Editing Cepstrum (section)

==Types==

The cepstrum is used in many variants. Most important are:
* power cepstrum: The logarithm is taken from the "power spectrum"
* complex cepstrum: The logarithm is taken from the spectrum, which is calculated via Fourier analysis

The following abbreviations are used in the formulas to explain the cepstrum:
{| class="wikitable"
|+
!Abbreviation
!Explanation
|-
|<math>f(t)</math>
|Signal, which is a function of time
|-
|<math>C</math>
|Cepstrum
|-
|<math>\mathcal{F}</math>
|[[Fourier transform]]: The abbreviation can stand i.e. for a [[continuous Fourier transform]], a [[discrete Fourier transform]] (DFT) or even a [[z-transform]], as the z-transform is a generalization of the DFT.<ref name="Childers_1977" />
|-
|<math>\mathcal{F}^{-1}</math>
|Inverse of the fourier transform
|-
|<math>\log(x)</math>
|[[Logarithm]] of ''x''. The choice of the base ''b'' depends on the user. In some articles the base is not specified, others prefer base 10 or&nbsp;''e''. The choice of the base has no impact on the basic calculation rules, but sometimes base ''e'' leads to simplifications (see "complex cepstrum").
|-
|<math>\left|x\right|</math>
|[[Absolute value]], or magnitude of a [[complex value]], which is calculated from real- and imaginary part using the [[Pythagorean theorem]].
|-
|<math>\left|x\right|^2</math>
|[[Absolute square]]
|-
|<math>\varphi</math>
|Phase angle of a [[complex value]]
|}

===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>

===Complex cepstrum===
The '''complex cepstrum''' was defined by Oppenheim in his development of homomorphic system theory.<ref name="AVOppenheim_1965">A. V. Oppenheim, "Superposition in a class of nonlinear systems" Ph.D. diss., Res. Lab. Electronics, M.I.T. 1965.</ref><ref name="AVOppenheim_1975">A. V. Oppenheim, R. W. Schafer, "Digital Signal Processing", 1975 (Prentice Hall).</ref> The formula is provided also in other literature.<ref name="Norton_2003" />

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

As <math>\mathcal{F}</math> is complex the log-term can be also written with <math>\mathcal{F}</math> as a product of magnitude and phase, and subsequently as a sum. Further simplification is obvious, if log is a [[natural logarithm]] with base&nbsp;''e'':

:<math>\log(\mathcal{F}) = \log(\mathcal{|F| \cdot e^{i\varphi}})</math>
:<math>\log_e(\mathcal{F}) = \log_e(\mathcal{|F|}) + \log_e(e^{i\varphi}) = \log_e(\mathcal{|F|}) + i\varphi</math>

Therefore: The complex cepstrum can be also written as:<ref name="Randall_2017">R.B. Randall:, [https://surveillance7.sciencesconf.org/conference/surveillance7/01_a_history_of_cepstrum_analysis_and_its_application_to_mechanical_problems.pdf "A history of cepstrum analysis and its application to mechanical problems"], (PDF) in: Mechanical Systems and Signal Processing, Volume 97, December 2017 (Elsevier).</ref>

:<math>C_{c}=\mathcal{F}^{-1}\left\{\log_e(\mathcal{|F|}) + i\varphi\right\}</math>

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:<ref name="Norton_2003" /><ref name="Childers_1977" />

:<math>f(t)=\mathcal{F}^{-1}\left\{b^\left(\mathcal{F}\{C_c\}\right)\right\},</math>

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.<ref name="Norton_2003" /><ref name="Childers_1977" /> An example is the suppression of echo effects by suppression of certain quefrencies.<ref name="Norton_2003" />

{{anchor|Phase cepstrum}}The '''phase cepstrum''' (after [[phase spectrum]]) is related to the complex cepstrum as
: phase spectrum = (complex cepstrum − time reversal of complex cepstrum)<sup>2</sup>.