Energy/power signals; periodicity of cos + sin

Energy and power signals

• Energy signal: finite energy E = Σ |x[n]|² (n from −∞ to ∞), 0 < E < ∞, and average power P = 0. Example: x[n] = (0.5)ⁿ u[n] has E = 1/(1 − 0.25) = 4/3.
• Power signal: finite non-zero average power P = lim(N→∞) 1/(2N+1) Σ(n=−N to N) |x[n]|², with infinite energy. Example: x[n] = cos(ω₀n) has P = ½; the unit step has P = ½.

Periodicity of x[n] = cos(2πn/5) + sin(πn/3)

A sum of periodic sequences is periodic if the individual periods have a common multiple. Check each term (each ω₀/2π must be a ratio of integers):

• cos(2πn/5): ω₁ = 2π/5, f₁ = 1/5, so N₁ = 5.
• sin(πn/3): ω₂ = π/3, f₂ = 1/6, so N₂ = 6.

Both frequencies are rational, so the sum is periodic. The fundamental period is the least common multiple:

N = LCM(5, 6) = 30 samples.

Check: x[n + 30] contains cos(2πn/5 + 12π) and sin(πn/3 + 10π), both equal to the original values.