Energy or power: unit step, impulse

Energy signal: 0 < E < ∞ (and P = 0), with E = ∫|x(t)|² dt over (-∞, ∞). Power signal: 0 < P < ∞ (and E = ∞), with P = lim (T→∞) (1/2T) ∫|x(t)|² dt over (-T, T).

x(t) = u(t)

Energy: E = ∫₀^∞ 1² dt = ∞

Power: P = lim (1/2T) ∫₀^T 1 dt = lim T/2T = 1/2

E is infinite and P is finite, so the unit step is a power signal (P = 0.5 W).

x(t) = δ(t)

The impulse has infinite amplitude at t = 0 and zero width, and |δ(t)|² = δ(t)·δ(t) is not defined, so the energy ∫δ²(t) dt is infinite (the integral does not converge). The power, P = lim (1/2T) · ∞, is also not a finite value, so the impulse is treated as neither an energy nor a power signal.

Many textbooks state this directly: δ(t) has infinite energy, and since it does not satisfy 0 < P < ∞ either, it is neither.