Nested Allpass Filters

Another common method for increasing the density of an allpass impulse response is to nest two or more allpass filters, as described in §2.8.2 and shown in Fig.2.32 on page . In general, a nested allpass filter is created when one or more of its delay elements is replaced by another allpass filter. As we saw in §2.8.2, first-order nested allpass filters are equivalent to lattice filters. This equivalence implies that any order $N$ transfer function (any $N$ poles and zeros) may be obtained from a linear combination of the delay elements of nested first-order allpass filters, since this is a known property of the lattice filter [299].

In general, any delay-element or delay-line inside a stable allpass-filter can be replaced with any stable allpass-filter, and the result will be a stable allpass.