# Phase spectrum range

I computed spectrum analysis of sine wave (without windowing and zero-padding) and I got plots as follows:

As far as I know they are correct. When I learnt about DSP, I got the huge problem with understanding what is the phase spectrum. Recently I realised that "phases" are translations along time axis of each sinusoidal component. So I don't see any point to get them out of range $\langle -\pi, \pi \rangle$. I computed my spectrums with python and to get the phase spectrum I used unwrap function from numpy. As I understand this documentation correctly, it is impossible to get the result larger than $\pi$ or less than $-\pi$ unless I set the discount parameter. It does make sense for me.

Could you explain me:

• whether I understand the phase spectrum correctly?

• why the values of phase spectrum are out of range $\langle -\pi, \pi \rangle$?

• how does the function unwrap work?

"Unwrap" just adds even multiples of $\pi$ so that any border discontinuities (at $-\pi$ and $\pi$) in a phase signal are removed, which may be a sensible thing to do or not, depending on the situation.