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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Introduction to Physical Signal Models
       Physical Models
          Modal Representation

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Modal Representation

One of the filter structures introduced in Book II [449, p. 209] was the parallel second-order filter bank, which may be computed from the general transfer function (a ratio of polynomials in $ z$) by means of the Partial Fraction Expansion (PFE) [449, p. 129]:

$\displaystyle H(z) \isdefs \frac{B(z)}{A(z)} \eqsp \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$ (2.12)

where

\begin{eqnarray*} B(z) &=& b_0 + b_1 z^{-1}+ b_2z^{-2}+ \cdots + b_M z^{-M}\\ A(z) &=& 1 + a_1 z^{-1}+ a_2z^{-2}+ \cdots + a_N z^{-N},\quad M<N \end{eqnarray*}

The PFE Eq.$ \,$(1.12) expands the (strictly proper2.10) transfer function as a parallel bank of (complex) first-order resonators. When the polynomial coefficients $ b_i$ and $ a_i$ are real, complex poles $ p_i$ and residues $ r_i$ occur in conjugate pairs, and these can be combined to form second-order sections [449, p. 131]:

\begin{eqnarray*} H_i(z) &=& \frac{r_i}{1-p_iz^{-1}} + \frac{\overline{r_i}}{1-\... ..._i-\theta_i)z^{-1}}{1-2R_i\,\cos(\theta_i)z^{-1}+ R_i^2 z^{-2}}. \end{eqnarray*}

where $ p_i\isdeftext R_ie^{j\theta_i}$ and $ r_i\isdeftext G_ie^{j\phi_i}$. Thus, every transfer function $ H(z)$ with real coefficients can be realized as a parallel bank of real first- and/or second-order digital filter sections, as well as a parallel FIR branch when $ M\ge N$.

As we will develop in §8.5, modal synthesis employs a ``source-filter'' synthesis model consisting of some driving signal into a parallel filter bank in which each filter section implements the transfer function of some resonant mode in the physical system. Normally each section is second-order, but it is sometimes convenient to use larger-order sections; for example, fourth-order sections have been used to model piano partials in order to have beating and two-stage-decay effects built into each partial individually [30,29].

For example, if the physical system were a row of tuning forks (which are designed to have only one significant resonant frequency), each tuning fork would be represented by a single (real) second-order filter section in the sum. In a modal vibrating string model, each second-order filter implements one ``ringing partial overtone'' in response to an excitation such as a finger-pluck or piano-hammer-strike.


Subsections
Previous: Transfer Functions
Next: State Space to Modal Synthesis

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About the Author: 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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