# 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 $$H(\exp(j*\Omega))=\cos(\Omega)$$ 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!

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 $$e^{j\pi}=e^{-j\pi}=-1$$.