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 ) by means of the Partial Fraction Expansion (PFE) [452, p. 129]:
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 and are real, complex poles and residues occur in conjugate pairs, and these can be combined to form second-order sections [452, p. 131]:
where and . Thus, every transfer function 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 .
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.