Maximum Versoria Criterion Algorithm with Adaptive Radius in Active Impulse Noise Control

Abstract

The conventional algorithm used in active noise control (ANC) performs poorly under non-Gaussian (impulse) noise. This is because the stability of a conventional adaptive controller is easily damaged by the impulse noise. For the purpose of eliminating the noise and enhancing stability, two improvements are listed in the paper. Firstly, based on the robustness of the Versoria function to impulsive noise, a filtered-x recursive maximum Versoria criterion (FXRMVC) algorithm is proposed to attenuate the impulsive noise. Secondly, the FXRMVC algorithm employs an adaptive radius strategy which is only subject to the estimated error signal. The strategy can adjust the radius of Versoria function online. The benefit of this method is that the FXRMVC with adaptive radius algorithm breaks step size limitation and avoids noise prior information. The computer simulation results manifest that the proposed algorithm has superior noise reduction, faster convergence rate, lower minimum steady state error and better power spectral density compared with other algorithms in different noise environments.

1. Introduction

Active noise control has been applied to noise reduction in aircraft, automobile cabin and pipeline noise reduction which can effectively suppress low-frequency noise [1,2]. The idea behind the technique is that the primary noise can be cancelled by the superposition of the secondary noise on amplitude and phase. Nevertheless, the characteristic of noise in many practical situations usually exhibits impulsive, with low probability but large amplitude, outliers [3]. The ANC system have been applyed successfully in many scenarios, but there are still some challenges for controlling impulse noise.

The noise reduction and tracking ability of the conventional algorithm become poor when the interference with impulse noise occurs in the actual environment. When the environmental noise changes abruptly, the impulse noise active control algorithm needs to have the tracking capability to follow the transformation of the surrounding environment in order to reduce noise effectively.

The filtering x-LMS [2] algorithm has been commonly used in ANC system due to its simplicity and convenience. However, the algorithm will be unstable due to the infinite second-order of impulsive noise. In 1995, Leahy et al. [4] proposed FxLMP algorithm by choosing a non-quadratic cost function, the mean p-norm of error. It requires the priori knowledge corresponding to the impulse noise characteristics. Sun et al. [5] and Ather et al. [6] each proposed a modified FXLMS algorithm. Both algorithms directly ignore the sample values of the reference signal in different degrees. In 2012, Wu et al. [7] proposed the FxlogLMS algorithm which uses the square of the logarithm of error as the cost function to improve the stability and robustness, but the small drawback of algorithm is the dead zone in weight update. Later, Zhou et al. [8] improved FxlogLMS algorithm and proposed FxgsnLMS algorithm which uses a new nonlinear function of the reference signal. The algorithms that add square or absolute of error to logarithmic of error are stable under impulsive interferences [9]. Moreover, refs. [10,11] combined arctangent function and logarithm function [9] to step-size scaler in the presence of impulse noise. Recent years, a large number of Gaussian kernel functions have emerged to improve the performance of the classical MCC algorithm [12,13,14,15,16]. In [17], Zhu et al. promoted FxGMCC algorithm which confirms the advantage of the generalized Gaussian density (GGD) function for non-Gaussian noise.

The RLS algorithm has faster convergence and smaller steady-state error. Moreover, it is insensitive to the characteristic root distribution of the autocorrelation matrix of the reference signal compared with the LMS algorithm. [18] uses filtered-x RLS algorithm to control impulse noise in ANC, but the algorithm performs poorly in terms of steady-state error. Improved FxRLS algorithms are applied in active control of non-Gaussian noise with high computational complexity in [19].

Deng et al. [20] developed a LMS adaptive algorithm using Versoria function. Huang [21] proposed a maximum Versoria criterion (MVC) algorithm based on the maximum Versoria criterion and tested its robustness under non-Gaussian noise in acoustic echo-cancellation scenarios. Recently, the application of MVC algorithm in system identification and channel equalization has also attracted much attention [22]. Few of the above algorithms and related literature focus the MVC algorithm on active noise control.

A new idea is presented in this paper, since the robustness of the Versoria to control the outliers in impulse noise. The proposed filtered-x recursive maximum Versoria criterion (FxRMVC) algorithm is more stable than the basic FxRLS algorithm in impulse noise environment. The performance of the FxRMVC algorithm in this paper is affected by the radius of the generating circle of Versoria. Therefore, an effective sliding window estimation strategy to update radius is further proposed.

In the second part, this paper reviews the ANC system and several basic algorithms. In the third part, the FxRMVC algorithm is deduced and the online estimation strategy is summarized. Simulation results illustrate the advantages of the new algorithm in forth part. Finally, the conclusion and discussion is presented.

2. ANC System and Algorithm Review

2.1. ANC Model

Figure 1 shows the structure of the ANC system. $P\left(z\right)$ denotes the primary path transfer function between two microphones and $S\left(z\right)$ is the secondary path transfer function from the output of the controller to the error microphone. $\widehat{S}\left(z\right)$ is the estimation model for the secondary path $S\left(z\right)$, which can be estimated in two ways [2]. The noise passes through $P\left(z\right)$ to reach the error microphone and the reference microphone picks up the reference signal $x\left(n\right)$, while the output signal $y\left(n\right)$ of filter $W\left(z\right)$ which passes through $S\left(z\right)$ also reaches the error microphone. The two cancel out to form the error signal $e\left(n\right)$.

The reference signal vector and FIR controller vector is represented as

$$\mathbf{x}\left(n\right)={[x\left(n\right),x(n-1),\cdots ,x(n-L+1)]}^T$$

(1)

$$\mathbf{W}\left(n\right)={[W_0\left(n\right),W_1\left(n\right),\cdots ,W_{L-1}\left(n\right)]}^T$$

(2)

The error $e\left(n\right)$ is calculated as

$$e\left(n\right)=d\left(n\right)-y^{\prime }\left(n\right)=d\left(n\right)-y\left(n\right)\ast s\left(n\right)$$

(3)

where $y\left(n\right)=W{\left(n\right)}^T\ast x\left(n\right)$, ∗ denotes the linear convolution calculation, $s\left(n\right)$ is the time-discrete representation of $S\left(z\right)$.

2.2. Algorithm Review

The cost function of FxLMS [2] algorithm is $e^2\left(n\right)$, and the weight of filter is updated by

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+2\mu e\left(n\right)\left[s\right(n)\ast \mathbf{x}(n\left)\right]$$

(4)

George and Panda [23] proposed the RFxLMS algorithm which modify the logarithm function as a cost function to suppress noise effectively.

The cost function and weight update is computed as

$$J=log(1+\frac{e{\left(n\right)}^2}{2b^2})$$

(5)

where J is the cost function.

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mu \frac{e\left(n\right)}{e^2\left(n\right)+2b^2}{\mathbf{x}}^{\prime }\left(n\right)$$

(6)

The combination of adaptive filtering and MCC in information theory raises much concern. The FxMCC algorithm is proposed in [24] to process non-Gaussian reference signals. The update rule is denoted as

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mu exp(-\frac{e^2\left(n\right)}{{\sigma }^2})e\left(n\right){\mathbf{x}}^{\prime }\left(n\right)$$

(7)

where $\mu$ is the step size.

The weight recurrence of Arc−NLMS algorithm [11] is shown as follows

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mu \frac{\mathbf{x}\left(n\right)e\left(n\right)}{{∥\mathbf{x}\left(n\right)∥}^2+e^4\left(n\right)/{∥\mathbf{x}\left(n\right)∥}^2}$$

(8)

The FxRLS algorithm focus on fast tracking recent data but suffers from stability issues. The recurrence formula is

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mathbf{K}\left(n\right)e\left(n\right)$$

(9)

where $\mathbf{K}\left(n\right)$ is the gain factor.

The MVC algorithm [21] which uses the Versoria as cost function guarantees the lower steady-state under impulse noise. The cost function can be denoted as

$$J_{MVC}=\frac{2r}{1+{\left(e\right(n)/2r)}^2}$$

(10)

where $J_{MVC}$ is the cost function of MVC algorithm.

3. Proposed Algorithm

3.1. Derivation of the FxRMVC Algorithm

The FxRMVC algorithm is a new solution for active impulse noise control. Figure 2 shows the standard Versoria function ($r=0.5$). It ensures the stability of weight vector and reduces the impact of error messages on weight updates. Versoria function is defined as

$$f\left(e\right)=\frac{8r^3}{4r^2+e^2}=2r\frac{1}{1+{(e/2r)}^2}$$

(11)

where $r>0$ is the radius of the generating circle of Versoria and the larger the radius, the steeper the function curve. The Versoria cost function performs better than the Gaussian kernel−based correlation entropy cost function. When the impulse interferences, Versoria cost function can reach the optimal faster [21].

The cost function of proposed FxRMVC algorithm is expressed by

$$J_{FxRMVC}\left(n\right)=\sum _{i=1}^n{\lambda }^{n-i}f\left(e\left(i\right)\right)=\sum _{i=1}^n{\lambda }^{n-i}2r\frac{1}{1+{\left(e\right(i)/2r)}^2}$$

(12)

where $0<\lambda <1$. Taking the gradient calculation of $J_{FxRMVC}\left(n\right)$ yields:

$$\frac{\partial J_{FxRMVC}\left(n\right)}{\partial \mathbf{W}\left(n\right)}=\sum _{i=1}^n{\lambda }^{n-i}\frac{1}{2r}\frac{1}{{(1+{\left(e\right(i)/2r)}^2)}^2}e\left(i\right){\mathbf{x}}^{\prime }\left(i\right)$$

(13)

Letting (13) be zero, we can obtain

$$\sum _{i=1}^n{\lambda }^{n-i}F\left(i\right){\mathbf{x}}^{\prime }\left(i\right){\mathbf{x}}^{\prime T}\left(i\right)\mathbf{W}\left(n\right)=\sum _{i=1}^n{\lambda }^{n-i}F\left(i\right){\mathbf{x}}^{\prime }\left(i\right)d\left(i\right)$$

(14)

where $F\left(i\right)=\frac{1}{2r}\frac{1}{{(1+{\left(e\right(i)/2r)}^2)}^2}$.

The FxRMVC algorithm is a derivation of the FxRLS algorithm. Inspired by the literature [25,26], the recursive form for the relevant vectors in this paper is obtained:

$$\mathbf{R}\left(n\right)\approx \lambda \mathbf{R}(n-1)+F\left(n\right){\mathbf{x}}^{\prime }\left(n\right){\mathbf{x}}^{\prime T}\left(n\right)$$

(15)

$$\mathbf{r}\left(n\right)\approx \lambda \mathbf{r}(n-1)+F\left(n\right){\mathbf{x}}^{\prime }\left(n\right)d\left(n\right)$$

(16)

Next, inverse operations using the matrix theorem (15) can be evolved as

$$\mathbf{Psi}\left(n\right)={\mathbf{R}}^{-1}\left(n\right)={\lambda }^{-1}\mathbf{Psi}(n-1)-{\lambda }^{-1}\mathbf{K}\left(n\right){\mathbf{x}}^{\prime T}\left(n\right)\mathbf{Psi}(n-1)$$

(17)

where the gain factor is

$$\mathbf{K}\left(n\right)=\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}$$

(18)

Finally, combining above equations, the update rule of $W\left(n\right)$ is denoted as

$$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mathbf{K}\left(n\right)e\left(n\right)$$

(19)

In FxRMVC algorithm, the output error tends to saturate whenever the input signal is disturbed by heavy impulse interference. From the analysis above, the radius of the generating circle of Versoria plays an important role in FxRMVC algorithm, affecting convergence performance. Therefore, an online strategy for choosing the radius is necessary. The strategy will be discussed in the following subsection. Next, the stability of the FxRMVC algorithm is analyzed.

3.2. Stability of FxRMVC Algorithm

After subtracting $W_o$ from $W\left(n\right)$, the weight deviation vector can be computed as

$$\mathbf{V}\left(n\right)=\mathbf{W}\left(n\right)-{\mathbf{W}}_o$$

(20)

where $W_o$ is the optimal weight vector. $W_o$ is subtracted by (19), then we can obtain

$$\mathbf{V}(n+1)=\mathbf{V}\left(n\right)+\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}e\left(n\right)$$

(21)

as well as

$$\begin{array}{cc}\hfill e\left(n\right)& =d\left(n\right)-{{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{W}\left(n\right)\hfill \\ \hfill & =e_o\left(n\right)-{{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{V}\left(n\right)\hfill \end{array}$$

(22)

where $e_o\left(n\right)$ is the estimation error when the filter weight is optimal. Taking the expectation of both sides of (21), then according to the independence assumption [25] and principle of orthogonality [26,27], equations can be obtained as follows

$$E[\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{e}_o\left(n\right)]=0$$

(23)

$$\begin{array}{cc}\hfill E[\mathbf{V}(n+1)]& =E[\mathbf{V}\left(n\right)]+E[\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}(e_o\left(n\right)-{{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{V}\left(n\right))]\hfill \\ \hfill & \approx E[\mathbf{V}\left(n\right)]-E[\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}]E[\mathbf{V}\left(n\right)]\hfill \end{array}$$

(24)

where $\mathsf{\Theta }=E[\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}]\approx 0$. Therefore, the FIR controller in the FXRMV algorithm can update stably only if

$$0<{\lambda }_{max}\left(\mathsf{\Theta }\right)<2$$

(25)

According to ${\lambda }_{max}\left(\mathbf{AB}\right)<Tr\left(\mathbf{AB}\right)$, we can get

$${\lambda }_{max}\left(\mathsf{\Theta }\right)<E[\frac{Tr\left({\mathbf{x}}^{\prime }\left(n\right)F\left(n\right)\mathbf{Psi}(n-1){{\mathbf{x}}^{\prime }}^T\left(n\right)\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}]<1$$

(26)

where $\lambda (·)$ is the largest eigenvalue of a matrix. From the above, it can be concluded that the FxRMVC algorithm can remain stable as long as a continuous input signal exists [28].

3.3. Adaptive Radius Strategy

Motivated by [21], an online sliding-window radius adaptive strategy is applied to the FxRMVC algorithm. The strategy uses an estimate of the error without other priori information. This not only reduces the complexity of traversal errors, but also efficiently calculates the appropriate values. The sliding window estimation method is computed below

$$\widehat{e}\left(n\right)=\gamma \widehat{e}(n-1)+(1-\gamma )min\left(A_e\left(n\right)\right)$$

(27)

where $0<\gamma <1$ is the smoothing factor. $A_e\left(n\right)=[e\left(n\right),e(n-1),\cdots ,e(n-N_w+1)]$, where $N_w$ is the length of window. The effect of different $N_w$ on the algorithm will be discussed in the simulations. The $min(·)$ avoids sudden radius changes due to impulsive interference.

The score function in the standard MVC algorithm [21] is given as

$$\psi \left(e\left(n\right)\right)=\frac{1}{{(1+{\left(2r\right)}^{-2}{e}^2\left(n\right))}^2}$$

(28)

As the weight decreases along the gradient direction, make the derivative of (28) equal to 0. After appropriate mathematical operations, the optimal radius at maximum decay is calculated as

$$r\left(n\right)=\beta \mid e\left(n\right)\mid$$

(29)

Introducing $\mid e\left(n\right)\mid$ to $A_e\left(n\right)$ for robustness to impulse noise yields

$$A_e\left(n\right)=[\mid e\left(n\right)\mid ,\mid e(n-1)\mid ,\cdots ,\mid e(n-N_w+1)\mid ]$$

(30)

Taking the above equation into (27), and we can obtain

$$r\left(n\right)=\beta \widehat{e}\left(n\right)$$

(31)

where $\beta$ is a positive constant. Complementing this strategy with the FxRMVC algorithm can obtain adaptive regulation. Larger the radius is, steeper the $J_{FxRMVC}$ is and convergence is faster. When impulsive disturbance appears, increasing the radius can adjust the weight quickly.

The new FxRMVC algorithm flow is summarized in

Table 1. Summary of the algorithms discussed in this paper are summarized as follows.

Initialization: $P\left(z\right)$, $S\left(z\right)$, $\lambda =0.999$

FxRMVC algorithm:length of filter L, $\mathbf{W}\left(0\right)=\mathbf{0}$, $\mathbf{Psi}\left(0\right)=\delta \mathbf{I}$, r

FxRMVC algorithm with adaptive radius: $Nw$

1:$y\left(n\right)=s\left(n\right)\ast \left[{\mathbf{W}}^{\mathbf{T}}\left(n\right)\mathbf{x}\left(\mathbf{n}\right)\right]$;

2:${\mathbf{x}}^{\prime }\left(n\right)=s\left(n\right)\mathbf{x}\left(\mathbf{n}\right)$;

3:$y^{\prime }\left(n\right)=y\left(n\right)\ast s\left(n\right)$;

4:$F\left(n\right)=\frac{8r^3}{4r^2+e{\left(n\right)}^2}=2r\frac{1}{1+{\left(e\right(n)/2r)}^2}$;

5:$A_e\left(n\right)=[\mid e\left(n\right)\mid ,\mid e(n-1)\mid ,\cdots ,\mid e(n-N_w+1)\mid ]$;

6:$\widehat{e}\left(n\right)=\gamma \widehat{e}(n-1)+(1-\gamma )min\left(A_e\left(n\right)\right)$;

7:$r\left(n\right)=\beta \widehat{e}\left(n\right)$;

8:$\mathbf{K}\left(n\right)=\frac{F\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}{\lambda +F\left(n\right){{\mathbf{x}}^{\prime }}^T\left(n\right)\mathbf{Psi}(n-1){\mathbf{x}}^{\prime }\left(n\right)}$;

9:$\mathbf{W}(n+1)=\mathbf{W}\left(n\right)+\mathbf{K}\left(n\right)e\left(n\right)$;

10:$\mathbf{Psi}\left(n\right)={\lambda }^{-1}\mathbf{Psi}(n-1)-{\lambda }^{-1}\mathbf{K}\left(n\right){\mathbf{x}}^{\prime T}\left(n\right)\mathbf{Psi}(n-1)$;

¹ Proposed FxRMVC algorithm is step 1–4 and step 8–10. ² Proposed FxRMVC algorithm using adaptive radius strategy is step 1–10.

.

3.4. Computational Complexity

The proposed FxRMVC algorithms are compared with other related algorithms on computational requirements in

Table 2. Computational requirement.

Algorithms	Multiplications	Additions/Subtractions	Divisions
FxMCC	$2L+2M+6$	$2L+2M-3$	-
RFxLMS	$2L+2M+5$	$2L+2M-1$	1
Arc-NLMS	$2L+2M+8$	$2L+2M-1$	2
FxRLS	$4L^2+4L+2M$	$3L^2+L+2M-2$	1
FxRMVC	$4L^2+5L+2M+1$	$3L^2+L+2M-2$	2
FxRMVC with adaptive radius	$4L^2+5L+2M+5$	$3L^2+L+2M-1$	2

. The RFxLMS algorithm has lower complexity. The FxMCC algorithm also increases the complexity due to the exponential operations in the cost function. Based on the FxRLS algorithm, the FxRMVC algorithm increases complexity. However, the increased complexity is within the appropriate range, and the concern of this paper is still on improving noise reduction.

4. Simulations

The proposed algorithms will be compared with other algorithms in noise reduction for impulse active noise control.

Table 3. The parameters in algorithms.

Algorithms	$\mathit{\alpha }=1.2$	$\mathit{\alpha }=1.4$	$\mathit{\alpha }=1.6$
FxMCC	$\sigma =2$	$\sigma =2$	$\sigma =4$
RFxLMS	$\mu =0.0003$	$\mu =0.0005$	$\mu =0.0005$
Arc-NLMS	$\mu =0.02$	$\mu =0.02$	$\mu =0.02$
FxRLS	$\lambda =0.999$	$\lambda =0.999$	$\lambda =0.999$
FxRMVC	$\lambda =0.999$, $r=0.5$, $\delta =\frac{1}{50}$
FxRMVC with adaptive radius	$\lambda =0.999$, $N_w=12$, $\delta =\frac{1}{50}$

lists the parameters for each algorithm in the simulation. We select primary and secondary acoustic paths from [2]. The frequency response of acoustic paths are shown in Figure 3. The lengths L of adaptive controller $W\left(z\right)$ is set to 192. The average noise reduction (ANR) was used to be the performance metric, defined as [22]

$$ANR=20{log}_{10}\left(\frac{A_e\left(n\right)}{A_d\left(n\right)}\right)$$

(32)

where $A_e\left(n\right)=\eta A_e(n-1)+(1-\eta )\mid e\left(n\right)\mid$,$A_d\left(n\right)=\eta A_d(n-1)+(1-\eta )\mid d\left(n\right)\mid$. Here $\eta =0.999$. All simulation results were obtained by taking average 100 ANR realizations.

The impulse noise in the computer simulation adopts standard symmetric alpha−stable distribution. Its probability density function is described by characteristic function [29]

$$\phi \left(t\right)=exp(-\mid t{\mid }^{\alpha })$$

(33)

where the parameter $\alpha$ indicates the impulsiveness degree of signal, and larger value will result in less impulsiveness. Specifically, $\alpha =2$, $\alpha =1$ corresponds to the Gaussian and the Cauchy distribution, respectively.

4.1. Effect of Sliding Window Size

In the online sliding window strategy, the window length $N_w$ represents the number of samples used in the update. Take different $N_w$ from 2 to 14 for the simulations in the case of $\alpha =1.6$. From the simulation results in Figure 4, Larger $N_w$ shows faster convergence speed and better noise reduction. However, the noise reduction is no longer elevated and the convergence speed slow down beyond $N_w=12$. In addition, the greater $N_w$ requires more computational complexity to calculate the radius. For the purpose of good performance and less complexity, we choose $N_w=12$ in the next simulations.

4.2. Alpha-Stable Noise

Figure 5 presents the reference signal of $\alpha =1.2$, $\alpha =1.4$, $\alpha =1.6$ in $S\alpha S$ process which correspond to strongly, moderately, and mildly impulsive noise, respectively. The aim of the experiment is to compare and verify the noise reduction of the proposed method under the above three cases. For the stability of the algorithm, the radius in the FxRMVC algorithm is chosen as 0.5. The ANR comparison curves is shown in Figure 6 under a different impulsive environment. During the update, the FxRLS algorithm diverges because the second−order or higher−order moments of the reference signal are infinite [30]. The FxMCC algorithm performs unstably because of the single kernel and the RFxLMS converging slowly. The FxRMVC algorithm has a greater advantage in convergence and noise reduction under heavy impulsive noise. By applying the adaption strategy in Section 3.3, the FxRMVC with adaptive radius algorithm inherits the superior performance and even has a faster convergence rate. Moreover, the proposed method breaks the performance limitation of step size compared with conventional algorithms. Figure 7 gives the process of radius adaption, which starts fluctuating and tends to stabilize after about 5000 iterations. The presentation is truncated to 5000 iterations for clear display.

A further analysis of the proposed algorithm is the power spectral density (PSD) illustrated in Figure 8. We can see that the new algorithms achieve lower PSD level compared to other algorithms and exhibits good stability in PSD performance in all cases. The new algorithms are insusceptible to different cases of impulse noise and they can demonstrate better control levels.

4.3. Mixture of Impulse and Gaussian Noise

In this experiment, the mixed noise is used to verify the performance of the algorithm. The mixed noise consists of Gaussian noise in intervals 0 $<n<$ 10,000 and impulse noise in 10,000 $<n<$ 35,000. From Figure 9, we can dissect that the FxRMVC algorithms converge faster than FxRLS algorithm and Arc−NLMS algorithm under white noise environment. Meanwhile, the FxRLS algorithm can reach the same level to proposed algorithms under white noise environment. However, the FxRLS algorithm and Arc−NLMS algorithm fail to reach the lower noise reduction and the tracking ability declined when the impulse noise appears. The FxMCC algorithm is divergent and does not converge easily under impulse noise. The Figure 10 exhibits the comparison of PSD level. It is clear that the PSD of proposed algorithms have greater advantage to other algorithms. The PSDs of proposed algorithms fluctuate from 50 Hz to 100 Hz and then can reduce to −80 dB at 250 Hz. As the intensity of impulse noise increases, the PSDs of proposed algorithms can still maintain the same level or even decrease.

5. Conclusions

Based on Versoria function, a new FxRMVC algorithm is proposed and applied to impulse active noise control. Furthermore, the new method is complemented with the online adaptive radius strategy. Algorithms perform independently and are self−motivated without selecting prior radius. The proposed algorithms exhibit improved resistance to impulse noise in spite of involved computational complications. Moreover, stability of FxRMVC algorithm has also been analyzed theoretically. Numerical simulation results show that proposed algorithms have gratifying robustness in increasing convergence rate and enhancing noise reduction to impulse noise. The proposed algorithm outperforms other comparative algorithms in terms of noise reduction and tracking performance, with the ability to follow changes in the surrounding environment and track quickly and consistently. The algorithm can improve the tracking speed by up to 51.3% and can achieve a maximum noise reduction of more than 30 dB.