# How to calculate Cepstral Peak Prominence (Smoothed) for speech signal - when to take logs

I understand the main steps of finding smoothed cepstral peak prominence, but am confused on some specifics:

1. You first take the log of the Fourier transform of the signal, getting the spectrum.

• I'm not sure if it's supposed to be the magnitude spectrum or power spectrum (which I believe is taking the magnitude vs. the magnitude squared) - is it correct that it doesn't matter because it'll just be off by a constant due to the log?
• Do I first convert the spectrum to decibels before taking the log? I think I'm not supposed to, because that would be a log of a log. But the papers I've been reading have gotten me confused about this.
• Also, I know it would just make things off by a constant, but is there a convention to use the natural log vs. log_10?
2. You take the inverse Fourier transform of the log-spectrum from step #1 to get the cepstrum.

• Same question here, only this time I'm more worried that it matters - do you take its magnitude or power?
• Do you "convert to decibels" here with a log_10? I put it in quotes, because at this point I'm extremely confused what the actual units of this are. For all I know, it's already in decibels since the previous step had a log of the fourier transform. I suspect magnitude vs. power doesn't matter if you are supposed to "convert to decibels" - off by a constant.
• There's a smoothing operation of the cepstrum over time and quefrency. If converting to decibels is correct, does that smoothing happen before or after the decibel conversion?
3. Then, you take a linear regression of #2. This is where it's important that I know the answers to the previous two questions, since squares and logs would mess with the linearity.

• (Not as important) I read something in one paper about only using quefrencies from ~1ms and higher for the regression. Is this standard practice?
4. Last, you find the maximum peak of the cepstrum (#2) and subtract it from the value of the regression (#3) at that same quefrency.

• Papers seem to indicate this final value shouldn't depend on the loudness of the signal, because subtracting from the regression line normalizes it. Is this actually true? Is there an intuitive explanation for that? If the units are actually decibels, I guess you'd expect the difference between the peak decibel and the average to be the same no matter how loud the signal is overall? I don't quite get that though, like why that wouldn't be higher for a louder signal.