A family of threshold based robust adaptive algorithms for active impulsive noise control

Applied Acoustics 97 (2015) 30–36

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Technical Note

A family of threshold based robust adaptive algorithms for active impulsive noise control Guohua Sun ⇑, Mingfeng Li, Teik C. Lim Vibro-Acoustics and Sound Quality Research Laboratory, College of Engineering and Applied Science, Department of Mechanical and Materials Engineering, 598 Rhodes Hall, P.O. Box 210072, University of Cincinnati, Cincinnati, OH 45221-0072, United States

a r t i c l e

i n f o

Article history: Received 8 December 2014 Received in revised form 22 February 2015 Accepted 8 April 2015

Keywords: Active noise control FxLMS algorithm Impulsive noise M-estimator FxLMM algorithm

a b s t r a c t The common active noise control (ANC) algorithm, namely the filtered-x least mean square (FxLMS) algorithm, becomes unstable for the non-Gaussian impulsive noise. This is because the typical FxLMS algorithm is based on the minimization of variance of the error signal (the second order moment in L2 space), which does not exist for the non-Gaussian impulsive noise. In this study, a family of threshold based algorithms is proposed by minimizing several robust objective error functions as well as thresholding the reference signal to further refine the robustness of the ANC system for impulsive noise. The proposed algorithms are also expected to generalize the existing adaptive algorithms for impulsive noise control. These robust error functions are typically represented by (1) robust space vectors: Lp and Log space; and (2) re-descending M-estimators: Huber, Fair and Hampel threshold functions. The threshold parameters in the reference signal and those M-estimators can be determined by using online and/or offline statistical estimation approaches. Numerical simulations are carried out to verify the performance of proposed algorithms by using synthesized impulsive noise following symmetric a-stable ðSaSÞ distribution. Results show the improved robustness and convergence performance of the proposed algorithms for ANC of impulsive noises as compared to the conventional algorithms. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Active noise control (ANC) is based on the principle of linear wave superposition through the generation of a controllable secondary sound, which is normally realized by using adaptive filter as a controller [1]. The adaptive filter is commonly adapted by the filtered-x least mean square (FxLMS) algorithm, which generally assumes that the reference signal is following a Gaussian distribution. In practice, however, there are many situations where the target noise is impulsive. Under these situations, the Gaussian assumption is not satisfied anymore. These noises are typically man-made noises with very high impact characteristic. For examples, intense impact sounds from punching factory and pile driving site [2,3], and the myriad of transient and impact noises generated by the powertrain and tire-road interaction of a motor vehicle system. Since the FxLMS algorithm is based on the h i minimization of the variance of error signal ðJ ðnÞ ¼ E eðnÞ2 Þ, the conventional FxLMS algorithm tends to exhibit degraded performance and have instability issue for these impulsive noises [4]. ⇑ Corresponding author. Tel.: +1 (513) 365 2909. E-mail address: [email protected] (G. Sun). http://dx.doi.org/10.1016/j.apacoust.2015.04.003 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

Several approaches have been proposed to deal with this problem as noted in the literature. All of the methods mainly fall under two primary categories of adaptive algorithms. One is based on the minimization of robust optimization criteria since the typical LMS criterion may not be suitable. For instance, Leahy et al. [5] proposed the filtered-x least mean p-norm (FxLMP) algorithm that is based on the minimization of fractional lower order moment   (p-norm) of error signal ðJ ðnÞ ¼ E j eðnÞjp Þ. The FxLMP algorithm was developed by assuming that most of the impulsive noises can be modeled as a standard symmetric a-stable ðSaSÞ model. Here, 0 < a 6 2 is the exponential characteristic parameter determining the impulsiveness of the noise, and when a ¼ 2 it is reduced to a normal Gaussian noise. Thanigai et al. [6] developed the filtered-x least mean M-estimator (FxLMM) algorithm for ANC of impulsive noise in infant incubators. The FxLMM algorithm is reliance on minimizing the cost function J ðnÞ ¼ E½qfeðnÞg. Here, qfeðnÞg is the M-estimator function of the error signal, and Hampel’s three-part M-estimator was used. Wu et al. [7] proposed the filtered-x logarithmic error LMS (FxLogLMS) algorithm that is h i 2 developed by minimizing the cost function J ðnÞ ¼ E log j eðnÞ j . Very recently, Wu and Qiu [8] proposed a new M-estimator algorithm called filtered-x least mean Fair M-estimator (FxLMFM) for

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impulsive noise control, which is based on the Fair M-estimator in robust statistics theory. The other category from the public literature is reliance on simple modifications of the conventional FxLMS algorithm by thresholding the reference and/or error signal in impulsive ANC [9–15]. Sun et al. [15] first developed a simple variant of the FxLMS algorithm by adding a threshold to ignore the impulsive samples in the reference signal path (denotes Sun’s algorithm in this study). Akhtar and Mitsuhashi [14] further enhanced the robustness of the Sun’s algorithm by thresholding both reference and error signals and replacing the impulsive samples with threshold values. Even though the current set of algorithms based on robust error criteria work well for most cases, when applying these conventional algorithms for a strong impact noise, the impulses in the reference signal may still have adverse influence on the filter weight update process. In the previous study [9], the modified FxLMM (MFxLMM) algorithm with an additional Hampel Mestimator three-part threshold in the reference signal path was proposed. The enhanced performance for impulsive noise control has been validated by extensive numerical simulations and experimental studies. In this paper, several simplified modifications of the FxLMP, FxLogLMS, FxLMM and FxLMFM algorithms are proposed by introducing a threshold (in the form of a two-part threshold) in the reference signal path to further enhance their robustness for impulsive noise. The proposed algorithms can be considered as a family of threshold based robust adaptive algorithms, which generalizes most of the existing adaptive algorithms for ANC of impulsive noise, as shown in Table 1. Extensive numerical simulations are carried out to verify the performance of these enhanced algorithms. The impulsive noise is synthesized by following the symmetric a-stable model with various exponential indices. Results demonstrate the improved robustness of the proposed algorithms. The rest of this paper is organized as follows: Section 2 reviews the conventional FxLMS algorithm. Section 3 describes the derivation of these proposed algorithms, and relationship to the existing algorithms is discussed. In Section 4, numerical simulations are performed to validate the effectiveness of the proposed algorithms for impulsive noise control. Conclusions are given in Section 5.

Adaptive Controller Fig. 1. Feed-forward control diagram with conventional FxLMS algorithm.

implementation of the FxLMS algorithm, it requires an accurate model ^ SðzÞ of the secondary transfer path from the control speaker to the error microphone, which can be estimated by using offline [1] or online system identification approach [16–20]. The residual error signal is expressed as:

eðnÞ ¼ dðnÞ  yðnÞ

ð1Þ

  yðnÞ ¼ sðnÞ  wT ðnÞxðnÞ

ð2Þ

where n is the time index, sðnÞ represents the impulse response of the secondary path, yðnÞ is the anti-phase secondary noise,  denotes the linear convolution, the filter weights and reference signal vectors of the controller are:

wðnÞ ¼ ½w0 ðnÞ w1 ðnÞ    wL1 ðnÞT

ð3aÞ

xðnÞ ¼ ½xðnÞ xðn  1Þ    xðn  L þ 1ÞT

ð3bÞ

where L is the order of the adaptive filter. The derivation of the FxLMS algorithm is based on the minimum mean square error (MMSE) criterion by assuming a mean   square cost function J ðnÞ ¼ E e2 ðnÞ , as shown in Ref. [21]. The filter weight update equation is obtained:

wðn þ 1Þ ¼ wðnÞ þ leðnÞ½^sðnÞ  xðnÞ 2. FxLMS algorithm The block diagram of the single-channel feedforward ANC system configured with the FxLMS algorithm is shown in Fig. 1, where xðnÞ is the reference signal, dðnÞ is the primary noise and eðnÞ is the error signal after superposition of the primary noise and secondary canceling noise. The reference signal xðnÞ and error signal eðnÞ are processed by the FxLMS algorithm to update the parameters of the adaptive filter to generate the anti-phase secondary noise. In the

ð4Þ

where l is the convergence step-size that determines the convergence and stability of the FxLMS algorithm. ^sðnÞ is the impulse response function of ^SðzÞ. From Eq. (4), one can see that the filter weight update equation may burst into a large value and system may diverge when there are peaky impulses occurring in the reference and/or error signal. This tends to make the typical FxLMS algorithm unstable for impulsive noise. To improve the robustness of the conventional FxLMS algorithm for impulsive samples, a general family of threshold based algorithms will be developed in the next

Table 1 Proposed family of threshold based robust algorithms for active impulsive noise control. Cost function qfeðnÞg

Robust estimator

Robust space

Huber Fair Hampel

Lp Log

p

j eðnÞj =p 2

log ðj eðnÞ jÞ  e2 ðnÞ=2 kðj eðnÞ j k=2Þ h  i jeðnÞj c2 jeðnÞj c  log 1 þ c 8 2 > < e ðnÞ=2 2 D2 Þ2 nðj eðnÞ j n=2Þ 2n ðD2 þ D1 Þ  n2 þ 2n ðjeðnÞj D1 D2 > 2 :n n 2 ðD2 þ D1 Þ  2

Threshold in the reference signal xðnÞ No

Yes

Leahy et al. [5] Wu et al. [7]

Akhtar and Mitsuhashi [13] This study

Wu and Qiu [8]

Akhtar and Mitsuhashi [12]

Wu and Qiu [8]

This study

Thanigai et al. [6]

This study

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section, which unifies most of the existing adaptive algorithms for impulsive noise control. 3. Proposed algorithms As seen from Table 1, many robust error criteria have been proposed for impulsive noise control. The purpose of this study is to give a unifying formulation of these robust estimator-based algorithms. In addition, an extra threshold in the reference signal is proposed to further enhance the robustness for impulsive samples. The robust error function qfeðnÞg is considered as a general formulation that gives a stable estimator for the outliers in the processed data (i.e., impulsive samples in eðnÞ). Similar derivation as the abovementioned FxLMS algorithm is illustrated, in which the robust cost function can be defined as [9,22]:

J ðnÞ ¼ E½qfeðnÞg  qfeðnÞg

ð5Þ

where qfeðnÞg is the family of robust cost functions as shown in Table 1. The first derivative of the objective cost function is:

@J ðnÞ @ qfeðnÞg @ qfeðnÞg @eðnÞ ¼ ¼ @wðnÞ @wðnÞ @eðnÞ @wðnÞ ¼ wfeðnÞg½^sðnÞ  xðnÞ

b ðnÞ ¼ r

ð6Þ

feðnÞg where the score function wfeðnÞg ¼ @ q@eðnÞ is shown in Table 2,

which controls the influence of the error signal by impulsive samples. Then applying the steepest decent algorithm, the general filter weight update equation is:

wðn þ 1Þ ¼ wðnÞ þ lwfeðnÞg½^sðnÞ  xðnÞ

ð7Þ

As reported in [9], the impulses in the reference signal show an adverse influence on the filter weight update process for these M-estimator based algorithms. Even though some of the score functions wfeðnÞg can restrict the impulsive samples in the error signal and prevent the whole term wfeðnÞg½^sðnÞ  xðnÞ from bursting too much at a certain time index, the impulsive samples in the reference signal may also adversely affect the updating process in Eq. (7). Hence, a family of threshold based robust algorithms (see Table 2) is proposed by further clipping the impulsive samples in xðnÞ (note xc ðnÞ denotes the clipped signal after the threshold). The block diagram of the single channel ANC system with the proposed algorithms for impulsive noise is shown in Fig. 2. The general filter weight update equation of the modified algorithms becomes:

Table 2 Adaptive filter weight update equation of the proposed family of threshold based algorithms. Robust estimator Robust space

Filter weight update equation Lp

wðn þ 1Þ ¼ wðnÞ þ lwLp feðnÞg½^sðnÞ  xc ðnÞ wLp feðnÞg ¼j eðnÞjp1 sign½eðnÞ

Log

wðn þ 1Þ ¼ wðnÞ þ lwLog feðnÞg½^sðnÞ  xc ðnÞ wLog feðnÞg ¼ logjeðnÞj jeðnÞj sign½eðnÞ

Huber

Fair

wðn þ 1Þ ¼ wðnÞ þ lwH feðnÞg½^sðnÞ  xc ðnÞ  eðnÞ 0 6j eðnÞ j6 k wH feðnÞg ¼ ksign½eðnÞ j eðnÞ j> k wðn þ 1Þ ¼ wðnÞ þ lwF feðnÞg½^sðnÞ  xc ðnÞ eðnÞ wF feðnÞg ¼ 1þjeðnÞj=c

Hampel

wðn þ 1Þ ¼ wðnÞ þ lwM feðnÞg½^sðnÞ  xc ðnÞ 8 0 6j eðnÞ j6 n > eðnÞ > > < nsign½eðnÞ h i n > > : 0 D2
Two-part threshold Robust estimator algorithm

Adaptive Controller Fig. 2. Feed-forward control diagram with proposed family of threshold based robust algorithms.

wðn þ 1Þ ¼ wðnÞ þ lwfeðnÞg½^sðnÞ  xc ðnÞ

ð8aÞ

8 xðnÞ P c2 > < c2 xc ðnÞ ¼ c1 xðnÞ 6 c1 > : xðnÞ otherwise

ð8bÞ

As discussed in Sun’s and Akhtar’s papers [14,15], the threshold parameters c1 and c2 can be estimated by offline-calculated statistics (such as choosing the 1th and 99th percentile of the original signal). The following paragraphs describe the relationship of the proposed algorithms to the existing adaptive algorithms. For the Lp space with p ¼ 2, the modified FxLMP (MFxLMP) algorithm reduces to Sun’s algorithm [15]. When 1 < p < a, it is a simplified version of the MFxLMP algorithm by Akhtar and Mitsuhashi [13], and their MFxLMP algorithm incorporates thresholds both in the reference and error signal paths. Here, only one threshold is included in the reference signal path. This simplified MFxLMP algorithm inherently exhibits its enhanced robustness since both the threshold and p-order exponential may restrict the burst of term xðnÞ and eðnÞ, respectively. On the other hand, when the robust error function is Log space, the proposed modified FxLogLMS (MFxLogLMS) algorithm is a modification of the conventional FxLogLMS algorithm by Wu et al. [7]. Huber and Hampel re-descending M-estimator functions are generally a concept of adding thresholds in the optimization cost function. Here, the Huber function has two-part thresholds and the Hampel function has three-part thresholds. If the Huber function is chosen as the cost function, the proposed algorithm is corresponding to the modified FxLMS (MFxLMS) algorithm proposed by Akhtar and Mitsuhashi [14]. Here, the threshold k in the Huber function can be split into lower and upper threshold values defined as c1 and c2 . For the Hampel M-estimator function, the proposed algorithm is modification of the conventional FxLMM algorithm [6]. For the case of Fair M-estimator [8], the proposed algorithm is a modified version on top of the Fair algorithm (MFxLMFM). Fig. 3 describes the score functions for all these robust error criteria. One can notice that there is no restriction on large impulsive samples when the second order space L2 is taken as the criterion. In contrast, the robust error functions put certain constraints on the outlier of the error signal eðnÞ. It is also implied that both FxLogLMS and FxLMM algorithms pose ‘‘hard’’ limits and the score functions descend to zero more sharply when large amplitude of impulses occur. These two algorithms can be very effective for large impulsive noises. However, the logarithmic and three-part threshold calculations increase the computational complexity. On the other hand, both the Lp space and Fair M-estimator do not offer

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since it is realizable and practical to do the preliminary measurement of the reference signal and the primary noise.

ψ {e(n)}

4. Numerical simulation

Fig. 3. Score functions for various robust error criteria (Keys: thin solid line L2; bold solid line , Lp; left-pointing triangle , Fair; dotted line Huber; dash dotted line , Log; circle , Hampel).

, ,

very hard bounds when the amplitude of the signal is very large. In addition, the Fair algorithm offers a harder constraint than the FxLMP algorithm. Hence, the Fair algorithm may show relatively better performance for highly impulsive noises, which will be demonstrated in the following simulation. It may also be noted that the Huber M-estimator offers two-part thresholds, where the impulsive samples are replaced by the upper and lower limit threshold values. The score function of the Huber function does not descend to zero; but it provides more strict restriction than the Lp space and Fair M-estimator for large samples in the error signal. Fig. 4 shows the comparison of various algorithms from the view point of complexity of the algorithms, and impulsiveness of the noise source. As discussed by the a-stable statistical model, the smaller the a value, the higher the impulsiveness. The exponential parameter a can be identified through prior statistical estimation of the noise. Thus, the preferred algorithm can be determined by the impulsiveness of the noise and computational complexity. For example, if the estimated impulsiveness index a is close to 1, then one can use Hampel function; while a is close to 2, Lp or Fair function can be chosen. Notice that when a ¼ 2, the noise is corresponding to a Gaussian type noise, the FxLMS algorithm is efficient and robust in this case. In contrast, the proposed MFxLMM algorithm can be applied for impulsive noise with a smaller a value. Note the major computational cost of the FxLMM and proposed MFxLMM algorithms as compared to the traditional FxLMS algorithm is the mean value function occurred in the online threshold method [9]. The adoption of various sorting algorithms for threshold identification leads to additional computational requirements. On the other hand, the threshold values can also be determined by off-line estimation (percentile calculation) before the ANC system is working. This is reasonable

Low

Hampel



 Ae ðnÞ Ad ðnÞ

ð9aÞ

Ae ðnÞ ¼ kAe ðn  1Þ þ ð1  kÞjeðnÞj

ð9bÞ

Ad ðnÞ ¼ kAd ðn  1Þ þ ð1  kÞjdðnÞj

ð9cÞ

where Ae ðnÞ and Ad ðnÞ are the recursive estimation of the magnitude of the error signal eðnÞ and primary disturbance dðnÞ; k is the forgetting factor. In following numerous simulations, ensemble of 25 realizations for each impulsive process is conducted, and the mean NR (MNR) by averaging all these realizations is used to evaluate the convergence performance. The proposed algorithm for the Huber Mestimator function can be considered as a general case of the MFxLMS algorithm (Akhtar’s algorithm) with two-part thresholds, and the later has been extensively validated in [14]. Also, the MFxLMM (Hampel) algorithm with additional three-part thresholds in the reference has been extensively validated in comparison to Sun’s and Akhtar’s algorithms [9]. Therefore, the following section mainly focuses on the performance validation of the proposed simplified MFxLMP ðLp Þ, MFxLMFM (Fair), MFxLogLMS (Log) and simplified MFxLMM (Hampel) algorithms for various impulsive noises with different a values. 4.1. Threshold estimation The determination of these threshold parameters is very important for yielding a balanced performance. These thresholds can be 30 20 10 0 -10 -20

Log

-30

Huber Fair

Phase (deg.)

Computational complexity

High

NRðnÞ ¼ 20log10

Magnitude (dB)

e(n)

In this section, several numerical simulations were performed to verify the performance of the proposed algorithms. Both the primary and secondary paths PðzÞ and SðzÞ are modeled as a FIR filter with 128 taps, where the data is being taken from [1]. The frequency response characteristics of the plant are shown in Fig. 5. The order of the adaptive filter WðzÞ is 128. The impulsive noise is modeled by using the symmetric a-stable ðSaSÞ noise model (using a MATLAB function stblrnd developed by McCulloch [23]). The control performance evaluation criterion is the noise reduction (NR) formula defined in paper [14]:

Lp L2

Impulsiveness

High

Fig. 4. A pictorial representation of comparison of various algorithms from the view point of complexity of the algorithms, and impulsiveness of the noise source.

1000 0 -1000 -2000 -3000

0

0.2

0.4

0.6

0.8

1

Normalized Frequency (×π rad/sample) Fig. 5. Frequency response functions of the primary and secondary paths (Keys: solid line , primary path Pðz); dotted line , secondary path SðzÞÞ.

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either determined via offline identification (i.e. percentile calculation) or online approaches. In this study, the two-part threshold in the reference signal path is using offline approach by prior percentile calculation. The thresholding in the robust error criteria is realized by online identification approaches presented in the following. For the Fair function, the threshold parameter c can be computed as 1, 1.5, 2 and 3 times of the average absolute value of the error signal as discussed by Wu and Qiu [8]. It has been found that the control performance is not sensitive to the value ofc, and the authors suggested the following online identification formula:

cðnÞ ¼

X 1 M1 jeðn  iÞj M i¼0

ð10Þ

For the Hampel three-part M-estimator function, the three threshold parameters n; D1 and D2 can be estimated through the online variance estimation of the ‘‘impulse-free’’ samples [9,22]. The ^ e ðnÞ: robust estimation formula of the variance r

^ ðnÞ ¼ ku ^ ðn  1Þ þ C 1 ð1  kÞeðnÞ u

ð11aÞ 



r^ 2e ðnÞ ¼ kr^ 2e ðn  1Þ þ C 1 ð1  kÞmed A0e ðnÞ

ð11bÞ

8 ^ e ðnÞ > < n ¼ 1:960r ^ e ðnÞ D1 ¼ 2:240r > : ^ e ðnÞ D2 ¼ 2:576r

ð11cÞ

where the impulse’s adverse effect on the variance estimation can be guaranteed by computing the median of the vector n ^ ðnÞ2 ; ½eðn  1Þ  u ^ ðn  1Þ2 ; . . . ; ½eðn  N w þ 1Þ A0e ðnÞ ¼ ½eðnÞ  u o ^ ðn  Nw þ 1Þ2 . The forgetting factor is k ¼ 0:99, and window u length is N w ¼ 100. 4.2. Results and discussions A set of simulations were conducted first to validate the performance of the proposed algorithms for impulsive noise with

μ=6.0×10-5

0

MNR (dB)

(b)

5

-5

0

μ=5.0×10-4

-10

μ=3.0×10-4 μ=5.0×10-4

-5

μ=1.0×10-3

-10 -15

-15 -20

5 μ=1.0×10-4

μ=1.0×10-4 μ=3.0×10-4

MNR (dB)

(a)

different a values. The impulsive noises were synthesized by the SaS model with a ¼ 1:6; 1:4 and1.2, respectively, which corresponds to a small, mild and heavy impulsiveness. In all these cases, the upper and lower limits of the threshold values in the reference signal were determined by off-line percentile calculation (99.97 and 0.03 percentiles). The thresholds of the error signal in the FxLogLMS, FxLMFM and FxLMM algorithms and the proposed modified versions were determined using online approaches as shown in Section 4.1. Fig. 6 shows the performance comparison, on the basis of MNR, between the proposed MFxLogLMS algorithm and the conventional FxLogLMS algorithm. Here, Fig. 6(a) and (b) study the effect of stepsize on the performance of individual algorithms. The best results for respective algorithms are used for the performance comparison given in Fig. 6(c). We observe that, as compared with the FxLogLMS algorithm (solid line), the proposed MFxLogLMS algorithm (dashed line) achieves a faster convergence rate, and a better steady-state NR. Due to the incorporated threshold values in the reference signal path, the impulsive samples with relatively large amplitudes were removed. Therefore, the step-size can be further tuned to get a faster convergence rate without losing stability. The corresponding curves for the residual error eðnÞ in comparison with the primary disturbance dðnÞ for a single realization are shown in Fig. 7. Similarly, other simulation cases with SaS impulsive noises of a ¼ 1:4 and a ¼ 1:2 were conducted to validate the effectiveness of the proposed MFxLogLMS algorithm. Fig. 8 shows the performance comparison of the conventional FxLogLMS and the MFxLogLMS algorithms on the best MNR curve. One can observe that the proposed MFxLogLMS algorithm demonstrates its improved performance for a very highly impulsive noise. Next, simulations with the impulsive noise of a ¼ 1:6 were performed to validate the performance of the MFxLMP, MFxLMFM, and MFxLMM algorithms as compared to the conventional versions. The MNR curves are shown in Fig. 9(a)–(c), respectively. Note the solid line is for the conventional algorithm and the dashed curve is for the modified algorithm. Fig. 9(a) shows the results for the FxLMP and proposed MFxLMP algorithms. It is noted that the MFxLMP algorithm can yield improved convergence rate and steady-state performance. Similar results can be referred to the

-20

0

2000

4000

6000

8000

10000

0

2000

Number of iterations, k

(c)

6000

8000

10000

5

FxLogLMS MFxLogLMS

0

MNR (dB)

4000

Number of iterations, k

-5 -10 -15 -20

0

2000

4000

6000

8000

10000

Number of iterations, k Fig. 6. Performance comparison on the basis of MNR for SaS impulsive noise with a ¼ 1:6: (a) FxLogLMS algorithm, (b) proposed MFxLogLMS algorithm, and (c) the best MNR comparison.

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G. Sun et al. / Applied Acoustics 97 (2015) 30–36

200

(a)

0

Amplitude

-200 200

(b)

0 -200 200

(c)

0 -200 0

2000

4000

6000

8000

10000

Samples Fig. 7. Comparison of the residual error noise for a single realization with SaS impulsive noise source of a ¼ 1:6: (a) control off, (b) FxLogLMS algorithm, and (c) proposed MFxLogLMS algorithm.

(a)

(b)

5 FxLogLMS MFxLogLMS

-5 -10 -15 -20

5 FxLogLMS MFxLogLMS

0

MNR (dB)

0

MNR (dB)

work done in the Ref. [13], where two thresholds were added in both reference and error signal paths simultaneously. In this study, only one threshold was added in the reference signal path, and the proposed MFxLMP algorithm shows comparable performance as that did in Ref. [13]. Here the p-norm of the error signal will restrict the burst of the error signal eðnÞ, and the threshold in the reference signal xðnÞ will enhance the reference signal xðnÞ. Fig. 9(b) shows the best MNR comparison of the FxLMFM and MFxLMFM algorithms. In these algorithms, the threshold parameter c for the error signal was determined by the on-line formula shown in Eq. (10). One can see the improved convergence rate and steady-state performance by the enhanced algorithm. Also, the MFxLMM algorithm shows improved performance as shown in Fig. 9(c), and the threepart threshold M-estimator was determined by on-line estimation as introduced in Eq. (10). Fig. 10 shows the performance comparison of the various algorithms on the basis of the best MNR curve. The conventional FxLMS algorithm shows poor performance and even becomes unstable when there are very large impulsive samples. In order to avoid the instability, it is suggested to reduce the step-size, however,

-5 -10 -15

0

2000

4000

6000

8000

-20

10000

0

2000

Number of iterations, k

4000

6000

8000

10000

Number of iterations, k

Fig. 8. Performance comparison of the conventional FxLogLMS and proposed MFxLogLMS algorithms based on the mean noise reduction (MNR) for SaS impulsive noise of (a) a ¼ 1:4 and (b) a ¼ 1:2.

(a)

(b)

5 FxLMP MFxLMP

-5 -10 -15 -20

FxLMFM MFxLMFM

0

MNR (dB)

MNR (dB)

0

5

-5 -10 -15

0

2000

4000

6000

8000

10000

-20

0

2000

Number of iterations, k

(c)

6000

8000

10000

Number of iterations, k

5 FxLMM MFxLMM

0

MNR (dB)

4000

-5 -10 -15 -20

0

2000

4000

6000

8000

10000

Number of iterations, k Fig. 9. Performance comparison of the conventional and proposed modified algorithms based on the mean noise reduction (MNR) for SaS impulsive noise of a ¼ 1:6: (a) FxLMP vs MFxLMP, (b) FxLMFM vs MFxLMFM, and (c) FxLMM vs MFxLMM.

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0

MNR (dB)

(1) (2)

-5

(7)

-10

(3)

(6) -15

(4) (5)

-20 0

2000

4000

6000

8000

10000

Number of iterations, k Fig. 10. Performance comparison of various algorithms on the basis of mean noise reduction (MNR) for SaS impulsive noise with a ¼ 1:6: (1) FxLMS algorithm, (2) Sun’s algorithm, (3) Akhtar’s algorithm, (4) proposed MFxLMFM algorithm, (5) proposed MFxLMP algorithm, (6) proposed MFxLogLMS algorithm, and (7) proposed MFxLMM algorithm.

smaller step-size will give very slow convergence rate and yield very small NR values. It is noted that the proposed algorithms by thresholding the reference signal as well as adopting robust functions for the error signal yield good and robust performance. As compared with the proposed algorithms, Sun’s algorithm with only thresholding the reference signal is stable for a small value of stepsize, and the steady-state NR is degraded when increasing the stepsize. It can also observed that both the FxLogLMS and FxLMM algorithms have relatively faster convergence rate and better NR performance than other algorithms. This is mainly because of the harder re-descending threshold incurred for the impulsive samples. This is also in consistent with the results shown in the score functions from Fig. 3. 5. Conclusions In this study, a family of threshold based robust algorithms has been proposed for treating of impulsive noise. The main idea of the proposed algorithms is to integrate the thresholding in reference signal and robust estimators for the error signal such that the instability issue of existing ANC algorithms due to impulsive samples in the processed data can be alleviated. The proposed algorithms unify the existing adaptive algorithms for impulsive noise control. The concept of M-estimators is developed from the perspective of robust statistics theory, where more robust optimization criteria are applied. The robust minimization criteria are either based on robust space vectors or reliance on re-descending M-estimators. Extensive numerical simulations have been conducted to validate the performance of the proposed algorithms. Various impulsive noises are synthesized by using the symmetric a-stable ðSaSÞ model where different impulsiveness indices a are

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