General Allpass Filters

• General, order $L$, allpass filter:
  $$\begin{eqnarray*} H_a(z) &\mathrel{\stackrel{\Delta}{=}}& z^{-L} \frac{A(z^{-1})... ...)} + z^{-L}}{1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_L z^{-L}} \end{eqnarray*}$$
  

• General order $L$, monic, minimum-phase polynomial:
  $$A(z) \mathrel{\stackrel{\Delta}{=}}1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_L z^{-L}$$
  where $A(z_i)=0\Rightarrow \left\vert z_i\right\vert<1$ (roots inside unit circle)
• Numerator polynomial = reverse of denominator
• First-order case:
  $$H_a(z) \mathrel{\stackrel{\Delta}{=}}\frac{a_1z^{-1}+1}{1+a_1z^{-1}}$$

• Each pole $p_i$ gain-compensated by a zero at $z_i=1/p_i$
• There are papers in the literature describing methods for designing allpass filters with a prescribed group delay (see reader for refs)
• For piano strings $L$ is on the order of $10$