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Recursive Allpass Filters

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

$\displaystyle 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*}


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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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