Nested Allpass Filter Design

Any delay-element or delay-line inside a stable allpass-filter can be replaced by any stable allpass-filter to obtain a new stable allpass filter:

$$z^{-1}\leftarrow H_a(z)\,z^{-1}$$

(The pure delay on the right-hand-side guarantees no delay-free loops are introduced, so that the original structure can be used)

Proof:

1. Allpass Property: Note that the above substitution is a conformal map taking the unit circle of the $z$ plane to itself. Therefore, unity gain for $\vert z\vert=1$ is preserved under the mapping.
2. Stability: Expand the transfer function in series form:
   $$S\left([H_a(z)z^{-1}]^{-1}\right) \;=\; s_0 + s_1 H_a(z)z^{-1}+ s_2 H_a^2(z)z^{-2}+ \cdots$$
   where $s_n=$ original impulse response. In this form, it is clear that stability is preserved if $H_a(z)$ is stable.