Discussion
Is the given discrete time signal periodic? If periodic, what is the fundamental period?
\(\cos(3\pi n)\)
Answer: A
A discrete time sinusoid \(\cos(\omega_0 n)\) is periodic only if \(\omega_0 / 2\pi\) is a rational number, and the fundamental period is the smallest integer \(N\) with \(\omega_0 N = 2\pi k\) for some integer \(k\).
\[ \frac{\omega_0}{2\pi} = \frac{3\pi}{2\pi} = \frac{3}{2} \]
This is a ratio of integers, so the signal is periodic, and the smallest \(N\) is 2, since \(3\pi \times 2 = 6\pi = 2\pi \times 3\). The signal is \(\cos(3\pi n) = (-1)^n\), which repeats every 2 samples. The answer is periodic with fundamental period 2, option A.
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