Periodicity of 5sin(3πn/8 − π/2) − 2cos(7πn/12)

A DT sinusoid $\sin(\omega_0 n+\theta)$ is periodic if $\omega_0/2\pi$ is rational; its period is the smallest integer $N = 2\pi m/\omega_0$. A sum is periodic if each term is periodic, and its period is the LCM of the individual periods.

Term 1: $5\sin\left(\frac{3\pi}{8}n - \frac{\pi}{2}\right)$, $\omega_1 = \frac{3\pi}{8}$

$$\frac{\omega_1}{2\pi} = \frac{3}{16}\ (\text{rational}) \;\Rightarrow\; N_1 = \frac{2\pi m}{3\pi/8} = \frac{16m}{3} \;\Rightarrow\; N_1 = 16\ (m = 3)$$

Term 2: $2\cos\left(\frac{7\pi}{12}n\right)$, $\omega_2 = \frac{7\pi}{12}$

$$\frac{\omega_2}{2\pi} = \frac{7}{24}\ (\text{rational}) \;\Rightarrow\; N_2 = \frac{24m}{7} \;\Rightarrow\; N_2 = 24\ (m = 7)$$

Both terms are periodic, so $x[n]$ is periodic with

$$N = \text{LCM}(16, 24) = 48$$

The phase $-\pi/2$ does not affect the period.

Fundamental frequency:

$$\omega_0 = \frac{2\pi}{N} = \frac{2\pi}{48} = \frac{\pi}{24}\ \text{rad/sample}, \qquad f_0 = \frac{1}{48}\ \text{cycles/sample}$$

Answer: $x[n]$ is periodic; $N = 48$ samples, $\omega_0 = \pi/24$ rad/sample ($f_0 = 1/48$).