When is $ x(n) = A \cos (\omega_0 n + \phi)\ ,\ n \in \mathbb{Z} $ periodic?

Question

I know that sometimes, discrete time sinusoids are not periodic.

But how can you figure that out?

Answer 1

So, for anyone wondering, as I found here, we're looking for the integer period $N\ ,\ N > 0$ such that

$$ \begin{align} x(n) &= x(n + N)\\ A \cos (\omega n + \phi) &= A \cos (\omega (n + N) + \phi )\\ A \cos (\omega n + \phi + 2\pi k) &= A \cos (\omega n + \phi + \omega N)\\ \therefore\ 2\pi k &= \omega N\\ N &= \frac{2\pi k}{\omega} \end{align} $$ , where $k \in \mathbb{Z}$.

Notice that there may not exist a $k \in \mathbb{Z}$ such that $\frac{2\pi k}{\omega}$ is an integer.