In Feynman's spirit of "If you truly understand a concept, you should be able to explain it clearly" and the explanation should make solid sense.

There are two main challenges:

  1. The axis of "Quefrency" is not a straightforward/natural concept, how to describe and make sense of what it is?

  2. Why is the cepstral view an efficieny way of "elevating" the main information contained in the signal? We can see that the vector is more succinct, but where does this efficiency come from?

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    $\begingroup$ Feynman also said "I think I can safely say that nobody understands quantum mechanics." $\endgroup$
    – Hilmar
    Jun 14 '19 at 6:32
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    $\begingroup$ I've been thinking about this, now adding some notes here. One basic idea is that physical phenomena may have a lot of data points, but the information these points reflect may be simpler, e.g. a sine wave has a lot of curves and cycles but it's just one simple formula. Sound can be understood as a composition of periodic waves and band noise, therefore spectral representation shows the more essential information. But spectra still contain redundant info, there may be harmonics -- more precisely, if we imagine the spectra as the results of the source and filter that produced them... <limit> $\endgroup$ Jun 22 '19 at 17:09

“efficiency” comes from a-priori assumptions. One assumption that makes a Cepstrum useful is that the signal contains something roughly periodic including a set of harmonics or overtones, possibly inside some sort of frequency spectrum envelope.

So a Cepstrum is mostly just an FFT of an FFT. with a few mods. On mod is to take the log magnitude after the first FFT to keep big overtones from drowning out the little ones.

So a DFT (slow FFT) correlates periodic happenings against some sinusoidal basis vectors. Unless the periodic time domain happenings are a single pure sinusoid, those periodic happenings will often correlate well against multiple equally spaced overtone frequency DFT basis vectors.

Thus you end up where you started, with a bunch of roughly equally spaced happenings, but now in the frequency domain instead of the time domain. The second FFT will again correlate these against sinusoidal basis vectors related to that spectral periodicity, which for audio signals turns out to be an interesting proxy for (overtone rich) pitch. The spectral envelope, since it was forced to be positive (by the log magnitude), ends up surrounding DC, instead of becoming sidebands of the spectral peaks as with time domain modulation. This helps separate envelope information (formant, etc.) from pitch information in the Cepstrum result.

A Cepstrum also works for a single pure sinusoid, as the two spectral lines (positive and negative frequency peaks) correlate against a negative cosine of the same width as the line separation.

Note a required trick: One often has to make zero and tiny FFT spectrum magnitudes bigger before the log to avoid the log from producing gigantic negative spikes or NaNs.


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