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 [371]. However, in this case, an excellent choice is the bilinear transform (see §L.4), 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 the analog and digital frequency axes follows immediately from Eq.(O.2) as

Thus, given a particular desired break frequency , we can set

Recall from Eq.(O.1) 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.(O.2) gives

where we have denoted the