Power cepstrum

The "cepstrum" was originally defined as power cepstrum by the following relationship:^[1][3]

${\displaystyle C_{p}=\left|{\mathcal {F}}^{-1}\left\{\log \left(\left|{\mathcal {F}}\{f(t)\}\right|^{2}\right)\right\}\right|^{2}}$

The power cepstrum has main applications in analysis of sound and vibration signals. It is a complementary tool to spectral analysis.^[2]

Sometimes it is also defined as:^[2]

${\displaystyle C_{p}=\left|{\mathcal {F}}\left\{\log \left(\left|{\mathcal {F}}\{f(t)\}\right|^{2}\right)\right\}\right|^{2}}$

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 ^[2] which can be applied afterwards. Some articles prefer the second formula.^[2][4]

Other notations are possible due to the fact that the log of the power spectrum is equal to the log of the spectrum if a scaling factor 2 is applied:^[5]

${\displaystyle \log |{\mathcal {F}}|^{2}=2\log |{\mathcal {F}}|}$

and therefore:

${\displaystyle C_{p}=\left|{\mathcal {F}}^{-1}\left\{2\log |{\mathcal {F}}|\right\}\right|^{2},{\text{ or}}}$

${\displaystyle C_{p}=4\cdot \left|{\mathcal {F}}^{-1}\left\{\log |{\mathcal {F}}|\right\}\right|^{2},}$

which provides a relationship to the real cepstrum (see below).

Further, it shall be noted, that the final squaring operation in the formula for the power spectrum ${\displaystyle C_{p}}$ is sometimes called unnecessary^[3] and therefore sometimes omitted.^[4][2]

The real cepstrum is directly related to the power cepstrum:

${\displaystyle C_{p}=4\cdot C_{r}^{2}}$

It is derived from the complex cepstrum (defined below) by discarding the phase information (contained in the imaginary part of the complex logarithm).^[4] It has a focus on periodic effects in the amplitudes of the spectrum:^[6]

${\displaystyle C_{r}={\mathcal {F}}^{-1}\left\{\log({\mathcal {|{\mathcal {F}}\{f(t)\}|}})\right\}}$

Complex cepstrum

The complex cepstrum was defined by Oppenheim in his development of homomorphic system theory.^[7][8] The formula is provided also in other literature.^[2]

${\displaystyle C_{c}={\mathcal {F}}^{-1}\left\{\log({\mathcal {F}}\{f(t)\})\right\}}$

As ${\displaystyle {\mathcal {F}}}$ is complex the log-term can be also written with ${\displaystyle {\mathcal {F}}}$ as a product of magnitude and phase, and subsequently as a sum. Further simplification is obvious, if log is a natural logarithm with base e:

${\displaystyle \log({\mathcal {F}})=\log({\mathcal {|F|\cdot e^{i\varphi }}})}$

${\displaystyle \log _{e}({\mathcal {F}})=\log _{e}({\mathcal {|F|}})+\log _{e}(e^{i\varphi })=\log _{e}({\mathcal {|F|}})+i\varphi }$

Therefore: The complex cepstrum can be also written as:^[9]

${\displaystyle C_{c}={\mathcal {F}}^{-1}\left\{\log _{e}({\mathcal {|F|}})+i\varphi \right\}}$

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:^[2][3]

${\displaystyle f(t)={\mathcal {F}}^{-1}\left\{b^{\left({\mathcal {F}}\{C_{c}\}\right)}\right\},}$

where b is the base of the used logarithm.

Main application is the modification of the signal in the quefrency domain (liftering) as an analog operation to filtering in the spectral frequency domain.^[2][3] An example is the suppression of echo effects by suppression of certain quefrencies.^[2]

The phase cepstrum (after phase spectrum) is related to the complex cepstrum as

phase spectrum = (complex cepstrum − time reversal of complex cepstrum)².