A cepstrum (pronounced /ˈkɛpstrəm/) 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. Not to be confused with the NATO phonetic alphabet, which has also informally been called the “International Phonetic Alphabet”. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The power spectrum is a plot of the portion of a signals power (energy per unit time) falling within given frequency bins. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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

• verbally: the cepstrum (of a signal) is the FT of the log (with unwrapped phase) of the FT (of a signal)

• 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 → 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. In signal processing, a general sinusoidal signal with constant frequency is defined to be where is the amplitude, is phase, is called the angular frequency (usually radians/second), and is the frequency (usually in hertz or cycles/second). ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... In signal processing, a general sinusoidal signal with constant frequency is defined to be where is the amplitude, is phase, is called the angular frequency (usually radians/second), and is the frequency (usually in hertz or cycles/second). ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Partial plot of a function f. ...

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.

There are many ways to calculate the cepstrum. Some of them need a phase-wrapping algorithm; others do not.

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. ECHO is a German music award granted every year by the Deutsche Phono-Akademie (an association of recording companies). ... An earthquake is a natural phenomenon that results from the sudden release of stored energy in the Earths crust that creates seismic waves. ... The Massive Ordnance Air Blast (MOAB) bomb produced in the United States. ... This long range Radar antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll[1]. Radar is a system that uses radio waves to determine and map the location, direction, and/or speed...

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 frequency bands. The result is called the 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. The mel scale, proposed by Stevens, Volkman and Newman in 1937 is a scale of pitches judged by listeners to be equal in distance one from another. ... Mel Frequency Cepstral Coefficients (MFCCs) are coefficients that represent audio, based on perception. ... An algorithm that attempts to detect the pitch of a periodic signal, usually digitized sound. ... Music information retrieval or MIR is the interdisciplinary science of retrieving information from music. ...

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 also related to the homomorphic sound theory.

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 the their cepstra. x₁ * x₂ − − > 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.

In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

Further reading

• D. G. Childers, D. P. Skinner, R. C. Kemerait, "The Cepstrum: A Guide to Processing," Proceedings of the IEEE, Vol. 65, No. 10, October 1977, pp. 1428-1443.