# Periodicity of a discrete time complex exponential signal?

I'm new to DSP and I'm struggling to solve a basic problem. I think I've solved it but I feel that something's off in my reasoning.

The problem involves investigating a signal of the form $$e^{jk\pi n}$$, where $$n$$ is an integer and $$k$$ is real. How do I confirm if this signal is periodic? I tried dividing the frequency by $$2\pi$$ as my classmate suggested but the how does the result $$\frac{k}{2\pi}$$ confirm whether a signal is periodic or not? $$k$$ isn't required to be an integer, so how does $$\frac{k}{2\pi}$$ being rational mean the signal is periodic? Even if the signal is periodic, how do I find the fundamental frequency? There's a formula given in Oppenheim that involves converting the exponential to an $$e^{jm\frac{2\pi}{N}n}$$ form but I don't quite understand how this would help find the fundamental period.

The second part of the problem involves finding the absolute value of $$e^{jk\pi n}$$. This should be straightforward enough - the absolute value is 1 because of Euler's formula, but I'm not missing out on anything, right?

The third part of the problem is finding out a value of $$n$$ to make the signal take on any value within its range. Now since the signal is a complex exponential, the only real values it takes on are $$1$$ and $$-1$$. But since the frequency depends on $$k$$, a real number, wouldn't we find a non-integer value of $$n$$ for the vast majority of cases?

You have a sequence of complex numbers

$$x[n]=e^{j\pi an},\quad a\in\mathbb{R},\quad n\in\mathbb{Z}\tag{1}$$

$$x[n]$$ is periodic with period $$N$$ if $$x[n]=x[n+N]$$ is satisfied for all $$n$$. For the given sequence, that means

$$e^{j\pi an}=e^{j\pi a(n+N)}=e^{j\pi an}e^{j\pi aN}\tag{2}$$

For $$(2)$$ to be satisfied we require $$e^{j\pi aN}=1$$, which is equivalent to

$$\pi aN=2\pi m,\qquad m\in\mathbb{Z}\tag{3}$$

Consequently,

$$a=2\frac{m}{N}\tag{4}$$

and

$$x[n]=e^{j2\pi n \frac{m}{N}}$$

If $$m$$ and $$N$$ in $$(4)$$ are coprime, then $$N$$ is the (smallest) period of $$x[n]$$.

As for the last question, if I understand correctly, it should be about the value of $$a$$ (not $$n$$) for which $$x[n]$$ assumes all possible values satisfying $$|x[n]|=1$$. This is the case if $$x[n]$$ is not periodic. I.e., any $$a$$ that does not satisfy Eq. $$(4)$$ results in $$x[n]$$ taking on all possible values with unity magnitude as $$n$$ is varied.

A signal $$x[n]$$ is periodic with period $$P$$ if $$x[n] = x[n+P]$$ for all $$n$$.

That means that some "sinusoidal" signals which are periodic when $$n$$ is real-valued and continuous are not necessarily periodic when $$n$$ is an integer.