# Verifying periodicity of a signal/sinusoid in the discrete case

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 $$\frac{l}{N} = \frac{1}{16}$$?

• I think in this case periodicity is not about a single period of the $\cos()$, so 'harmonics' are meaningless here, but any integer $n$ where $x(n)$ starts repeating itself. In case of $x(n) = \cos{\left(6.5 n \pi + \frac{\pi}{3}\right)}$ you simply seek $6.5 n \pi=k2\pi, \quad \{k,n\}\in \mathbb{N}$. That solution exists: $n=4,k=13$. For the second one, if $x$ is periodic, then $f(x)$ also (with restrictions I guess, integrating is not allowed but a mathematician knows better than me...). Jan 25 at 21:04
• I understand what you mean for the first signal. For the second signal are you saying if the function input is periodic then the function output is also periodic (for most things)? Jan 25 at 21:37
• Yes, if the cosine is periodic, then $e^{\cos}$ is also periodic since the $f(x)=e^x$ simply maps its input $x$ to a function value. If you have an $f(x)$ which does not have such a static mapping (can't come up with the mathematical name...), for example an integrating function, and is therefore not memoryless, then periodicity of $f(x)$ does not occur automatically. Jan 26 at 7:51