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Periodicity of sin 5n and cos sum

Short answer

For sin5n, ω₀=5 gives ω₀/(2π)=5/(2π), irrational, so it is not periodic. For cos(2πn/5)+cos(2πn/7), ω₁/(2π)=1/5 (N₁=5) and ω₂/(2π)=1/7 (N₂=7), both rational; the overall signal is periodic with fundamental period N = LCM(5,7) = 35.

Determine whether the following discrete time signals are periodic or not (i) sin 5n (ii) cos(2πn/5) + cos(2πn/7)

A DT sinusoid sin⁡ω0n\sin\omega_0 n is periodic only if ω0/2π\omega_0/2\pi is rational; then N=2πm/ω0N = 2\pi m/\omega_0 (smallest integer).

(i) x[n]=sin⁡5nx[n] = \sin 5n

ω02π=52π(irrational)\frac{\omega_0}{2\pi} = \frac{5}{2\pi} \quad\text{(irrational)}

No integer NN satisfies 5N=2πm5N = 2\pi m. Not periodic.

(ii) x[n]=cos⁡(2πn5)+cos⁡(2πn7)x[n] = \cos\left(\frac{2\pi n}{5}\right) + \cos\left(\frac{2\pi n}{7}\right)

  • Term 1: ω1/2π=1/5\omega_1/2\pi = 1/5, rational, N1=5N_1 = 5.
  • Term 2: ω2/2π=1/7\omega_2/2\pi = 1/7, rational, N2=7N_2 = 7.
N=LCM(5,7)=35N = \text{LCM}(5, 7) = 35

Answer: (i) aperiodic; (ii) periodic with fundamental period N=35N = 35 samples.

✓ Answer checked by Claude Opus 5.5 (high effort)

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