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  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 , and is obtained by replacing in by to get
Figure 2.31a depicts the first-order allpass in direct form II. Figure 2.31b shows the same filter redrawn as a two-multiplier lattice filter section [305,299]. 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 [299,317].
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.