4.6 Evaluation of Discrete Cepstrum Estimation

The evaluation of discrete cepstrum spectral envelope estimation described in section 3.4 serves to make sure that the developed algorithm performs well. Especially the property that the input points don't have to be regularly spaced can lead to large gaps and thus too much freedom for the envelope, such that there is a risk of bogus humps where the envelope rises several dB over the desired level. If such a hump occurs in one time frame only, it can be audible as an artefact (an impulse) in resynthesis.

To check that such humps don't occur with the developed algorithm, a large corpus of recordings of the singing voice has been evaluated. It comprises 42 minutes of recordings of a female coloratura soprano and a male counter tenor made for the film Farinelli [DGR94]. The corpus covers the upper frequency range, where the risk for humps is greatest.

The difficulty in evaluation is that the test if the estimated spectral envelope is well-behaved has to be automated to be feasible, but how can this property of being well-behaved be described to the automaton? If we knew a formal description of a well-behaved spectral envelope, we would immediately implement it as an estimation algorithm. We can, however, compare the spectral envelope to something that is not eligible to be a spectral envelope, but is close, and is guaranteed to produce no humps. This something is the curve obtained by linear interpolation of the points in the frequency-amplitude plane. It is not a good spectral envelope because it has corners.

The simplest approach is to take the vertical distance between the linear interpolation and the estimated spectral envelope, as shown in figure 3.13. The histogram of absolute differences, which shows how their number is distributed, and the point of maximum difference are recorded and can be examined later, to see how the deviation can be avoided.

**Figure 3.13:** Principle of evaluation of the discrete cepstrum method by vertical deviation (left), and difference in area (right)
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A more sophisticated approach takes the area between the linear interpolation and estimated spectral envelope, as demonstrated in figure 3.13 (right). Because, for the human hearing, upward deviations are very annoying, while downward deviations are hardly noticed at all, it would suffice to consider positive areas.

$\begin{figure}\centerline{\epsfbox[114 282 540 515]{pics/evalvertical.eps}} <\end{figure}$

The evaluation showed some deviations first, which could be reduced by choosing a regularization factor of $\lambda = 0.0005$ , after this, no humps in sensitive frequency ranges and at high amplitudes could be observed. Figure 3.14 shows such a case, where a deviation of over 16 dB occurs close to 8000 Hz, but it comes from a sensible interpolation of the algorithm between extremely placed partials at low amplitudes, and can thus be ignored.