Digital Signal Processing Basics
Which of the following correctly relates the unit impulse δ[n] (1 at n = 0, 0 elsewhere) and the unit step u[n] (1 for n ≥ 0, 0 for n < 0)?
- Aδ[n] = u[n] + u[n − 1]
- Bδ[n] = u[n] × u[n − 1]
- Cδ[n] = u[n] − u[n − 1]
Show answer and explanation
Answer: C. δ[n] = u[n] − u[n − 1]
Subtracting u[n − 1], the step delayed by one sample, from u[n] gives 1 − 1 = 0 for n ≥ 1, 1 − 0 = 1 at n = 0 and 0 for n < 0, which is δ[n]. Conversely, u[n] is the running sum of δ[n] up to n. This expresses the continuous-time relation "the derivative of a step is an impulse, and the integral of an impulse is a step" with a difference and a running sum.
- A. The sum is 1 at n = 0 and 2 for n ≥ 1.
- B. The product is 1 for n ≥ 1, which is the same as u[n − 1].
- C. Correct. The difference from the step delayed by one sample remains only at n = 0.