Modal Representation, Cont'd

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

• 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 $M\ge N$

• 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

• Each section (mode) is typically second-order, but fourth-order sections are sometimes used as well (Chant, piano partials)

• In modal synthesis of vibrating strings, each second-order filter implements one ``ringing partial overtone'' in response to an excitation such as a finger-pluck or piano-hammer-strike