The cepstrum was defined in a 1963 paper (Bogert et al.). It may be defined
The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the reconstruction of the signal. The real cepstrum uses only the information of the magnitude of the spectrum.
Many texts 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 it is widespread. Note that the Fourier inversion theorem inherently relates the two processes.
There are many ways to calculate the cepstrum. Some of them need a phase-wrapping algorithm; others do not.
Operations on cepstra are labelled quefrency analysis, quefrency alanysis, liftering, or cepstral analysis.
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.
The autocepstrum is defined as the cepstrum of the autocorrelation. The autocepstrum is more accurate than the cepstrum in the analysis of data with echoes.
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 scale. The result is called the mel-frequency cepstrum or MFC (its coefficients are 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.
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