Cepstral Sorcery

[Kurt A]

My understanding of the cepstrum is that it’s useful because it separates the source and the filter since the high frequencies that give the spectrum its comb-like appearance appear on one side of the cepstrum and the low frequencies responsible for the shape of the envelop appear on the other side. So essentially, the cepstrum gives us the frequency composition of the spectrum (J&M call the cepstrum the “spectrum of the log of the spectrum”).

I have a few questions:
How is this achieved by taking the IDFT of the log of the spectrum?
Is it done by the omission of phase?
Shouldn’t you instead take the DFT of the spectrum if you want its frequencies?

[Simon]

Your summary of how the cepstrum separates source and filter is good.

Omitting the phase of the speech signal is only a small part of the story – this happens right in the first step after windowing, when we retain only the magnitude spectrum.

The key ideas to understand are:

The magnitude spectrum of speech is equal to the product of the magnitude spectrum of the source and the magnitude spectrum of the filter.

The log magnitude spectrum of speech is equal to the sum of the log magnitude spectrum of the source and the log magnitude spectrum of the filter.

We perform a series expansion of the log magnitude spectrum of the speech. Whether we use the DFT, inverse DFT or DCT (Discrete Cosine Transform) isn’t important conceptually.

This series expansion expresses the log magnitude spectrum of speech as sum of simple components (e.g., cosines). Some of those simple components are representing the filter (the low order ones) and one or two of the higher order components represent the source. They are additive in the log spectral domain.

[Author]

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