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Chapters

Physical Audio Signal Processing
    Virtual Analog Synthesis and Effects
       Phasing
          Phasing with 2nd-Order Allpass Filters
             Allpass Phaser Architecture

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Allpass Phaser Architecture

The architecture of the allpass-based notch filter is shown in Fig.O.5. 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.