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: and therefore: 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 is sometimes called unnecessary and therefore sometimes omitted. The real cepstrum is directly related to the power cepstrum: It is derived from the complex cepstrum (defined below) by discarding the phase information (contained in the imaginary part of the complex logarithm). It has a focus on periodic effects in the amplitudes of the spectrum: Complex cepstrum The complex cepstrum was defined by Oppenheim in his development of homomorphic system theory. The formula is provided also in other literature. As is complex the log-term can be also written with 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: Therefore: The complex cepstrum can be also written as: 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: 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. An example is the suppression of echo effects by suppression of certain quefrencies. The phase cepstrum (after phase spectrum) is related to the complex cepstrum as phase spectrum = (complex cepstrum − time reversal of complex cepstrum)2. Related concepts The independent variable of a cepstral graph is called the quefrency. The quefrency is a measure of time, though not in the sense of a signal in the time domain. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a fundamental frequency that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic and the period corresponds to the fundamental frequency, since harmonics are integer multiples of the fundamental frequency. The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has