Active Impulsive Noise Control with Missing Input Data Based on FxImdMCC Algorithm

Abstract

In this study, we address the challenge of noise reduction in environments characterized by impulsive noise and missing input data in active noise control (ANC) systems, where existing algorithms often fail to deliver satisfactory results. Background noise, especially impulsive noise, poses a significant obstacle to signal processing and noise reduction. Furthermore, data loss during transmission or acquisition further complicates the noise reduction task. In this paper, a filtered-x imputation of the missing data maximum correntropy criterion (FxImdMCC) algorithm is proposed based on an imputation model, least mean square, and the maximum correntropy criterion (MCC), which can effectively reduce the impact of outliers and missing input data. The simulation results demonstrate the efficacy of the proposed FxImdMCC algorithm, which significantly outperforms existing algorithms in the context of active impulsive noise control.

1. Introduction

Active noise control (ANC) is an innovative noise reduction technology [1,2] designed to significantly reduce or eliminate disturbing noise by generating sound waves that are opposite in phase and equal in amplitude to external ambient noise [3]. The core of this technology is that it can analyze the frequency and amplitude of noise in real time, quickly generate the corresponding reverse sound wave and realize the noise cancellation through the principle of sound wave superposition [4]. Compared with passive noise control (PNC), the ANC method has better noise reduction performance at low frequencies [5].

Due to its low computational complexity and ease of implementation, the filtered-x least-mean-square (FxLMS) algorithm is the most widely used adaptive algorithm in ANC techniques [6]. The FxLMS algorithm assumes that the error has a Gaussian distribution and finds the optimal weight by minimizing the mean square error (MSE); however, in fact, the real-world noise always contains outliers (such as impulsive noise) [7,8]. When faced with the impulsive noise, the MSE-based FxLMS algorithms tend to have poor performance and do not converge stably [9]. Furthermore, the filtered-x logarithmic least-mean-square (FxlogLMS) algorithm [10] was proposed by minimizing the logarithmic transformation of the error signal. Simulation results demonstrate the effectiveness of the filtered-x-least mean p-power (FxLMP) algorithm and FxlogLMS algorithm in the impulsive-noise environment. In order to accelerate the convergence of the algorithm, the filtered-x recursive-least-square (FxRLS) algorithm [11,12] and the filtered-x recursive-least-p-power (FxRLP) algorithm [13] were proposed, both of which may have stability problems while accelerating convergence at the same time.

In addition to the methods mentioned above, the adaptive filtering method combined with the maximum entropy criterion is also studied. The information entropy (IE) method could include all the possible higher-order information of a random variable under the condition of non-Gaussian distribution [14] so the IE method could extract possible latent information from the signal. Due to this characteristic, the maximum correntropy criterion (MCC) of IE has been investigated, such as the filtered-x recursive maximum correntropy (FxRMC) algorithm [15]. By using the MCC method for impulsive noise, the FxRMC algorithm outperforms the FxLMS algorithm in terms of the averaged noise reduction (ANR), which is a measure of the performance of active noise reduction technology indicating the average reduction in noise level over a period of time. Furthermore, a filtered-x generalized maximum correntropy criterion (FxGMCC) algorithm is proposed [16], which has exceptional robustness to non-Gaussian environments. Based on the Bessel function of the first kind to constrain outliers, the filtered-s Bessel CG (FsBCG)-II algorithm is proposed [17], which can handle a noise source when it is impulsive, Gaussian, logarithmic, and time-varying. In these works and other similar references on the subject [18,19,20,21], many active noise control algorithms are proposed.

However, all ANC algorithms designed for reducing impulsive noise have poor performance when the data used for adaptive algorithms are absent at random instants. In the system identification area, the imputation-based missing data LMS (ImdLMS) algorithm considers the impact of missing data for adaptive filtering [22]; however, the ImdLMS algorithm cannot directly deal with abnormal data, e.g., impulsive noise.

In this work, a new algorithm based on the MCC of the IE method considering the missing data is proposed. In order to reduce the impact of outliers and missing input data on the effectiveness of active noise control, we combine the maximum correntropy criterion and imputation model correspondingly. Therefore, a new approach named the filtered-x imputation of missing data maximum correntropy criterion (FxImdMCC) algorithm is proposed, which achieves more stable performance in the case of outliers and missing input data than existing algorithms.

The main contributions of this paper are summarized as follows:

• By using an imputation model, we reset the data at the missing data moments to a constant multiple of the data available at the previous moment. In other words, if the input data are 0 at a time moment n, it is replaced by a constant multiple of the previous moment’s non-zero input data;
• We use the imputation model to construct an unbiased estimation of the true stochastic gradient that combines the FxLMS algorithm and MCC, thus deriving a new algorithm that still performs outstandingly in the presence of outliers and random missing values in the input noise, which is validated by simulation experiments.

The remainder of this paper is organized as follows. Section 2 presents the derivation of the FxLMS algorithm and the filtered-x maximum correntropy criterion (FxMCC) algorithm. Section 3 describes the missing data interpolation model and presents the derivation of the proposed FxImdMCC algorithm. Simulations and experimental studies are performed to evaluate the performance of proposed algorithms in Section 4. Finally, Section 5 presents the discussions and conclusions of this work.

2. Problem Statement

2.1. FxLMS Algorithm

The block diagram of the basic single-channel feed-forward ANC system is illustrated in Figure 1, where P(z) represents the primary path transfer function between the reference signal x(n) and the primary noise d(n), S(z) denotes the secondary path transfer function between the output of the adaptive filter $\mathsf{\theta }$(z) and the output of the error microphone e(n), and y(n) is the output of the secondary path. The transfer function $\widehat{S}(z)$ can be estimated by an offline or online adaptive filter [23,24]. The digital filter’s coefficients are adjusted in real time by an adaptive algorithm, which uses the error signal e(n) to update the weights of the filter. The active loudspeaker emits the anti-noise signal, which is intended to destructively interfere with the original noise signal x(n), thus reducing the overall noise level at the error microphone’s location. The output of the error microphone e(n) is calculated by the following:

$$e(n)=d(n)-y(n)=d(n)-\mathrm{s}(n)*[{\mathsf{\theta }}^T(n)\mathit{x}(n)]$$

(1)

where $\mathit{x}\left(n\right)={\left[x\left(n\right),x\left(n-1\right),\dots ,x\left(n-L+1\right)\right]}^T$, s(n) is the impulse response of S(z), $*$ represents the discrete convolution operator, and superscript T denotes transposition.

By minimizing the mean square error signal ${\left|e(n)\right|}^2$, the FxLMS algorithm achieves the update process, and the update process is denoted as follows:

$$\mathsf{\theta }(n+1)=\mathsf{\theta }(n)+\mu e(n){\mathit{x}}^{\prime }(n)$$

(2)

where

$${\mathit{x}}^{\prime }(n)=s(n)*\mathit{x}(n)$$

(3)

2.2. FxMCC Algorithm

Based on MCC, the FxMCC algorithm is proposed [25]. The correntropy is a partial measure of the similarity between two random variables $\mathcal{U}$ and $\mathcal{Z}$ [26], which is defined as follows:

$$V\left(\mathcal{U},\mathcal{Z}\right)=\mathbb{E}\left[\varrho (u,z)\right]={\displaystyle \int \varrho }(u,z)d{\gamma }_{\mathcal{U},\mathcal{Z}}(u,z)$$

(4)

where $\mathbb{E}$[·] is the expectation operator, $\varrho (u,z)$ is the shift-invariant Mercer kernel, and ${\gamma }_{\mathcal{U},\mathcal{Z}}$ is the joint distribution function of $(u,z)$. Here, the kernel function is chosen by the most commonly used Gaussian kernel $\varrho (u,z)=\exp (-{\displaystyle \frac{|u-z{|}^2}{2{\sigma }^2}})$, where $\sigma$ is the kernel width. Maximizing the correntropy (i.e., maximizing the correlation between the desired signal and the adaptive filter output), then, the cost function that the FxMCC algorithm used is derived, as outlined below:

$$\begin{array}{ll}{J}_{\mathrm{FxMCC}}(n)& =\varrho \left(d\left(n\right),y\left(n\right)\right)\\ & =\exp \left\{-{\displaystyle \frac{e^2(n)}{2{\sigma }^2}}\right\}\end{array}$$

(5)

Taking the gradient $J_{FxMMC}(n)$ with respect to weight vector $\theta (n)$ yields

$${\displaystyle \frac{\partial J_{\mathrm{FxMMC}}(n)}{\partial \mathsf{\theta }(n)}}=\exp \left\{-{\displaystyle \frac{e^2(n)}{2{\sigma }^2}}\right\}{\displaystyle \frac{e(n){\mathit{x}}^{\prime }(n)}{2{\sigma }^2}}$$

(6)

and using the gradient descent method to update the estimated parameter, then,

$$\begin{array}{ll}\mathsf{\theta }(n+1)& =\mathsf{\theta }(n)+\eta {\displaystyle \frac{\partial J_{\mathrm{FxMCC}}(n)}{\partial \mathsf{\theta }(n)}}\\ & =\mathsf{\theta }(n)+\eta \exp \left\{-{\displaystyle \frac{e^2(n)}{2{\sigma }^2}}\right\}{\displaystyle \frac{e(n){\mathit{x}}^{\prime }(n)}{2{\sigma }^2}}\end{array}$$

(7)

To make the expression more concise, one can let

$$\mu ={\displaystyle \frac{\eta }{2{\sigma }^2}}$$

(8)

The weight update rule given by Equation (8) is the FxMCC algorithm. Previous studies [27] have shown that in order to ensure the convergence of the algorithm, the range of the learning rate should be satisfied

$$0<\mu <{\displaystyle \frac{2\mathbb{E}\left[\exp (-\frac{e^2(n)}{2{\sigma }^2})e^2(n)\right]}{\mathbb{E}\left[\exp (-\frac{e^2(n)}{{\sigma }^2})e^2(n){‖\mathit{x}\prime ‖}^2\right]}}$$

(9)

where ${‖\cdot ‖}^2$ denotes the square of the L2 norm.

In order to solve the problem of frequent abnormal input data such as impulsive noise and random loss due to some problems, e.g., sensor limitations and communication, the FxImdMCC algorithm is proposed in this paper. The specific derivation process of the proposed FxImdMCC algorithm is shown in Section 3.

3. Proposed FxImdMCC Algorithm

3.1. Imputation Model

Considering that there is a random loss of noise during the data transfer process, we assume that the data acquired by the filter at moment n as follows:

$$x_f=f(n)x(n)$$

(10)

where $f(n)$ is a sequence of Bernoulli random variables independent of $x(n)$. The probability that the value of each $f(n)$ takes 1 is p and takes 0 is 1 − p. When the input data are correlated, in order for the sequence of filter weight coefficients $\theta (n)$ to achieve mean convergence and thus ensure better active noise control, we need to make better use of the data available at previous moments to construct a new data sequence.

Therefore, an imputed data sequence is created through the imputation model, where as long as the data acquired at any time moment are 0, it is reset to a constant multiple of the data available at the previous moment.

Since the noisy data sequence $\left\{x(n)\right\}$ almost satisfies that $x(n)\ne 0$, it can be inferred from Equation (10) that $\mathbb{P}(f(n)=0)+\mathbb{P}(f(n)=1)=1$. Consequently, the imputed input data sequence can be described as outlined below:

$$\overline{x}(n)=f(n)x(n)+\gamma (1-f(n))\overline{x}(n-1)$$

(11)

where $\gamma$ is a constant real number. By setting its specific value, different imputed variables can be obtained, which allow the adaptive filter to update an unbiased sequence of weight coefficients, $\mathsf{\theta }(n)$, thus achieving better active noise control effects.

3.2. Derivation of the FxImdMCC Algorithm

Considering the case of dealing with both impulsive noise and random missing values, the FxImdMCC algorithm is proposed, which adopts the following objective function:

$$J_{FxImdMCC}(n)=\mathbb{E}[\exp (-{\displaystyle \frac{e^2(n)}{2{\sigma }^2}})]$$

(12)

Taking the gradient of $J_{FxImdMCC}(n)$ gives the following:

$$\nabla J_{FxImdMCC}(n)={\displaystyle \frac{\partial J_{FxImdMCC}(n)}{\partial \mathsf{\theta }(n)}}=-\exp (-{\displaystyle \frac{e^2(n)}{2{\sigma }^2}}){\displaystyle \frac{e(n)x^{\prime }(n)}{2{\sigma }^2}}$$

(13)

In order for the iteration to produce an unbiased update sequence when the input noisy data are not fully available, we need to find an unbiased estimation of the true gradient using the imputed data sequence $\left\{\overline{x}(n)\right\}$. In other words, the coefficient vector $\theta (n)$ of the adaptive filter is updated based on the following form:

$$\mathsf{\theta }(n+1)=\mathsf{\theta }(n)-\mu P_u(n)$$

(14)

where $P_u(n)$ is an unbiased estimation of the true gradient $\nabla J_{FxImdMCC}(n)$.

In deriving the specific update expression with respect to Equation (14), we assume that the input data sequence is missing at random moments, without assuming its probability distribution, in order to ensure that the update formula is more widely applicable. According to Equation (11), it can be seen that the imputed data sequence has the following form:

$$\overline{\mathit{x}}(n)=F(n)\mathit{x}(n)+\gamma (I-F(n))\overline{\mathit{x}}(n-1)$$

(15)

where $F(n)$ is a diagonal matrix, which can be described as $F(n)=diag(f(n),f(n-1),\dots ,f(n-L+1))$. Therefore, similarly, $F(n)$ is statistically independent of $\overline{\mathit{x}}(n)$. In addition, we assume that $F(n)$ is also statistically independent of $\mathsf{\theta }(n)$.

Thus, defining ${\overline{\mathit{x}}}^{\prime }(n)=s(n)\ast \overline{\mathit{x}}(n)$, we can obtain that $\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n))=p{\mathit{x}}^{\prime }(n)$$+\gamma (1-p)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))$. Now,

$$\begin{array}{l}\mathbb{E}(pd(n)-{\mathsf{\theta }}^T(n){\overline{\mathit{x}}}^{\prime }(n))\\ =pd(n)-{\mathsf{\theta }}^T(n)[p{\mathit{x}}^{\prime }(n)+\gamma (1-p)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))]\\ =pe(n)-\gamma (1-p){\mathsf{\theta }}^T(n)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))\end{array}$$

(16)

To make the results of the following expressions more concise, we define that $\widehat{\mathit{x}}(n)={\overline{\mathit{x}}}^{\prime }(n)-\gamma (1-p){\overline{\mathit{x}}}^{\prime }(n-1)$, from which the following can be obtained:

$$\begin{array}{ll}e(n)& ={\displaystyle \frac{1}{p}}[\mathbb{E}(pd(n)-{\mathsf{\theta }}^T(n){\overline{\mathit{x}}}^{\prime }(n))+\gamma (1-p){\mathsf{\theta }}^T(n)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))]\\ & ={\displaystyle \frac{1}{p}}[\mathbb{E}(pd(n)-{\widehat{\mathit{x}}}^T(n)\mathsf{\theta }(n)-\gamma (1-p){{\overline{\mathit{x}}}^{\prime }}^T(n-1)\mathsf{\theta }(n))+\gamma (1-p){\mathsf{\theta }}^T(n)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))]\\ & ={\displaystyle \frac{1}{p}}\mathbb{E}[pd(n)-{\widehat{\mathit{x}}}^T(n)\mathsf{\theta }(n)]\end{array}$$

(17)

Let $\widehat{e}(n)=pd(n)-{\widehat{\mathit{x}}}^T(n)\mathsf{\theta }(n)$, from which we obtain $e(n)={\displaystyle \frac{1}{p}}\mathbb{E}[\widehat{e}(n)]$. Defining ${\overline{R}}_{m,n}=\mathbb{E}[{\overline{\mathit{x}}}^{\prime }(m){{\overline{\mathit{x}}}^{\prime }}^T(n)]$ and ${R^{\prime }}_{m,n}={\mathit{x}}^{\prime }(m){{\mathit{x}}^{\prime }}^T(n)$, one obtains the calculations below:

$$\begin{array}{l}\mathbb{E}[{\overline{\mathit{x}}}^{\prime }(n)(pd(n)-{\mathsf{\theta }}^T(n){\overline{\mathit{x}}}^{\prime }(n))]\\ =pd(n)(p{\mathit{x}}^{\prime }(n)+\gamma (1-p)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1)))-\mathsf{\theta }(n){\overline{R}}_{n,n}\\ =p^{2}d(n){\mathit{x}}^{\prime }(n)+\gamma p(1-p)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))-\mathsf{\theta }(n){\overline{R}}_{n,n}\end{array}$$

(18)

We define $A_n={\overline{\mathit{x}}}^{\prime }(n-1){{\mathit{x}}^{\prime }}^T(n)$ and $B_n={\mathit{x}}^{\prime }(n){{\overline{\mathit{x}}}^{\prime }}^T(n-1)$. Using the fact that

$$\mathbb{E}[F(n){\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)F(n)]=\left\{\begin{array}{l}p{[{\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)]}_{j,k},j\ne k\\ p^2{[{\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)]}_{j,k},j=k\end{array}\right.$$

(19)

where ${[{\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)]}_{j,k}$ represents the ${(j,k)}^{th}$ element of ${\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)$, one obtains

$$\begin{array}{ll}{\overline{R}}_{n,n}& =\mathbb{E}[{\overline{\mathit{x}}}^{\prime }(n){{\overline{\mathit{x}}}^{\prime }}^T(n)]\\ & =\mathbb{E}[(F(n){\mathit{x}}^{\prime }(n)+\gamma (I-F(n)){\overline{\mathit{x}}}^{\prime }(n-1))\cdot {(F(n){\mathit{x}}^{\prime }(n)+\gamma (I-F(n)){\overline{\mathit{x}}}^{\prime }(n-1))}^T]\\ & =\mathbb{E}[F(n){\mathit{x}}^{\prime }(n){{\mathit{x}}^{\prime }}^T(n)F(n)+\gamma F(n){\mathit{x}}^{\prime }(n){{\overline{\mathit{x}}}^{\prime }}^T(n-1)(I-F(n))+\gamma (I-F(n))\cdot \\ & \hspace{1em}{\overline{\mathit{x}}}^{\prime }(n-1){{\mathit{x}}^{\prime }}^T(n)F(n)+{\gamma }^2(I-F(n)){\overline{\mathit{x}}}^{\prime }(n-1){{\overline{\mathit{x}}}^{\prime }}^T(n-1)(I-F(n))\\ & =p^2{R^{\prime }}_{n,n}+p(1-p)\mathrm{diag}({R^{\prime }}_{n,n})+\gamma p(1-p)\mathbb{E}(A_n+B_n-\mathrm{diag}(A_n+B_n))\\ & \hspace{1em}+{\gamma }^2{(1-p)}^2{\overline{R}}_{n-1,n-1}+{\gamma }^{2}p(1-p)\mathrm{diag}({\overline{R}}_{n-1,n-1})\end{array}$$

(20)

Due to

$${\overline{R}}_{n,n-1}=p\mathbb{E}(B_n)+\gamma (1-p){\overline{R}}_{n-1,n-1}$$

(21)

it can be seen that

$$\mathbb{E}(B_n)={\displaystyle \frac{1}{p}}({\overline{R}}_{n,n-1}-\gamma (1-p){\overline{R}}_{n-1,n-1})$$

(22)

The same reasoning can be used to obtain

$$\mathbb{E}(A_n)={\displaystyle \frac{1}{p}}({\overline{R}}_{n-1,n}-\gamma (1-p){\overline{R}}_{n-1,n-1})$$

(23)

Then, one can obtain from Equations (22) and (23) that

$$\mathbb{E}(A_n+B_n)={\displaystyle \frac{1}{p}}({\overline{R}}_{n-1,n}+{\overline{R}}_{n,n-1}-2\gamma (1-p){\overline{R}}_{n-1,n-1})$$

(24)

therefore, Equation (20) can be rewritten as follows

$$\begin{array}{ll}{\overline{R}}_{n,n}=& p^2{R^{\prime }}_{n,n}+p(1-p)\mathrm{diag}({R^{\prime }}_{n,n})-{\gamma }^2{(1-p)}^2{\overline{R}}_{n-1,n-1}\\ & +{\gamma }^2(1-p)(2-p)\mathrm{diag}({\overline{R}}_{n-1,n-1})+\gamma p(1-p)({\overline{R}}_{n-1,n}+{\overline{R}}_{n,n-1}-diag({\overline{R}}_{n-1,n}+{\overline{R}}_{n,n-1}))\end{array}$$

(25)

Taking diagonal matrices of Equation (25), it follows that

$$\mathrm{diag}({\overline{R}}_{n,n})=p\mathrm{diag}({R^{\prime }}_{n,n})+{\gamma }^2(1-p)\mathrm{diag}({\overline{R}}_{n-1,n-1})$$

(26)

Noting that $-e(n){\mathit{x}}^{\prime }(n)=-(d(n){\mathit{x}}^{\prime }(n)-{R^{\prime }}_{n,n}\mathsf{\theta }(n))$, Equation (18) can be rewritten as outlined below:

$$\begin{array}{l}\mathbb{E}[{\overline{\mathit{x}}}^{\prime }(n)(pd(n)-{\mathsf{\theta }}^T(n){\overline{\mathit{x}}}^{\prime }(n))]\\ =p^{2}e(n){\mathit{x}}^{\prime }(n)+\gamma p(1-p)d(n)\mathbb{E}({\overline{\mathit{x}}}^{\prime }(n-1))\\ \hspace{1em}-(1-p)\mathrm{diag}({\overline{R}}_{n,n}+{\gamma }^2{\overline{R}}_{n-1,n-1}-\gamma ({\overline{R}}_{n-1,n}+{\overline{R}}_{n,n-1}))\mathsf{\theta }(n)\\ \hspace{1em}-\gamma (1-p)({\overline{R}}_{n-1,n}+{\overline{R}}_{n,n-1}-\gamma (1-p){\overline{R}}_{n-1,n-1})\mathsf{\theta }(n)\end{array}$$

(27)

Defining ${\mathit{x}}_s(n)={\overline{\mathit{x}}}^{\prime }(n)-\gamma {\overline{\mathit{x}}}^{\prime }(n-1)$, one obtains

$$-e(n){\mathit{x}}^{\prime }(n)=-{\displaystyle \frac{1}{p^2}}\mathbb{E}[\widehat{e}(n)\widehat{\mathit{x}}(n)]-{\displaystyle \frac{1-p}{p^2}}\mathbb{E}[\mathrm{diag}({\mathit{x}}_s(n){\mathit{x}}_s^T(n))]\mathsf{\theta }(n)$$

(28)

Thus,

$$\nabla J_{F\mathit{x}ImdMCC}(n)=-\exp (-{\displaystyle \frac{\mathbb{E}[{\widehat{e}}^2(n)]}{2p^2{\sigma }^2}}){\displaystyle \frac{\mathbb{E}[\widehat{e}(n)\widehat{\mathit{x}}(n)]+(1-p)\mathbb{E}[\mathrm{diag}({\mathit{x}}_s(n){\mathit{x}}_s^T(n))]\mathsf{\theta }(n)}{2p^2{\sigma }^2}}$$

(29)

which means we propose the unbiased estimation of the true gradient. Moreover, letting $H(n)=\mathrm{diag}({\mathit{x}}_s(n){\mathit{x}}_s^T(n))$ and combining Equation (14), $\mathsf{\theta }(n)$ can be updated by the following:

$$\mathsf{\theta }(n+1)=\mathsf{\theta }(n)+\mu \exp (-{\displaystyle \frac{{\widehat{e}}^2(n)}{2p^2{\sigma }^2}}){\displaystyle \frac{\widehat{e}(n)\widehat{\mathit{x}}(n)+(1-p)H(n)\mathsf{\theta }(n)}{p^2}}$$

(30)

It is worth noting that factors such as the probability $p$ of the input data being correctly acquired, the kernel size $\sigma$ in the MCC, the iteration step size $\mu$, and the impulsive noise all affect the performance of the proposed FxImdMCC algorithm, so we will discuss each of these issues in the next section, respectively (Algorithm 1).

Algorithm 1 Summary of the proposed FxImdMCC algorithm.

Initialization and parameter selection length of the primary path P(z), length of the secondary path S(z) FxImdMCC algorithm: L, μ, p, γ, ${\sigma }^2$$,\mathsf{\theta }(0)=0$$,\left\{\overline{x}(n)\right\}$

$\mathbf{while}\left\{x_f(n),e(n)\right\}$ available do $\hspace{1em}1.e(n)=d(n)-\mathrm{s}(n)*[{\mathsf{\theta }}^T(n)\mathit{x}(n)]$ $\hspace{1em}2.{\overline{\mathit{x}}}^{\prime }(n)=s(n)\ast \overline{\mathit{x}}(n)$ $\hspace{1em}3.\widehat{\mathit{x}}(n)={\overline{\mathit{x}}}^{\prime }(n)-\gamma (1-p){\overline{\mathit{x}}}^{\prime }(n-1)$ $\hspace{1em}4.{\mathit{x}}_s(n)={\overline{\mathit{x}}}^{\prime }(n)-\gamma {\overline{\mathit{x}}}^{\prime }(n-1)$ $\hspace{1em}5.H(n)=\mathrm{diag}({\mathit{x}}_s(n){\mathit{x}}_s^T(n))$ $\hspace{1em}6.\widehat{e}(n)=pd(n)-{\widehat{\mathit{x}}}^T(n)\mathsf{\theta }(n)$ $\hspace{1em}7.\mathsf{\theta }(n+1)=\mathsf{\theta }(n)+\mu \exp (-{\displaystyle \frac{{\widehat{e}}^2(n)}{2p^2{\sigma }^2}}){\displaystyle \frac{\widehat{e}(n)\widehat{\mathit{x}}(n)+(1-p)H(n)\mathsf{\theta }(n)}{p^2}}$ end while

4. Simulation Results

Simulations are carried out to examine the performance of the proposed algorithms. This section comprises three parts. For the first part, the basic parameters of the simulation and the model of impulsive noise are introduced. For the second part, the ANR performance of the proposed FxImdMCC algorithm is shown. For the last part, the ANR of the proposed algorithm with varying step size and noise parameter α is compared with the previous algorithms. All the simulations are averaged over 100 independent trials.

4.1. Parameters of the Simulation Systems

The sample frequency of the ANC systems is 8000 Hz. The primary path transfer function P(z) and the secondary path transfer function S(z) are modeled as finite impulse response (FIR) filters with lengths of 256 and 100, respectively. In the simulation, the estimated secondary path transfer function $\widehat{S}(z)$ was exactly identified as S(z). The magnitude and the phase responses of the primary and secondary paths are shown in Figure 2. The adaptive filter $\theta \left(z\right)$ was taken as an FIR filter of order 128. To compare the performance of different algorithms, the ANR is used, as outlined below [28]:

$$\mathrm{ANR}(n)=20\log ({\displaystyle \frac{A_e(n)}{A_d(n)}})$$

(31)

where $A_e(n)=\xi A_e(n-1)+(1-\xi )|e(n)|, A_d(n)=\xi A_d(n-1)+(1-\xi )|d(n)|$, $A_e(0)=A_d(0)=0$, and $\xi =0.999$.

The impulsive noise used in this article is modeled as a symmetric α-stable (SαS) process [29] having the characteristic function of the form

$$\phi (t)=\exp \{-|t{|}^{\alpha }\}$$

(32)

where $0<\alpha <2$ is a characteristic exponent, which determines the degree of impulse characterization of the SαS distribution. The SαS distribution is characterized by significant pulse spikes. A small α implies a peaky and heavy-tailed distribution with more outliers. If $\alpha =1$, the SαS process is a Cauchy distribution; and if $\alpha =2$, the abovementioned process becomes a Gaussian distribution. The probability density function (PDF) of the SαS process for different α values is shown in Figure 3.

4.2. Simulation Results of the Proposed FxImdMCC Algorithm

The proposed algorithm’s performance depends on the kernel width and the step size. When the kernel width and the step change, the performance of the proposed algorithm will also vary. In order to find out the effect of changing the kernel width and step size on algorithm performance, the simulation experiments are performed, which are carried out in the absence of impulsive noise with 10% of the data missing.

As shown in Figure 4, the ANR of the proposed algorithm is best when ${\sigma }^2=8$, followed by ${\sigma }^2=64$, ${\sigma }^2=1$, and ${\sigma }^2=512$ (the worst). This shows that there is a minimum value for the influence of kernel width on the ANR algorithm, and the minimum value of the kernel width is moderate (not too large and not too small), which is consistent with previous studies [30,31].

Like other adaptive filtering algorithms, the ANR of the FxImdMCC algorithm is also affected by the step size of the algorithm. As the step size increases, the ANR of the FxImdMCC algorithm becomes worse, as illustrated in Figure 5. For the sake of the convergence of the proposed algorithm, the step size needs to satisfy the relationship with the eigenvalue of the input data matrix like the FxMCC algorithm.

Simulation results show that the FxImdMCC algorithm can achieve active noise control under impulsive noise and its effect is related to the kernel width of the Gaussian kernel and influenced by step size. In order to verify the effectiveness of the algorithm, the comparison experiment with other algorithms needs to be further carried out.

4.3. Comparison with Other Algorithms

In the comparison with different algorithms, two kinds of noise situations are considered.

Firstly, the ANR of the proposed algorithm and other previous ANC algorithms under varying impulsive noise environments (i.e., α varies) with the occurrence of data missing are compared, as illustrated in Figure 6. When α varies, the probability of data not being lost is fixed at 0.8. From Figure 6, it is evident that the proposed algorithm (the FxImdMCC algorithm) outperforms the previous algorithms (followed by the FxMCC algorithm, the FxLMS algorithm, and the filtered-x imputation-based missing data LMS (FxImdLMS) algorithm) under the impulsive noise environment with data missing in terms of the ANR. The ANR of the FxMCC algorithm and the FxLMS algorithm are close to each other when the number of iterations is small. As the number of iterations increases, the ANR of the FxMCC algorithm is better than that of the FxLMS algorithm. Due to its increasingly poor ANR performance, the FxImdLMS algorithm, which could not deal with impulsive noise, does not converge in fact. It can also be observed that the poor ANR performance of the FxMCC and FxImdMCC algorithm does not change with the variations in α, i.e., the change in the impulsive noise.

Secondly, the ANR of the proposed algorithm and other previous ANC algorithms is compared when the probability of data missing varies while α is fixed at 1.55. As illustrated in Figure 7, it is evident that the proposed algorithm (the FxMCC algorithm) performs better than the other algorithms. It is observed that the ANR of the FxImdLMS algorithm becomes better when the probability of the data missing increases, while it still does not converge at any data missing probability. That is because the noise used in this experiment is impulsive noise that the FxImdLMS algorithm could not deal with. It is also shown that the ANR of the FxMCC algorithm and the FxImdMCC algorithm is almost the same under the different missing probabilities of impulsive noise conditions.

Finally, we compare the computation time of the proposed FxImdMCC algorithm with that of the existing algorithms, which is shown in

Table 1. Computation time of algorithms.

Algorithms	Computation Time (Seconds)
FxLMS	0.3799
FxMCC	0.5214
FxImdLMS	2.4157
FxImdMCC	2.7283

. The computation time is measured using MATLAB R2022b on a 2.40 GHz Intel Pentium processor with 16 GB of RAM. In this measurement, $1\times {10}^5$ samples are considered and the number of independent runs is 1. Due to the increase in computational complexity, the proposed algorithm takes more time for computing and simulation experiments than the other three algorithms used for comparison.

5. Conclusions

In order to solve the problem that within the mean square error criterion-based adaptive filtering algorithm in active noise control it is difficult to eliminate the impulsive noise frequently encountered in practical applications, along with the available data loss during the adaptive weight process, the FxImdMCC algorithm is proposed. Specifically, based on the MCC of IE and an imputation model, we interpolate the missing data and then construct an unbiased estimation of the true stochastic gradient that combines the FxLMS algorithm and MCC method. The simulation results and the comparison experiments prove the feasibility and effectiveness of the proposed algorithm, which remains stable under impulse noise and has better noise cancellation performance compared to other algorithms. Thus, the proposed algorithm has a positive impact on the practical applications of ANC.