### Dispersion Filter Design

A pure *dispersion filter* is an ideal allpass filter. That is,
it has a gain of 1 at all frequencies and only delays a signal in a
frequency-dependent manner. The need for such filtering in
*piano string models* is discussed in §9.4.1.

There is a good amount of literature on the topic of *allpass
filter design*. Generally, they fall into the categories of
*optimized parametric*, *closed-form parametric*, and
*nonparametric* methods. Optimized parametric methods can
produce allpass filters with optimal group-delay characteristics in
some sense [272,271]. Closed-form parametric
methods provide coefficient formulas as a function of a desired
parameter such as ``inharmonicity'' [368].
Nonparametric methods are generally based on measured signals and/or
spectra, and while they are suboptimal, they can be used to design
very large-order allpass filters, and the errors can usually be made
arbitrarily small by increasing the order
[551,369,42,41,1],
[428, pp. 60,172]. In music applications, it is usually the
case that the ``optimality'' criterion is unknown because it depends
on aspects of sound perception (see, for example,
[211,384]). As a result, perceptually
weighted nonparametric methods can often outperform optimal parametric
methods in terms of cost/performance [2].

In historical order, some of the allpass filter-design methods are as follows: A modification of the method in [551] was suggested for designing allpass filters having a phase delay corresponding to the delay profile needed for a stiff string simulation [428, pp. 60,172]. The method of [551] was streamlined in [369]. In [77], piano strings were modeled using finite-difference techniques. An update on this approach appears in [45]. In [340], high quality stiff-string sounds were demonstrated using high-order allpass filters in a digital waveguide model. In [384], this work was extended by applying a least-squares allpass-design method [272] and a spectral Bark-warping technique [459] to the problem of calibrating an allpass filter of arbitrary order to recorded piano strings. They were able to correctly tune the first several tens of partials for any natural piano string with a total allpass order of 20 or less. Additionally, minimization of the norm [271] has been used to calibrate a series of allpass-filter sections [42,41], and a dynamically tunable method, based on Thiran's closed-form, maximally flat group-delay allpass filter design formulas (§4.3), was proposed in [368]. An improved closed-form solution appears in [1] based on an elementary method for the robust design of very high-order allpass filters.

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Damping Filter Design