Question

The period of the discrete-time signal, \( \sin\left( \frac{6\pi n}{14} \right) \), is _______.

• A. 7
• B. 14
• C. 28
• D. 42

Hint:

Use the rule \( \omega/2\pi = p/q \) → period \( = q \) for checking periodicity of discrete-time sinusoids.

Answer 1

The Correct Option is B

Step 1: Standard form of DT sinusoid
A discrete-time sinusoidal signal of form \( \sin(\omega n) \) is periodic if: \[ \frac{\omega}{2\pi} = \frac{p}{q} \text{(rational)} \] and then period \( N = q \) (the smallest integer that satisfies this).
Step 2: Apply to this problem
Given: \[ \omega = \frac{6\pi}{14} = \frac{3\pi}{7} \] Then: \[ \frac{\omega}{2\pi} = \frac{3\pi}{7 \cdot 2\pi} = \frac{3}{14} \] which is rational.
Step 3: Determine period
Thus, the fundamental period is: \[ N = \frac{2\pi}{\gcd(\omega, 2\pi)} = 14 \] Therefore, the signal is periodic with period 14.