Circuits and Systems, 2013, 4, 6-10
http://dx.doi.org/10.4236/cs.2013.41002 Published Online January 2013 (http://www.scirp.org/journal/cs)
An Adaptive Howling Canceller Using 2-Tap Linear
Predictor
Akira Sogami, Yosuke Sugiura, Arata Kawamura*, Youji Iiguni
Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, Japan
Received October 1, 2012; revised November 1, 2012; accepted November 8, 2012
ABSTRACT
This paper proposes an adaptive howling canceller using notch filter and 2-tap linear predictor, where howling consists
of a single sinusoidal signal whose magnitude is much greater than other frequency’s magnitudes. The employed 2-tap
linear predictor can quickly detect howling due to its high convergence speed. Although the output signal of the 2-tap
linear predictor cannot be directly used as one of a howling canceller, we can obtain the frequency of howling from the
filter coefficient. We utilize the filter coefficient of the 2-tap linear predictor to design a notch filter which achieves a
very narrow elimination band. The designed notch filter removes only howling and retains other desired signals. Simu-
lation results show that the proposed adaptive howling canceller can quickly detect and effectively remove howling.
Keywords: Howling Canceller; Linear Predictor; Notch Filter; Single Sinusoidal Signal; Convergence Speed
1. Introduction
Howling is an annoying and persistence problem which
is mainly caused in public address system. The public
address system amplifies a signal observed at a micro-
phone and transmits the amplified signal with a loud-
speaker. Then, there exists a feedback path from the
loudspeaker to the microphone. This feedback path forms
an acoustical closed loop. When a frequency amplitude
response of the closed loop is greater than 1 and its phase
response is 2π, howling is caused, i.e., a single sinusoidal
signal rapidly becomes large. In this case, audience can-
not avoid perception of this unpleasant sound, and also it
is difficult to receive a desired signal. Moreover, the
public address system often breaks down due to howling.
To avoid this undesired phenomenon, many approaches
have been studied. One of the most famous and effective
methods is an echo canceller [1]. The echo canceller
adaptively estimates the acoustical impulse response
from the loudspeaker to the microphone, and provides a
replica of the feedback signal. When the echo canceller
perfectly estimates the feedback impulse response, sub-
tracting the replica of the feedback signal from the ob-
served signal gives perfect suppression of the feedback
signal and thus howling is not caused. Since the acousti-
cal impulse response is time-variant, the echo canceller is
required to achieve both of high estimation accuracy and
high convergence speed. Unfortunately, a tread-off exists
between the convergence speed and the estimation accu-
racy. Actually, in many practical environments, the echo
canceller cannot work well and hence it often causes
degradation of sound quality, echo, or howling. Sogami
et al. have proposed a simple howling canceller [2-3]
which estimates howling by utilizing only the distance
information between the loudspeaker and the microphone.
This method can estimate the frequency of howling faster
than the conventional echo canceller, under the assump-
tion that howling is depending only on the direct distance
from the loudspeaker to the microphone. Although howl-
ing often depends on the direct distance, the above as-
sumption cannot hold for howling which depends on the
distance including reflections. Actually, all the distances
including reflections cannot be calculated. These con-
ventional researches imply that the occurrence of howl-
ing may not be avoided and thus we have to prepare an-
other howling canceller which adaptively removes howl-
ing as fast as possible after it is caused.
To remove howling after its occurrence, adaptive
notch filter techniques are useful [4-10]. They achieve
very narrow elimination bandwidth, and automatically
estimate and remove a single sinusoidal signal like howl-
ing. Unfortunately, it also has a trade-off problem be-
tween the convergence speed and the estimation accuracy.
Efficient notch filters which can solve the trade-off
problem have been proposed [11-13]. Their main idea is
to employ two or more notch filters, where one notch
filter achieves high estimation accuracy at an expense of
convergence speed, and the other notch filter has high
convergence speed with low estimation accuracy. In such
*Corresponding author.
C
opyright © 2013 SciRes. CS

A. SOGAMI ET AL. 7
techniques, addition to an increase of their computational
cost, it may not achieve an accurate adaptation because
the notch filter’s impulse responses are long basically.
In this paper, we propose an adaptive howling cancel-
ler which achieves both of high convergence speed and
high estimation accuracy to remove howling. The pro-
posed method consists of a notch filter and a 2-tap linear
predictor. Although the proposed method utilizes the
additional filter, the 2-tap linear predictor updates only
one filter coefficient to detect howling. With its short
impulse response, the 2-tap linear predictor can quickly
evaluate the prediction error signal in the steady-state for
adaptation. Since the filter coefficient has the informa-
tion of the frequency of howling after convergence, we
easily design the notch filter whose elimination fre-
quency is identical to the frequency of howling. Then, we
can remove howling quickly and effectively. The addi-
tional computational cost of the proposed method is
minimal among conventional adaptive notch filters
which employ two or more additional notch filters.
Simulation results show that the proposed howling can-
celer effectively removes howling.
2. Howling Canceller with Notch Filter
Let consider a public address system shown in Figure 1,
where

s
n

denotes the source signal produced by hu-
man in general, and
y
n

is the feedback signal from
the loudspeaker to the microphone. The observed signal
is represented as



x
nsnyn
 
. (1)
The observed signal is amplified by the attenuator so
that

s
nan
a

x

, where is a constant. The ampli-
fied signal
s
n


is produced from the loudspeaker. The
signal
s
n


Pz
is received at the microphone after passing
through the unknown acoustical feedback path whose
transfer function is . Then, the closed loop is
formed as shown in Figure 1. When the amplitude fre-
quency response of the closed loop is greater than 1 and
the phase frequency response is 2π for a certain fre-
quency, howling will occur at the corresponding fre-
quency. Figure 2 shows an example of the occurrence of
howling. Here, we set the acoustical impulse response as


Pz


yn

x
n

s
n


s
n

attenuator
Figure 1. Public address system.
uniform random variables. The top panel shows the
waveform of


s
n, and the middle panel shows the
transmitted signal


s
n
. In this simulation, we gradu-
ally increased a. As a result, howling occurred at
around 30,000 samples. An expanded waveform is
shown in the bottom panel. We see from this result that
howling explosively increases and we should remove it
as fast as possible.
First, we explain the standard adaptive notch filter to
remove howling. The notch filter passes all frequencies
expect of the narrow frequency band whose center fre-
quency is called as the notch frequency. The elimination
bandwidth and the notch frequency can be individually
designed [4-10]. The transfer function of the notch fil-
ter


Nz

is given by [6-8]
12
12
11
21
rzz
Nz zrz




 




, (2)

is a parameter to design the notch frequency and where


11rr

 determines the elimination bandwidth.
The relation between

and the notch frequency is
given as

1cos2π
S
F
r
F


 

, (3)


HzF denotes the notch frequency and where


HzF

S denotes the sampling frequency. The relation of
r and the elimination bandwidth

HzK is represented
as



1cos2πsin 2π
.
1cos2πsin 2π
SS
SS
K
FKF
r
K
FKF


Figure 3 shows the structure of the notch filter, where


n

is the input signal, and
x
x
n
 is the output signal.
The notch filter includes the IIR unit, and hence its im-
pulse response is infinite. Figure 4 shows the frequency
amplitude response of the notch filter


Nz when


0 2FF   0.8, 0.9, 0.99
S with r, where the
vertical axis denotes the amplitude response and the


s
n


s
n



s
n

Figure 2. Example of howling.

A. SOGAMI ET AL.
8
r

x
n

D
D
r



x
n

1
2
Figure 3. Structure of notch filter.
Figure 4. Frequency amplitude response of notc h filte r.
horizontal axis denotes the normalized frequency. We
see from Figure 4 that the elimination bandwidth be-
comes narrow with increasing r toward to 1. Thus, we
can remove only howling when setting r close to 1.
The frequency of howling is usually unknown. Hence,
we have to adaptively estimate and remove it. When the
observed signal includes a sinusoidal signal whose mag-
nitude is much greater than the other frequency’s magni-
tudes, an adaptive notch filter can automatically estimate
and remove the single sinusoidal signal when its coeffi-
cient

is updated by a gradient method [4]. The coef-
ficient

converges so that the notch filter’s output
power is minimized. Since howling can be approximated
such as the sinusoidal signal, the adaptive notch filter can
automatically detect and remove howling.
However, the gradient method includes an annoying
trade-off problem between convergence speed and esti-
mation accuracy. To accelerate convergence speed with
remaining high estimation accuracy, in many approaches,
an additional adaptive notch filter is introduced [11-13],
where the main notch filter has high estimation accuracy
and the other notch filter has high convergence speed.
Comparing their output signals, we can choose better one.
This method is useful when the notch filter is appropri-
ately updated by using the output signal in the steady
state. Unfortunately, such update is difficult, because
notch filter’s impulse responses are generally long.
3. Howling Canceller with Notch Filter and
2-Tap Linear Predictor
In this section, we propose an adaptive howling canceller
which utilizes an adaptive notch filter and an additional
2-tap linear predictor, where the proposed method
achieves both of high convergence speed and high esti-
mation accuracy.
As mentioned above sections, howling can be ap-
proximated as a single sinusoidal signal whose magni-
tude is much greater than other frequency’s magnitudes.
Our purpose is to detect the frequency of howling as fast
as possible. A linear predictor, often called as an adaptive
line enhancer [1], is useful to extract a sinusoidal signal
from the sinusoidal signal embedded in a wideband sig-
nal. Hence, the estimation error signal of the linear pre-
dictor does not include the sinusoid. Intuitively, we note
that the estimation error signal can be utilized as the
output signal of a howling canceller. Unfortunately, to
accurately remove howling only, the filter length of the
linear predictor must be long. In actual, the notch filter is
the most effective filter to remove howling, because of
its low computational complexity and steep frequency
response. In the use of the adaptive notch filter, we en-
counter the trade-off problem again.
To solve this problem, we propose a combination
method of a linear predictor and a notch filter. The em-
ployed linear predictor has the minimum impulse re-
sponse which is just 2 samples (called 2-tap linear pre-
dictor) to detect the frequency of howling. The notch
filter is adaptively designed from the 2-tap linear predic-
tor’s coefficient to align the notch frequency with the
frequency of howling. Since the 2-tap linear predictor
can achieve very short impulse response, we can quickly
obtain its prediction error signal in the steady state for
adaptation. It means that the 2-tap linear predictor
achieves high convergence speed and we obtain the fre-
quency of howling quickly.
Figure 5 shows the 2-tap linear predictor, where


ˆ

x
n and

en denote the predicting signal and the
prediction error signal, respectively. The 2-tap linear
predictor provides a replica of the present input signal by
linear combination of past two input samples with two
coefficients as



 


ˆ12xnh nxnhnxn
12

 .
Here,



hn
1 and

2
h denote the 1st and 2nd filter
coefficients, respectively, and the replica has inverse
phase of the present input signal. We update the 2-tap
linear predictor so that the power of the following pre-
diction error signal is minimized.
n




en
x
n
D
D


1
hn


x

n


2
hn
Figure 5. 2-tap linear predictor.

A. SOGAMI ET AL. 9
 


2n xn 
12
ˆ11enxnxnh nxnh 
(5)
Under the assumption that howling can be expressed
as



n
111S
cos 2πxnpF F

, where 111 , , Fp

are
the frequency of howling, amplitude, and phase, respec-
tively, the 2-tap linear predictor can estimate the single
sinusoidal signal by using a gradient method [3-10]. It is
well known that the 2-tap linear predictor for such single
sinusoidal input signal converges so that
 
12
,1 hn
 



hn
1
2cos 2π
S
F
hn F  . (6)
Here, we note that the 1st coefficient 1 depends
on the frequency of howling, while the 2nd coefficient


2 is independent with the input signal. It implies
that we do not need to update the 2nd coefficient when
predicting howling. Specifically, we can fix the 2nd coef-
ficient as 2. On the other hand, we have to up-
date 1 to obtain the frequency of howling. After
convergence, 1 is used to design the notch filter to
remove howling. When the notch frequency is accurately
designed, the notch filter successfully cancels howling.
Comparing (3) and (6), we have the following simple
relation.
hn

1hn

hn

hn

1
1
2
rhn 


hn
 (7)
The accurate 1 gives the accurate

. Then, the
notch filter with (7) completely cancels howling.
Figure 6 shows the proposed howling canceller, where
LP denotes the 2-tap linear predictor. In the proposed
system,


1 is updated by using the NLMS type al-
gorithm given as
hn


 

11
1hn hn   2
1
1
enxn
Ex n






, (8)
where

is the step-size for adaptation. The notch fil-
ter’s coefficient

is updated with (7), where r and
are fixed.

n
2
h

s
n


s
n

x
n


yn


Pz

Nz


en
attenuator
LP
Figure 6. Proposed system.
puter simulation to confirm the effec-
4. Simulatio
We carried out com
n
tiveness of the proposed method. Here, we set feedback
pass P(z) as an FIR filter whose impulse response is set
as a uniform random signal, where the order of the FIR
filter is 50. We also set the attenuator as 4


to sat-
isfy the condition for occurrence of howand the
step-size used in (8) as 0.05
ling,


. The source signal was
a female voice signal sa 16 kHz. Since the ex-
pectation value
mpled at


21Ex n




 shown in (8) cannot be
calculated, we useraged value defined as d a time ave
 
0
1
m
1
22
1M
nxnm
M

x


, (9)
instead of


21Ex n


, where we set 50M


. To
howling can of the
llation,
w
compare the celation capability pro-
posed method, we also performed the computer simula-
tion for the conventional adaptive notch filter [4].
Figure 7 shows the results of howling cance
here Figure 7(a) shows the source signal


s
n,
Figure 7(b) shows the output signal

s
n
 witha
howling canceller, Figure 7(c) denotes tutput signal
of the conventional adaptive notch filter, and Figure 7(d)
shows the output signal of the proposed method. We see
from Figure 7(b) that the waveform is explosively in-
creasing when the howling canceller did not exist.
Figure 7(c) shows that the conventional adaptive notch
filter can remove howling. But, its convergence speed is
comparatively slow, and thus we did not avoid the per-
ception of howling. On the other hand, Figure 7(d)
shows that the proposed method can effectively cancel
howling. The difference between Figures 7(c) and (d)
expresses the difference of respective convergence
speeds. The proposed method removed howling faster
than the conventional notch filter.
out
he o


s
n


s
n



s
n



s
n

Figure 7. Waveforms of simulation result. (a) Source signal;
(b) Loudspeaker output without howling canceller; (c) Loud-
speaker output with conventional notch filter; (d) Loud-
speaker output with proposed howling canceller.

A. SOGAMI ET AL.
10
posed a howling canceller which em-
[1] S. Haykin, “Inilters,” Macmillan
i, “Improvement
mura and Y. Iiguni, “A High
e Notch Filter for
5. Conclusion
This paper has pro
ploys a 2-tap linear predictor and a notch filter. The 2-tap
linear predictor is for estimating the frequency of howl-
ing, and the notch filter is for eliminating howling. Since
the coefficient of the 2-tap linear predictor is simply
transformed to the coefficient of the notch filter, we can
easily update the notch filter according to the linear pre-
dictor. The simulation result showed the effectiveness of
the proposed method.
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