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. Origin The concept of the cepstrum was introduced in 1963 by B. P. Bogert, M. J. Healy, and J. W. Tukey. It serves as a tool to investigate periodic structures in frequency spectra. 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: transformation of a signal from the time domain to the frequency domain computation of the logarithm of the spectral amplitude transformation to frequency domain, where the final independent variable, the quefrency, has a time scale. Types The cepstrum is used in many variants. Most important are: power cepstrum: The logarithm is taken from the "power spectrum" complex cepstrum: The logarithm is taken from the spectrum, which is calculated via Fourier analysis The following abbreviations are used in the formulas to explain the cepstrum: Power cepstrum The "cepstrum" was originally defined as power cepstrum by the following relationship: 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: Due to this formula, the cepstrum is also sometimes called the spectrum of a spectrum. It can be shown that both formulas are consistent with each other as the frequency spectral distribution remains the same, the only difference being a scaling factor which can be applied afterwards. Some articles prefer the second formula.