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  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
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.
More General Allpass Filters
Allpass from Two Combs