Find the period of a signal with the DTFT plot I have an exercise and I'm struggling to resolve it. Here it is : 



My problem is about the DTFT. I've always been taught that we use DTFT for infinite-lenght signal that are not periodic (if the signal were periodic I would have used DFS)... So, how am I supposed to find the period of x[n] and y[n] ? I must have missed or misunderstood something. Can somebody explain it to me ? (obviously without giving the explicit answer). Thanks !
The discrete-time Fourier transform (DTFT) can be used for general infinite length signals, and it can also be used for periodic signals if we allow Dirac delta impulses in the expression for the DTFT. You have the correspondence

$$\textrm{DTFT}\{e^{jn\omega_0}\}=2\pi\delta(\omega-\omega_0)\tag{1}$$

where it is understood that the expression $(1)$ is valid in the interval $[ -\pi,\pi)$, because the DTFT is always $2\pi$-periodic.

An $N$-periodic signal has Dirac impulses at multiples of $2\pi/N$. Note that not all of these Dirac impulses must be present, some of them can have a weight of zero. I'm sure you can take it from here.