Dictionary:Cepstrum

(sep’ strum) more commonly (kep’ strum) The inverse Fourier transform of a logarithmic frequency domain representation of a function. Let $\leftrightarrow$ indicate a Fourier transform operation. If $g(t)\leftrightarrow G(\omega )=|G(\omega )|e^{i\phi (\omega )}$, the cepstrum $g(\zeta )$ is

$g(\zeta )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left[\ln \left(|G(\omega )|\right)+i\phi (\omega )\right]e^{-i\omega \zeta }\;d\omega .$

Here, $|G(\omega )|$ and $\phi (\omega )$ are, respectively the modulus and the unwrapped phase of $G(\omega )$. Here we use $i={\sqrt {-1}}$, though electrical engineers may use $j$ for this quantity.

The phase must be unwrapped, because the complex valued function $G(\omega )$ must be analytic for further analysis to be valid.

Forward Cepstral transform

The transform is usually carried out in three steps:

1.) The forward Fourier transform of $g(t)\leftrightarrow G(\omega )$

$G(\omega )=\int _{0}^{\infty }g(t)e^{i\omega t}\;dt$

where we have selected the positive sign for the exponent in the exponential of the forward temporal transform. Here the limits of integration reflect that fact that most time varying signals in exploration geophysics are causal functions, having nonzero values only for $t>0.$

2.) The natural log is taken of the $G(\omega )$ and the phase $\phi (\omega )$ is the phase after it is unwrapped,

$\ln[G(\omega )]=\ln |G(\omega )|+i\phi (\omega ).$

These constitute the real and imaginary parts of the function in the log frequency domain.

3.) The (complex to real) inverse Fourier transform is applied to the log frequency domain representation of $g(t)$

$g(\zeta )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left[\ln[G(\omega )]=\ln |G(\omega )|+i\phi (\omega )\right]e^{-i\omega \zeta }\;d\omega .$

The quantity $g(\zeta )$ is called the cepstrum of $g(t)$ and the quantity $\zeta$ is called the quefrency. We note that the cepstrum is the complex cepstrum and is therefore the sum of a real and imaginary part, and may be written in terms of a modulus and a phase $g(\zeta )={\mbox{Re}}\;g+i{\mbox{Im}}\;g=|g(\zeta )|e^{i\psi (\zeta )}.$ Here, the phase $\psi (\zeta )$ is called the saphe.

We note that the exponent sign conventions may be reversed, with a minus sign on the forward Fourier transform, and a positive sign on the phase of the inverse Fourier transform.

Inverse Cepstral transform

The inverse cepstral transform is the reverse operation

1.) The forward transform $g(\zeta )\leftrightarrow \Gamma (\omega ))$

$\Gamma (\omega )=\int _{-\infty }^{\infty }g(\zeta )e^{i\omega \zeta }\;d\zeta .$

Here ${\mbox{Re}}\Gamma (\omega )+i{\mbox{Im}}\Gamma (\omega )=\ln |G(\omega )|+i\phi (\omega )$

2.) $\Gamma (\omega )$ is exponentiated to recover $G(\omega )$

$\exp(\Gamma (\omega ))=|G(\omega )|e^{i\phi (\omega )}=G(\omega )$

3.) and, finally, the inverse transform $G(\omega )\leftrightarrow g(t)$ is performed

$g(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }G(\omega )e^{-i\omega t}\;d\omega$

The cepstral domain is often indicated by the hat. The transform can also be expressed as z-transforms; see Sheriff and Geldart (1995, 298–299; 554–555).

Applications of the cepstral representation include digital filtering echoes out of human speech, and homomorphic signal processing.