# Questions on Cepstral Analysis

I have a few questions regarding cepstral analysis, that the numerous articles and papers I've read on the topic didn't answer.

What I understood: The cepstrum captures the periodicity of harmonics in a spectrum.

My questions: In articles treating about default detection in gearboxes, they say that the presence of sidebands around the ToothMesh frequency (corresponding to +- both shafts rotating frequencies) in the spectral domain leads to the presence of two peaks at corresponding quefrencies for both of these rotating frequencies on the cepstrum.

• Are there also peaks in the cepstrum corresponding to the presence of toothmesh harmonics? (this is never addressed). They show TM, 2xTM, 3xTM frequencies on the spectrum and I would expect to see a peak at TM quefrency on the cepstrum but this is never shown.
• I struggle to understand why the sidebands are "used" by the cepstrum when they are not spaced by int (let's say you have sidebands at +-85Hz and +-15Hz and TM=1000Hz so for example TM-85Hz=915Hz and 2xTM-85=1915Hz but 1915/915 is not an integer). I would expect to also have on the spectrum proper peaks at 85Hz and 15Hz together with some harmonics of these frequencies. Is that periodicity also used by the cepstrum? Is this adding with a periodicity found in the repetition of sidebands?

For applications linked to speech recognition, they say that the use of the log in the cepstrum calculation formula is very interesting as it allows to separate the source excitation from the vocal tract transfer function.

• How is that true? I get that log(axb)=log(a)+log(b) but I don't see how it is leveraged to the purpose of separating two contributions. The inverse FFT is applied to the entire log.

Sorry if I used terms incorrectly. Also, my thoughts on the topic are a bit fuzzy, to say the least. Many thanks in advance for your help.

Cheers

Antoine

• @Antoine101 That’s where $\log(X\cdot H) = \log(X) + \log(H)$ comes in play. Speech is $(x * h)(t)$ in the time domain. The source excitation in this case is $x(t)$ (harmonics, etc) and the vocal tract is $h(t)$(envelope).