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 $g_i(n)$ controlling the pole and zero location over time, plus a feedforward path through gain $g$ which is a fixed depth control.

Figure O.1: Structure of a phaser based on four first-order allpass filters.

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

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

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

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

We now consider the analog and digital cases, respectively.