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 alanysis), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with kepstrum.
right|thumb|Steps in forming cepstrum from time history
Origin
The concept of the cepstrum was introduced in 1963 by B. P. Bogert, M. J. Healy, and J. W. Tukey. Such effects are related to noticeable echos or reflections in the signal, or to the occurrence of harmonic frequencies (partials, overtones). Mathematically it deals with the problem of deconvolution of signals in the frequency space.
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.
General definition
The cepstrum is the result of following sequence of mathematical operations:
- Fourier transformation of a signal from the time domain to the frequency domain
- computation of the logarithm of the spectral amplitude
- Inverse Fourier transformation to time domain, where the final independent variable, the quefrency, has a time scale.
:<math>C_{r}=\mathcal{F}^{-1}\left\{\log(\mathcal{|\mathcal{F}\{f(t) \}|})\right\}</math>
Complex cepstrum
The complex cepstrum was defined by Oppenheim in his development of homomorphic system theory. The formula is provided also in other literature.
:<math>C_{c}=\mathcal{F}^{-1}\left\{\log_e(\mathcal{|F|}) + i\varphi\right\}</math>
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: 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 fundamental frequency 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 fundamental frequency, since harmonics are integer multiples of the fundamental frequency.
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.
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.
Interpretation
The cepstrum can be seen as information about the 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 determine the fundamental frequency of human speech and to analyze radar signal returns. Cepstrum pitch determination is particularly effective because the effects of the vocal excitation (pitch) and vocal tract (formants) are additive in the logarithm of the power spectrum and thus clearly separate.
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.
A short-time cepstrum analysis was proposed by Schroeder and Noll in the 1960s for application to pitch determination of human speech.
References
Further reading
- "Speech Signal Analysis"
- "Speech analysis: Cepstral analysis vs. LPC", www.advsolned.com
- "A tutorial on Cepstrum and LPCCs"