[music-dsp] Digital Coding

[Ethan Duni]

Actually, I think I spoke too quickly. It's important to distinguish between
complex and real cepstra here. In practice, people tend to use real cepstra
for speech recognition. In this case, the cepstral coefficients are often
computed via the FFT. The common modification is to use what's called
"mel-scale" cepstra. The mel-scale is a perceptually motivated frequency
scale (basically you sample less finely as frequency increases). This in
turn, is accomplished by taking a long FFT and then averaging together
adjacent bins according the the mel-scale.

However, there is a recursion that allows computation of the complex cepstra
from LPC coefficients. I'm not sure if this gets used in practice (as
opposed to in analytical developments), and I haven't seen a version for
computing real cepstra.  I don't have a good source for this recursion, but
it is obtained by assuming that the true spectrum has the form of your LPC
approximation and then plugging this into the definition of complex
cepstrum. Then, with a little bit of algebra and the application of some
Z-transform properties, you can get the recursion. The actual form of the
recursion is in terms of a convolution of the cepstral coefficients with the
LPC coefficients; this is usable because you know ahead of time that the LPC
coefficients are equal to 0 outside {0,...,M}, where M is the LPC model
order.

Ethan

> Now that is *truly* interesting. How does it happen? From my (naïve and
> perhaps a bit outdated) Oppenheim&Schafer background, it'd seem more
> natural to do an FFT+log+FFT cycle, instead of going with LPC. Is there
> some nifty trick that gives us cepstra from LPC? Does it per chance help
> us avoid the nasty unwrapping problems with the complex logarithm?
> -- 
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