# Cepstrum calculus disambiguation

Learning the cepstrum analysis for speech recognition, I have met two different definitions of cepstrum (for discrete signals):

1. $F^{-1}(ln|F(x[n])|)$ . That is, the cepstrum is the inverse Fourier transform of the logarithm of the magnitude of the Fourier transform of the signal $x[n]$. So, cepstrum coefficients have the dimension of time.
2. $F(ln|F(x[n])|)$. That is, the cepstrum coefficients are the Fourier transform of the logarithm of the magnitude of the Fourier transform of the signal $x[n]$. Hence, the spectrum of the spectrum. From here, the name of quefrency to indicate the variable of the cepstrum coefficients.

The first equation seems the most used, and intuitively makes more sense to me w.r.t the source separation for speech signal processing. However, I met the second definition quite few times, causing me a great confusion.

How the conceptually differ these two definitions? And which is the most correct one? Always in regards to speech signal processing.

• Hi, just a nudge. If you are happy with the answer, could you mark it as accepted?
– jojeck
Nov 26, 2019 at 10:07

## 1 Answer

A common technique for computing the inverse fft is to invert the imaginary part of the input array, perform a forward fft and then invert the imaginary part of the output array. In the case of the Cepstrum, the input to the ifft is real-valued (due to the absolute value function) and also symmetric. In this case there is no imaginary part on the input to invert, and the symmetry implies that the ifft will produce no imaginary components on the output that would need to be inverted. Hence in this case the fft and the ifft are fully equivalent.