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Pole Mapping with Optimal Zeros

We saw in the preceding sections that both the impulse-invariant and the matched-$ z$ transformations map poles from the left-half $ s$ plane to the interior of the unit circle in the $ z$ plane via

$\displaystyle z_i = e^{s_i T} \protect$ (9.8)

where $ s_i$ is the location of the $ i$ th pole in the $ s$ plane (assumed to lie in the strip $\vert\mbox{im\ensuremath{\left\{s\right\}}}\vert<\pi/T$ 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-$ z$ 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 [452, p. 393],[275,299].

It is hard to do better Eq.(8.8) as a pole mapping from $ s$ to $ z$ , when aliasing is avoided, because it preserves both the resonance frequency and bandwidth for a complex pole [452]. 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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University