Phase information of a signal when creating a spectrogram

Question

Sorry for the uninformed question but I need some help understanding this. I'm trying to understand what phase information is and it would be very helpful if someone could correct my understanding of it.

Fourier transforms are used to create spectrograms (I understand the vanilla DFT). They allow us to see the different frequencies that make up a signal, and with this knowledge we can plot create an amplitude spectrogram. During this process, phase information is discarded. Correct?

Therefore we can not reconstruct the original audio signal because we are missing phase information. Now what I don't understand is what this phase information is telling us. The way I understand it, is that phase tells us how the different frequencies are aligned/arranged in the original signal? Is this correct?

Answer 1

Yes, you are correct. If you take a DFT of square wave and only look at the amplitudes, doing an inverse DFT but using different or random phases for the sine components, it does not look like square wave in time domain any more. But it will have a matching spectra. Kind of like two racecars on a circular track, they might always have same velocity, but without knowing the initial position, or even the direction they are going, it will be impossible to know if they side by side or on the opposite side of track.

Answer 2

Consider a real, discrete-time signal

$$ x[n] = A_1 \cos(\omega_1 n + \phi_1) + A_2 \cos(\omega_2 n + \phi_2) $$

Assuming that $\omega_1$ and $\omega_2$ are known, then in order to describe $x[n]$ completely and uniquely, you need both the amplitudes $A_1$, $A_2$, and also the phase angles, $\phi_1$ and $\phi_2$. That's what the various Fourier transforms are giving you.

If you are only interested in the relative signal powers, for comparing the power in the first cosine to the power of the second, then the amplitudes $A_1$ and $A_2$ alone might suffice.

But if you are intersted in the exact waveform, then you need to know the phase angles $\phi_1$, $\phi_2$, in addition to the amplitudes, in order to reconstruct the unique wave that gave those coefficients.

Answer 3

Fourier Transform (FT) contains both the amplitude information and phase information. So the FT coefficients are complex numbers. But when you plot the FT results, you and most commercial software just plot the amplitude or the squared amplitude in the graph. Actually there is no nice way to plot the complex numbers in the 2D graph. This might make you misunderstand that the phase information is lost. Don't worry. The complex coefficients of FT still contain the phase information!