of Microprocessors

574 MUSICAL ApPLICATIONS OF MICROPROCESSORS
the zeroes, an all-pole model is used. The filter resulting from all-pole linear
prediction analysis gives the "best" (least-square error) approximation possible
when constrained to the specified number of poles (twice the number of
resonances). Often, the appropriate number of resonances to use can be
determined from a physical knowledge of the natural sound source. In such
cases, the linear prediction model can be expected to give excellent results. In
cases in which the filter is arbitrary, a balance between resonance count and
result quality will have to be determined by experiment. If vocals are being
analyzed, three or four resonances are quite sufficient to capture the intelligence
of the words. Many more are necessary to characterize the timbre in
sufficient detail for recognition of the singer's identity. Figure 16-15 shows
the results of linear prediction analysis of a vowel sound with different
resonance counts.
An all-poles digital linear predierion calculation usually results in the
set of multiplier coefficients necessary for implementation of the filter in cannonical
form. These coefficients apply only to the feedback paths; the feedforward
coefficients, which implement zeroes, are zero for an all-poles model.
The number of coefficients from the calculation will be 2N+ 1, where N is
the number of resonances. The extra coefficient is an overall gain factor. The
cannonical form of the filter can be converted into cascade form with N
sections. The conversion is desirable because multiplier accuracy requirements
of the cascade form are much less than the cannonical form.
Note that linear predicti0n just gives the filter design, not the actual
formant frequencies and bandwidths. Although the latter can be determined
by analyzing the filter, they are not needed to utilize the filter for synthesis
with an arbitrary excitation function.
Homomorphic Analysis
In digital music synthesis using the natural sound model of Fig.
16-14, all that is really needed is a plot of the filter's amplitude response.
With such a plot, the methods of arbitrary filter implementation discussed in
Chapter 14 can be used to apply the filter to a different excitation funerion.
However, a typical spectrum plot such as the one in Fig. 16-1GA shows
effects due to discrete harmonics of the excitation function as well as general
trends due'to the system function. In homomorphic analysis, the goal is to
obtain an amplitude response curve of the system function independent of the
characteristics of the excitation function.
From examination of the overall spectrum, it is obvious that what is
desired is a "smoothed" plot of the spectrum that retains the general spectral
shape but suppresses the individual harmonic "noise." This smoothing may
be accomplished by a moving average, which is actually a digital low-pass
transversal filter, or by other low-pass filters applied to the frequency sample
sequence just like they would be normally applied to a time sample sequence.