Pole Mapping with Optimal Zeros
We saw in the preceding sections that both the impulse-invariant and
the matched-
transformations map poles from the left-half 
plane
to the interior of the unit circle in the plane via
where
is the location of the 
th pole in the plane (assumed
to lie in the strip
to avoid aliasing). The
zeros, on the other hand, were different because the impulse-invariant
method started with the partial fraction expansion while the
matched- transformation started with the factored form of the
transfer function.
Therefore, an obvious generalization is to map the poles according to
Eq.
(8.8), but compute the zeros in some optimal way, such as by
Prony's method [449, p. 393],[273,297].
It is hard to do better Eq.(8.8) as a pole mapping from to
, when aliasing is avoided, because it preserves both the resonance
frequency and bandwidth for a complex pole [449]. Therefore,
good practical modeling results can be obtained by optimizing the
zeros (residues) to achieve audio criteria given these fixed poles.
Alternatively, only the least-damped poles need be constrained in this
way, e.g., to fix and preserve the most important resonances of a
stringed-instrument body or acoustic space.
Next Section:
Modal Expansion
Previous Section:
Matched Z Transformation
