Phasing with First-Order Allpass Filters

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Phasing with First-Order Allpass Filters

The block diagram of a typical inexpensive phase shifter for guitar players is shown in Fig.O.1.^O.1It consists of a series chain of first-order allpass filters,^O.2 each having a single time-varying parameter controlling the pole and zero location over time, plus a feedforward path through gain which is a fixed depth control.

**Figure O.1:** Structure of a phaser based on four first-order allpass filters.
$\begin{figure}\input fig/allpass1phaser.pstex_t \end{figure}$

In analog hardware, the first-order allpass transfer function [460, Appendix C, Section 8]^O.3is

$\displaystyle \hbox{AP}_{1}^{\,\omega_b} \isdef \frac{s-\omega_b}{s+\omega_b}. \protect$

(O.1)

In discrete time, the general first-order allpass has the transfer function

$\displaystyle \hbox{AP}_{1}^{\,g_i} \isdef \frac{g_i + z^{-1}}{1 + g_i z^{-1}}.$

We now consider the analog and digital cases, respectively.

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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.

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Physical Audio Signal Processing
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          Phasing with First-Order Allpass Filters