Could you please confirm my method or correct me on computing the phases of my signals?

Question

I did some measurements, and I have two received signals in two different distances.

By doing the fft on the signals, I have the spectrum, then by using the angle() function in MATLAB on my fft, I computed the phase for each frequency.

I understand that the angle() function just give me the angle of each point of fft. I'm confused, is this the phase of my signal? Is this the phase relative to what signal?

As far as I know, phase of a signal, a Line-of-sight signal, after traveling a distance of L is computed as this:

-2*pi* f*(L/c)

c: is the speed of light.

The number that the angle(fft()) gives me is very different. Not even close to the above formula.

And I did all of this in MATLAB.

Answer 1

As Marcus Müller says, phase is relative. Consider the Discrete Fourier Transform equation:

$$X[k] = \sum_{n=0}^{N-1}x[n]e^{-j2 \pi nk/N}$$

The DFT at element $k$ is the inner product of the signal with a complex exponential with frequency $2{\pi}k/N$ cycles/sample and an initial phase of 0. Thus the phase of element $k$ is the phase difference of a particular frequency component of the signal relative to a particular complex exponential.

The equation you referenced is the phase shift of a single tone that travels a distance $L$.