# Phase information of a signal when creating a spectrogram

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?

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