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Cepstrum

A cepstrum (pronounced "kepstrum") is the result of taking the Fourier transform of the decibel spectrum as if it were a signal. See quefrency, lifter, gamnitude[?], shape. There is the complex cepstrum and the real cepstrum.

The cepstrum is:

the FT of the log of the FT

then

cepstrum of signal = FT(log(FT(the signal)))

or

Signal -> FT -> log -> FT -> cepstrum

The real cepstrum uses the real log function, while the complex cepstrum uses the complex log function.

The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the recontruction of the signal. The real cepstrum only uses the information of the magnitude of the spectrum.

Many texts indicate that the process is FT->log->IFT. This is incorrect. (Are we sure? Is the result still meaningful either way?)

There are many ways to calculate the cepstrum, some of them need a phase-warping algorithm, others do not. (fixme which one one is?)

The cepstrum can be seen as information about rate of change in the different spectrum bands. It is thus an excellent feature vector for representing the human voice and musical signals. Usually the spectrum is first transformed using the Mel Frequency bands[?]. The result is called Mel Frequency Cepstral Coefficients[?] or MFCCs[?]. It is used for voice identification, pitch detection and much more. Recently it has also been getting a lot of attention from Music Information Retrieval researchers.

This is a result of the cepstrum separating the energy resulting from vocal cord vibration from the "distorted" signal formed by the rest of the vocal tract.

The cepstrum is also related to the so called homomorphic sound theory (<-fix this).

("FT" is used to indicate the Fourier transform function, rather than "FFT", since the fast fourier transform isn't specifically required.)

Etymology: "cepstrum" is an anagram of "spectrum".

Similarly:

  • "frequency" -> "quefrency",
  • "magnitude" -> "gamnitude",
  • "convolution" -> "novcolution",
  • "phase" -> "saphe".



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