Recursive Allpass Filters

In general, (finite-order) allpass filters can be written as

$$H(z) = e^{j\phi} z^{-K} \frac{\tilde{A}(z)}{A(z)}$$

where

$$\begin{eqnarray*} A(z) &=& 1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_N z^{-N}\\ [10pt] \tilde{A}(z)&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& z^{-N}\overline{A}(z^{-1})\\ [10pt] &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \overline{a}_N + \overline{a}_{N-1} z^{-1}+ \cdots + \overline{a}_1 z^{-(N-1)} + \cdots + z^{-N} \end{eqnarray*}$$

• The polynomial $\tilde{A}(z)$ can be obtained by reversing the order of the coefficients in $A(z)$ and conjugating them

• The problem of dispersion filter design is typically formulated as an allpass-filter design problem