4.4 Discrete Cepstrum Spectral Envelope

Contrary to the previous two methods, LPC, which is computed from the signal, and cepstrum, which is computed from a spectral representation of the signal with points spaced regularly on the frequency axis, the discrete cepstrum spectral envelope is computed from discrete points in the frequency/amplitude plane. These points, which do not have to be regularly spaced in frequency, are the spectral peaks of a sound, which will most often be the sinusoidal partials found by additive analysis (see section 2.2).

As described at the end of sections 3.2 and 3.3, the LPC or cepstrum spectral envelopes will both exhibit the problem to descend down to the level of residual noise between partials which are spaced too far apart, as can be seen in figures 3.3 and 3.6.

The discrete cepstrum , to the contrary, will not care for anything going on in the signal except the partials. It will generate a smoothly interpolated curve which tries to link the partial peaks, as in figure 3.7.

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The following method to estimate the discrete cepstrum was developed
by Thierry Galas and Xavier Rodet in [GR90,GR91a,GR91b]. A
given set of spectral peaks (partials) with amplitudes *x*_{i} at
frequencies
,
defines a magnitude spectrum *X* as

We consider

where

and

Assuming a flat source spectrum
for all ,
all
we need to do is find the filter parameters *c*_{i} which minimize the
quadratic error *E* between the log spectra. This error criterion is
developed from the idea of a spectral distance.

To achieve this, one has to simply solve the matrix equation

where A is a matrix of order

and

The matrix *A* can be computed very efficiently by using an
intermediate vector *R* given by

so that

The matrix equation (3.18) can be efficiently solved applying the
Cholesky algorithm , which factorises *A* such that

where

The asymptotic complexity of the discrete cepstrum method described
above is
*O*(*np* + *p*^{3}), which means that the number of partials *n* is
not of a big concern, since the order is linear in *n*, but that the
order *p* has to be kept as small as possible, because of its cubic
influence.