of Microprocessors

SOURCE-SIGNAL ANALYSIS 573
and some, such as the violin, can have significant antiresonant valleys as
well.
In technical terms, the resonant peaks are called poles of the system
function while the notches, if present, are termed zeroes. In speech and
music work, the zeroes are frequently ignored and the poles are called formants.
The goal of formant estimation is determination of the number of
formants, their resonant frequencies, and possibly their bandwidth or Q
factors. This information represents the characteristics of the filter, which is
usually a majority portion of the analyzed sound's timbre. Musical applications
of formant analysis seek the design of a filter with the same formant
structure as the analyzed sound. Different synthesized excitation functions
are then modified by the filter. The resulting sound has the most prominent
characteristics of both. For example, if vocal vowel sounds are used for the
formant analysis and the resultant filter is driven with a square-wave excitation
function, the vowel quality would be retained but with the characteristically
hollow timbre of square waves!
Perhaps the most straightforward method of formant analysis is to scan
the amplitude spectrum and find the center points of high-amplitude clusters
of harmonics. The center points are the formant frequencies. The bandwidths
may be estimated by noting the 3-dB points on either side ofeach peak. This
method works well only if the harmonics are dense compared to the
bandwidths of the formants, which means a low excitation frequency. This is
because the excitation function harmonics effectively sample the filter response
curve and insufficiently dense sampling leads to aliasing and incorrect
conclusions regarding the response curve. Even if the sampling is theoretically
dense enough, interpolation may have to be used to increase it further so
that the center point and 3-dB points of the peaks can be accurately determined.
Note that the original spectral analysis need not resolve the harmonics
themselves to be useful for this method of formant analysis.
Linear Prediction
A more refined method of formant analysis is called linear prediction
because, by specifying the filter involved in the original sound generation,
one is able to predict what future time samples of the sound are likely to be.
Although originally developed for the efficient transmission of speech signals
over limited bandwidth channels, linear prediction is useful in music synthesis
because it results in an actual design of the filter represented by the system
function. Unforrunately, the calculations are too involved to be covered here
in simple terms but the general characteristics of linear prediction can be
discussed. Additional reference material is listed in the bibliography.
In linear prediction, the actual filter used to generate the sound being
analyzed is approximated by a filter having a specified number of poles and
zeroes. Often, because of computational difficulties encountered in handling