Nested Allpass Filters

Transfer function:

$$H_2(z) = S_1\left((z^{-1}S_2(z))^{-1}\right) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{k_1+z^{-1}S_2(z)}{1+k_1z^{-1}S_2(z)}$$

(a)

Nested direct-form-II structures

(b)

Two-multiply lattice-filter structure (equivalent)