74
D. Rocchesso: Sound Processing
state space description
delay matrix
feedback matrix
lossless prototype
normal modes
s
i
(n + m
i
) =
N
j=1
a
i,j
s
j
(n) + b
i
x(n)
(41)
where s
i
(n), 1 i N , are the delay outputs at the n-th time sample. If m
i
= 1
for every i, we obtain the well known state space description of a discrete-time
linear system [46]. In the case of FDNs, m
i
are typically numbers on the orders
of hundreds or thousands, and the variables s
i
(n) are only a small subset of the
system state at time n, being the whole state represented by the content of all
the delay lines.
From the state-variable description of the FDN it is possible to find the
H(z) =
Y (z)
X(z)
= c
T
[D(z
-1
) - A]
-1
b + d.
(42)
The diagonal matrix D(z) = diag (z
-m
1
, z
-m
2
, . . . z
-m
N
) is called the delay
matrix, and A = [a
i,j
]
N ŚN
is called the feedback matrix.
The stability properties of a FDN are all ascribed to the feedback matrix.
The fact that A
n
decays exponentially with n ensures that the whole structure
The poles of the FDN are found as the solutions of
det[A - D(z
-1
)] = 0 .
(43)
In order to have all the poles on the unit circle it is sufficient to choose a
unitary matrix. This choice leads to the construction of a lossless prototype but
this is not the only choice allowed.
In practice, once we have constructed a lossless FDN prototype, we must
insert attenuation coefficients and filters in the feedback loop (blocks G
i
in
figure 15). For instance, following the indications of Jot [45], we can cascade
every delay line with a gain
g
i
=
m
i
.
(44)
This corresponds to replacing D(z) with D(z/) in (42). With this choice of the
attenuation coefficients, all the poles are contracted by the same factor . As
a consequence, all the modes decay with the same rate, and the reverberation
time (defined for a level attenuation of 60dB) is given by
T
d
=
-3T
s
log
.
(45)
In order to have a faster decay at higher frequencies, as it happens in real en-
closures, we must cascade the delay lines with lowpass filters. If the attenuation
coefficients g
i
are replaced by lowpass filters, we can still get a local smoothness
of decay times at various frequencies by satisfying the condition (44), where g
i
and have been made frequency dependent:
G
i
(z) = A
m
i
(z),
(46)
where A(z) can be interpreted as per-sample filtering [43, 45, 98].
It is important to notice that a uniform decay of neighbouring modes, even
though commonly desired in artificial reverberation, is not found in real en-
closures. The normal modes of a room are associated with stationary waves,