If h[n] has finite length (nonzero for 0 ≤ n ≤ N), what happens to s[n] for n ≥ N?

• A. It becomes constant
• B. It continues to grow
• C. It oscillates
• D. It becomes negative

Answer: (A)

Convolution with a finite-length impulse response means only a fixed number of past input samples contribute to the current output. If the input is a unit step, then once n reaches N, every term in the sum uses an input value of 1, so the whole output stops changing. Mathematically, s[n] = sum_{k=0}^N h[k] x[n−k], and for n ≥ N with a step input, x[n−k] = 1 for all k in 0…N. Therefore s[n] = sum_{k=0}^N h[k], a constant independent of n. This constant is the DC gain of the system.