### Phasing with First-Order Allpass Filters

The block diagram of a typical inexpensive*phase shifter*for guitar players is shown in Fig.8.23.

^{9.20}It consists of a series chain of first-order allpass filters,

^{9.21}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*. 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.)

In analog hardware, the first-order allpass transfer function [449, Appendix E, Section 8]

^{9.22}is

(In classic phaser circuits such as the Univibe, 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

#### Classic Analog Phase Shifters

Setting in Eq.(8.19) gives the frequency response of the analog-phaser transfer function to be*break frequency*of the allpass section.

^{9.23}Figure 8.24a shows the phase responses of four first-order analog allpass filters with set to . Figure 8.24b shows the resulting normalized amplitude response for the phaser, for (unity feedfoward gain). The amplitude response has also been normalized by dividing by 2 so that the maximum gain is 1. Since there is an even number (four) of allpass sections, the gain at dc is . Put another way, the initial phase of each allpass section at dc is , so that the total allpass-chain phase at dc is . As frequency increases, the phase of the allpass chain decreases. When it comes down to , the net effect is a sign inversion by the allpass chain, and the phaser has a notch. There will be another notch when the phase falls down to . Thus, four first-order allpass sections give two notches. For each notch in the desired response we must add two new first-order allpass sections.

*i.e.*, . Since the phase of a first-order allpass section at its break frequency is , the sum of the other three sections must be approximately . Equivalently, since the first section has ``given up'' radians of phase at , the other three allpass sections

*combined*have given up radians as well (with the second section having given up more than the other two). In practical operation, the break frequencies must

*change dynamically*, usually periodically at some rate.

#### Classic Virtual Analog Phase Shifters

To create a*virtual analog*phaser, following closely the design of typical analog phasers, we must translate each first-order allpass to the digital domain. Working with the transfer function, we must map from plane to the plane. There are several ways to accomplish this goal [362]. However, in this case, an excellent choice is the

*bilinear transform*(see §7.3.2), defined by

where is chosen to map one particular frequency to exactly where it belongs. In this case, can be chosen for each section to map the

*break frequency*of the section to exactly where it belongs on the digital frequency axis. The relation between analog frequency and digital frequency follows immediately from Eq.(8.20) as

*analog*allpass filter is given by

*oversampling*; for example, the slight visible deviation in Fig.8.26 below 10 kHz can be largely eliminated by increasing the sampling rate by 15% or so. See the case of digitizing the Moog VCF for an example in which the presence of feedback in the analog circuit leads to a delay-free loop in the digitized system [479,477].

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Phasing with 2nd-Order Allpass Filters

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Body Factoring Example