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 [274,273]. Closed-form parametric methods provide coefficient formulas as a function of a desired parameter such as ``inharmonicity'' [371]. 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 [555,372,42,41,1], [432, 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, [212,387]). 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 [555] was
suggested for designing allpass filters having a phase delay
corresponding to the delay profile needed for a stiff string
simulation [432, pp. 60,172]. The method of
[555] was streamlined in [372]. In
[77], piano strings were modeled using
finite-difference techniques. An update on this approach appears in
[45]. In [343], high quality stiff-string
sounds were demonstrated using high-order allpass filters in a digital
waveguide model. In [387], this work was extended by
applying a least-squares allpass-design method [274]
and a spectral Bark-warping technique [461] 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 [273] 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 [371]. An improved closed-form
solution appears in [1] based on an elementary
method for the robust design of very high-order allpass filters.
