Cepstrum

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

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

Operations on cepstra are labelled quefrency alanysis, liftering, or cepstral analysis.

Contents

Origin and definition

The power cepstrum was defined in a 1963 paper by Bogert et al.[1] It may be defined

  • verbally: the power cepstrum (of a signal) is the squared magnitude of the Fourier transform of the logarithm of the squared magnitude of the Fourier transform of a signal[2]
  • mathematically: power cepstrum of signal =\left|\mathcal{F}\left\{\mbox{log}(\left|\mathcal{F}\left\{ f(t)  \right\}\right|^2)\right\}\right|^2
  • algorithmically: signal → FT → abs() → square → log → FT → abs() → square → power cepstrum

The complex cepstrum was defined by Oppenheim in his development of homomorphic system theory.[3] It may be defined

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

The real cepstrum uses the logarithm function defined for real values. The real cepstrum is related to the power via the relationship (4 * real cepstrum)^2 = power cepstrum, and is related to the complex cepstrum as real cepstrum = 0.5*(complex cepstrum + time reversal of complex cepstrum).

The complex cepstrum uses the complex logarithm function defined for complex values. The phase cepstrum is related to the complex cepstrum as phase spectrum = (complex cepstrum - time reversal of complex cepstrum).^2

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.

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