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Cepstrum

A cepstrum is the result of taking the Fourier transform (FT) of the decibel spectrum as if it were a signal. Its name was derived by reversing the first four letters of "spectrum". There is a complex cepstrum and a real cepstrum.

The cepstrum was defined in a 1963 paper (Bogert et al.). It may be defined

verbally: the cepstrum (of a signal) is the Fourier transform of the logarithm (with unwrapped phase) of the Fourier transform (of a signal). Sometimes called the spectrum of a spectrum.

mathematically: cepstrum of signal = FT(log(|FT(the signal)|)+j2πm) (where m is the integer required to properly unwrap the angle or imaginary part of the complex log function)

algorithmically: signal → FT → abs() → log → phase unwrapping → FT → cepstrum

The "real" cepstrum uses the logarithm function defined for real values. The complex cepstrum uses the complex logarithm function defined for complex values.

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.

Applications

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.

The cepstrum is a representation used in homomorphic signal processing, to convert signals (such as a source and filter) combined by convolution into sums of the their cepstra, for linear separation.

Cepstral concepts

The independent variable of a cepstral graph is called the quefrency. The quefrency is a measure of time, though not in the sense of a signal in the time domain. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a pitch that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic, and the period corresponds to the pitch.

Liftering

Playing further on the anagram theme, a filter that operates on a cepstrum might be called a lifter. A low pass lifter is similar to a low pass filter in the frequency domain. It can be implemented by multiplying by a window in the cepstral domain and when converted back to the time domain, resulting in a smoother signal.

Convolution

A very important property of the cepstral domain is that the convolution of two signals can be expressed as the addition of their cepstra:

$x_1 * x_2 rightarrow x'_1 + x'_2$

References

B. P. Bogert, M. J. R. Healy, and J. W. Tukey: "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, 1963.