# Plotting the Phase Response

I would appreciate it very much if someone would be able to provide some clarity on plotting phase responses.

For instance, given that the frequency response of a filter can be written as H(exp(j*Ω))=cos(Ω), how would I go about determining the phase response?

The answer key demonstrates a piecewise solution: This is what I understand so far:

-cos(omega) is strictly real, so the phase carries values of 0, or +/- pi

-between the values of -pi/2 and pi/2, cos(ω) is positive, so the phase is 0

-I understand since the function cos(ω) is strictly real, phases where the function is not positive should either be pi/-pi (as opposed to -pi/2/pi/2, for when it's imaginary), but how do you know for which range it's pi, and for which range it's -pi?

This might be a stupid question, but I've spent a lot of time thinking it through, so I'm thinking I'm missing something, and would appreciate any clarity.

Thank you so much!

Your understanding is correct. There is no difference between a phase of $$\pi$$ and a phase of $$-\pi$$. You can always add or subtract integer multiples of $$2\pi$$ to the phase without changing anything because $$e^{j2\pi n}=1$$ for $$n\in\mathbb{Z}$$. Clearly you have $$e^{j\pi}=e^{-j\pi}=-1$$.