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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 [365]. 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 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.
Figure 8.25:
(a) Phase responses of first-order
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.
![\includegraphics[width=\twidth]{eps/phaser1d}](img1942.png) |
Figure 8.26:
Phase response of four first-order
allpass sections in series -- analog and digital cases overlaid.
![\includegraphics[width=\twidth]{eps/phaser1ad}](img1943.png) |
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
[482,480].
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![\begin{eqnarray*}
j\omega_a &=& c\frac{e^{j\omega_d T}-1}{e^{j\omega_d T}+1}
\eqsp c\frac{e^{j\omega_d T/2}\left(e^{j\omega_d T/2}-e^{-j\omega_d T/2}\right)}{e^{j\omega_d T/2}\left(e^{j\omega_d T/2}+e^{-j\omega_d T/2}\right)}\\ [5pt]
&=& jc\frac{\sin(\omega_dT/2)}{\cos(\omega_dT/2)}
\eqsp jc\tan(\omega_dT/2).
\end{eqnarray*}](img1935.png)
