To create a
virtual analog phaser, following closely the design
of typical analog phasers, we must translate each firstorder
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

(9.20) 
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
Thus, given a particular desired breakfrequency
, we can set
Recall from Eq.
(
8.19) that the transfer function of the
firstorder
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 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.
Figure 8.25:
(a) Phase responses of firstorder
digital allpass sections with break frequencies at 100, 200, 400,
and 800 Hz, with the sampling rate set to 20,000 Hz. (b)
Corresponding phaser amplitude response.

Figure 8.26:
Phase response of four firstorder
allpass sections in series  analog and digital cases overlaid.

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
delayfree loop in the digitized system
[
479,
477].
Next Section: Phaser Notch ParametersPrevious Section: Classic Analog
Phase Shifters