Say I have the signal, $$x(n) = \cos{\left(6.5 n \pi + \frac{\pi}{3}\right)}$$
Periodicity in the discrete case is given by, $\alpha = \frac{2 \pi l}{N}$ where if it is a rational multiple of $2 \pi$ the signal is periodic.
One can find that $\frac{l}{N} = \frac{6.5}{2}$, this is sufficient to say the signal is not periodic or so I thought. I am told that the signal is periodic, the argument is that you can multiple both $l$ and $N$ by $2$. This is confusing because for one $l$ is supposed to represent the $l$-th harmonic of $x(n)$ hence $l$ not being an integer initially should be problematic or is it valid to make $l \in \mathbb{N}$. What stops one from multiplying by 4 in that case? Is this why $l$ and $N$ must have no common factors?
What about in the case of, $$\exp{\left(\cos{\left(\frac{\pi n}{8}\right)}\right)}$$
Is the signal not periodic because it is not a complex exponential or because the curve is dictated by a sinusoidal function one can use the $\alpha$ test and say it is periodic due to $\frac{l}{N} = \frac{1}{16}$?
