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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. It consists of a series chain of first-order allpass filters,9.21 each having a single time-varying parameter $ g_i(n)$ controlling the pole and zero location over time, plus a feedforward path through gain $ g$ which is a fixed depth control. 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.)

Figure 8.23: Structure of a phaser based on four first-order allpass filters.

In analog hardware, the first-order allpass transfer function [452, Appendix E, Section 8]9.22is

$\displaystyle \hbox{AP}_{1}^{\,\omega_b} \isdef \frac{s-\omega_b}{s+\omega_b}. \protect$ (9.19)

(In classic phaser circuits such as the Univibe, $ -\hbox{AP}_{1}^{\,\omega_b}$ is used, but since there is an even number (four) of allpass stages, there is no difference.) 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University