinear or time-varying systems don’t guarantee that the impulse response is simply the derivative of the step response. In discrete time, you’d use a difference, not a derivative, which further points to this being a continuous-time LTI scenario. Hence the described system is Linear Time-Invariant.