A Robust Filtered-x Least Mean Square Algorithm with Adjustable Parameters for Active Impulsive Noise Control

Abstract

In active noise control (ANC) systems, the traditional filtered-x least mean square (FxLMS) algorithm has poor control effect on impulsive noise. To overcome this drawback, a robust cost function was designed in this paper by embedding the cost function of the FxLMS algorithm into the framework of hyperbolic tangent function; this paper thus proposes a robust filtered-x least hyperbolic tangent (FxLHT) algorithm in ANC systems. Moreover, the value of λ in the FxLHT algorithm greatly affects the robustness and convergence performance of the algorithm. Therefore, a variable λ-parameter was proposed to enhance the performance of the FxLHT algorithm. Simulation results show that in the active control of impulsive noise, compared with the FxLMS algorithm and other robust ANC algorithms, the proposed FxLHT algorithm and variable λ-parameter FxLHT algorithm not only exhibit good robustness and noise reduction performance but also have a better tracking ability.

1. Introduction

In recent years, due to the rapid development of the economy and the widespread use of various machinery and equipment, noise has become an increasingly serious problem in our daily life and work. Active noise control (ANC) has developed into a key technique for low-frequency noise control, which generates anti-noise through the ANC controller to achieve noise cancelation [1], and its working principle is shown in Figure 1.

Due to the least mean square (LMS) algorithm using mean square error (MSE) as the cost function and due to its advantages of simple calculation and implementation, it has been widely applied in Gaussian noise environments. However, the convergence performance of the LMS algorithm is severely degraded under the interference of impulsive/non-Gaussian noise environments [2,3]. To overcome this drawback, researchers have designed some robust ANC algorithms [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In 2009, an improved filtered-x LMS (FxLMS) algorithm [5] was proposed by setting threshold parameters for the input and error signals. In 2011, a filtered-x logarithmic error LMS (FxlogLMS) algorithm [7] was designed by using the squared logarithm of the absolute value of the error signal as the cost function. In 2012, a robust filtered-s LMS (RFsLMS) algorithm [8] was applied to the active control of impulsive noise in nonlinear ANC systems, using an adaptive controller with functional link artificial neural networks. In 2015, Sun et al. proposed a robust filtered-x least mean M-estimation (FxLMM) algorithm [10] using the M-estimation function, which requires online estimation of the threshold parameter for impulsive noise cancelation. Recently, a variable tap-length FxLMM algorithm based on sub-band adaptive filtering technology [11] has been designed. In [13], a modified FxLMS (MFxLMS) algorithm was designed by employing the technique of data reuse, which provides better robustness and faster convergence compared to the FxLMS algorithm with normalized step size [6]. In 2017, a robust filtered-s MCC (FsMCC) algorithm [14] was proposed for nonlinear ANC systems, due to the better robustness of maximum correntropy criterion (MCC) for dealing with non-Gaussian noise. Two robust filtered-x recursive least square (RFxRLS) algorithms [15] have been presented for ANC systems with impulsive noise interference. In 2018, a robust filtered-x generalized mixed norm (FxGMN) algorithm [16] was proposed by using a convex mixing of l_p-norm and l_q-norm as the cost function for error signals, and the convex combination method was used to further improve the performance of the system. In 2020, a filtered-x generalized MCC (FxGMCC) algorithm [17] was designed for impulse noise control in ANC systems. Moreover, a robust filtered-x Champernowne adaptive filter (FxCMAF) [18] was designed for an adaptive room equalizer, which showed lower MSE compared to the FxGMCC algorithm. A filtered-x least cosine hyperbolic (FxLCH) algorithm was designed in [19] by minimizing the logarithmic hyperbolic cosine cost function. By setting thresholds for reference and error signals, the stability performance of the FxLCH algorithm can be further improved. An improved filtered-x arctangent LMS (FxatanLMS) algorithm was proposed in [20] by minimizing the arctangent square of the error signal. Henceforth, the performance of the FxatanLMS algorithm is further improved by using an improved normalized step-size method, and it shows better control performance compared to the improved FxLMS algorithm [5] and the FxLMM algorithm [10]. The filtered-x least mean absolute third (FxLMAT) algorithm and its improved algorithms [22] have been developed in impulsive noise ANC systems. Compared with the improved FxLCH algorithm [19], they show better noise reduction performance. In 2023, a robust filtered-x tanh LMS (FxtanhLMS) algorithm [23] has been presented, which minimizes the squared hyperbolic tangent of the error signal. It shows better performance compared to the FxlogLMS algorithm. Recently, a robust Swish filtered-x sign algorithm (Swish-FxSA) has been proposed for diffusion ANC systems based on a Swish function framework [24].

In this paper, we define a robust cost function by embedding the cost function of the traditional FxLMS algorithm into the hyperbolic tangent framework; hence, a filtered-x least hyperbolic tangent (FxLHT) algorithm is proposed to eliminate the interference of impulsive noise in ANC systems. In addition, in order to further improve the convergence performance of the algorithm, an FxLHT algorithm with variable λ-parameter is designed. Simulation tests show that the proposed algorithms not only have good robustness and noise reduction performance but also have good tracking performance.

The rest of the paper has the following structure. In Section 2, we introduce the basic FxLMS algorithm. In Section 3, we present FxLHT and VFxLHT algorithms. Section 4 analyses the range of convergence of the step-size and the computational complexity of the algorithm. The noise reduction performance of the proposed algorithms is verified in Section 5, and the conclusion is presented in Section 6.

2. Traditional FxLMS Algorithm

Figure 2 provides a block diagram of a single-channel feed-forward ANC system, where $u(i)$ is the input signal collected by a reference sensor near the noise source, $P(z)$ represents the primary channel, and $d(i)$ is the noise to be eliminated. $H(z)$ denotes the adaptive filter, $y(i)$ represents the output signal of the ANC controller, $S(z)$ is the secondary channel, and the residual noise $e(i)$ can be observed by the error sensor. $\widehat{S}(z)$ is the estimated value of the secondary channel [26].

The output signal $y(i)$ of the filter is

$$y(i)=h^T(i)u(i)$$

(1)

where $h(i)={\left[h_1(i),h_2(i),\dots ,h_M(i)\right]}^T$ represents the weight vector of the ANC controller, and $u(i)={\left[u(i),u(i-1),\dots ,u(i-M+1)\right]}^T$ denotes the input vector.

The residual error $e(i)$ of the ANC system can be calculated as

$$e(i)=d(i)-y(i)\ast s(i)$$

(2)

where $d(i)=u(i)\ast p(i)$, $\ast$ is the discrete convolution operator.

The traditional cost function of the FxLMS algorithm is given by

$$J_{LMS}(i)=\mathbb{E}\left[e^2(i)\right]\approx e^2(i)$$

(3)

Therefore, its weight vector update formula is

$$\begin{array}{r}h(i+1)=h(i)-{\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\partial J_{LMS}(i)}{\partial h(i)}}\\ =h(i)+\eta e(i)u^{\prime }(i)\end{array}$$

(4)

where $\eta$ is the step-size, $u^{\prime }(i)=u(i)\ast \widehat{s}(i)$.

3. Proposed Algorithms

3.1. FxLHT Algorithm

When the amplitude of the reference input signal has a pulse characteristic, it will result in a very large error signal, and the stability of the FxLMS algorithm will be reduced. By utilizing the saturation property of the hyperbolic tangent function, a robust cost function is derived by embedding the cost function of the FxLMS algorithm into the hyperbolic tangent function framework.

The mathematical expression for the tanh function is

$$\tanh (x)={\displaystyle \frac{\exp (x)-\exp (-x)}{\exp (x)+\exp (-x)}}$$

(5)

where $\tanh \left(x\right)$ is the hyperbolic tangent function; its corresponding derivative is

$${\displaystyle \frac{d\left[\tanh (x)\right]}{dx}}=1-{\tanh }^2(x)$$

(6)

Embedding the cost function in Equation (3) into the tanh function framework, a robust cost function is obtained as follows:

$$J(i)\triangleq \mathbb{E}\left[{\displaystyle \frac{1}{\lambda }}\tanh \left(\lambda J_{\mathrm{L}\mathrm{M}\mathrm{S}}(i)\right)\right]\approx {\displaystyle \frac{1}{\lambda }}\tanh \left(\lambda J_{\mathrm{L}\mathrm{M}\mathrm{S}}(i)\right)$$

(7)

where $\lambda$ is a positive parameter, which is the key parameter to suppress the impulsive noise interference.

In Figure 3, when the error signal exceeds the threshold, the gradient of (7) is equal to zero, which improves the robustness of the design algorithm.

The derivative of the cost function with regard to the weight vector $h(i)$ defined in (7) based on the tanh function framework is

$$\begin{array}{ll}\hfill \nabla J(i)& ={\displaystyle \frac{\partial J(i)}{\partial h(i) }} ={\displaystyle \frac{\partial \tanh \left(\lambda J_{\mathrm{L}\mathrm{M}\mathrm{S}}(i)\right)}{\partial \lambda J_{\mathrm{L}\mathrm{M}\mathrm{S}}(i)}}\times {\displaystyle \frac{\partial J_{\mathrm{L}\mathrm{M}\mathrm{S}}(i)}{\partial h(i)}}\\ & =-2\left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)u^{\prime }(i)\end{array}$$

(8)

Hence, the update formula for the filtered-x least hyperbolic tangent (FxLHT) algorithm can be obtained as

$$\begin{array}{ll}\hfill h(i+1)& =h(i)-{\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\partial J(i)}{\partial h(i)}}\\ & =h(i)+\eta \left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)u^{\prime }(i)\end{array}$$

(9)

where the parameter $\lambda$ affects the FxLHT algorithm’s convergence rate and residual error.

As shown in Figure 4, when the residual error $e(i)$ is very large (impulsive noise interference) and the nonlinear error function $\left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)$ approaches 0, the FxLHT algorithm is not updated, thereby suppressing the interference of impulsive noise. When the residual error $e(i)$ is very small (without impulsive noise interference), the nonlinear error function $\left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)$ approaches $e(i)$, and the FxLHT algorithm becomes the FxLMS algorithm, performing normal algorithm updates.

3.2. VFxLHT Algorithm

If there is impulsive noise in the input signal captured by the reference sensor, we expect the parameter $\lambda$ to be taken as a larger value so that the algorithm has good robustness. If there is no impulsive noise interference in the input signal, the parameter $\lambda$ takes a smaller value, resulting in a faster convergence speed.

Using a gradient-based method, we propose the FxLHT algorithm with a variable λ-parameter, which is updated as follows:

$$g(i)=\kappa g(i-1)+(1-\kappa )\left[1-{\tanh }^2\left(\lambda (i)e^2(i)\right)\right]e(i)u^{\prime }(i)$$

(10)

$$\lambda (i)=\gamma \exp \left(-{\displaystyle \frac{1}{\beta ‖g(i)‖}}\right)$$

(11)

where $g(i)$ is the smoothed gradient vector, the parameter $\kappa$ is set to be very close to 1, $\gamma$ and $\beta$ are positive parameters.

In order to avoid the instability of the λ-parameter, a constraint on $\lambda (i)$ is used.

$$\lambda (i)=\left\{\begin{array}{l}{\lambda }_{\max },\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} \lambda (i)>{\lambda }_{\max }\\ {\lambda }_{\min },\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} \lambda (i)<{\lambda }_{\min }\\ \lambda (i),\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{otherwise}\end{array}\right.$$

(12)

where $0<{\lambda }_{\min }<{\lambda }_{\max }$, and the proposed VFxLHT algorithm is listed in Algorithm 1.

Algorithm 1: Proposed VFxLHT algorithm.

Initializations:    $h(0)=0$, $g(0)=0$, $\lambda (0)=0$

Parameters:      $\gamma$, $\kappa$, $\beta$, $\eta$, $0<{\lambda }_{\min }<{\lambda }_{\max }$

Adaptive process:      for i = 0, 1, 2, …        $d(i)=u(i)\ast p(i)$        $u(i)={\left[u(i),u(i-1),\dots ,u(i-M+1)\right]}^T$         $y(i)=h^T(i)u(i)$         $e(i)=d(i)-y(i)\ast s(i)$         $u^{\prime }(i)=u(i)\ast \widehat{s}(i)$         $g(i)=\kappa g(i-1)+(1-\kappa )\left[1-{\tanh }^2\left(\lambda (i)e^2(i)\right)\right]e(i)u^{\prime }(i)$         $\lambda (i)=\gamma \exp \left(-{\displaystyle \frac{1}{\beta ‖g(i)‖}}\right)$         $\lambda (i)=\left\{\begin{array}{l}{\lambda }_{\max },\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} \lambda (i)>{\lambda }_{\max }\\ {\lambda }_{\min },\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} \lambda (i)<{\lambda }_{\min }\\ \lambda (i),\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{otherwise}\end{array}\right.$          $h(i+1)=h(i)+\eta \left[1-{\tanh }^2\left(\lambda (i)e^2(i)\right)\right]e(i)u^{\prime }(i)$      end

4. Performance Analysis

4.1. Stability Analysis

In this section, we analyze the convergence range of the step-size of the FxLHT algorithm. According to (9), rewrite the equation of the FxLHT algorithm as

$$h(i+1)=h(i)+\eta \phi \left[e(i)\right]u^{\prime }(i)$$

(13)

where $\phi \left[e(i)\right]=\left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)$. Assuming that $h^{\ast }$ is the filter optimal weight vector, the weight error vector is

$$\epsilon (i)=h^{\ast }-h(i)$$

(14)

By using the weighted error vector for Equation (13), it follows that

$$\epsilon (i+1)=\epsilon (i)-\eta \phi \left[e(i)\right]u^{\prime }(i)$$

(15)

Taking the square of the l₂-norm for Equation (15) and then taking the expected value yields

$$\delta (i+1)=\delta (i)+{\eta }^2\mathbb{E}\left\{{\phi }^2\left[e(i)\right]{‖u^{\prime }(i)‖}^2\right\}-2\eta \mathbb{E}\left\{\phi \left[e(i)\right]{\epsilon }^T(i)u^{\prime }(i)\right\}$$

(16)

where $\delta (i)=\mathbb{E}\left[{‖\epsilon (i)‖}^2\right]$.

When the FxLHT algorithm converges monotonically, i.e., $\delta (i+1)-\delta (i)<0$, the step size $\eta$ should therefore meet

$$0<\eta <{\displaystyle \frac{2\mathbb{E}\left\{\phi \left[e(i)\right]e(i)\right\}}{\mathbb{E}\left\{{\phi }^2\left[e(i)\right]{‖u^{\prime }(i)‖}^2\right\}}}$$

(17)

where ${\epsilon }^T(i)u^{\prime }(i)=e(i)$.

4.2. Analysis of Computational Complexity

The designed ANC algorithms should be simple and effective, not only meeting real-time requirements but also being easy to implement on hardware. The computational complexity of the algorithms involved in this simulation is shown in

Table 1. Computational complexity.

Algorithms	Multiplications/Divisions	Additions/Subtractions
FxLMS	$2M+2N+1$	$2M+2N-2$
FxMCC	$2M+2N+6$	$2M+2N-2$
Swish-FxSA	$2M+2N+8$	$2M+2N+1$
FxtanhLMS	$2M+2N+9$	$2M+2N+4$
FxLHT	$2M+2N+7$	$2M+2N+2$
VFxLHT	$5M+2N+16$	$4M+2N+2$

M denotes the adaptive filter of length; N is the length of the secondary path.

. From Table 1, it can be seen that the proposed FxLHT algorithm has a lower computational complexity than the FxtanhLMS algorithm [23]. Although the proposed VFxLHT algorithm has higher computational complexity, its noise reduction performance and robustness are relatively better.

5. Simulation Experiments

In the simulation section, the primary channel $P(z)$ and secondary channel $S(z)$ are modeled by FIR filters of lengths, $L=256$ and $N=100$, respectively, and their amplitude response and phase response are shown in Figure 5, where the adaptive filter of length is $M=128$. In this paper, we use the average noise reduction (ANR) [27,28,29,30] to validate the convergence performance of these ANC algorithms, which is expressed as

$$\mathrm{ANR}=20{\log }_{10}\left({\displaystyle \frac{{\mathrm{\Phi }}_e(i)}{{\mathrm{\Phi }}_d(i)}}\right)$$

(18)

where ${\mathrm{\Phi }}_e(i)=\xi {\mathrm{\Phi }}_e(i-1)+\left(1-\xi \right)\left|e(i)\right|$, ${\mathrm{\Phi }}_d(i)=\xi {\mathrm{\Phi }}_d(i-1)+\left(1-\xi \right)\left|d(i)\right|$, and their initial values, ${\mathrm{\Phi }}_e(0)=0$, ${\mathrm{\Phi }}_d(0)=0$, and $\xi =0.999$, is the forgetting factor close to 1.

5.1. Impulsive Noise Control

Impulsive noise usually appears in control system disturbances with low probability and high amplitude characteristics, and it is shown that α-stable distribution noise can effectively simulate impulsive noise. In the simulation experiments, the impulsive noise is subjected to standard symmetric α-stable (SαS) distribution noise [17,26], whose characteristic function is defined as

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

(19)

In the following simulations, we verify the noise reduction performance of the proposed ANC algorithms by taking the impulsive noise obeying the standard SαS distribution with α = 1.6, α = 1.7, α = 1.8, and α = 1.9, respectively. The simulation curves of these ANC algorithms are obtained by averaging after 100 independent runs.

Table 2. Algorithm’s weight vector update formula.

Algorithms	Weight Vector Update
FxLMS	$h(i+1)=h(i)+\eta e(i)u^{\prime }(i)$
FxMCC [14]	$h(i+1)=h(i)+\eta \exp \left(-{\displaystyle \frac{e^2(i)}{2{\sigma }^2}}\right)e(i)u^{\prime }(i)$
Swish-FxSA [24]	$h(i+1)=h(i)+\eta \mathrm{sign}\left(e(i)\right){\displaystyle \frac{\left[1+\left(1-\beta \left|e(i)\right|\right)\exp \left(-\beta \left|e(i)\right|\right)\right]}{{\left[1+\exp \left(-\beta \left|e(i)\right|\right)\right]}^2}}{u}^{\prime }(i)$
FxtanhLMS [23]	$h(i+1)=h(i)+\eta \tanh \left(\lambda e(i)\right)\left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]u^{\prime }(i)$
FxLHT	$h(i+1)=h(i)+\eta \left[1-{\tanh }^2\left(\lambda e^2(i)\right)\right]e(i)u^{\prime }(i)$

shows the weight vector update formulas for these ANC algorithms, and the simulation parameters for these algorithms are set as follows: FxLMS algorithm with $\eta ={10}^{-4}$; FxMCC algorithm with $\eta ={10}^{-4}$, $\sigma =0.5$; Swish-FxSA algorithm with $\eta ={10}^{-4}$, $\beta =0.7$; FxtanhLMS algorithm with $\eta =3\times {10}^{-4}$, $\lambda =0.3$; FxLHT algorithm with $\eta ={10}^{-4}$, $\lambda =0.1$; and VFxLHT algorithm with $\eta ={10}^{-4}$, $\gamma =1$, $\beta =10^{-2}$.

In order to verify the noise reduction performance of the proposed FxLHT algorithm with respect to the value of λ, as shown in Figure 6, when the value of λ is larger, the residual error of the FxLHT algorithm is smaller, but its convergence speed decreases. By contrast, when λ is small, the convergence rate of the FxLHT algorithm increases, but its residual error increases. In addition, when the impulsive noise intensity is small (i.e., α = 1.9), then the FxLHT algorithm has similar residual errors for different values of λ. However, the smaller the value of λ, the faster the convergence rate of the FxLHT algorithm. From Figure 6a–d, it can be seen that the proposed VFxLHT algorithm has better noise reduction performance and a faster convergence rate for different values of α.

As shown in Figure 7a–d, the noise reduction performance of the FxLMS algorithm is easily affected by the impulsive noise, and the convergence performance of the FxLMS algorithm is poorer when α takes a smaller value, and even divergence occurs. As the value of α gradually increases, the convergence performance of the FxLMS algorithm gradually becomes better. Compared with FxLMS algorithm, FxMCC, Swish-FxSA and FxtanhLMS algorithms, the proposed FxLHT and VFxLHT algorithms have good robustness, fast convergence rate, and small residual error under impulsive noise interference. In addition, the proposed VFxLHT algorithm with a variable λ-parameter has a faster convergence rate and a smaller residual error as compared to the FxLHT algorithm under different impulsive noise intensities.

5.2. Sinusoidal Impulsive Noise Control

To further verify the noise reduction performance of the proposed FxLHT algorithm, the reference signal $u(i)$ of the ANC system uses mixed noise [17], which is represented as

$$u(i)=\sqrt{2}\sin \left({\displaystyle \frac{2\pi \times i\times 500}{8000}}\right)+v(i)$$

(20)

where $v(i)$ is the impulsive noise obeying the standard SαS distribution with α taken to be set to 1.8.

As shown in Figure 8, the noise reduction performance of the proposed FxLHT algorithm is verified at different λ values. We can see that the noise reduction performance of the FxLHT algorithm is related to the value of λ. The smaller the value of the parameter λ is, the faster the convergence of the FxLHT algorithm is. The larger the value of the parameter λ is, the smaller the residual error of the FxLHT algorithm is. In addition, a variable λ-parameter FxLHT algorithm, i.e., the VFxLHT algorithm, is proposed in this paper, which not only shows faster convergence but also has a smaller residual error.

The noise reduction performance of the proposed FxLHT and VFxLHT algorithms is verified as shown in Figure 9. It can be seen that the FxLMS algorithm has poor convergence performance when it is disturbed by impulsive noise. The convergence performance and noise reduction performance of the proposed FxLHT and VFxLHT algorithms are superior to those of the FxLMS algorithm, the FxMCC algorithm, Swish-FxSA algorithm, and the FxtanhLMS algorithm.

5.3. Verification of Tracking Capability

To evaluate the tracking capability of the proposed ANC algorithms, the impulse response of the primary path $P(z)$ is multiplied by −1 at iteration $i=2\times {10}^4$. As shown in Figure 10, the proposed algorithms achieve the best tracking performance compared to other ANC algorithms. As can be seen in Figure 11, Swish-FxSA algorithm shows faster tracking performance compared to FxMCC and FxtanhLMS algorithms under sinusoidal impulsive noise. The proposed FxLHT algorithm has good tracking performance. However, the VFxLHT algorithm has better tracking capability in case of sudden changes in the primary path.

6. Conclusions

For active control of impulse noise, the traditional FxLMS algorithm exhibits poor noise reduction performance. To improve this disadvantage, this paper proposes a robust FxLMS algorithm based on the hyperbolic tangent function framework, namely the FxLHT algorithm. In addition, since the value of the parameter λ in the FxLHT algorithm greatly affects the convergence speed and noise reduction performance of the algorithm, in order to further enhance the performance of the FxLHT algorithm, the FxLHT algorithm with a variable λ-parameter is proposed. Finally, the computer simulation verifies that the proposed FxLHT algorithm has good robustness, faster convergence speed, smaller residual error, and better tracking performance in impulse noise active control.