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Allpass Phaser Architecture

The architecture of the allpass-based notch filter is shown in Fig. 3.14. It consists of a series connection of allpass filters with a feed-around. Thus, the delay line of the flanger is replaced by a string of allpass filters. (A delay line is of course an allpass filter itself.) The phaser will have a notch wherever the phase of the allpass chain is at $ \pi$ (180 degrees). It can be shown that these frequencies occur very close to the resonant frequencies of the allpass chain. It is therefore convenient to use a single conjugate pole pair in each allpass section, i.e., use second-order allpass sections of the form

$\displaystyle H(z) = \frac{a_2 + a_1 z^{-1} + z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}
$

where

\begin{eqnarray*}
a_1 &=& -2R\cos(\theta)\\
a_2 &=& R^2
\end{eqnarray*}

and $ R$ is the radius of each pole in the complex-conjugate pole pair, and pole angles are $ \pm\theta$. The pole angle can be interpreted as $ \theta=\omega_c T$ where $ \omega_c$ is the resonant frequency and $ T$ is the sampling interval.


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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