Delay Lines and Effects
73
Feedback Delay Networks
FDN
FDN
Delay Networks (FDNs). The Stautner-Puckette FDN was obtained as a vector
generalization of the recursive comb filter (20), where the m-sample delay line
was replaced by a bunch of delay lines of different lengths, and the feedback
gain g was replaced by a feedback matrix G. Stautner and Puckette proposed
the following feedback matrix:
G = g
0
1
1
0
-1 0
0
-1
1
0
0
-1
0
1 -1
0
/
2 .
(40)
Due to its sparse special structure, G requires only one multiply per output
channel.
1
1,3
1
2
2
1
-m
-m
-m
1
2
3
1,1
1,4
3,4
4,4
4,3
3,3
3,2
2,2
3,2
4,2
4,1
3,1
1,2
2,1
4,2
-m
4
2
3
4
3
4
3
4
z
z
z
z
+
+
+
+
H
a
a
a
b
b
c
+
+
+
+
H
H
H
a
a
a
a
a
a
a
a
a
a
a
a
a
b
c
c
c
x
y
b
d
Figure 15: Fourth-order Feedback Delay Network
More recently, Jean-Marc Jot investigated the possibilities of FDNs very
thoroughly. He proposed to use some classes of unitary matrices allowing efficient
implementation. Moreover, he showed how to control the positions of the poles of
the structure in order to impose a desired decay time at various frequencies [44].
His considerations were driven by perceptual criteria with the general goal x to
obtain an ideal diffuse reverb. In this context, Jot introduced the important
design criterion that all the modes of a frequency neighborhood should decay at
the same rate, in order to avoid the persistence of isolated, ringing resonances
in the tail of the reverb [45]. This is not what happens in real rooms though,
where different modes of close resonance frequencies can be differently affected
by wall absorption [63]. However, it is generally believed that the slow variation
of decay rates with frequency produces smooth and pleasant impulse responses.
Referring to fig. 15, an FDN is built starting from N delay lines, each being
i
= m
i
T
s
seconds long, where T
s
= 1/F
s
is the sampling interval. The FDN is
completely described by the following equations:
y(n) =
N
i=1
c
i
s
i
(n) + dx(n)