Thanks for the notes! Another idea I've been thinking about is to capture the phase by splitting ACF into multiple complementary parts at the spectral density s

Thanks for the notes! Another idea I've been thinking about is to capture the phase by splitting ACF into multiple complementary parts at the spectral density step, i.e. when we draw the spectrogram, we use |r exp(i phi)|^2 of the FFT output and that |...|^2 drops the phi. However, we could split this |...|^2 into a sum of a few terms that would add up to the same spectral density, but would separately capture the phase. In the simplest case, |x + iy|^2 = x^2 + y^2, can be interpreted as two half-spectrograms: one for x^2 and another for y^2. If colored the same, we'd get the original spectrogram, but if colored differently, we'd know values of x^2 and y^2 and that's enough to know the phase up to a quadrant.