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