Re: Cepstrum computation

["Alain de Cheveigne'" <Alain.de.Cheveigne@xxxxxxxx>]

Hi Eric,

Thanks for your follow-up.

The point I tried to make was that the singularities are harmless if they
have "finite weight".  The cepstrum can then be defined even if the log
spectrum is not (in places).  I suspect this to be true in practice, as a
consequence of the spectrum being derived from a portion of signal limited
in time, but I don't know how to prove it...   If anyone does, it would be
a useful contribution.

But this applies only for a continuous representation.  It doesn't solve
the problem for a digital representation, as you pointed out, because a
sample that is unlucky enough to fall at the singularity carries "infinite"
weight.

The thing that troubles me, supposing the above conjecture is true, is that
the problem can be avoided with a continuous representation, but not with a
discrete (sampled) representation.  That implies a fundamental difference
between digital and continuous representations.  Normally we expect the two
to match if the sampling rate is sufficient.  How is such a mismatch
possible?

What I suspect is that the condition for adequate sampling of the signal
(that it be band-limited to half the sampling rate) might not necessarily
guarantee adequate sampling of the log spectrum.  If so, the digital
cepstrum is not a well defined beast.  For example, there's no guarantee
that values of the digital cepstrum correspond to samples of the continuous
cepstrum of the same signal.  Comments, anyone?


Alain

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Alain de Cheveigne'
CNRS/IRCAM, 1 place Stravinsky, 75004, Paris.
phone: +33 1 44784846, fax: 44781540, email: cheveign@ircam.fr
http://www.ircam.fr/equipes/perception/cheveign
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