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
) by means of the *Partial Fraction Expansion*
(PFE) [452, p. 129]:

where

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
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.

- State Space to Modal Synthesis
- Force-Driven-Mass Diagonalization Example
- Typical State-Space Diagonalization Procedure
- Efficiency of Diagonalized State-Space Models

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University