Delay Lines and Effects
61
recursive comb filter
IIR comb
one-dimensional resonator
plucked string synthesis
If g < 1, it is easy to verify that the amplitude of the resonances is
P =
1
1 - g
,
(23)
while the amplitude of the points of minimum (halfway between contiguous
resonances) is
V =
1
1 + g
.
(24)
An important parameter of this filtering structure, called recursive comb
filter (or IIR comb), is the peak-to-valley ratio
P
V
=
1 + g
1 - g
.
(25)
Fig. 6 shows the frequency response of a recursive comb filter having a delay
line of m = 11 samples and feedback attenuation g = 0.9. The shape of the
magnitude response justifies the name comb given to the filter.
0
1
2
3
0
2
4
6
8
10
frequency [rad/sample]
magnitude
0
1
2
3
0
20
40
60
80
100
120
frequency [rad/sample]
phase delay [samples]
Figure 6: Magnitude and phase delay response of the recursive comb filter having
coefficient g = 0.9 and delay length m = 11
The poles of the comb filter are evenly distributed along the unit circle at
the m-th roots of g, as shown in figure 7.
In sound synthesis by physical modeling, a recursive comb filter can be in-
terpreted as a simple model of lossy one-dimensional resonator, like a string, or
a tube. This model can be used to simulate several instruments whose resonator
is not persistently excited. In fact, if the input is a short burst of filtered noise,
we obtain the basic structure of the plucked string synthesis algorithm due to
Karplus and Strong [47].
3.4.1
The Comb-Allpass Filter
The filter given by the difference equation (20) has a frequency response char-
acterized by evenly-distributed resonances. With a slight modification of its
structure, such filter can be made allpass. In other words, the magnitude re-
sponse of the filter can be made flat even though the impulse response remains
almost the same (20). The modification is just a direct path connecting the
input of the delay line to the filter output, as it is depicted in fig. 8. It is easy