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
![$\displaystyle S_i(z) = \frac{k_i+z^{-1}}{1+k_iz^{-1}}.
$](http://www.dsprelated.com/josimages_new/pasp/img606.png)
![$ H_1(z)\isdef S_1(z)$](http://www.dsprelated.com/josimages_new/pasp/img607.png)
![$ H_2(z)$](http://www.dsprelated.com/josimages_new/pasp/img601.png)
![$ z^{-1}$](http://www.dsprelated.com/josimages_new/pasp/img128.png)
![$ H_1(z)$](http://www.dsprelated.com/josimages_new/pasp/img600.png)
![$ z^{-1}S_2(z)$](http://www.dsprelated.com/josimages_new/pasp/img608.png)
![$\displaystyle H_2(z) \isdef S_1\left([z^{-1}S_2(z)]^{-1}\right) \isdef \frac{k_1+z^{-1}S_2(z)}{1+k_1z^{-1}S_2(z)}.
$](http://www.dsprelated.com/josimages_new/pasp/img609.png)
![$ S_1(z)$](http://www.dsprelated.com/josimages_new/pasp/img610.png)
![$ z^{-1}$](http://www.dsprelated.com/josimages_new/pasp/img128.png)
![$ z^{-1}S_2(z)$](http://www.dsprelated.com/josimages_new/pasp/img608.png)
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.
Next Section:
More General Allpass Filters
Previous Section:
Allpass from Two Combs