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Unit step signal and relation to impulse

Short answer

Unit step u[n]=1 for n≥0, 0 otherwise. Unit impulse δ[n]=1 at n=0, 0 elsewhere. Relation: δ[n]=u[n]−u[n−1] (first difference); u[n]=Σ_{k=-∞}^{n} δ[k] (running sum).

Define and plot a discrete time unit step signal. Explain its relation with unit impulse signal.

Unit step signal

The discrete-time unit step is

u[n] = 1 for n ≥ 0, and 0 for n < 0.

 u[n]
  1 |           o  o  o  o  o  ...
    |           |  |  |  |  |
  0 +--o--o--o--+--+--+--+--+--- n
      -3 -2 -1  0  1  2  3  4

(The samples are 0 for n < 0 and 1 for n ≥ 0; the step starts at n = 0.)

Unit impulse

δ[n] = 1 for n = 0, and 0 otherwise.

Relation between u[n] and δ[n]

  1. Impulse as the first difference of the step:

    δ[n] = u[n] − u[n − 1]

    For n = 0: 1 − 0 = 1; for n ≥ 1: 1 − 1 = 0; for n < 0: 0 − 0 = 0.

  2. Step as the running sum of the impulse:

    u[n] = Σ(k = −∞ to n) δ[k] = Σ(m = 0 to ∞) δ[n − m]

    The sum is 0 until n reaches 0 and 1 from then on.

These are the discrete counterparts of the continuous relations δ(t) = du(t)/dt and u(t) = ∫δ(τ)dτ: the difference replaces the derivative and the sum replaces the integral.

✓ Answer checked by Claude Opus 5.5 (high effort)

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