Modal Representation

One of the filter structures introduced in Book II [452, 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) [452, p. 129]:

$$H(z) \isdefs \frac{B(z)}{A(z)} \eqsp \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$$ (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 proper^2.10) 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 [452, p. 131]:

$$\begin{eqnarray*} H_i(z) &\!=\!& \frac{r_i}{1-p_iz^{-1}} + \frac{\overline{r_i}}{1-\overline{p_i}z^{-1}} \eqsp \frac{r_i-r_i\overline{p_i}z^{-1}+\overline{r_i}-\overline{r_i} p_iz^{-1}}{(1-p_iz^{-1})(1-\overline{p_i}z^{-1})}\\ [5pt] &\!=\!& \frac{2\mbox{re}\left\{r_i\right\}-2\mbox{re}\left\{r_i\overline{p_i}\right\}z^{-1}}{1-2\mbox{re}\left\{p_i\right\}z^{-1}+ \left\vert p_i\right\vert^2 z^{-2}} \eqsp 2G_i\frac{\cos(\phi_i)-\cos(\phi_i-\theta_i)z^{-1}}{1-2R_i\,\cos(\theta_i)z^{-1}+ R_i^2 z^{-2}}. \end{eqnarray*}$$

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.