Active impulsive noise control using maximum correntropy with adaptive kernel size

Mechanical Systems and Signal Processing #vol# (xxxx) xxxx–xxxx

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Active impulsive noise control using maximum correntropy with adaptive kernel size ⁎

Lu Lua,b, Haiquan Zhaoa,b, a b

Key Laboratory of Magnetic Suspension Technology and Maglev Vehicle, Ministry of Education, China School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China

A R T I C L E I N F O

ABSTRACT

Keywords: Active noise control Impulsive noise Recursive algorithm Maximum correntropy Kernel size

The active noise control (ANC) based on the principle of superposition is an attractive method to attenuate the noise signals. However, the impulsive noise in the ANC systems will degrade the performance of the controller. In this paper, a filtered-x recursive maximum correntropy (FxRMC) algorithm is proposed based on the maximum correntropy criterion (MCC) to reduce the effect of outliers. The proposed FxRMC algorithm does not requires any priori information of the noise characteristics and outperforms the filtered-x least mean square (FxLMS) algorithm for impulsive noise. Meanwhile, in order to adjust the kernel size of FxRMC algorithm online, a recursive approach is proposed through taking into account the past estimates of error signals over a sliding window. Simulation and experimental results in the context of active impulsive noise control demonstrate that the proposed algorithms achieve much better performance than the existing algorithms in various noise environments.

1. Introduction Active noise control (ANC) has been widely used in many applications, such as road noise [1], active headphones [2], and transformer noise [3]. The ANC technique is based on the principle that a noise can be cancelled by another noise with the same amplitude but an opposite phase [4,5]. Compared with the traditional approach (enclosures, barriers, and silencers etc.), the ANC method is very low-cost and can achieve high attenuation at low frequencies. The filtered-x least-mean-square (FxLMS) algorithm [4] is the most widely used adaptive algorithm for the feed-forward ANC systems. However, it may become unstable in cases where the primary noise is impulsive noises. To surmount this problem, several variants of FxLMS were proposed [6–13]. In 1995, Leahy et al. proposed the filtered-x least mean p-power (FxLMP) algorithm by minimizing the mean p-power of error instead of the mean-square error [6]. Later, Wu et al. developed the filtered-x logarithmic error LMS (FxLogLMS) algorithm [7], which minimizes the logarithmic error. The FxLogLMS algorithm overcomes the disadvantages of Amplitude Threshold (AT)-based methods [14], and it doesn’t need any prior knowledge or estimation of noise. In 2012, George and Panda developed a robust filtered-s LMS (RFsLMS) algorithm for the functional link artificial neural network (FLANN) based adaptive controller [9], which performs with similar levels of efficiency for Gaussian as well as non-Gaussian noise. Alternatively, to achieve fast convergence rate of the adaptive filter, an interesting and effective way is to use the filtered-x recursive least square (FxRLS) algorithm [4]. But few algorithms aimed at enhancing the stability of FxRLS have been investigated. Reddy et al. proposed the new FxRLS algorithm which employs the hybrid scheme and is capable of obtaining fast convergence rate and stable steady-state error under non-impulsive noise environments [15]. However, the abovementioned algorithms have the risk

⁎

Corresponding author at: School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China. E-mail addresses: [email protected] (L. Lu), [email protected] (H. Zhao).

http://dx.doi.org/10.1016/j.ymssp.2016.10.020 Received 11 March 2016; Received in revised form 9 October 2016; Accepted 19 October 2016 Available online xxxx 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Lu, L., Mechanical Systems and Signal Processing (2016), http://dx.doi.org/10.1016/j.ymssp.2016.10.020

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Fig. 1. Block diagram of the single channel feed-forward ANC system based on adaptive algorithm.

of instability in cases where the primary noise is impulsive disturbances. Under the impulsive (non-Gaussian) noises, the conventional mean square error (MSE) criterion may fail to work since it cannot extract all possible information from the signals. In contrast, the information entropy (IE) method provides a more comprehensive description of the signals, that is, it can include all the possible higher order information of a random variable under the condition of non-Gaussian [16,17]. Due to its simplicity and robustness, the maximum correntropy criterion (MCC) of IE has been developed in the previous studies [18,19]. In these works and other similar references on the topic, the algorithms typically rely on the use of MCC and less attention is paid to the kernel size in MCC. In this work, a new algorithm based on the MCC of the IE method is proposed to enhance the performance of the existing ANC algorithms. The new algorithm achieves more stable performance than the FxRLS algorithm in the presence of impulsive noise. The filtered-x recursive maximum correntropy (FxRMC) algorithm does not need priori information of the noise characteristics, but the selection of kernel size of Gaussian kernel largely affects the performance of the filtering. To deal with this problem, a kernel adaptive version of FxRMC based on the sliding window approach is further proposed. As the kernel size is updated, this algorithm can remain stable in the presence of impulsive noise. The remainder of this paper is organized as follows. Section 2 introduces the ANC system model along with the review of the FxLMP and RFxLMS algorithms, then the proposed FxRMC algorithm and adaptive kernel size strategy are presented together with the stability condition. Simulations and experimental studies are performed to evaluate the performance of proposed algorithms in Sections 3 and 4. Finally, Section 5 presents the discussions and conclusions of this work. 2. Proposed algorithms 2.1. Previous work The block diagram of ANC system is illustrated in Fig. 1, where d(n) represents the primary noise to be cancelled, the transfer function P(z) denotes the primary path from the reference signal u(n) to the error microphone e(n), S(z) is the secondary path transfer function between the output of the adaptive filter Θ(z ) and the output of the error microphone e(n), and y (n ) is the output of the secondary path. The transfer function Sˆ (z ) can be estimated by an adaptive filter using either off-line or on-line secondary path modelling techniques [4]. The error microphone e(n) is given by

e (n ) = d (n ) − y (n ) = d (n ) − s (n )*[ΘT (n ) u (n )]

(1) T

where * denotes the discrete convolution operator, s(n) is the impulse response of S(z), and u(n)=[u(n), u(n−1),…, u(n−L+1)] is the reference signal vector. By minimizing a mean square of error signal |e (n )|2 , the FxLMS algorithm has been developed in [4]. The weight of this controller is given by

Θ (n + 1) = Θ (n ) + μe (n ) u′(n )

(2)

u (n ) = s (n )*u (n ).

(3)

where

The robust FxLMS (RFxLMS) algorithm [9] is a modified version of the FxLogLMS algorithm, which employs a new logarithmic cost function. The coefficient vector Θ(n ) of the algorithm is updated according to the following update equation:

Θ (n + 1) = Θ (n ) + μ

e (n ) u′(n ) e 2 ( n ) + 2β 2

(4)

where β is the standard deviation of e(n) which is obtained by using a sliding window approach in [20]. 2.2. Derivation of the FxRMC algorithm Based on MCC, the FxRMC algorithm is proposed to provide a robust solution for active impulsive noise control. The MCC 2

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generalizes the conventional correlation function to nonlinear spaces, making the controller robust against the outliers. Additionally, an exponentially weighted mechanism tends to put more emphasis on current data and to de-emphasize data from the remote past. The correntropy between two random variables D and Y, is defined as follows [16]

V (D, Y ) = E {κ (D, Y )} =

∫ κ (d , y ) d R d , y (d , y )

(5)

where E (⋅) is the expectation, κ (⋅,⋅) is a shift-invariant Mercer Kernel, and R d , y (d , y) is the joint distribution function of (d, y). The commonly used Gaussian kernel is defined as

⎧ e2 ⎫ κ (d , y) = exp ⎨− 2 ⎬ ⎩ 2σ ⎭

(6)

where e = d−y,1 and σ is the kernel size (kernel bandwidth). The FxRMC algorithm adopts the following cost function [19] n

JFxRMC (n ) = ∑i =1 λn − iκ (d (i ), y (i )) ⎧ e 2 (i ) ⎫ n = ∑i =1 λn − i exp ⎨− 2 ⎬ ⎩ 2σ ⎭

(7)

where 0 ≪ λ < 1 is the forgetting factor. Taking the gradient of JFxRMC (n ) with respect to weight vector Θ(n ) yields:

∂JFxRMC (n ) = ∂Θ (n ) Defining ψ (i ) = n

n

⎧ e 2 (i ) ⎫ e (i ) u′(i ) ⎬ . 2σ 2 ⎭ 2σ 2

∑ λn −i exp ⎨⎩− i =1

exp{−e 2 (i )/2σ 2}

(8)

and letting (8) to zero, we obtain n

∑ λn −iψ (i ) u′(i ) u′T (i ) Θ (n) = ∑ λn −iψ (i ) u′(i ) d (i ). i =1

(9)

i =1

For ψ (i ) = 1, the algorithm reduces to the FxRLS algorithm. Then, the expression of Θ(n ) is obtained from (9) as follows:

Θ (n ) = F (n ) π (n ), F (n ) = R−1(n ).

(10)

In the FxRMC algorithm, ψ (i ) ≠ 1, making the coefficients R(n ) and π(n ) need to be recalculated in each iteration. To overcome such drawback, R(n ) and π(n ) can be obtained by using a finite temporal window [21]. However, the algorithm in [21] is not a truly online algorithm. To obtain a truly online algorithm, R(n ) is adapted by a recursive form:

R (n ) ≈ λ R (n − 1) + ψ (n ) u′(n ) u′T (n )

(11)

π (n ) ≈ λ π (n − 1) + ψ (n ) u′(n ) d (n ).

(12)

and

Using the matrix inversion lemma [4], F(n ) can be updated as

F (n ) = λ−1F (n − 1) − λ−1Φ (n ) u′T (n ) F (n − 1)

(13)

where the gain factor Φ(n ) is given by

Φ (n ) =

ψ (n ) F (n − 1) u′(n ) . λ + ψ (n ) u′T (n ) F (n − 1) u′(n )

(14)

Combining (10)–(14), Θ(n ) can be updated by

Θ (n + 1) = Θ (n ) + Φ (n )[d (n ) − u′T (n ) Θ (n )].

(15)

Two points in the proposed algorithm need to be highlighted. First, the FxRMC algorithm is nearly blind since it does not require any priori information of the noise characteristics, and its performance depends only on σ and λ. Second, it is generally known that the inappropriate selection of kernel size σ will deteriorate the performance of the controller. When σ in MCC is larger than some values, the maximum correntropy estimation will achieve an optimal solution [22]. Since the kernel size σ in FxRMC is set manually, an online scheme for selecting the kernel size is warranted. Besides, the stability of the FxRMC algorithm needs to be analysed. In the following subsections, the results regarding above problems are discussed separately. 2.3. Mean weight behaviour analysis of the FxRMC algorithm Here, the mean weight behaviour of the FxRMC algorithm is analysed. To simplify the analysis, the following assumptions are adopted. 1) The reference signal u(n) and signal u′(n ) are independent, and identically distributed (i.i.d.) with zero-mean.

1

The notation d denotes the primary noise, and y denotes the output signal, respectively.

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2) u′(n ) is approximately independent of the a priori excess errors at steady state. The weight deviation vector is defined as follows: (16)

Ω (n ) = Θo − Θ (n )

where Θo is the optimal weight vector of the controller. Subtracting (15) from Θo , and considering Eq. (14), the following equation is obtained:

Θ (n + 1) = Θ (n ) +

ψ (n ) F (n − 1) u′(n ) [d (n ) − u′T (n ) Θ (n )]. λ + ψ (n ) u′T (n ) F (n − 1) u′(n )

(17)

Using (16), the update formulation of the weight deviation vector of the proposed algorithm can be expressed as

Ω (n + 1) = Ω (n ) −

ψ (n ) F (n − 1) u′(n ) [d (n ) − u′T (n ) Θ (n )]. λ + ψ (n ) u′T (n ) F (n − 1) u′(n )

(18)

Taking expectation of both sides of (18), the mean convergence behaviour of the coefficient vector is given by

⎧ ⎫ ψ (n ) F (n − 1) u′(n ) E [Ω (n + 1)] = E [Ω (n )] − E ⎨ [d (n ) − u′T (n ) Θ (n )] ⎬. T λ + ψ ( n ) u ′ ( n ) F ( n − 1) u ′( n ) ⎩ ⎭ The error ζ (n ) = d (n ) −

ζ (n ) ≈

u′T (n ) Θ (n )

(19)

can be approximately calculated by

u′T (n ) Ω (n ).

(20)

Introducing (20) to (19), and supposing the independence between d (n ) − u′T (n ) Θ (n ) and

ψ (n ) F (n − 1) u ′ (n ) λ + ψ (n ) u ′T (n ) F (n − 1) u ′ (n )

, (19) can be

written as

⎡ ψ (n) F (n − 1) u ′ (n) u ′T (n) ⎤ E [Ω (n + 1)] = E [Ω (n )] − E ⎢ Ω (n ) ⎥ ⎣ λ + ψ (n) u ′T (n) F (n − 1) u ′ (n) ⎦ ⎡ ψ (n) F (n − 1) u ′ (n) u ′T (n) ⎤ ≈E [Ω (n )] − E ⎢ ⎥ E [Ω (n )] ⎣ λ + ψ (n) u ′T (n) F (n − 1) u ′ (n) ⎦

⎡ ψ (n) F (n − 1) u ′ (n) u ′T (n) ⎤ where E ⎢ ⎥ ≈ 0 . Therefore, the weight vector in the FxRMC algorithm converges if and only if ⎣ λ + ψ (n) u ′T (n) F (n − 1) u ′ (n) ⎦ ⎧ ⎡ ψ (n ) F (n − 1) u′(n ) u′T (n ) ⎤ ⎫ 0 < λ max ⎨E ⎢ ⎥⎬ < 2 ⎩ ⎣ λ + ψ (n ) u′T (n ) F (n − 1) u′(n ) ⎦ ⎭

(21)

(22)

where λ max {⋅} represents the largest eigenvalue of a matrix. According to the fact that λ max (AB) < Tr(AB) in (22), we obtain

⎧ ⎡ ψ (n ) F (n − 1) u′(n ) u′T (n ) ⎤ ⎫ ⎡ ⎤ Tr(u′(n ) Λ (n ) u′T (n )) λ max ⎨E ⎢ ⎥⎬ < E ⎢ ⎥ <1 T (n ) F (n − 1) u′(n ) ⎦ T (n ) F (n − 1) u′(n ) ⎦ ⎣ ⎣ λ + ψ ( n ) u ′ λ + ψ ( n ) u ′ ⎩ ⎭

(23)

where Λ (n ) = ν (n ) F (n − 1). Therefore, the mean error weight vector of the proposed FxRMC algorithm is convergent if the input signal is persistently exciting [[23], Chapter 14]. 2.4. Adaptive kernel size Motivated by [20], the FxRMC algorithm is complemented by an on-line recursive scheme for kernel size update. There are few observations available to the designer for kernel size selection. In general, we only have access to the error microphone signals e(n) of ANC system. Therefore, it is reasonable to make use of e(n) to adapt the kernel size. The sampled mean of σ is computed recursively as (24)

mσ (n + 1) = mσ (n ) + Δmσ (n + 1).

At the nth iteration, the feedback information Δmσ (n + 1) is the estimated variety of e(n), which can be estimated from the following equation.

Δmσ (n + 1) =

1 [e (n ) − e (n − Nw + 1)] Nw

(25)

where Nw is the length of window. It should be noted that we estimate e(n) by using only the varieties of window edge. Actually, such estimation is rough, and we neglect the variation data from n−1th iteration to n−Nw+2th iteration. Belge et al. developed a sliding window approach to minimize the averaged p-norm of the error [21]. Although this approach defines a robust function for the adaptation of filter coefficients, it suffers from a practical problem: the scheme is not truly online. Because all data in a window of past inputs to the filter are processed in batch mode, requiring several iterations of the algorithm at each iteration, therefore increasing the memory and computational requirements of the filter [24]. A more practical scheme can be obtained by simply discarding samples as we used in (25). 4

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2

σ =1 2

σ =2

0.8

2

σ =4

J FxRMC ( n )

σ 2=8 0.6

0.4

0.2

0 -10

-8

-6

-4

-2

0

2

4

6

8

10

e(n) Fig. 2. The cost functions with different σ2. (α=1.8).

During the adaptation of the algorithm, the appropriate kernel sizes are in the range of observation errors [25]. The rationality of this selection is based on the fact that half of the cost function has the observation error and a similar kernel size will put us in the convex hull [25]. Hence, we heuristically choose the kernel size at each time instant to be the estimation of the observation error. The mean value of σ2 can be recursively computed according to the approach in [20]

σ 2 (n + 1) = σ 2 (n ) + Δmσ2 (n + 1) +

1 1 [e (n ) − mσ (n + 1)]2 − [e (n − Nw + 1) − mσ (n + 1)]2 . N N w  

w 

I

II

(26)

Note that I and II can be considered as the compensation for estimation of e(n). To further reduce the computational burden, the previous expression can be expressed by using the alternative equation2:

σ 2 (n + 1) = σ 2 (n ) + Δmσ2 (n + 1).

(27)

When the FxRMC algorithm is with different kernel bandwidths, its cost function will change, as illustrated in Fig. 2. It is observed that JFxRMC (n ) with small kernel size is less steep than the MCC with large kernel size, that is, the kernel size has a significant influence on the convergence rate. With the same forgetting factors, the FxRMC with small kernel size converges faster than the others, but it tends to be unstable in the presence of impulsive noise. By using the adaptive kernel size scheme, the FxRMC algorithm has a large initial kernel size, and stable performance for active impulsive noise control. Summary of the proposed algorithms for ANC is given in Table 1. 2.5. Computational complexity The computational requirements for the five adaptive filters are summarized in Table 2. The FxLMS-based algorithms (FxLMS, RFxLMS) have lower computational complexity than the FxRLS-based algorithms. Owing to the use of recursive scheme, the FxRLS increases the computational burden with improved convergence rate. The increase in the complexity of the proposed algorithms compared with that of the FxRLS algorithm is moderate, since the additional operations are much smaller than L. 3. Simulation results The performance of the proposed algorithms was evaluated and compared with the existing algorithms in MATLAB for active impulsive noise control. The sample frequency of the ANC system is 8000 Hz. The primary path P(z) and secondary path S(z) were modelled as FIR filters of length 256 and 100, respectively. It is assumed that the estimated secondary path Sˆ (z ) was exactly identified as S(z). The frequency response of the primary path and secondary path were shown in Fig. 3. The ANC filter Θ(z ) was taken as an FIR filter of order 128. The averaged noise reduction (ANR) was used to compare the performance, which was defined as [9]

⎛ A (n ) ⎞ ANR (n ) = 20 log ⎜ e ⎟ ⎝ A d (n ) ⎠

(28)

where Ae (n ) = ξAe (n − 1) + (1 − ξ )|e (n )|, Ad (n ) = ξAd (n − 1) + (1 − ξ )|d (n )|, and ξ=0.999. 2 The adaptation in (27) has a quite similar performance to Eq. (26). For paper length optimization, we decided to only use (26) to FxRMC algorithm in following simulations/experiments.

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Table 1 Summary of the algorithms discussed in this paper. Initialization and parameter selection length of the primary path P (z), length of the secondary path S (z), FxRMC algorithm: L, λ, Θ (0) = 0 ,F (0) = δ I ,σ2, δ=1/50 FxRMC algorithm with adaptive kernel size: Nw, while {u(n), e(n)} available do 1: y (n ) = s (n )*[ΘT (n ) u (n )]; 2: u′(n ) = s (n )*u (n ); 3: e (n ) = d (n ) − y (n ); 4: ψ (n ) = exp{−e 2 (n )/2σ 2}; 5: mσ (n + 1) = mσ (n ) + Δmσ (n + 1); 6: Δmσ (n + 1) =

1 [e (n ) Nw

− e (n − Nw + 1)];

7: σ 2 (n + 1) = σ 2 (n ) + Δmσ2 (n + 1) + 8: Φ(n ) =

1 [e (n ) Nw

− mσ (n + 1)]2 −

1 [e (n Nw

− Nw + 1) − mσ (n + 1)]2 ;

ψ (n ) F (n − 1) u ′ (n ) ; λ + ψ (n ) u ′T (n ) F (n − 1) u ′ (n )

9: Θ (n + 1) = Θ (n ) + Φ (n )[d (n ) − u′T (n ) Θ (n )]; 10: F (n ) = λ−1F (n − 1) − λ−1Φ (n ) u′T (n ) F (n − 1) ; end while Note: 1. The parameter δ is a regularization parameter of the recursive algorithm, which is related to a soft initialization [26,27], and I represents an identity matrix. 2. Proposed FxRMC algorithm =1–4, 8–10. 3. Proposed FxRMC algorithm with adaptive kernel size =1–10.

Table 2 Computational complexity of the algorithms for ANC systems. Algorithms

Multiplications

Divisions

Additions/Subtractions

Exponential operations

FxLMS RFxLMS FxRLS FxRMC FxRMC with adaptive kernel size

2L+1 2L+6 4L2+3L 4L2+3L+5 4L2+3L+7

No 3 1 2 5

2L 2L+7 3L2 3L2 3L2+7

No No No 1 1

Magnitude, dB

10 0 -10 -20 -30 -40 -50 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.8

0.9

Normalized Frequency (×π rad/sample)

Phase, radians

0 -50 -100 -150 -200 -250 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized Frequency (×π rad/sample)

Fig. 3. Frequency response of acoustic paths used in computer simulations. (Keys: dashed line, primary path P(z); solid line, secondary path S(z)).

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L. Lu, H. Zhao

3000

3000

3000

2000

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0

0

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-1000

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-2000

0

2 x 10

-2000

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Iterations

x 10

c 0

2

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x 10

4

Fig. 4. Original primary noise of case 1 (a) α=1.2, case 2 (b) α=1.6, and case 3 (c) α=1.8. 10

10 FxLMS( μ=0.00001) RFxLMS( Nw=5, μ=0.0001)

8 6

6

FxRMC( λ=0.999, σ 2 =8) FxRMC with adaptive kernel size( λ=0.999, Nw=2)

4 2 0 -2

FxRLS( λ=0.999)

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FxRMC( λ=0.999, σ 2=8) FxRMC with adaptive kernel size( λ=0.999, Nw=2)

0 -2

-4

-4

-6

-6

a

-8 -10

FxLMS( μ=0.00001) RFxLMS( Nw=5, μ=0.0001)

4

ANR,dB

ANR,dB

8

FxRLS( λ=0.999)

0

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b

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-10

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10 8

FxLMS( μ=0.00002) RFxLMS( N w=5, μ=0.0005)

6

FxRLS( λ=0.999)

4

FxRMC( λ=0.999, σ 2=8) FxRMC with adaptive kernel size( λ=0.999, N w=2)

2 0 -2 -4 -6

c

-8 -10

0

0.5

1

1.5

2

Iterations

2.5

3

3.5

4 x 10

4

Fig. 5. Performance comparison of the algorithms under impulsive noise condition. (a) α=1.2, (b) α=1.6, (c) α=1.8.

3.1. Impulsive noise The performance comparison for the FxLMS, RFxLMS, FxRLS and the proposed algorithms in impulsive noise environments are showed in Figs. 4–7. Here, we consider the impulsive noise with symmetric α-stable (SαS) distribution. The SαS process has no closed probability density function expression, and can be described by the characteristic function as follows [6]

φ (t ) = exp{−|t|α }

(29)

where 0 < α < 2 is a characteristic exponent that a small value indicates a peaky and heavy tailed distribution with more outliers. In particular, for α=1, the stable distribution is a Cauchy distribution and becomes a Gaussian distribution for α=2. Fig. 4 shows the original primary noise of case 1–3, which corresponds to the three cases in Figs. 5 and 6. Fig. 5 depicts the ANR 7

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160 140 120 100 80 60 40

40

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0

b

-20

a

20 0

FxLMS RFxLMS FxRLS FxRMC FxRMC with adaptive kernel size

60

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FxLMS RFxLMS FxRLS FxRMC FxRMC with adaptive kernel size

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40 FxLMS RFxLMS FxRLS FxRMC FxRMC with adaptive kernel size

Power Spectrum,dB

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c

-20

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Frequency,Hz Fig. 6. PSD comparison of the algorithms under impulsive noise condition. (a) α=1.2, (b) α=1.6, (c) α=1.8.

curves of algorithms. For stability of the FxRMC algorithm, σ2 of this example and second example are selected as 8. As can be seen, the FxLMS and FxRLS have fluctuations during the adaptation, since for α < 2 the second- or higher-order moments are infinite [24]. The FxRMC algorithm, in contrast, has more stability than that of the FxRLS in all cases. By using the adaptation rule in Section 3.3, the FxRMC with adaptive kernel size algorithm has a faster initial convergence rate than that of the FxRMC. To further demonstrate the effectiveness of proposed algorithms, Fig. 6 illustrates the power spectral density (PSD) of the error microphone of five algorithms.3 It is found that the proposed algorithms outperform other algorithms in terms of noise attenuation, and have quite similar PSD performances in all cases. To further illustrate the proposed adaptive kernel size approach, Fig. 7 shows the evolution of the kernel size for each case.4 It can be seen that the FxRMC algorithm has large initial kernel size, and the adaptive kernel size σ converges to a value between 1 and 2. After about iteration 1000, the kernel sizes in all cases tend to be stable.

3.2. Noise mixed with Gaussian and impulsive noise In this example, we consider two different noises, white Gaussian noise (WGN) and impulsive noise, for intervals 0 < n ≤ 15000 and 15000 < n ≤ 40000 , respectively. Besides, three cases (a) α=1.3, (b) α=1.5, (c) α=1.7 are also considered for impulsive noise environments. Fig. 8 shows the simulation results of WGN mixed with impulsive noise. It is demonstrated that the proposed algorithms converge faster than the FxLMS algorithm at 1–15,000 points, and outperform the RFxLMS algorithm under impulsive noise environments. The FxRLS algorithm has the quite similar performance to FxRMC algorithm under WGN noise environment. However, as we expected, the FxLMS and FxRLS algorithms fail to work in the presence of impulsive noise owing to MSE criterion. Fig. 9 illustrates a comparison of PSD of the error microphone e(n) from the proposed algorithms and existing algorithms. It is found that the proposed algorithms reach low PSD level than the other algorithms. Furthermore, we evaluate the performance of algorithms when the adaptive method is applied to update the kernel size. The curves of kernel sizes for three cases are demonstrated in Fig. 10. In WGN environment, the adaptive kernel size method has stable performance, with the kernel size between 1 and 2. The 3 The PSD of the error microphone is obtained by using Welch’s Method with the 256-point hanning window, and the psd function is employed to estimate thePSD in Matlab. 4 To demonstrate the kernel sizes clearly, the number of iterations is truncated to 1500 points.

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FxRMC algorithm has some fluctuations when noise gets severe in α=1.3 (Case 1), and obtains relatively stable performances in other cases. 4. Experiment results The FxLMS, RFxLMS, FxRLS and the proposed algorithms are implemented on a dSPACE real time control board. We design and compile the ANC controllers for the five algorithms in MATLAB/Simulink environment and upload to the dSPACE system. The secondary path identification was estimated off-line by using LMS algorithm [4]. The Sampling frequency is 5000 Hz. Adaptive filter 9

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length is 32 bits. In this experiment, the noise source is impulsive noise (500 Hz) combined with the sinusoidal waves (450 Hz, 480 Hz, 500 Hz), as shown in Fig. 11(a). The set of results for impulsive noise environment is obtained for the following algorithms: the FxLMS algorithm that employs the step size μ=0.01; the RFxLMS algorithm [9] uses the step size μ=0.01, with Nw=5; the RLSbased algorithms use the same forgetting factor λ=0.999 and σ2=2; and, the FxRMC with adaptive kernel size algorithm uses Nw=2 for kernel update equation. The first experiment is to compare the average ANR learning curves of the proposed algorithms with the state-of-the-art ANC algorithms. Results in Fig. 11(b) show that the FxLMS and FxRLS algorithms diverge in the presence of impulsive noise scenario. The proposed algorithms converge as fast as the RFxLMS, and outperform other ANC algorithms. The experimental PSD performance is illustrated in Fig. 11(c). As can be seen, the RFxLMS and the proposed algorithms have stable performance. In 10

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particular, the proposed algorithm has an improved performance when compared with the RFxLMS algorithm. It should be noted that the FxRLS algorithm is seldom used in practice, because of its heavy computation complexity and low robustness. The proposed FxRMC algorithm and its adaptive kernel size version improve the robustness of the FxRLS algorithm. However, they are still inherit the heavy computation of the FxRLS algorithm. Hence, the two proposed algorithms may use the setmembership (SM) scheme [28,29] for significant reduction in the overall computational complexity and memory. The central idea of the SM scheme is that the tap-weight vector is updated only when the magnitude of the estimated error exceeds a predetermined bound. There are several SM methods available for the adaptive filters [28–30], and these methods can be also directly combined with the FxRMC algorithm.

5. Conclusions Based on the MCC of IE, a new FxRMC algorithm was developed for active impulsive noise control, which does not require any prior knowledge or estimation of the characteristic exponent. In the analysis, it has been proved that the proposed algorithm ⎧ ⎡ ψ (n) F (n − 1) u ′ (n) u ′T (n) ⎤ ⎫ converges in the mean sense for λ max ⎨E ⎢ ⎥ ⎬ < 1. Nevertheless, its performance largely depends on the kernel T ⎩ ⎣ λ + ψ (n) u ′ (n) F (n − 1) u ′ (n) ⎦ ⎭ bandwidth. To select a proper kernel size of the FxRMC algorithm on-line, an adaptive selection kernel strategy was proposed. The extensive numerical simulations and experimental tests were performed, which showed that the proposed algorithms have better noise cancelling performance than the other algorithms.

Acknowledgment The authors would like to thank Dr. Xiaoqiang Zhang (Southwest Jiaotong University) and Dr. Jianming Liu (Fortemedia Inc. USA) for polishing the language. This work was partially supported by National Science Foundation of P.R. China (Grant: 61571374, 61271340, 61433011). 11

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