Question about quadrature signals

Question

My question is related to this article: https://www.dsprelated.com/showarticle/192.php

I think I understand mostly everything until this sentence:

"The directions in which the impulses are pointing show the relative phases of the spectral components".

Please, note that I understand (and tested in Matlab) that the DFT of a pure sine wave of frequency f has a negative peak at the positive freq f and a positive peak at freq -f in the imaginary part Im(fft(x)) and no peaks in Re(fft(x)). Cosine has only positive peaks in Re(.) and nothing in Im(.) etc.

However, I have trouble understanding this story about phases, can someone help me understand this sentence:

"The directions in which the impulses are pointing show the relative phases of the spectral components".

Answer 1

This is basically just about the phase of a complex number. If you have a sinusoid

$$x(t)=\cos(\omega_0t+\phi)\tag{1}$$

then it can be written in terms of complex exponentials:

$$x(t)=\frac{e^{j\omega_0t}e^{j\phi}+e^{-j\omega_0t}e^{-j\phi}}{2}\tag{2}$$

So the signal has two components, one spectral line at frequency $\omega_0$ with phase $\phi$, and another spectral line at $-\omega_0$ with phase $-\phi$.

Note that if you had a pure cosine, i.e., if $\phi=0$ in $(1)$ then you would get two spectral lines with a real-valued and positive weight. If you had a pure sinusoid, i.e., $\phi=-\pi/2$ in $(1)$, then you would get two purely imaginary weights with opposite signs. These are the two special cases that you've already figured out by yourself.