An Adaptive Howling Canceller Using 2-Tap Linear Predictor

doi:10.4236/cs.2013.41002

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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

Email: *[email protected]

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.

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



denotes the source signal produced by hu-

man in general, and



is the feedback signal from

the loudspeaker to the microphone. The observed signal

is represented as







nsnyn

 

. (1)

The observed signal is amplified by the attenuator so

that



nan

a





, where is a constant. The ampli-

fied signal





is produced from the loudspeaker. The

signal





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

















attenuator

Figure 1. Public address system.

uniform random variables. The top panel shows the

waveform of





n, and the middle panel shows the

transmitted signal





. 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







is given by [6-8]

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





 



, (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







1cos2πsin 2π

FKF





Figure 3 shows the structure of the notch filter, where







is the input signal, and

 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

















Figure 2. Example of howling.

A. SOGAMI ET AL.











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







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



 .

Here,





1 and



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.





















Figure 5. 2-tap linear predictor.

A. SOGAMI ET AL. 9

 





2n xn 

ˆ11enxnxnh nxnh 

(5)

Under the assumption that howling can be expressed







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

 

 







2cos 2π

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.



1hn



rhn









(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





 



1hn hn



 2

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.





















attenuator

Figure 6. Proposed system.

puter simulation to confirm the effec-

4. Simulatio

We carried out com

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

 

nxnm

M





, (9)

instead of





21Ex n





, where we set 50M





. To

howling can of the

llation,

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





Figure 7(b) shows the output signal



 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























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.

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.

REFERENCES

troduction to Adaptive F

Publishing Company, New York, 1984.

[2] A. Sogami, A. Kawamura and Y. Iigun

of Speech Quality in Distance-Based Howling Canceller,”

IEICE Transactions on Fundamentals, Vol. 92, No. 4,

2009, pp. 1039-1046.

[3] A. Soagami, A. Kawa

Speech Quality Distance-Based Howling Canceller with

Adaptive Cascade Notch Filter and Silent Pilot Signal,”

IEICE Transactions on Fundamentals, Vol. 94, No. 11,

2011, pp. 2306-2314.

[4] S. Nishimura, “An Improved Adaptiv

Detection of Multiple Sinusoids,” IEICE Transactions on

Fundamentals, Vol. 77, No. 6, 1994, pp. 950-955.

[5] A. Nehorai, “A Minimal Parameter Adaptive Notch Filter

with Constrained Poles and Zeros,” IEEE Transactions on

Acoustics, Speech and Signal Processing, Vol. 33, No. 4,

1985, pp. 983-996. doi:10.1109/TASSP.1985.1164643

[6] H. C. Chong and U. L. Sang, “Adaptive Line Enhance-

ment by Using an IIR Lattice Notch Filter,” IEEE Trans-

ac tions on Ac oust ics , Speech and Signal Processing, Vol.

37, No. 4, 1989, pp. 585-589. doi:10.1109/29.17543

[7] C. C. Tseng and S. C. Pei, “IIR Multiple Notch Filter

nalytical Two-

y, “Design of IIR Multiple

Filter

. Zou and N. Zheng, “A Piloted Adaptive

Design Based on Allpass Filter,” IEEE Transactions on

Circuits and Systems II: Analog and Digital Signal Proc-

essing, Vol. 44, No. 2, 1997, pp. 133-136.

[8] S. C. Pei, W. S. Lu and C. C. Tseng, “A

Dimensional IIR Notch Filter Design Using Outer Prod-

uct Expansion,” IEEE Transactions on Circuits and Sys-

tems II: Analog and Digital Signal Processing, Vol. 44,

No. 9, 1997, pp. 765-768.

[9] Y. V. Joshi and S. C. D. Ro

Notch Filters Based on All-Pass Filters,” IEEE Transac-

tions on Circuits and Systems II: Analog and Digital Sig-

nal Processing, Vol. 46, No. 2, 1999, pp. 134-138.

[10] V. DeBrunner, “An Adaptive, High-Order, Notch

Using All Pass Sections,” IEEE International Conference

on Acoustics, Speech and Signal Processing, Vol. 3, 1998,

pp. 1477-1480.

[11] Y. C. Lim, Y. X

Notch Filter,” IEEE Transactions on Signal Processing,

Vol. 53, No. 4, 2005, pp. 1310-1323.

doi:10.1109/TSP.2005.843742

[12] A. Kawamura, Y. Itoh, J. Okello, M. Kobayashi and Y.

go-

Fukui, “Parallel Composition Based Adaptive Notch Fil-

ter: Performance and Analysis,” IEICE Transactions on

Fundamentals, Vol. 87, No. 7, 2004, pp. 1747-1755.

[13] A. Kawamura, Y. Iiguni and Y. Itoh, “An Adaptive Al

rithm with Variable Step-Size for Parallel Notch Filter,”

IEICE Transactions on Fundamentals, Vol. 89, No. 2,

2006, pp. 511-519.