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Physical Audio Signal Processing
    Transfer Function Models
      
             Modal Expansion

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Parallel Second-Order Expansion

The general parallel second-order model can be written

$\displaystyle H(z)=\sum_{k=1}^N \frac{b_k + c_k z^{-1}}{1 - 2 r_k \cos(\theta_k) z^{-1} + r_k^2 z^{-2}}, $

where typically $ r_k = \exp(-\pi b_k T)$, $ \theta_k = 2\pi f_k T$, and $ T$ is the sampling period in seconds. The numerator coefficients are related to the fundamental mode parameters by

\begin{eqnarray*} b_k & = & 2 a_k \cos(\phi_k) \nonumber \\ c_k & = & -2 a_k r_k \cos(\phi_k - \theta_k) \nonumber \end{eqnarray*}

While the direct modal expansion technique is fully general, it does not take advantage of structure in the modal tunings common in musical instruments.

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Next: Delay Loop Expansion
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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