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In Fourier analysis, the cepstrum (; plural cepstra, adjective cepstral) is the result of computing the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. The method is a tool for investigating periodic structures in frequency spectra. The power cepstrum has applications in the analysis of human speech.
The term cepstrum was derived by reversing the first four letters of spectrum. Operations on cepstra are labelled quefrency analysis (or quefrency alanysisB. P. Bogert, M. J. R. Healy, and J. W. Tukey, The Quefrency 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.), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with kepstrum.
References to the Bogert paper, in a bibliography, are often edited incorrectly. The terms "quefrency", "alanysis", "cepstrum" and "saphe" were invented by the authors by rearranging the letters in frequency, analysis, spectrum, and phase. The invented terms are defined in analogy to the older terms.
The following abbreviations are used in the formulas to explain the cepstrum:
| + !Abbreviation !Explanation | |
| Signal, which is a function of time | |
| Cepstrum | |
| Fourier transform: The abbreviation can stand i.e. for a continuous Fourier transform, a discrete Fourier transform (DFT) or even a z-transform, as the z-transform is a generalization of the DFT. | |
| Inverse of the fourier transform | |
| Logarithm of x. The choice of the base b depends on the user. In some articles the base is not specified, others prefer base 10 or e. The choice of the base has no impact on the basic calculation rules, but sometimes base e leads to simplifications (see "complex cepstrum"). | |
| Absolute value, or magnitude of a complex value, which is calculated from real- and imaginary part using the Pythagorean theorem. | |
| ^2 | Absolute square |
| Phase angle of a complex value |
The power cepstrum has main applications in analysis of sound and vibration signals. It is a complementary tool to spectral analysis.
Sometimes it is also defined as:
Other notations are possible due to the fact that the log of the power spectrum is equal to the log of the spectrum if a scaling factor 2 is applied:
which provides a relationship to the real cepstrum (see below).
Further, it shall be noted, that the final squaring operation in the formula for the power spectrum is sometimes called unnecessary and therefore sometimes omitted.
The real cepstrum is directly related to the power cepstrum:
It is derived from the complex cepstrum (defined below) by discarding the phase information (contained in the imaginary part of the complex logarithm). It has a focus on periodic effects in the amplitudes of the spectrum:
As is complex the log-term can be also written with as a product of magnitude and phase, and subsequently as a sum. Further simplification is obvious, if log is a natural logarithm with base e:
Therefore: The complex cepstrum can be also written as:R.B. Randall:, "A history of cepstrum analysis and its application to mechanical problems", (PDF) in: Mechanical Systems and Signal Processing, Volume 97, December 2017 (Elsevier).
The complex cepstrum retains the information about the phase. Thus it is always possible to return from the quefrency domain to the time domain by the inverse operation:
where b is the base of the used logarithm.
Main application is the modification of the signal in the quefrency domain (liftering) as an analog operation to filtering in the spectral frequency domain. An example is the suppression of echo effects by suppression of certain quefrencies.
The phase cepstrum (after phase spectrum) is related to the complex cepstrum as
The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the cepstrum.
"Predictive decomposition of time series with applications to seismic exploration", E. A. Robinson MIT report 1954; Geophysics 1967 vol. 32, pp. 418–484;
"Use of the kepstrum in signal analysis", M. T. Silvia and E. A. Robinson, Geoexploration, volume 16, issues 1–2, April 1978, pages 55–73.
"A kepstrum approach to filtering, smoothing and prediction with application to speech enhancement", T. J. Moir and J. F. Barrett, Proc. Royal Society A, vol. 459, 2003, pp. 2957–2976.
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.
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 quefrency domain and then converting back to the frequency domain, resulting in a modified signal, i.e. with signal echo being reduced.
The cepstrum is a representation used in homomorphic signal processing, to convert signals combined by convolution (such as a source and filter) into sums of their cepstra, for linear separation. In particular, the power cepstrum is often used as a 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. The cepstrum is useful in these applications because the low-frequency periodic excitation from the and the formant filtering of the vocal tract, which convolve in the time domain and multiply in the frequency domain, are additive and in different regions in the quefrency domain.
Note that a pure sine wave can not be used to test the cepstrum for its pitch determination from quefrency as a pure sine wave does not contain any harmonics and does not lead to quefrency peaks. Rather, a test signal containing harmonics should be used (such as the sum of at least two sines where the second sine is some harmonic (multiple) of the first sine, or better, a signal with a square or triangle waveform, as such signals provide many overtones in the spectrum.).
An important property of the cepstral domain is that the convolution of two signals can be expressed as the addition of their complex cepstra:
Recently, cepstrum-based deconvolution was used on surface electromyography signals, to remove the effect of the stochastic impulse train, which originates an sEMG signal, from the power spectrum of the sEMG signal itself. In this way, only information about the motor unit action potential (MUAP) shape and amplitude was maintained, which was then used to estimate the parameters of a time-domain model of the MUAP itself.G. Biagetti, P. Crippa, S. Orcioni, and C. Turchetti, “Homomorphic deconvolution for muap estimation from surface emg signals,” IEEE Journal of Biomedical and Health Informatics, vol. 21, no. 2, pp. 328– 338, March 2017.
A short-time cepstrum analysis was proposed by Schroeder and Noll in the 1960s for application to pitch determination of human speech. A. Michael Noll and Manfred R. Schroeder, "Short-Time 'Cepstrum' Pitch Detection," (abstract) Journal of the Acoustical Society of America, Vol. 36, No. 5, p. 1030A. Michael Noll (1964), “Short-Time Spectrum and Cepstrum Techniques for Vocal-Pitch Detection”, Journal of the Acoustical Society of America, Vol. 36, No. 2, pp. 296–302.A. Michael Noll (1967), “Cepstrum Pitch Determination”, Journal of the Acoustical Society of America, Vol. 41, No. 2, pp. 293–309.
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