Phase and Magnitude of a Cosine

fruitloops

1. The problem statement, all variables and given/known data

Graph the magnitude and phase of the function: H(w) = cos(3w)

2. Relevant equations

None

3. The attempt at a solution

So here's the thing, I understand how to graph the phase and magnitude of any sort of function like X(w) = A*exp(wt). In that case, the magnitude would just be |X(w)| = A and the phase would be w.

However, I'm not sure how to apply that to graphing a cosine. I know that it doesn't have any imaginary portions (I think), so the magnitude would be just the absolute value of the function, which would be the cos function with a period of (2*pi)/3 but with all the negative parts flipped over the x-axis. Using Matlab, I see that this is indeed the case:



However, I don't understand how to get the phase graph or why it is like it is:



Could someone explain this to me?

Thanks in advance!

NascentOxygen

Hi fruitloops.

It seems you are thinking of the case where the exponent is imaginary. But it isn't imaginary in the example you show here.
Phase is always relative. Are you wanting to plot 𝟇 of cos(3w) relative to cos(w)? To see their phase relationship, sketch a cosine cos(w), and superimpose on it a cosine(3w). You can see that cos(3w) starts off in phase with cos(w) at w=0, and when w=Pi they are again in phase.

This sort of phase comparison — of signals of differing frequencies — is relevant to PLLs.

aralbrec

H(w) changes sign at some w. The magnitude only shows size so sign change has to be reflected in the phase.