Editing Cepstrum (section)

===Power cepstrum===
The "cepstrum" was originally defined as '''power cepstrum''' by the following relationship:<ref name="Bogert_19632"/><ref name="Childers_1977" />

:<math>C_{p}=\left|\mathcal{F}^{-1}\left\{\log\left(\left|\mathcal{F}\{f(t)\}\right|^2\right)\right\}\right|^2</math>

The power cepstrum has main applications in analysis of sound and vibration signals. It is a complementary tool to spectral analysis.<ref name="Norton_2003" />

Sometimes it is also defined as:<ref name="Norton_2003" />

:<math>C_{p}=\left|\mathcal{F}\left\{\log\left(\left|\mathcal{F}\{f(t)\}\right|^2\right)\right\}\right|^2</math>
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 <ref name="Norton_2003" /> which can be applied afterwards. Some articles prefer the second formula.<ref name="Norton_2003" /><ref name="Randall_2002" />

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:<ref name="Beckhoff" />
:<math>\log |\mathcal{F}|^2  = 2 \log |\mathcal{F}| </math> 
and therefore:
:<math>C_{p}=\left|\mathcal{F}^{-1}\left\{2\log |\mathcal{F}|\right\}\right|^2, \text{ or} </math>
:<math>C_{p}=4\cdot\left|\mathcal{F}^{-1}\left\{\log |\mathcal{F}| \right\}\right|^2,</math>

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 <math>C_{p}</math> is sometimes called unnecessary<ref name="Childers_1977" /> and therefore sometimes omitted.<ref name="Randall_2002" /><ref name="Norton_2003" />

{{anchor|Real cepstrum}}The '''real cepstrum''' is directly related to the power cepstrum:

:<math>C_{p}=4\cdot C_{r}^2</math>

It is derived from the complex cepstrum (defined below) by discarding the phase information (contained in the [[imaginary part]] of the [[complex logarithm]]).<ref name="Randall_2002" />  It has a focus on periodic effects in the amplitudes of the spectrum:<ref>{{cite web|url=https://www.mathworks.com/help/signal/ref/rceps.html|title=Real cepstrum and minimum-phase reconstruction - MATLAB rceps}}</ref>

:<math>C_{r}=\mathcal{F}^{-1}\left\{\log(\mathcal{|\mathcal{F}\{f(t) \}|})\right\}</math>