![](bg2.png)
ally assumed to be linear and defined as an FIR filter
G(z,n) = g
0
(n) + g
1
(n)z
−1
+ . .. + g
L
G
−1
(n)z
−(L
G
−1)
= g
T
(n)z (2)
with length L
G
and has g
0
(n) = 0, ∀n, for closed-loop
analysis.
Let the system input signal u(n) be the source sig-
nal v(n) added to the ambient noise signal r(n), i.e.,
u(n) = v(n) + r(n), and, for simplicity, also include
the characteristics of the microphone and A/D con-
verter. The system input signal u(n) and the loud-
speaker signal x(n) are related by the system closed-
loop transfer function as
X(z) =
G(z,n)
1− G(z,n)F(z,n)
U(z). (3)
According to the Nyquist’s stability criterion, the
closed-loop system will become unstable if there is at
least one frequency ω such that
(
G(e
jw
,n)F(e
jw
,n)
≥ 1
∠G(e
jw
,n)F(e
jw
,n) = 2kπ, k ∈ Z.
(4)
It means that if at least one frequency compo-
nent is reinforced after traversing the system open-
loop transfer function G(z,n)F(z,n) and is added to
the input signal u(n) with no phase shift, this fre-
quency component will never disappear from the sys-
tem even if there is no more input signal. After each
loop through the system, its amplitude will increase
causing a howling at that frequency, a phenomenon
known as Larsen effect (Rombouts et al., 2006; van
Waterschoot and Moonen, 2011). This howling will
be very annoying for all listeners present and the sys-
tem gain generally has to be reduced. As a conse-
quence, the maximum stable gain (MSG) of the PA
system is limited by the occurrence of acoustic feed-
back (Rombouts et al., 2006; van Waterschoot and
Moonen, 2011).
In order to eliminate or, at least, to control the
Larsen effect, several methods have been developed
over the past decades (van Waterschoot and Moonen,
2011). The acoustic feedback cancellation (AFC)
methods identify and track the acoustic feedback path
F(z,n) using an adaptive filter that is generally de-
fined as an FIR filter
H(z,n) = h
0
(n) + h
1
(n)z
−1
+ . .. + h
L
H
−1
(n)z
−(L
H
−1)
= h
T
(n)z (5)
with length L
H
. Then, the feedback signal f(n) ∗ x(n)
is estimated as h(n) ∗ x(n) and subtracted from the
microphone signal y(n) so that, ideally, only the sys-
tem input signal u(n) is processed by the forward path
G(z,n). Such a scheme is shown in Fig. 2.
Forward
Path
_
+
Adaptive Feedback
Path
Filter
P P
G(z, n)
P
H(z, n)
y(n) u(n) v(n)
r(n)
F (z, n)
e(n)
x(n)
Figure 2: Typical AFC system.
But, because of the presence of the forward path
G(z,n), the estimation noise (system input u(n)) and
input (loudspeaker x(n)) signals to the adaptive filter
are highly correlated. Then, if the traditional adap-
tive filtering algorithms based on the Wiener theory
or least squares are used, a bias is introduced in the
estimate of the acoustic feedback path (Siqueira and
Alwan, 2000; Hellgren and Forssell, 2001). As un-
desired consequences, the adaptive filter H(z,n) will
only partially cancel the feedback signal f(n) ∗ x(n)
and will also apply distortions in the system input sig-
nal u(n).
Mostly, the solutions existing in the literature to
overcome the bias in the estimate of the feedback
path try to, somehow, decorrelate the loudspeaker
x(n) and system input u(n) signals. Among them, the
PEM-AFROW stands out for having the best perfor-
mance (Rombouts et al., 2006; van Waterschoot and
Moonen, 2011). Recently, one method that extracts
well-defined information from the cepstrum of the mi-
crophone signal to update the adaptive filter was pro-
posed (Bispo et al., 2013). However, until nowadays,
only results in PA systems with a single feedback path
were presented.
The present work evaluates the performance of
some state-of-art AFC methods in a more realistic
scenario with multiple feedback paths. The paper
is organized as follows: Section 2 presents a typical
PA system with multiple feedback paths; Section 3
briefly presents the AFC methods under evaluation;
Section 4 describes the configuration of the simulated
experiments; in Section 5, the obtained results are pre-
sented and discussed. Finally, Section 6 concludes the
paper emphasizing its main contributions.
2 AFC SYSTEM WITH
MULTIPLE FEEDBACK PATHS
Typically, aiming to be heard by a large audience in
the same acoustic environment, a speaker uses a PA
system with one microphone, responsible for picking
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