An Improved Active Noise Control Algorithm Without Secondary Path Identification Based on the...

IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 16, NO. 8, NOVEMBER 2008 1409

An Improved Active Noise Control AlgorithmWithout Secondary Path Identification Based on the

Frequency-Domain Subband ArchitectureMing Wu, Guoyue Chen, and Xiaojun Qiu

Abstract—Common active noise control (ANC) algorithmsneed to identify the secondary path transfer functions betweenthe output of the adaptive control filters and the error sensors,and then use the information to guide the direction of controlfilter coefficient updating. Recently, Zhou et al. proposed an ANCalgorithm without secondary path identification, and we improvetheir algorithm in this paper. For single-tone and narrowbandnoise control, the direction of control filter coefficient updatinghas four choices 180 , 0 , and 90 . We test the four updatedirections and select the one that works the best. If for all fourupdate directions, the system converges slowly or diverges, weadjust the step size and test again with the new step size. Themultitone and broadband noise control problems are convertedinto several single-tone and narrowband noise control problemsby means of a frequency-domain delayless subband architecture.Compared to Zhou’s algorithm, our proposed method yields goodperformance and converges quickly. Simulation results confirmthe effectiveness of our proposed algorithm.

Index Terms—Active noise control (ANC), adaptive filters, de-layless subband algorithm.

I. INTRODUCTION

A CTIVE noise control (ANC) is a method for attenuatingunwanted disturbances by the introduction of controllable

secondary sources, whose outputs are arranged to interfere de-structively with the original primary source [1]–[3]. For mostANC algorithms, it is necessary to identify the secondary pathtransfer functions between the output of the adaptive control fil-ters and the error sensors, which includes the D/A converter,power amplifier, actuator, physical path, error sensor, and othercomponents.

When the secondary path is time-varying, an online sec-ondary path identification technique is adopted. The mostpopular online secondary path identification technique isproposed by Eriksson et al. [4], where an additional noise isinjected into the actuators as a modeling signal. However, thisonline secondary path identification technique will bring someproblems to the control system. For example, as the secondary

Manuscript received December 13, 2007; revised August 07, 2008. This workwas supported by the NSFC and NCET under Project 10674068. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Sen M. Kuo.

M. Wu and X. Qiu are with the Key Laboratory of Modern Acoustics andInstitute of Acoustics, Nanjing University, Nanjing 210093, China (e-mail:[email protected]; [email protected]).

G. Chen is with Department of Electronics and Information Systems, Facultyof Systems Science and Technology, Akita Prefectural University, Akita 015-0055, Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TASL.2008.2005027

path is identified under the disturbance, error between the truesecondary path and the estimated secondary path is inevitable,and if the phase error of the estimation exceeds 90 , theadaptive algorithm might diverge. Furthermore, the modelingsignal injected into the actuator contributes to the residualnoise. To obtain the precise secondary path model under thedisturbance, some new algorithms have been proposed [5]–[8].The modeling signal should also be designed elaborately toreduce its contribution to the residual noise [9]–[11]. However,each of these algorithms significantly increases the controlsystem complexity.

To avoid the problems caused by online secondary pathidentification, several algorithms that do not require secondarypath identification have been proposed [12]–[19]. Among allthese algorithms, the algorithm proposed by Zhou et al. [19] isthe most attractive due to its simple implementation and goodperformance. In his algorithm, instead of the filtered-x LMS(FXLMS) algorithm, the standard LMS algorithm is adoptedto update the adaptive filter coefficients, where the referencesignal does not need to pass through the secondary path. Theadaptive filter coefficients converge to ideal values if the phaseangle of the secondary path is within 90 . If the phase angle ofthe secondary path is outside of the range of 90 , by changingthe sign in front of the step size, the adaptive filter coefficientswill still converge. The proper update direction of the adaptivefilter coefficients (the proper sign in front of the step size) ischosen automatically by monitoring the excess noise power.

Previously, we proposed a new frequency-domain delaylesssubband architecture, where an individual step size is adoptedfor each frequency bin [20]. Based on that architecture, we im-proved the ANC algorithm proposed by Zhou et al. Comparedto Zhou’s algorithm, the improved algorithm has the followingmerits.

1) In Zhou’s algorithm, there are only two choices, 180and 0 , for the update direction, which is implemented bychanging the sign in front of the step size. If the phaseresponse of the secondary path is close to 90 , the algo-rithm will converge very slowly. To solve this problem,Zhou et al. suggest adding a delay in the reference signalto push the phase away from 90 . However, they donot give a method to judge whether the phase response ofthe secondary path is close to 90 . Also, for differentfrequencies, the same delay corresponds to different phaseshifts, so it is hard to decide how much delay shouldbe added in the reference signal. In our proposed algo-rithm, there are four choices 180 , 0 , and 90 for theupdate direction. The update direction that satisfies the

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1410 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 16, NO. 8, NOVEMBER 2008

convergence condition is used to guide the control filtercoefficient updating. The convergence condition is stricterthan that in Zhou’s algorithm in order to guarantee thatthe system has a faster convergence rate. For example, ifthe phase response of the secondary path is close to 90 ,though the system will converge if the update directionis 180 or 0 , the convergence rate is very slow and theconvergence condition is not satisfied. To avoid slowconvergence, the update direction 90 is selected in thissituation.

2) According to the strict positive real property [19], the stepsize for adaptive filter coefficient updating should satisfy

(1)

where is the phase response of the secondary path,is the magnitude response, and is the power

spectrum of the input signal. Since the secondary path isunknown, the step size in Zhou’s algorithm should be suf-ficiently small to guarantee system stability, which leadsto a decreased convergence rate. In our algorithm, a properstep size can be chosen automatically without any infor-mation on the secondary path.

3) For narrowband noise control, the secondary path phaseresponse in Zhou’s algorithm should satisfy

(2)

where is an arbitrary integer. Usually, this constraint issatisfied if the bandwidth is small. However, if the noisebandwidth exceeds frequency band over which the sec-ondary path phase response is 90 , (2) cannot be sat-isfied. So such narrowband noise cannot be reduced nomatter what the update direction is. In our algorithm, theconstraint for narrowband noise control is that the dynamicrange of the secondary path phase response is less than 90

(3)

This constraint is always satisfied provided that the band-width is small enough.

4) In Zhou’s algorithm, a subband technique is adopted toconvert the broadband ANC problem into several nar-rowband problems [21]–[23]. In each narrow band, thesecondary path phase response should satisfy the con-straint of (2). However, it is found that in some cases,to ensure that the phase response of each narrowbandsecondary path meets the constraint of (2), a large numberof subbands is needed, which can degrade the systemperformance. To solve this problem, an adaptive subbandselection method was proposed in [19] to seek the re-quired number of subbands that must be used. However,this method is somewhat complicated and especially itwill introduce unevenly distributed subband filters, whichleads to higher computational complexity for subbandfiltering. In our algorithm, by means of the frequency-do-main delayless subband architecture, narrow bands ofsufficiently narrow bandwidth within each subband can beindependently processed without increasing the number

Fig. 1. Frequency-domain delayless subband algorithm.

of subbands. The adaptive subband selection method isnot needed in our algorithm and the unevenly distributedsubband filter is also avoided, which significantly reducesthe computation load of the algorithm.

5) For low-reverberation ANC cases, a simple secondary pathphase estimation technique is also proposed in this paperbased on our proposed algorithm. Unlike conventionalmodeling, no auxiliary modeling noise is needed in thistechnique. For conventional modeling, the modeling erroris affected by the disturbance signal, while the modelingerror of the proposed technique is immune to the distur-bance signal.

II. FREQUENCY-DOMAIN DELAYLESS

SUBBAND ALGORITHM

In this section, our frequency-domain delayless subbandANC system based on [21] is introduced briefly. As shown inFig. 1, it can be divided into five parts.

1) Subband Filtering. The input signal and error signalare fed to contiguous single-sideband bandpass

filters and then downsampled by a factor toproduce complex subband input signals andcomplex subband error signals , where the con-tiguous single-sideband bandpass filters are usually real-ized by means of the polyphase FFT technique. For realsignals, only half of the subbands, corresponding to thepositive frequency components of the wideband filter re-sponse, need to be processed. The other half of the responseis formed in complex conjugate symmetry [21], [22].

2) Adaptive Weight Update. For the conventional delaylesssubband algorithm, the subband adaptive filter coefficientsare updated in the time domain [21], [22]

(4)

where is the th subband adaptive weights, is thestep size,is the filtered reference input vector, is thelength of the wideband controller, denotes transpose, and

WU et al.: IMPROVED ACTIVE NOISE CONTROL ALGORITHM WITHOUT SECONDARY PATH IDENTIFICATION 1411

superscript * denotes complex conjugate. By means of theblock update technique, (4) can be rewritten as

(5)

where

(6)

is a Toeplitz matrix and

(7)

is the subband error vector. It is well knownthat the Toeplitz matrix can be transformed to acirculant matrix by doubling its size

(8)

where is also a Toeplitz matrix andis a circulant matrix. By using (8),

(5) can be expanded as

(9)

where are adaptive weights that are discarded.Multiplying both sides of (9) by theFourier transform matrix , the adaptive weights in thefrequency domain are updated as

(10)

where

(11)

is the frequency-domain subband adaptiveweight vector

(12)

is the frequency-domain subband error vector,and

(13)

is a diagonal matrix. The elements of are the dis-crete Fourier transform of the first column of de-fined as the vector

(14)

3) Filtered Reference Signal Generation. The filtered refer-ence input is the linear convolution of with

. Therefore, can be calculated approximately bythe FFT technique [2]

(15)

where

(16)

is the frequency-domain subband referencesignal vector and is a diagonal matrix whose elementsare given by the transformed vector

(17)

where is the estimated subband-decomposed sec-ondary path response vector.

4) Subband/Wideband Weight Transformation. In this part,the wideband weight is recovered from the fre-quency-domain subband adaptive weights . Afterfrequency stacking, the vector are converted to a

wideband weight vector . Since

(18)

it is obvious that the first components of corre-spond to the vector after subband/wideband weighttransformation, and the last components ofcorrespond to the vector after subband/widebandweight transformation. So the -length adaptive weight

can easily be obtained by discarding the last com-ponents of .

5) Signal Path Convolution with Wideband Filter . Thecontrol signal for the secondary source is the output of thewideband control filter .

The frequency-domain delayless subband algorithm hasfaster convergence since an individual step size can be adoptedfor each frequency bin, and the computational complexity isalso reduced. For example, assume that the length of the con-troller , the length of the secondary path ,the length of the polyphase prototype filter , thenumber of subbands , and decimation factor .Then, 6144 real multiplications per iteration are required forthe FXLMS algorithm, 4439 for Morgan’s algorithm, 2655for Park’s algorithm (time-domain implementation), and only2188 for the algorithm of Fig. 1. Park et al., also proposed afrequency-domain delayless subband algorithm [22], wherethe subband adaptive weights are updated by the sliding DFTalgorithm. In Park’s algorithm, only the decimated samples ofthe error signal for are used to update the subbandweights, which will result in large error variance at steady-stateor reduce the convergence speed. In our algorithm [20], thesubband adaptive weights are updated by the overlap-save


TABLE ISINGLE-TONE NOISE ANC WITHOUT IDENTIFYING THE SECONDARY PATH

frequency-domain algorithm, which yields better performancethan Park’s method.

III. ANC WITHOUT SECONDARY PATH IDENTIFICATION

Based on the architecture shown in Fig. 1, an ANC algorithmwithout secondary path identification is proposed in this section.For simplicity, (10) can be rewritten as

(19)

where is the actual fre-quency and is the sampling frequency. For example, assumethe sampling rate is 2048 Hz, , and .Then, according to the relationship between the subband signaland the wideband signal [21], at Hz, we have that

, and correspond tothe 73rd element of , and respec-tively. According to the geometric analysis [19], the update di-rection for is the same as . If the phase errorbetween the update direction and the phase responseof the secondary path is larger than 90 will diverge.In Zhou’s algorithm, is assumed to be or , whichmeans the update direction has only two choices 0 and 180 .When the phase response of the secondary path is close to 90 ,the system will converge very slowly since the phase error is al-most 90 . In our algorithm, is assumed to be

, or , where is the imaginary unit, which means the updatedirection has four choices 0 , 180 90 , and 90 . If the phaseresponse of the secondary path is close to 90 , the update di-rection 90 is selected and slow convergence is avoided. Of

Fig. 2. Flowchart of the proposed algorithm.

course, can be absorbed into the step size and no ad-ditional computation is needed for filtering the reference signal

(20)

A. Single-Tone Noise

The algorithm for single-tone noise control proposed in thispaper is also divided into four stages as shown in Fig. 2, and theprocedure for each stage is detailed in Table I. The initializationstage will be discussed in detail in Section III-E, and the perfor-mance monitoring stage is the same as that in Zhou’s algorithm[19]. In the updating stage, (20) is used. The direction searchstage is significantly different from that in Zhou’s algorithm. Inthe direction search stage, the convergence condition, which isstricter than that in Zhou’s algorithm, is

(21)

where a new parameter is introduced to guarantee thatthe system has faster convergence, and are the estimatedreference signal power and error signal power without update,

and are the estimated reference signal power and errorsignal power with update, is the maximum error signalamplitude without update, and is a positive number to


Fig. 3. Conventional single-tone ANC system.

make the algorithm tolerate estimation errors. The divergencecondition is

(22)

If for all four choices , the convergencecondition is not satisfied, then the step size is considered im-proper. If for all four choices of , the diverge condition issatisfied, then the previous step size is considered too large andis decreased to , where . Otherwise, the step size isconsidered too small to make the system have fast convergenceand is increased to . After adjusting the step size, theproper direction is recalculated.

In addition to the subband algorithm, there are many otheralgorithms for single-tone noise control. The above methodwithout secondary path identification can be easily extendedto these algorithms. The most used algorithm for single-tonecontrol is shown in Fig. 3 [2]. A nonacoustic sensor such as atachometer or an accelerometer provides frequency informa-tion for a signal generator to synthesize internally generatedreference signals

(23)

and

(24)

where is the frequency of the primary noise. The referencesignals are then processed by a two-tap adaptive filter

to generate the input signal of the secondarysource

(25)

The error sensor measures the residual noise and uses it toupdate the coefficients of the adaptive filter by the FXLMSalgorithm

(26)

(27)

where and are the filtered reference signals. As-sume the frequency response of the estimated secondary path is

(28)

Then, the filtered reference signals can be expressed as

(29)

and

(30)

where denotes the real part of the complex and Im denotesthe imaginary part. In our method, the update direction has fourchoices 0 , 180 , and 90 , which correspond to

, and , respectively. The properupdate direction can be obtained in the same way as shown inTable I.

Based on the algorithm for single-tone noise control, algo-rithms for multitone noise, narrowband noise, and broadbandnoise control are developed next.

B. Multitone Noise

The multitone noise control problem is converted into severalsingle-tone ANC problems. For each single-tone ANC problem,independent parameters are adopted. The update direction foreach single-tone is judged by monitoring its own error signal.

C. Narrowband Noise

For narrowband noise that satisfies the constraint (3), the al-gorithm is similar to the algorithm for single-tone noise. Themain difference between the algorithms for narrowband noisecontrol and single-tone noise control is in the power measure-ment. For example, with single-tone noise control, the meannoise power is

(31)

while for narrowband noise control, the mean noise power issummed over frequency

(32)

where is the lower cutoff frequency, is the upper cutofffrequency, and is the number of samples for estimating thenoise power.

D. Broadband Noise

The broadband noise control problem is converted intoseveral narrowband ANC problems. By the frequency-domaindelayless subband architecture, the bandwidth of each narrowband can be selected narrow enough to satisfy (3) withoutincreasing the number of subbands. For example, assume thesampling frequency is 2048 Hz and the primary noise is in therange [0 1024] Hz. The broadband noise control problem isdivided into five narrowband noise control problems. The five


Fig. 4. (a) Phase response of secondary path. (Solid line is for the true sec-ondary path, dotted line is for the estimated secondary path, “o” are the updatedirections for each narrow band). (b) Phase error between the true secondarypath and the estimated secondary path.

narrowband ranges are [0 128] Hz, [128 384] Hz, [384 640]Hz, [640 896] Hz, and [896 1024] Hz. For the conventionalsubband architecture, the required number of subbands is

. However, by means of the frequency-domain subbandarchitecture, it can also be achieved with . When ,the three subband ranges are [0 256] Hz, [256 768] Hz, and[768 1024] Hz. In the frequency-domain subband architecture,the adaptive weights are updated in the frequency domain andeach has its own step size. Therefore, we can make the adaptiveweights ( Hz) have one update direction,

( Hz) have another update direction,and so on. By using this method, the broadband noise controlproblem is divided into five narrowband noise control problemswith a smaller number of subbands . Based on theupdate directions for each narrow band, a simple method isgiven to roughly estimate the phase response of the secondarypath for low-reverberation ANC cases. For low-reverberationANC cases, the phase response of the secondary path decreasesalmost linearly from 180 to 180 due to the delay of theA/D and D/A converters and the physical path. For example,the phase response of the secondary path shown in Fig. 4(a)(solid line) is measured in an actual ANC system. So, if thephase responses at two frequencies are obtained, the phaseresponse at the other frequencies can be estimated roughly bylinear interpolation. For example, assume the primary noise is[200 800] Hz. The broadband noise control problem is divideduniformly into 30 narrowband noise control problems and thebandwidth of each narrow band is 20 Hz. The update directionsfor each narrow band is judged independently and the resultsare shown in Fig. 4(a) (denoted by “o”). It can be seen that whenthe phase response of the secondary path is close to 90 , theupdate direction is changed to 90 to avoid slow convergence.Denote the center frequency of the band where the updatedirections are 90 by . It is reasonable to suppose that at

, the phase response of the true secondary path is 90 . In thesame way, is obtained, where the phase responses of thetrue secondary path are supposed to be 90 . In this example,

Hz and Hz. The phase response at otherfrequencies is estimated by linear interpolation

(33)

The estimated phase response of the secondary path is shownin Fig. 4(a) (dotted line), and the estimation error is shown inFig. 4(b). It can be seen that the phase response of the secondarypath can be estimated roughly by this simple method. Compared

Fig. 5. Impulse response of (a) the primary path and (b) the secondary path.

to the online secondary path modeling techniques, this methoddoes not require an additional modeling noise source and is im-mune to the disturbance signal.

E. Parameter Initialization

As shown in Table I, there are a total of eight user-dependentparameters that need to be initialized: and

. The first five parameters are initialized in the same way asZhou’s algorithm [19]. The forgetting factor is usually chosenas a value between 0.9 and 0.99. The number of samples for esti-mating the noise power is set according to the reference signalas well as the variance of the additive noise. The parameters

, and are introduced to make the algorithm tolerate estima-tion errors while remaining sensitive to changes in the secondarypath. The variation factor is given by

(34)

where and represent the maximum and the minimuminstantaneous power of the additive noise . The choice ofsmall positive numbers and depends on and the dis-tribution of the additive noise. Although in our algorithm thestep size is adjusted automatically in the direction search stage,a proper initial step size can significantly reduce the time fordirection search and increase the convergence rate. Assume thetime-varying secondary path magnitude response is in the range

. The bandwidth is narrowenough to make the phase difference between the update direc-tion and the secondary path less than . In this situation, a properinitial step size is given by

(35)

Parameter is a positive number introduced to guarantee thatthe system has faster convergence, and its choice depends onthe convergence rate. Finally, the parameter is a factor foradjusting the step size, which is usually chosen as a value 0.5.

IV. SIMULATIONS

Computer simulations are carried out to validate the proposedalgorithm. The primary path and the secondary path used forthe simulations are measured in the listening room of NanjingUniversity and modeled by a 256-tap FIR filter and a 128-tapFIR filter, respectively, as shown in Fig. 5. A 128-tap FIR low-pass filter is designed for the polyphase prototype using themethod proposed by M. Harteneck [24]. All signals are sampledat 2048 Hz and the measurement signal-to-noise ratio (SNR) is30 dB. The parameters


Fig. 6. Learning curve for single-tone noise control.

Fig. 7. Secondary path after change. (a) Impulse response. (b) Phase response.

, and the initial step sizes are se-lected. The length of the adaptive filter , the numberof subbands , and the decimation factor . The al-gorithm performance is evaluated by the residual noise power,based on an ensemble average of 100 runs.

A. Single-Tone ANC for Stationary Secondary Path

In this simulation, the phase response of the secondary pathis shown in Fig. 4(a). The primary noise is a single tone of fre-quency 340 Hz. The number of samples for estimating the noisepower is . In Zhou’s algorithm, the step size is set to thelargest possible value while still assuring that the adaptive filterconverges. Since the secondary path is stationary, the adaptivefilter is updated after determining the correct update direction,and the performance monitoring stage is not used. Fig. 6 showsthe learning curves of different algorithms. The phase responseof the secondary path at 340 Hz is 82 , which makes the adaptivefilter in Zhou’s algorithm converge slowly, while our algorithmsconverge much faster.

B. Multitone ANC for Changing Secondary Path

In this simulation, at time 23 s, the secondary path is changedto the 128-tap FIR filter shown in Fig. 7. The primary noise is amulti-tone signal with frequencies 40 Hz, 340 Hz, and 700 Hz.The phase responses of the secondary path before change at40 Hz, 340 Hz, and 700 Hz are 48 , 82 , and 109 , respec-tively. After direction search, the update directions at 40 Hz,340 Hz, and 700 Hz are adjusted automatically to 0 , 90 , and

90 . The phase responses of the secondary path after changeat 40 Hz, 340 Hz, and 700 Hz are 86 , 4 , and 159 , respec-tively. After direction search, the update directions at 40 Hz,340 Hz, and 700 Hz are adjusted automatically to 90 , 0 , and180 . It can be seen that after direction search, the proper update

Fig. 8. Frequency response magnitude of the secondary path, (a) before and(b) after change.

Fig. 9. Learning curve of the proposed algorithm for mutiltone noise controlwith a change at 23 s.

direction is selected to make the phase error less than 90 . Atfrequencies where the phase responses of the secondary path isclose to 90 , the update direction is adjusted to 90 . Fig. 8shows the frequency responses magnitude of the secondary pathbefore and after change. At 40 Hz, the response magnitude israther small, so sufficiently fast convergence rate cannot be ob-tained with the initial step size. Therefore, after direction search,the step size is adjusted to 0.4. At 700 Hz, the initial step sizeseems to exceed the stability bound, so after direction search,it is automatically decreased to 0.1. The learning curve of theproposed algorithm is shown in Fig. 9. It can be seen the perfor-mance monitor detects the change of secondary path, and thenthe update directions are changed according to the new envi-ronment. (Note, at the beginning there is a peak in the learningcurve, which is caused by the initial step size at 700 Hz beingtoo large; the peak in the learning curve at 23 s is due to thesudden change of the secondary path.)

C. Narrowband ANC for Stationary Secondary Path

In this simulation, the primary disturbance is narrowbandnoise in the range [660 700] Hz. The number of samples forestimating the noise power is . It can be seen fromFig. 4(a) that the frequency where the phase response of thesecondary path is 90 is just within this narrow band. So forZhou’s algorithm, whatever the sign in front of the step size is,the system diverges as shown in Fig. 10, while for our proposedalgorithm, the system works well since the constraint (3) issatisfied.

D. Broadband ANC

In this simulation, the primary noise is broadband in the range[200 700] Hz. This broadband noise control problem is divided


Fig. 10. Learning curve for narrowband noise control of Zhou’s algorithm(� sign), Zhou’s algorithm (� sign), and proposed algorithm.

Fig. 11. Learning curve of broadband noise control for stationary secondarypath, showing Zhou’s algorithm, the proposed algorithm without phase estima-tion, the proposed algorithm with phase estimation, and FXLMS algorithm.

uniformly into 25 narrowband noise control problems and thebandwidth of each narrow band is 20 Hz. Fig. 11 shows thelearning curves of different algorithms when the secondary pathis stationary. For convenience, Zhou’s algorithm is also realizedbased on the frequency-domain delayless subband algorithm.With a precise secondary path model, the FXLMS algorithmconverges faster than the others. In Zhou’s algorithm, to guar-antee stability for each narrow band (especially where the phaseresponse of the secondary path is close to 90 ), a sufficientlysmall step size is used, which decreases the convergence speedsignificantly. For the proposed algorithm, using linear interpo-lation to estimate the phase response of the secondary path givesbetter performance, however, at cost of more computational forfiltering the reference signal. Fig. 12 shows the learning curvesfor a sudden change of the secondary path. At time 23 s, the sec-ondary path is changed to the 128-tap FIR filter shown in Fig. 7.It can be seen that our proposed algorithm is robust with respectto a sudden change in the secondary path. (Note that for the sec-ondary path after change, the phase estimation technique is notused, since depends on the update direction below 200 Hz.).

Fig. 12. Learning curve of broadband noise control for the secondary path withsudden change, showing the proposed algorithm without phase estimation andwith phase estimation.

Fig. 13. Transition of the update direction wth (a) � � �, and (b) � � �.

E. Effect of Parameter

In this simulation, the primary noise is broadband in the range[200 440] Hz. This broadband noise control problem is divideduniformly into three narrowband noise control problems. Thethree narrowband ranges are [200 280] Hz, [280 360] Hz, and[360 440] Hz. The parameters

, and the initial step sizesare selected. The secondary path is shown in Fig. 4. Accordingto the geometric analysis of [19], the system will converge if theupdate directions at narrow bands [200 280] Hz, [280 360] Hz,and [360 440] are 180 90 , and 0 , respectively.

Fig. 13 shows the transition of the update directions for dif-ferent . For , it takes about 2 s to search the updatedirections. After direction search, the correct update directions180 90 , and 0 are obtained. For , it only takes 0.5 s tosearch the update directions. However, at narrowband [280 360]Hz, it generates a false update direction result because of thelarger estimation error.

F. Changes in Primary Noise and Additive Noise Power

In this simulation, at time 50 s (after the adaptive filter con-verges) and 100 s, there are 6-dB increases in the primary noisepower and the additive noise power, respectively. As a result,we choose from (34), , and . The otherparameters are the same as in Simulation E. It can be seen fromFig. 14 that our proposed algorithm tolerates the changes in boththe primary noise power and the additive noise power.


Fig. 14. Learn curve of the proposed algorithm with changes in primary noiseand additive noise power.

Fig. 15. Update direction (left) and step size (right) at narrow bands [200 280]Hz, [280 360] Hz, and [360 440] Hz (top to bottom) for initial step size � � ��.

G. Effect of Initial Step Size

In this simulation, the parameters, and are selected.

Fig. 15 shows the transition of the update directions and thestep sizes with the initial step size . It can be seen thatwith the initial step sizes , sufficiently fast convergencehas been obtained. So after direction search, the step sizes areunchanged. Fig. 16 shows the transition of the update directionsand the step sizes with the initial step size . It canbe seen that with the initial step sizes , more time is

Fig. 16. Update direction (left) and step size (right) at narrow bands[200 280] Hz, [280 360] Hz, and [360 440] Hz (top to bottom) for initial stepsize � � ��.

Fig. 17. Learning curve of the proposed algorithm with different values of theparameter � .

taken to search the update direction since the step sizes have tobe adjusted to obtain sufficiently fast convergence in advance.

H. Effect of Parameter

In this simulation, the parameters, and the initial step size

are selected. As shown in (21), the parameter isintroduced in the convergence condition to ensure that the dis-turbance will be reduced by at least dB within

samples. So smaller leads to faster convergence. Fig. 17


Fig. 18. Transition of the update direction with � � �� at narrow band(a) [200 280] Hz, (b) [280 360] Hz, and (c) [360 440] Hz.

shows the learning curves of the proposed algorithm with dif-ferent value of . It can be seen that the system withconverges faster than with . However, if is too small,the convergence condition is not satisfied no matter what thestep sizes and the update directions are. In this situation, thesystem will search the update direction all the time as shown inFig. 18.

V. CONCLUSION

In this paper, we improved the algorithm proposed by Zhou,and thereby achieve better performance. In the proposed algo-rithm, the update direction has four choices: 180 , 0 , and 90 .If the secondary path phase response is closed to 90 , the up-date direction 90 is selected so as to avoid slow convergence.The step size is selected automatically in the direct search stageto achieve fast convergence and also to ensure system stability.For narrowband ANC, noise reduction can be achieved if the dy-namic range of the secondary path phase response in the bandis less than 90 . This constraint is always satisfied if the band-width is small enough. The broadband noise problem is con-verted into several narrowband problems. By means of the fre-quency-domain delayless subband algorithm, the bandwidth ofeach narrow band can be selected narrow enough to satisfy theconstraint without increasing the number of subbands. Whenthe ANC system is working in a low-reverberation situation, asimple method for the secondary path phase response estima-tion can be used based on the proposed algorithm. Unlike con-ventional methods, no additional modeling signal is needed, andthe algorithm is inherently immune to the disturbance signal.

ACKNOWLEDGMENT

M. Wu would like to thank W. W. Gao for encouragementand support.

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Ming Wu graduated in electronics from NanjingUniversity, China, in 2002 and received the Ph.D.degree from Nanjing University, China, in 2007 fora dissertation on active noise control.

He has been with the Institute of Acoustics,Nanjing University, as a Postdoctoral Researchersince 2007. His main research areas include noiseand vibration control, electro-acoustics, and audiosignal processing.

Guoyue Chen received the B.S. degree from EastChina Normal University, Shanghai, China, in 1983and the M.S. and Ph.D. degrees from Tohoku Univer-sity, Sendai, Japan, in 1993 and 1996, respectively.

He was a Research Associate in the Departmentof Computer Science, East China Normal University,until 1989. He worked as a Research Associate atTohoku University from 1996 to 1999. Currently, heis an Associate Professor with Akita Prefectural Uni-versity, Akita, Japan. His research interests includedigital signal processing and its applications to active

noise control, adaptive control algorithms, and medical image processing.

Xiaojun Qiu graduated in electronics from PekingUniversity, Beijing, China, in 1989 and receivedthe Ph.D. degree from Nanjing University, Nanjing,China, in 1995 for a dissertation on active noisecontrol.

He was with the University of Adelaide, Adelaide,Australia, as a Research Fellow from 1997 to 2002.He has been with the Institute of Acoustics, NanjingUniversity, as a Professor on acoustics and signal pro-cessing since 2002. His main research areas includenoise and vibration control, room acoustics, electro-

acoustics, and audio signal processing. He has authored and coauthored morethan 100 technique papers on audio acoustics and audio signal processing.

Prof. Qiu is member of Audio Engineering Society and International Instituteof Acoustics and Vibration.

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