### Nested Allpass Filters

An interesting property of allpass filters is that they can be
*nested* [412,152,153].
That is, if and
denote unity-gain allpass transfer functions, then both
and
are allpass filters. A proof can be
based on the observation that, since
, can
be viewed as a conformal map
[326] which maps the unit circle in the plane to itself;
therefore, the set of all such maps is closed under functional
composition. Nested allpass filters were proposed for use in artificial
reverberation by Schroeder [412, p. 222].

An important class of nested allpass filters is obtained by nesting first-order allpass filters of the form

*two-multiplier lattice filter*section [297]. In the lattice form, it is clear that replacing by just extends the lattice to the right, as shown in Fig.2.32.

The equivalence of nested allpass filters to lattice filters has computational significance since it is well known that the two-multiply lattice sections can be replaced by one-multiply lattice sections [297,314].

In summary, *nested first-order allpass filters are equivalent to
lattice filters made of two-multiply lattice sections*. In
§C.8.4, a one-multiply section is derived which is not
only less expensive to implement in hardware, but it additionally has
a direct interpretation as a physical model.

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More General Allpass Filters

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Allpass from Two Combs