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 . However, in this case, an excellent choice is the bilinear transform (see §7.3.2), defined by
Thus, given a particular desired break-frequency , we can set
Recall from Eq.(8.19) that the transfer function of the first-order analog allpass filter is given by
where is the break frequency. Applying the general bilinear transformation Eq.(8.20) gives
where we have denoted the pole of the digital allpass by
Figure 8.25 shows the digital phaser response curves corresponding to the analog response curves in Fig.8.24. While the break frequencies are preserved by construction, the notches have moved slightly, although this is not visible from the plots. An overlay of the total phase of the analog and digital allpass chains is shown in Fig.8.26. We see that the phase responses of the analog and digital allpass chains diverge visibly only above 9 kHz. The analog phase response approaches zero in the limit as , while the digital phase response reaches zero at half the sampling rate, kHz in this case. This is a good example of when the bilinear transform performs very well.
In general, the bilinear transform works well to digitize feedforward analog structures in which the high-frequency warping is acceptable. When frequency warping is excessive, it can be alleviated by the use of 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 [481,479].