An interesting property of allpass filters is that they can be
nested [415,153,154].
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
[329] which maps the unit circle in the
plane to itself;
therefore, the set of all such maps is closed under functional
composition.
An important class of nested allpass filters is obtained by nesting first-order allpass filters of the form
The nesting begins with
Figure 2.31a depicts the first-order allpass
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 [299,317].
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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.