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 $$H(\exp(j*\Omega))=\cos(\Omega)$$ 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 $\pm\pi$
Between the values of $-\pi/2$ and $\pi/2$, $\cos(\Omega)$ is positive, so the phase is $0$
I understand since the function $\cos(\Omega)$ is strictly real, phases where the function is not positive should either be $\pi$ or $-\pi$ (as opposed to $-\pi/2$ or $\pi/2$, 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!