Editing Cepstrum (section)

===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>.