In discrete time, how is the step response s[n] related to the impulse response h[k]?

• A. s[n] = ∑_{k=−∞}^{n} h[k]
• B. s[n] = h[n]
• C. s[n] = ∑_{k=0}^{n} h[k]
• D. s[n] = ∑_{k=0}^{n-1} h[k]

Answer: (C)

The step response builds up the effects of the impulse response over time. For a discrete-time LTI system, the output to any input is the convolution of the input with the impulse response. A unit step can be viewed as the accumulation of impulses up to the current time, so the step response is the cumulative sum of the impulse response values up to the present index.

Mathematically, s[n] = (h * u)[n]. For a causal system, h[k] = 0 for k < 0 and u[n - k] = 1 for n − k ≥ 0, i.e., k ≤ n. This collapses the convolution to s[n] = ∑_{k=0}^{n} h[k]. Hence the step response at time n is the sum of the impulse response samples from k = 0 up to n.