Cepstrum
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A cepstrum (pronounced "kepstrum") is the result of taking the Fourier transform of the decibel spectrum as if it were a signal. There is a complex cepstrum and a real cepstrum.
The cepstrum was defined in a 1963 paper (see ref. [1]). It may be defined
- verbally: the cepstrum is the FT of the log of the FT
- mathematically: cepstrum of signal = FT(log(FT(the signal)))
- algorithmically: signal → FT → log → FT → cepstrum
The real cepstrum uses the logarithm function defined for real values, while the complex cepstrum uses the complex logarithm function defined for complex values also.
The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the reconstruction of the signal. The real cepstrum only uses the information of the magnitude of the spectrum.
Many texts incorrectly state that the process is FT → log → IFT, i.e. that the cepstrum is the "inverse Fourier transform of the log of the spectrum". This is not the definition given in the original paper, but unfortunately is widespread.
There are many ways to calculate the cepstrum, some of them need a phase-warping algorithm, others do not.
The cepstrum can be seen as information about rate of change in the different spectrum bands. It was originally invented for characterizing the seismic echoes resulting from earthquakes and bomb explosions. It has also been used to analyze radar signal returns.
It is now also used as an excellent feature vector for representing the human voice and musical signals. For these applications, the spectrum is usually first transformed using the Mel Frequency bands. The result is called the 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", formed by reversing the first four letters.
Similarly:
- "frequency" → "quefrency"
- "magnitude" → "gamnitude"
- "convolution" → "novcolution"
- "phase" → "saphe"
- "filter" → "lifter"
References
[1] Tukey, J. W., B. P. Bogert and M. J. R. Healy: "The quefrency alanysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe-cracking". Proceedings of the Symposium on Time Series Analysis (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley.