stepresponse

Step response refers to the time-domain output of a dynamic system when the input is a step function, typically a unit step. It is a fundamental characterization of how systems react to sudden changes in input and is widely used in control theory, signal processing, and system identification. For linear time-invariant systems, the step response is the convolution of the input with the system’s impulse response. Since the unit step is the integral of a delta impulse, the step response is the time-domain integral of the impulse response. In continuous time, if h(t) is the impulse response, the step response s(t) = ∫0^t h(τ) dτ for a causal system. In discrete time, y[n] = ∑_{k=0}^n h[k], assuming a causal system.

The steady-state value of the step response equals the DC gain of the system when the input is unity. For a stable system, the step response tends to a finite value; unstable or marginally stable systems may exhibit unbounded growth or sustained oscillations. Common step responses include the exponential approach of first-order systems to a final value, and the various damping cases—underdamped, overdamped, or critically damped—seen in second-order systems.

Practically, the step response is used to assess stability, time constants, overshoot, settling time, and bandwidth. It also supports system identification and controller design, where the observed response informs model parameters or compensator requirements. Related concepts include impulse response, transfer function, unit step, and frequency response.