Notes · Speech from first principles
Cepstrum
The trick for undoing the source–filter model’s multiplication. Taking logs turns the product into a sum, \(\log P = \log U_g + \log V\): the source’s harmonic comb becomes a fast ripple across frequency (its spacing is the pitch), while the vocal-tract envelope carrying the formants is slow.
A second Fourier transform of the log spectrum then sorts the two by their rate of wiggle: the envelope lands in the low quefrency region, the pitch as a sharp peak further out. Keep the low coefficients and you have the formant envelope with the source discarded — exactly the quantity a recogniser wants.
The mel filterbank computes a cheaper, blurrier version of the same envelope; MFCCs are literally the cepstrum of a mel-warped spectrum. Where the all-pole filter assumes the filter’s shape, the cepstrum makes no such assumption — it just separates fast from slow.