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 [340]. However, in this case, an excellent choice is the bilinear transform (see §J.4), defined by
Thus, given a particular desired break frequency , we can set
Recall from Eq. (3.4) that the transfer function of the first-order analog allpass filter is given by
where we have denoted the pole of the digital allpass by
Figure 3.12 shows the digital phaser response curves corresponding to the analog response curves in Fig. 3.11. 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. 3.13. We see that the phase responses of the analog and digital alpass 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.