A robust evolutionary feedforward active noise control system using Wilcoxon norm and particle swarm optimization algorithm

Expert Systems with Applications 39 (2012) 7574–7580

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Short communication

A robust evolutionary feedforward active noise control system using Wilcoxon norm and particle swarm optimization algorithm Nithin V. George ⇑, Ganapati Panda School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 751 013, India

a r t i c l e

i n f o

Keywords: Active noise control Wilcoxon norm FxLMS algorithm Particle swarm optimization

a b s t r a c t The conventional filtered-x least mean square (FxLMS) algorithm commonly employed for active noise control (ANC) is sensitive to disturbances acquired by the error microphone and yields poor performance in such scenario. To circumvent this problem, in this paper, a Wilcoxon FxLMS (WFxLMS) algorithm is proposed and used in the design of an efficient ANC which is robust to outliers in the secondary path and immune to burst noise acquired by the error microphone. It is demonstrated through simulation study that under such situation the proposed algorithm outperforms the traditional FxLMS algorithm. A particle swarm optimization (PSO) algorithm based robust ANC system, which does not require the modeling of the secondary path is also derived in the paper. Improved performance of the robust evolutionary ANC system over L2 norm based evolutionary ANC system is also shown. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Advancement in low cost DSP and development of efficient control theory over the past couple of decades have led to enhanced interest in active noise cancellation for control of low frequency noise. The traditional single-input single-output feedforward ANC setup consists of an adaptive controller which drives a loudspeaker to generate a suitable antinoise that suppresses the undesired noise (Kuo & Morgan, 1996). An error microphone is used to detect the level of residual noise, which is the superposition of sound waves from the noise source and that from the active loudspeaker. A reference microphone placed near the noise source picks up a reference signal about the noise to be controlled. The controller weights are adapted using least mean square (LMS) algorithm (Widrow & Stearns, 1985), which uses the signals from the reference and error microphones. The presence of an electro-acoustic path from the input of the active loudspeaker to the output of the error microphone, hereafter called the secondary path, affects the convergence of the learning algorithm. In order to compensate the effect of the secondary path, the reference signal to the LMS algorithm is filtered through a model of the secondary path, giving rise to the well known FxLMS algorithm. The residual noise not only consists of the superposition of the two signals but also contains disturbance signals which are always present in practical implementations. Significant disturbances might occur due to movement of error microphone or active loudspeaker (Lan, Zhang, & Ser, 2002). The temporary or permanent

⇑ Corresponding author. Tel.: +91 9438227079; fax: +91 674 2301983. E-mail addresses: [email protected] (N.V. George), [email protected] (G. Panda). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.01.038

breakdown of sensors, analog-to-digital conversion errors, failure of transducers, measurement noise etc also introduce outliers in the secondary path. These disturbances not only increase the residual noise, but also severely affect the convergence performance of FxLMS algorithm. At times these may lead to divergence of the weights of the controller and hence poor performance. Presence of such disturbances necessitates the development of a learning algorithm which is robust against outliers in the secondary path as well as burst noise at the error microphone and would provide proper convergence of controller weights. Some research studies in this direction have been reported, but they have some limitations in achieving the desired performance. Before reviewing on similar work carried out in the literature a brief overview of an FxLMS based ANC system is presented. Fig. 1, excluding the serial-to-parallel blocks, outliers and burst noise sources represents a basic FxLMS algorithm based ANC system. Each of the blocks in this structure is modeled as a finite impulse response (FIR) filter. The primary noise at the error sensor, d(n) is the reference noise x(n) filtered through the primary path P(z), which is the path from the reference microphone to the error sensor. The active loudspeaker is driven by an adaptive controller, the weights, w(n) of which is updated according to the FxLMS algorithm given by

wðn þ 1Þ ¼ wðnÞ þ leðnÞ½^sðnÞ  xðnÞ

ð1Þ

where e(n) is the residual noise, n denotes the time index and l represents the step size. ^sðnÞ is the impulse response of the model of the secondary path S(z) and ‘‘⁄’’ denotes the linear convolution operation.

N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580

Fig. 1. Block diagram of the proposed WFxLMS algorithm based single-channel feedforward ANC system.

A weight constrained FxLMS (CFxLMS) algorithm is proposed in Lan et al. (2002), which is more robust compared to normal FxLMS algorithm. In this case, the tap-weights are updated as

wðn þ 1Þ ¼ wðnÞ þ leðnÞ½^sðnÞ  xðnÞ ( wðn þ 1Þ ¼

wðn þ 1Þ if kwðn þ 1Þk 6 b bwðnþ1Þ kwðnþ1Þk

if kwðn þ 1Þk > b

ð3Þ

ð4Þ

where

8 for eðnÞ 6 c1 > < c1 ~eðnÞ ¼ c2 for eðnÞ P c2 > : eðnÞ otherwise

for an ANC system (Russo & Sicuranza, 2006, 2007; Yim, Kim, Lee, & Ahn, 1999). An adaptive genetic algorithm based ANC system which does not require secondary path identification has been developed in Chang and Chen (2010). The particle swarm optimization (PSO) algorithm has gained much attention in the recent past owing to its faster and stable convergence over genetic algorithm (GA). PSO algorithm is a population based stochastic evolutionary algorithm (Kennedy & Eberhart, 1995, 2001). It employs a population (swarm) of individuals (particles) which move in a search space with an adaptable velocity to search for possible solutions. A large number of variants of PSO are available in literature. PSO has found applications in a wide range of fields and has been employed for updation of weights of a multilayer neural network in a nonlinear ANC system (Modares, Ahmadyfard, & Hadadzarif, 2006). A PSO based robust ANC system, which does not require the estimation of the secondary path is developed and its performance is accessed in this paper. The paper is organised as follows. The proposed robust FxLMS algorithm is derived in Section 2. A PSO based robust active noise control approach is developed in Section 3. Extensive simulation study is conducted in Section 4 followed by concluding remarks in Section 5. 2. Proposed robust learning algorithm

ð2Þ

where b is the constraining factor selected through a series of experiments and k  k denotes Euclidean Norm. Proper selection of the constraining factor, which is difficult in most practical applications, is essential for robust performance of this algorithm. Another robust FxLMS algorithm suggested in Akhtar and Mitsuhashi (2009) is given by

wðn þ 1Þ ¼ wðnÞ þ l~eðnÞ½^sðnÞ  ~xðnÞ

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A robust learning algorithm using Wilcoxon norm is derived in this section. The Wilcoxon norm is a rank based statistical parameter (Hogg, McKean, & Craig, 2005). The computation of ranks in a Wilcoxon norm based learning algorithm, requires the adoption of an epoch or block learning strategy. Epoch based learning techniques are time consuming and are difficult to implement for real time applications like the ANC. On the other hand, block learning approach is faster as the weights of the controller are updated after every block of input samples. Hence a block learning technique is adopted in this paper. The complete ANC system in presence of burst noise acquired by the error microphone and outliers in the secondary path is depicted inFig.  1. If L is the block length and k is the block index (0 < k < TL , for T samples of input noise), the controller signal at any instant kL + i is expressed as

ð5Þ yðkL þ iÞ ¼

M1 X

xðkL þ i  jÞwj ðkÞ for 0 6 i 6 L  1

ð7Þ

j¼0

and

8 for xðnÞ 6 c1 > < c1 ~xðnÞ ¼ c2 for xðnÞ P c2 > : xðnÞ otherwise

ð6Þ

are the transformed error and reference signals respectively. This algorithm is based on the principle of replacing large amplitude samples by an appropriate threshold value and thus introduces saturation nonlinearity to the reference and error signals. The performance is directly dependent on the estimated threshold magnitudes c1 and c2 which is difficult to estimate in a practical scenario. A robust LMS algorithm has been recently reported in Majhi, Panda, and Mulgrew (2009), using Wilcoxon norm (Hsieh, Lin, & Jeng, 2008) and is shown to offer robust performance (Ban & Kim, 2009). Accordingly in this paper we propose a Wilcoxon norm based FxLMS (WFxLMS) algorithm for ANC application and its performance is evaluated and compared with that obtained using the conventional FxLMS algorithm. The proposed algorithm as well as FxLMS algorithm require the estimation of the secondary path. Errors in the estimation of the secondary path could lead to divergence in the adaptation of ANC algorithms. The algorithms may also converge to a local minima solution. In order to overcome the above problems, genetic algorithm has been recently proposed

where M is the length of the impulse response of the controller. wj is the jth element of the weight vector w(k) = [w0(k), w1(k), . . . , wj(k), . . . , wM1(k)]T and x(n) = 0 for n < 0. The secondary noise near the error microphone, y0 (n) is given by

y0 ðkL þ iÞ ¼

N1 X

yðkL þ i  jÞwSj

ð8Þ

j¼0

where wSj is the jth element of the impulse response of the secondary path, S(z). The residual noise measured by the error microphone, e(n) is

eðkL þ iÞ ¼ dðkL þ iÞ  y0 ðkL þ iÞ

ð9Þ

To derive the proposed algorithm, the steepest-descent method (Widrow & Stearns, 1985) is employed. The weight vector w(k) is updated at every Lth sample as

^ ðkÞ wðk þ 1Þ ¼ wðkÞ  l$

ð10Þ

where l represents a control parameter which determines the ^ ðkÞ is the instantaneous estispeed and stability of convergence. $ mate of the gradient of the robust cost function E(n(k)), where E(.) represents the expectation operator. Assuming E(n(k))  n(k),the cost function is written as

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nðkÞ ¼

N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580 L1 X

eðkL þ iÞti

ð11Þ

i¼0

where

ti

 pffiffiffiffiffiffi r i ¼ 12  0:5 Lþ1

ð12Þ

The term ri(1 6 ri 6 L) denotes the rank of the error sample e(kL + i) in the error vector [e(kL), e(kL + 1), . . . , e(kL + i), . . . , e(kL + L  1)]T. Since ti depends only on ri, it is independent of the controller weights. The primary path output d(kL + i), being a measured output of a physical system, is also independent of the controller weights. Thus the estimate of the gradient of the cost function becomes

^ ðkÞ ¼ $

o owj

( 

L1 X

) y0 ðkL þ iÞti

ð13Þ

i¼0

Using (10) and (13), the WFxLMS update equation is obtained as

wðk þ 1Þ ¼ wðkÞ þ

l

L1 X

L

x0 ðkL þ iÞti

ð14Þ

i¼0

where x (kL + i) = [x (kL + i), x0 (kL + i  1), . . . , x0 (kL + i  M + 1)]T. x0 (kL + i), the filtered input signal, is computed as 0

x0 ðkL þ iÞ ¼

0

N1 X

^ Sj xðkL þ i  jÞw

3. Development of an evolutionary algorithm based robust ANC system The implementation of the proposed algorithm as given in (16) requires the estimation of the secondary path. A PSO based robust ANC system, which does not require identification of secondary path is developed in this section. PSO is a population based stochastic algorithm. For each generation of a PSO algorithm, the same set of input data is to be applied for position and velocity update of the particles. This is practically not possible in real time systems like ANC, which works on a sample-by-sample ‘measure’, ‘control’, ‘adjust’ principle. In order to overcome the above problem, the PSO algorithm is modified for ANC application. The ANC setup using a robust evolutionary algorithm is depicted in Fig. 2. The arrangement consists of R controllers, each represented by a particle of a PSO algorithm. The complete controller with all the particles is represented as

2

w10 6 2 6 w0 6 6 . 6 .. 6 W¼6 r 6 w0 6 6 . 6 . 4 . wR0

ð15Þ

w11 w21 ... wr1 .. . wR1

   w1M1

3

7    w2M1 7 7 7 .. ... 7 . 7 7    wrM1 7 7 .. .. 7 7 . . 5 R    wM1

ð18Þ

j¼0

^ Sj represents the weights of b where w SðzÞ. In vector notation, the weight update Eq. (14) may be expressed as

wðk þ 1Þ ¼ wðkÞ þ

l L

X0 ðkÞt

ð16Þ

where t = [t0, t1, . . . , tL1]T and

2

x0 ðkLÞ

6 x0 ðkL þ 1Þ 6 X0 ðkÞ ¼ 6 .. 6 4 .

x0 ðkL  1Þ 0

x ðkLÞ .. .

   x0 ðkL  M þ 1Þ

3T

   x ðkL  M þ 2Þ 7 7 7 .. .. 7 5 . .

x0 ðkL þ L  1Þ x0 ðkL þ L  2Þ   

where r is the controller index. At any instant of time n, only one controller is connected between the reference microphone and the active loudspeaker, which is ensured using a Demultiplexer/ Multiplexer arrangement controlled by a processor. The controllers are selected rotationally after every Lth sample and the controller index is given by

r¼



n mod ðR  LÞ L

 ð19Þ

0

x0 ðkL þ L  MÞ ð17Þ

with x0 (j) = 0 for j < 0. The set of Eqs. (7)–(9), (12), (16) and (17) represent the proposed robust WFxLMS algorithm.

A serial-to-parallel converter is included in the path between the error microphone and the processor. After accumulating L error samples, the fitness function is evaluated as in (11). Similar to a regular PSO algorithm, the evaluated fitness function for a particular controller is compared with the personal best fitness value of the particle corresponding to that controller. The personal best position wpbestr is replaced with the current position of the particle if the

Fig. 2. Robust evolutionary ANC system in a long narrow duct.

4.1. Case A: Burst noise at the error microphone The effect of burst noise near the error microphone on the convergence of block FxLMS (BFxLMS) and WFxLMS algorithms are studied in this section. The performance of the algorithms are measured using a relative error (Lan et al., 2002) werr(k), given by

wpbest

wgbest

Update personal best position for R particles and the global best position using samples from (i-1)LR+1 to iLR

werr ðkÞ ¼

Fig. 3. PSO updation scheme.

particle being at the current position offers a better solution. The process is repeated for other controllers. Once the fitness function is evaluated for all the particles in the population, the global best position wgbest is updated after comparing the evaluated fitness of each particle with the global best fitness value. The velocity and position of all the particles in the population are updated after the above operation as



wpbestr  wr ðkÞ

þ k2 c2 wgbest  wr ðkÞ

v r ðk þ 1Þ ¼ /vr ðkÞ þ k1 c1

wr ðk þ 1Þ ¼ wr ðkÞ þ v r ðk þ 1Þ

kwðkÞ  wo k kwo k

ð22Þ

where w(k) is the weight vector after the kth weight update and wo is the optimal weight vector when the interference g(n) and outliers q(n) are assumed to be zero. The optimal weights of the controller are computed as the average of the weights achieved during last fifty iterations of the training period.

wpbest

wgbest

Update personal best position for R particles and the global best position using samples from iLR+1 to (i+1)LR



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used as the reference signal x(n) for all the experiments. Both minimum and non-minimum phase transfer functions are considered for secondary path and its estimate. An additive white Gaussian noise of 40 dB SNR is assumed as the background noise for all the cases.

wpbest

wgbest

N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580

ð20Þ ð21Þ

where / represents an inertia weight which is used to maintain a balance between global and local searches, k1 and k2 are two random numbers in the range [0,1] and c1, c2 are positive constants. A set of (R  L) samples makes a generation. The inertia weight is varied from a large value /max to a relatively smaller value /min in succeeding generations. In contrast to the regular PSO algorithm, the process of evaluation of fitness function, updation of particle position and velocity are continued using new samples of noise signal instead of repeating the operation with the same input data set. The learning technique employed in the paper is shown in Fig. 3. 4. Simulation study In this section the performance of the proposed WFxLMS algorithm based ANC system of Fig. 1 is evaluated through simulation study and is compared with the corresponding performance achieved with the conventional FXLMS algorithm. To have the same basis of comparison, both structures have been simulated in block form (Das, Panda, & Kuo, 2007). The proposed robust evolutionary ANC approach is also simulated for performance evaluation. In an ANC scenario, two types of disturbances may occur near the error sensor. The first is the burst noise acquired by the error microphone and the second is the outlier samples produced in the secondary path. The effect of these two situations on the overall performance of the ANC system are studied separately in this section. A zero-mean white uniform noise with unity variance is

4.1.1. Experiment I In this experiment, the primary path transfer function is chosen as P(z) = z5  0.3z6 + 0.2z7 and a minimum phase secondary path S(z) = z2 + 0.5z3 is considered (Das & Panda, 2004). The various parameters used in this study for BFxLMS and WFxLMS algorithms are L = 10, l = 0.15 and M = 16. The disturbance signal g(n) is simulated as a burst noise for the first 50 samples. The amplitude of the disturbance is randomly selected in the range [10, 10]. Two additional bursts of 10 samples, each with amplitude uniformly distributed between [5, 5], are added after the 2000th and 3000th samples. The relative error obtained for both the algorithms are plotted in Fig. 4(b). The introduction of burst noises near the error microphone has resulted in surges in the relative error for BFxLMS algorithm. Meanwhile in the case of WFxLMS algorithm, the relative error is robust against such surges. The relative errors obtained when the burst noise just ends at the three locations for WFxLMS algorithm are found to be around 7, 10 and 10 dB lower than that obtained by BFxLMS algorithm. The superior performance of WFxLMS algorithm over the normal BFxLMS algorithm is also evident from the residual noise plotted in Fig. 4(c) and (d). 4.1.2. Experiment II In this case a non-minimum phase secondary path S(z) = z2 + 1.5z3  z4 is chosen. The primary path transfer function,the disturbance signal as well as all the simulation parameters are same as that of the previous experiment. The relative error for BFxLMS and WFxLMS algorithms are depicted in Fig. 5(b). The proposed WFxLMS algorithm demonstrates an improvement of 13, 14 and 13 dB over the conventional BFxLMS algorithm at the three locations where the burst noise just ends. The residual error plotted in Fig. 5(c) and (d) also reaffirms improved robustness of WFxLMS algorithm. 4.2. Case B: Outliers in the secondary path A couple of experiments are conducted to study the effect of outliers in the secondary path. Outliers, q(n) are samples of data which vary from the rest of the data by a large value. Due to the presence of outliers, the mean square error (MSE) is not an appropriate measure for comparison of algorithm performance. The presence of even a single outlier sample deviates the MSE. Therefore in this paper, the median square error (MDSE) is employed for comparison purpose. The MDSE is computed as the median of the square of the last 100 samples of the residual error. The residual error is averaged over 100 independent iterations. The outliers

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N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580

(c) Residual Noise (BFxLMS) 20

5

10 dB

Amplitude

(a) Burst noise acquired by the error microphone 10

0 −5 −10

0 −10

0

1000

2000 Sample n

3000

−20

4000

0

(b) werr(k)

3000

4000

(d) Residual Noise (WFxLMS) BFxLMS WFxLMS

10 dB

5 dB

2000 Sample n

20

10

0 −5 −10

1000

0 −10

0

100

200 Block Index k

300

−20

400

0

1000

2000 Sample n

3000

4000

Fig. 4. Experiment I: (a) and (b) Performance comparison of BFxLMS and WFxLMS algorithms. (c) and (d) Residual noise of the BFxLMS and WFxLMS algorithms.

(a) Burst noise acquired by the error microphone

(c) Residual Noise (BFxLMS)

10

30 20 10 dB

Amplitude

5 0

0 −10

−5 −20 −10

0

1000

2000 Sample n

3000

−30

4000

0

(b) werr(k)

1000

2000 Sample n

3000

4000

(d) Residual Noise (WFxLMS)

15

20 BFxLMS WFxLMS

10

10

5 dB

dB

0

0

−10

−5

−20

−10

0

100

200 Block Index k

300

400

−30

0

1000

2000 Sample n

3000

4000

Fig. 5. Experiment II: (a) and (b) Performance comparison of BFxLMS and WFxLMS algorithms. (c) and (d) Residual noise of the BFxLMS and WFxLMS algorithms.

are considered to occur at exactly same locations for BFxLMS and WFxLMS algorithms for fairness of comparison. 4.2.1. Experiment III The primary path and secondary path transfer functions as well as the simulation parameters used in this experiment are same as that of Experiment I. The MDSE values are compared in Table 1 for different percentage and magnitude of outliers. It is observed that for all the experiments conducted, the MDSE for WFxLMS algorithm based ANC is better than that of BFxLMS algorithm. The dif-

ference between the largest and smallest MDSE for this experiment are 9.21 and 0.11 dB, respectively for BFxLMS and WFxLMS based algorithms. 4.2.2. Experiment IV In this experiment, the primary path transfer function, the secondary path transfer function and all the simulation parameters are same as that of Experiment II. The robust performance of WFxLMS algorithm is evident from the MDSE values listed in Table 1. For extreme values of outliers, if the BFxLMS algorithm

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N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580 Table 1 Comparison of performance of BFxLMS and WFxLMS algorithms in the presence of outliers in the secondary path. Outlier percentage

Outlier value

BFxLMS MDSE (dB)

WFxLMS MDSE (dB)

Experiment III 1 5 10 1 5 10

[2, 2] [2, 2] [2, 2] [5, 5] [5, 5] [5, 5]

15.76 15.17 12.97 14.68 12.40 6.55

15.57 15.55 15.48 15.58 15.53 15.47

Experiment IV 1 5 10 1 5 10

[2, 2] [2, 2] [2, 2] [5, 5] [5, 5] [5, 5]

13.00 4.31 3.47 6.81 3.16 4.59

17.80 17.97 17.62 17.85 17.98 17.53

are selected to act as the controller rotationally after 50(L) samples of reference signal. The various parameters used in the study are: /max = 0.9, /min = 0.3, c1 = 0.4, c2 = 0.6. For the purpose of comparison, a PSO based ANC system which minimises the L2 norm is also simulated. 4.3.1. Experiment V The primary and secondary path for this experiment is same as that of Experiment I. The variation of mean square error (MSE) corresponding to the global best position, with generations for various levels of outliers in the secondary path is shown in Fig. 6. Wilcoxon norm based PSO algorithm provides better mean square error performance compared to L2 norm based algorithm. The difference in MSE is 0.56, 3.47 and 3.12 dB, respectively for 1%, 5% and 10% outliers of magnitude [5, 5] in the secondary path.

is employed to design the ANC, the adaptation process tends to diverge. The difference between the largest and smallest MDSE for BFxLMS and WFxLMS algorithms are 18 and 0.45 dB, respectively. 4.3. Case C: Outliers in the secondary path – evolutionary algorithm based robust ANC system

4.3.2. Experiment VI In this experiment, the primary and secondary path are same as that of Experiment II. The MSE variation with respect to generations of a PSO algorithm is depicted in Fig. 7. The improvement in MSE after 100 generations is 2.15, 2.02 and 2.52 dB, respectively for Wilcoxon norm based PSO over L2 norm based PSO algorithm in the presence of 1%, 5% and 10% outliers of magnitude [5, 5] in the secondary path. 5. Conclusion

Two experiments are conducted to evaluate the effectiveness of the proposed PSO algorithm based robust ANC system. A population size of 200 is selected for both the experiments. The particles

The paper proposes a robust WFxLMS algorithm for efficient active noise control in the presence of burst noise acquired by

−2 L2 Norm Wilcoxon Norm

MSE (dB)

−4 −6 −8 −10 (a) −12

0

10

20

30

40

50 60 Generations

70

80

90

100

−2 L2 Norm Wilcoxon Norm

MSE (dB)

−4 −6 −8 −10 (b) −12

0

10

20

30

40

50 60 Generations

70

80

90

100

0

MSE (dB)

L2 Norm Wilcoxon Norm −5

−10 (c) −15

0

10

20

30

40

50 60 Generations

70

80

90

100

Fig. 6. Variation of mean square error (MSE) with generations of a PSO algorithm for an ANC system with a minimum phase secondary path and (a) 1% outliers of magnitude in the range [5, 5] in the secondary path, (b) 5% outliers of magnitude in the range [5, 5] in the secondary path, (c) 10% outliers of magnitude in the range [5, 5] in the secondary path.

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N.V. George, G. Panda / Expert Systems with Applications 39 (2012) 7574–7580

5 L2 Norm Wilcoxon Norm MSE (dB)

0

−5 (a) −10

0

10

20

30

40

50 60 Generations

70

80

90

100

2 L2 Norm Wilcoxon Norm

MSE (dB)

0 −2 −4 −6 (b) −8

0

10

20

30

40

50 60 Generations

70

80

90

100

2 L2 Norm Wilcoxon Norm

MSE (dB)

0 −2 −4 −6 (c) −8

0

10

20

30

40

50 60 Generations

70

80

90

100

Fig. 7. Variation of mean square error (MSE) with generations of a PSO algorithm for an ANC system with a non minimum phase secondary path and (a) 1% outliers of magnitude in the range [5, 5] in the secondary path, (b) 5% outliers of magnitude in the range [5, 5] in the secondary path, (c) 10% outliers of magnitude in the range [5, 5] in the secondary path.

the error microphone and outliers in the secondary path. The improvement in performance is obtained by the incorporation of Wilcoxon norm into FxLMS algorithm. A robust ANC algorithm, which provides effective noise control and on the other hand does not require the estimation of the secondary path is also developed in this paper. Simulation study demonstrates that the robust algorithms introduced in this paper provide significant improvement in ANC performance in presence of strong disturbances at the error microphone compared to L2 norm based ANC algorithms. The proposed algorithms are also shown to be robust against outliers in the secondary path. Role of the funding source One of the authors, Nithin V. George, acknowledge the generous funding received from Ministry of Human Resource Development, Government of India for carrying out this work. References Akhtar, M. T., & Mitsuhashi, W. (2009). Improving performance of FxLMS algorithm for active noise control of impulsive noise. Journal of Sound and Vibration, 327(3– 5), 647–656. Ban, S. J., & Kim, S. W. (2009). Wilcoxon adaptive algorithms for robust identification. Electronics Letters, 45(18), 958–959. Chang, C. Y., & Chen, D. R. (2010). Active noise cancellation without secondary path identification by using an adaptive genetic algorithm. IEEE Transactions on Instrumentation and Measurement, 59(9), 2315–2327.

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