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0165-1684/$ - se

doi:10.1016/j.sig

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Signal Processing 86 (2006) 2318–2325

www.elsevier.com/locate/sigpro

A note on cancellation path modeling signal inactive noise control

Ming Wu�, Xiaojun Qiu, Boling Xu, Ningrong Li

State Key Laboratory of Modern Acoustics and Institute of Acoustics, Nanjing University, Nanjing 210093, China

Received 23 December 2004; received in revised form 6 July 2005; accepted 24 October 2005

Available online 7 December 2005

Abstract

This communication analyzes the performance of using band stop (BS) signal for on-line cancellation path modeling in

active noise control when the disturbance is a tonal or narrow band signal. It is shown theoretically and experimentally

that the modeling error is mainly caused by the corresponding spectral components of the modeling signal close to the

central frequency of the disturbance signal. Thus using BS random signal as the modeling signal can significantly reduce

the modeling error than using random signal if the frequency components of the BS random signal around the frequency of

the disturbance signal is much less than that of random signal.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Active noise control; Adaptive filters; Least mean square methods

1. Introduction

When implementing active noise control (ANC),the filtered-X LMS algorithm (FXLMS) [1–6] isoften used. While applying this algorithm, it isnecessary to identify the cancellation path transferfunctions (CPTF) between the output of theadaptive control filter and the error sensor. TheFXLMS algorithm keeps stable if the phase bias ofthe estimation is within 901, and a fast and goodestimation of the CPTF does benefit both theperformance and convergence speed of an ANCsystem [5–7].

A number of methods have been proposed for on-line CPTF modeling [8–14], and one of the mostpopular methods is injecting an additional modeling

e front matter r 2005 Elsevier B.V. All rights reserved

pro.2005.10.019

ing author. Tel.: +86 25 835 945 01.

ess: mingwu@nju.edu.cn (M. Wu).

signal into the system. Besides the conventionalrandom signal, many signals can be used as themodeling signals, such as swept sine, multi-sine,periodic noise, maximum length, binary sequence,multi-frequency binary sequence and pulse [6]. Insome applications, such as active control of fannoise, engine noise and power transformer noise,the primary noise mainly consists of large tonalcomponents. Under this situation, it is commonlybelieved that more energy should be put in thefrequency band of disturbance signal to increase theSNR. However, it is not correct and this is going tobe investigated in this paper.

2. Analysis

Fig. 1 shows a block diagram for on-line CPTFmodeling proposed by Eriksson [8]. A zero-meanrandom noise v(n), which is uncorrelated with the

.

ARTICLE IN PRESS

x(n)P

d(n)

+

+

e(n)y'(n)+v'(n)

u(n)^ +

−S(n)^

LMSes(n)v(n)White Noise

Generator

W(n)y(n)

++

S(n)^

copy

x'(n) LMS

S

Fig. 1. Block diagram of the on-line CPTF modeling.

M. Wu et al. / Signal Processing 86 (2006) 2318–2325 2319

primary noise x(n), is injected to the input of thesecondary loudspeaker. The vector P denotes theimpulse response vector of the primary path and S

denotes the impulse response of the secondary path.A LMS-based adaptive filter with weight vectorW(n) is employed as the ANC controller. The signald(n) is the output of P due to x(n). SðnÞ is thecoefficient vector of an M-tap FIR filter, which isused to model S. The LMS algorithm updates theweights of SðnÞ according to

Sðnþ 1Þ ¼ SðnÞ þ mesðnÞv�ðnÞ, (1)

where SðnÞ ¼ ½s0ðnÞ s1ðnÞ . . . sM�1ðnÞ�T and ½��T de-

notes transpose. vðnÞ ¼ ½vðnÞ vðn� 1Þ . . . vðn�M þ

1Þ�T is a vector comprising the M most resentsamples of modeling signal superscript * denotescomplex conjugation, and esðnÞ is error signal to theLMS algorithm estimating the CPTF satisfying

esðnÞ ¼ yTðnÞSþ vðnÞTS� vðnÞTSðnÞ þ dðnÞ, (2)

where it is assumed that S ¼ ½s0 s1 . . . sM�1�T is an

M-tap FIR filter and yðnÞ ¼ ½yðnÞ yðn� 1Þ . . .yðn�M þ 1Þ�T. Substituting (2) into (1) yields

Sðnþ 1Þ ¼ SðnÞ þ m½vTðnÞS� vðnÞTSðnÞ�v�ðnÞ

þ mfðnÞv�ðnÞ, ð3Þ

where fðnÞ ¼ dðnÞ þ yTðnÞS is the interference to thecancellation path modeling, named the interferencesignal. When ANC system is in operation, theinterference signal consists of the same narrowbandcomponents with the disturbance signal d(n) [10].For off-line modeling, since the primary noise isabsent, the interference signal f(n) becomes zero and

Eq. (3) can be rewritten as

Soff ðnþ 1Þ

¼ Soff ðnÞ þ m½vTðnÞS� vðnÞTSoff ðnÞ�v�ðnÞ. ð4Þ

It can be seen that due to the interference signalf(n), SðnÞ deviates from Soff ðnÞ which is modelingwithout any disturbance. Assume this mismatch isDSðnÞ ¼ SðnÞ � Soff ðnÞ. Subtracting Eq. (3) fromEq. (4),

DSðnþ 1Þ ¼ DSðnÞ � m½vTðnÞDSðnÞ�v�ðnÞ

þ mfðnÞv�ðnÞ. ð5Þ

Suppose at sample N, DSðNÞ ¼ 0 (the estimatedimpulse response of the secondary path for on-linemodeling coincides with that for off-line modeling).So at sample N+1, DSðN þ 1Þ ¼ mfðNÞv�ðNÞ and atsample N+2,

DSðN þ 2Þ ¼ � mvTðN þ 1Þv�ðNÞmfðNÞv�ðN þ 1Þ

þXn¼Nþ1

n¼N

mfðnÞv�ðnÞ. ð6Þ

So if mvTðN þ 1Þv�ðNÞ51 (this condition is usuallysatisfied for on-line modeling), DSðN þ 2Þ �Pn¼Nþ1

n¼N mfðnÞv�ðnÞ. Substituting this into Eq. (5),one has

DSðN þ 3Þ � �XNþ1n¼N

mvTðN þ 2Þv�ðnÞmfðnÞv�ðN þ 2Þ

þXn¼Nþ2

n¼N

mfðnÞv�ðnÞ. ð7Þ

ARTICLE IN PRESSM. Wu et al. / Signal Processing 86 (2006) 2318–23252320

So if mvTðN þ jÞv�ðN þ 2Þ51, j ¼ 0; 1, DSðN þ 3Þ �Pn¼Nþ2n¼N mf ðnÞv�ðnÞ. (Summing over several samples

also makes the first term smaller because f(n) is

uncorrelated with vTðN þ 2Þv�ðnÞ.) In a similar way,

one has DSðN þ kÞ �Pn¼Nþk�1

n¼N mfðnÞv�ðnÞ if

mvTðN þ jÞv�ðN þ k � 1Þ51, 0pjpk � 2. So the

norm of mismatch vector DSðN þ kÞ due to theinterference signal is

jjDSðN þ kÞjj �XNþk�1

n¼N

mfðnÞv�ðnÞ

����������

�ffiffiffiffiffiffiMp XNþk�1

n¼N

mfðnÞv�ðnÞ

����������. ð8Þ

Here we assume the coefficient error is spreaduniformly over the coefficients of the filter. Suppose

vðnÞ ¼ V ðoÞejðnoþjÞ and fðnÞ ¼ ejno0 , substitutingthese into Eq. (8) yields

jjDSðN þ kÞjj ¼ mffiffiffiffiffiffiMp

V ðoÞ ejjXNþk�1

n¼N

ejnðo�o0Þ

����������

¼ mffiffiffiffiffiffiMp

V ðoÞXNþk�1

n¼N

ejnðo�o0Þ

����������

¼ mffiffiffiffiffiffiMp

V ðoÞ ejNðo�o0Þ1� ejkðo�o0Þ

1� ejðo�o0Þ

��������

¼ mffiffiffiffiffiffiMp

V ðoÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos½kðo� o0Þ�

1� cosðo� o0Þ

s. ð9Þ

Form (9) it can be seen that if the spectrum of

modeling signal is close to o0, jjDSðN þ kÞjj

becomes very large. Therefore, if the band stop(BS) random signal is used as the modeling signalwith stop band ½o0 � B=2;o0 þ B=2�, where B is theband width, the mismatch will be much smaller thanthat of using random signal.

When using the BS random signal as themodeling signal, although it is lack of frequencyinformation around o0, obtained SðnÞ is stillidentical with S in spectrum character at o0. Thiswill be shown below.

Assume FB/2, F�B/2 and Fo0are the values of

frequency response of SðnÞ at o0 � B=2, o0 þ B=2and o0,

FB=2 ¼XM�1i¼0

siðnÞe�jiðo0þB=2Þ, (10)

F�B=2 ¼XM�1i¼0

siðnÞe�jiðo0�B=2Þ, (11)

Fo0¼XM�1i¼0

siðnÞe�jio0 , (12)

where siðnÞ is the ith element of the vector SðnÞ.Subtracting Fo0

from both sides of (9) yields

FB=2 � Fo0¼XM�1i¼0

siðnÞe�jio0ðe�jiB=2 � 1Þ. (13)

If MB/2 is small, e�jiB=2 � 1 � �jiB=2. Substitutingthis into Eq. (13) yields

FB=2 � Fo0¼ �jB=2

XM�1i¼0

isiðnÞe�jio0 . (14)

In a similar way for (11), one has

F�B=2 � Fo0¼ jB=2

XM�1i¼0

isiðnÞe�jio0 . (15)

Adding (14) and (15) yields

Fo0¼ ðF�B=2 þ FB=2Þ=2. (16)

Assume F0�B=2, F 0

þB=2 and F0o0

are the values offrequency response of S at o0�B/2, o0+B/2 ando0. In a similar way, one has

F0o0¼ ðF 0

�B=2 þ F0B=2Þ=2. (17)

Because the BS random signal has the spectruminformation of o0�B/2, o0+B/2, after conver-gence, F�B=2 � F0

�B=2 and FB=2 � F 0B=2. Therefore

if MB/2 is small, from (16) and (17) the value of thespectrum of SðnÞ at frequency o0 approaches that ofS. In practical applications, the frequency of thedisturbance signal can be obtained by finding themaximum value in its spectrum. By using anadaptive notch filter, the BC modeling signal withstop band around the frequency of the disturbancesignal can also be obtained automatically, and thefrequency resolution of the notch filter should belarger than that of the cancellation path estimate.

3. Simulations

The secondary path S used in simulation wasmeasured from a real testing setup in which the pathwas modeled as a finite impulse response filter oforder 128. The order of the filter SðnÞ was chosen tobe 128 because S is assumed to be of the order 128.Two modeling signals are compared below, one iszero-mean random signal with a variance of 0.33and the other is the BS random signal which isobtained using the former signal through a bandstop filter with stop band ½0:195 0:205�p. The

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interference signal f(n) used in this simulation was asinusoidal signal of frequency 0.2p.

3.1. Using the standard LMS algorithm to update

SðnÞ

The amplitude of the interference signal f(n) was0.1. The same step size m ¼ 0:01 was selected forthese two signals as the stop band is so small thatthe energy of the BS random signal is almost thesame as that of random signal. The impulseresponse of the reference CPTF S was obtainedwhen there is no disturbance signal which is shownin Fig. 2(a). It can be seen from that the first 60coefficients of the reference CPTF are almost zero.This is because of the acoustic delay from thespeaker to the microphone. However when thedisturbance signal exists and the random noise isused as modeling signal, the first 60 coefficients ofCPTF are not zero but fluctuate around zero asshown in Fig. 2(b). The maximum deviation fromzero is proportional to the amplitude of interferencesignal. When using the BS random signal asmodeling signal, it can be seen from Fig. 2(c) thatthe deviation is much less as expected. As a whole,the impulse response of the obtained CPTF whenusing BS random signal as modeling signal is moreclose to the reference CPTF. Fig. 2 also shows thespectrum of S where (d) is the reference spectrum,(e) is the amplitude difference between the referenceCPTF and the obtained CPTF using random signalas modeling signal and (f) is the amplitudedifference between the reference CPTF and theobtained CPTF using BS random signal as model-ing signal. They show that the deviation aroundfrequency 0.2p with BS random signal is less thanwith random signal. Fig. 3 shows the phasedifference at frequency 0.2p between the referenceand the obtained CPTF with different modelingsignal for 100 trials. It can be seen that when usingrandom noise, the maximum phase difference canreach 1801 as shown in Fig. 3(a) whereas using BSrandom noise the maximal phase difference is only591 as shown in Fig. 3(b).

No matter what kind of modeling signal, thesmaller the step size coefficient, the smaller themodeling error. This also holds for the BS randomsignal. Table 1 shows that with the same step sizecoefficient, the modeling error of using the BSrandom signal is much smaller than that of usingthe random signal. The values listed in the table arethe maximum value of 100 trials for each case.

3.2. The selection of band width

The amplitude difference at frequency 0.2p isshown in Fig. 4(a) (average with 100 trials), whenusing BS random noise with different stop bandwidth B as modeling signal. The standard LMSalgorithm is adopted to update SðnÞ and step sizem ¼ 0:002. It can be seen that the amplitudedifference is least when the BS random noise withthe stop band width B ¼ 0:01 is used as modelingsignal. The least phase difference is achieved whenthe BS random noise with the stop band width B ¼

0:02 is adopted which is shown in Fig. 4(b). Whenthe stop band width B is larger than 0.02, both theamplitude difference and the phase differencebecome very large at frequency 0.2p. So thefrequency response at frequency 0.2p of S cannotbe modeling if the BS random signal with stop bandwidth B40:02 is used as modeling signal.

3.3. Using the step-size control NLMS algorithm to

update SðnÞ

The optimum step-size is calculated using

moptðnÞ �E½ðuðnÞ � uðnÞÞ2�

E½e2s ðnÞ�¼

E½e2ðnÞ�E½e2s ðnÞ�

�jjSðnÞ � Sjj2jjvðnÞjj2

ME½e2s ðnÞ�, ð18Þ

where uðnÞ ¼ vTðnÞS and uðnÞ ¼ vTðnÞSðnÞ. The valueof the optimum step size is dependent on the short-term power of the modeling signal, on the short-term power of the error signal, and on the currentnorm of the system mismatch vector jjS� SðnÞjj.Since the latter is not known, it has to be estimated.One method to estimate the norm of systemmismatch vector is based on delay coefficients.Because of the delay between speaker and micro-phone, the first MT coefficients of the adaptive filterdenoted ST ðnÞ should converge to zero. Assume thatthe coefficient error is spread uniformly over thecoefficients of the filter then

jjSðnÞ � Sjj2 ¼M

MT

jjST ðnÞjj2. (19)

The expectation E½e2s ðnÞ� can be calculated by

E½e2s ðnÞ� ¼ 0:95E½e2s ðn� 1Þ� þ 0:05e2s ðnÞ (20)

Taking (18) together with ð19; 20Þ, a rule for thestep-size can be specified. More details can be foundin Ref. [15].

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Fig. 2. (a) Reference impulse response of CPTF; (b) modeled impulse response of CPTF obtained by using random signal as modeling

signal; (c) modeled impulse response of CPTF obtained by using BS random signal as modeling signal; (d) reference amplitude response of

CPTF; (e) amplitude difference between the reference CPTF and the obtained CPTF using random signal as modeling signal; and (f)

amplitude difference between the reference CPTF and the obtained CPTF using BS random signal as modeling signal.

M. Wu et al. / Signal Processing 86 (2006) 2318–23252322

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Fig. 3. The phase difference at frequency 0.2p between the

reference and the obtained CPTF with different modeling signal

(a) using random noise; and (b) using BS random signal.

Table 1

Maximum phase differences of 100 trials with different step size

at frequency 0.2p

Step size 0.001 0.002 0.004 0.006 0.008 0.01

Phase

diff.

(deg.)

Random

signal

62 82 104 161 175 180

BS random

signal

4 9 17 26 44 59

Fig. 4. Difference at frequency 0.2p between the reference and

the obtained CPTF with different stop band width B. (a) Mean

magnitude difference; and (b) maximum phase difference.

M. Wu et al. / Signal Processing 86 (2006) 2318–2325 2323

Fig. 5 shows the convergence of tap-weight errornorm when using the optimum step size NLMSalgorithm. The number of delay coefficientsMT : ¼ 30. It is shown in Fig. 5(a) that both randomnoise and BS random noise can converge to idealvalue when the disturbance signal is absent and the

speed of convergence using the BS random signal islittle slower than using random signal. When thedisturbance signal is present, it can be seen fromFig. 5(b) that using BS random signal as modelingsignal results in much faster convergence than usingrandom signal.

3.4. Using the subband LMS algorithm to update

SðnÞ

The subband adaptive filter (closed loop) isadopted in this simulation [16]. The modeling signaland the error signal are decomposed into M ¼ 32subbands and then decimated by a factor D ¼ 16. A256-tap FIR lowpass filter was designed for thepolyphase prototype using the method proposed byHarteneck et al. [17]. In each subband, the NLMSalgorithm was adopted. For the subbands without

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Fig. 5. Convergence of tap-weight error norm using the step-size

control NLMS algorithm when the amplitude of interference

signal f(n) was (a) 0, and (b) 10.

Fig. 6. Norm of system mismatch using the subband NLMS

algorithm when the amplitude of interference signal f(n) was 1. (a)

Step size m ¼ 0:1 for subband with disturbance, and (b) m ¼ 0:01for subband with disturbance.

M. Wu et al. / Signal Processing 86 (2006) 2318–23252324

disturbance the step size is one and for the subbandwith disturbance the step size is much less. It can beseen from Fig. 6 that even using the subband LMSalgorithm to make the error spreading only in singlesubband, using BS random signal is not worse thanusing random signal as modeling signal. It also canbe seen that the system is more robust using BSrandom signal than random signal at the same stepsize.

4. Conclusions

This communication compared different model-ing signals when the disturbance signal has a largepure tone or narrow band component. Boththeoretical analyzes and experiments show thatusing the BS signal can reduce the modeling error

at the frequency of interest. This is different fromthe common sense in the cancellation path modelingin ANC where it is believed that more energy shouldbe put in the frequency of interest.

Acknowledgments

Project 10304008 supported by NSFC. The workwas also sponsored by SRF for ROCS, SEM andSRFDP.

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