Active impulsive noise control algorithm based on an arctangent function

Abstract

To address the issue that the filtered-x least mean square (FxLMS) algorithm loses its stability in impulsive noise environments resulting in failure of noise reduction, a modified reference signal filtered-x arctangent least mean square (MRFxatanLMS) algorithm is proposed. A nonlinear function is utilized to constrain abnormally large values in the reference signal without requiring complex threshold estimation. By embedding the cost function of the traditional FxLMS algorithm into the framework of the arctangent function, a novel cost function is formed to control the impulsive noise and improve the robustness of the noise reduction system. A compression factor is introduced into the cost function to adjust the compression degree of the error signal, thereby balancing the convergence speed and robustness of the algorithm. Simulation results demonstrate that compared with several other algorithms, the proposed algorithm performs well in different intensities of impulsive noise environments.

Keywords

Active noise control impulsive noise modified reference signal arctangent function compression factor

Introduction

The advancement of modern industry has intensified noise pollution, with prolonged exposure posing serious risks to human health, such as sleep disturbances, cognitive problems and cardiovascular disease.^1–3 Consequently, implementing effective noise control measures is essential. Traditional passive noise control (PNC) methods typically employ sound-absorbing materials, barriers, and similar approaches to mitigate noise. However, these methods often fail to achieve ideal noise reduction for low-frequency noise and are hindered by bulky and costly equipment, thus limiting their practical applications.⁴ In contrast, active noise control (ANC) technology generates a signal with the same amplitude and opposite phase as the incoming noise signal through a controller, effectively suppressing various noises through sound attenuation.⁵ Compared with the PNC method, ANC has better control effect on low-frequency noise, reduces the size of noise control system, simplifies system design, and has demonstrated significant success in transportation, healthcare, and architecture.^6–9

The filtered-x least mean square (FxLMS) algorithm¹⁰ is the most widely utilized ANC algorithm, known for its real-time adjustment of filter coefficients through an adaptive algorithm. It is straightforward to implement and involves low computational costs. However, the FxLMS algorithm faces instability in environments with impulsive noise due to its reliance on a bounded second-order moment of the error signal.^11,12 Impulsive noise is prevalent in many practical applications, such as the noise generated by punching and stamping machines, combustion engines and pile drivers in factories,^13,14 and its probability density does not follow a Gaussian distribution, which is typically modeled using a Symmetric α Stable (SαS) distribution.¹⁵ The probability density function (PDF) curves for the standard SαS distribution with different α values are depicted in Figure 1. It is evident that the SαS distribution exhibits significant tailing, the smaller the value of α, the stronger the impact, while as α approaches 2, the noise increasingly resembles a Gaussian distribution.

Figure 1.

Standard SαS distribution PDF with different α.

To adapt the FxLMS algorithm for impulsive noise, various modified ANC algorithms have been proposed. Leahy et al.¹⁶ introduced a filtered-x least mean p norm (FxLMP) algorithm, which employs the fractional order moment of the error signal as a cost function to enhance system stability in controlling impulsive noise. However, the algorithm’s stability depends upon the selection of the parameter p, necessitating prior knowledge of the characteristic parameter α for effective implementation. Sun et al.¹⁷ proposed a MRFxLMS algorithm, which improves stability by disregarding the reference signal when its amplitude exceeds a predefined threshold. Akhtar et al.¹⁸ further enhanced the MRFxLMS algorithm by imposing amplitude limitations on both the reference and error signals during filter coefficient updates, replacing signal amplitudes with a predefined threshold when exceeded, thereby enhancing noise reduction performance. It is evident that the selection of threshold parameters in these algorithms requires priori knowledge of impulsive noise, complicating their application.

Various nonlinear transformation algorithms have been developed to avoid a threshold setting in the control of impulsive noise. Wu et al.¹⁹ proposed a filtered-x logarithmic error LMS (FxlogLMS) algorithm, which utilizes the square of a logarithmic transform of the error signal as the cost function to regulate filter coefficient updates. Although this approach eliminates the need for threshold setting, it introduces a dead zone in the coefficient update process. Subsequently, several algorithms have been developed to perform nonlinear transformations on the error signal using various nonlinear functions, such as the sigmoid function, Fair function and trigonometric function.^20–25 These algorithms solve the dead zone issue and enhance the system’s capability to suppress impulsive noise. Furthermore, adaptive algorithms based on information theory learning represent another class of nonlinear transformation methods in ANC systems. Liu et al.²⁶ first proposed the concept of correntropy, providing a theoretical analysis for the advantages of algorithms based on the maximum correntropy criterion (MCC) in controlling impulsive noise. Singh et al.²⁵ employed the MCC for system identification and noise cancellation in the presence of impulsive noise, thus enhancing the robustness of the control system. In summary, while these nonlinear transformation algorithms improve system stability by limiting outliers in the error signal, they fail to account for the influence of exceptionally large values in the reference signal on system stability.

In this paper, a modified reference signal filtered-x arctangent least mean square (MRFxatanLMS) algorithm is proposed, which utilizes nonlinear functions to simultaneously limit outliers in both the reference and error signals, thereby enhancing system stability in the control of impulsive noise. This algorithm operates without the need of prior knowledge for threshold setting and maintains relatively low computational complexity. Simulation results demonstrate that compared to several other algorithms, the MRFxatanLMS algorithm performs well in various intensities of impulsive noise environments.

The proposed algorithm

Figure 2 depicts the block diagram of the MRFxatanLMS algorithm proposed in this paper. $P (z)$ is the impulse response of the primary path from the reference microphone to the error microphone, $S (z)$ describes the secondary path between the controller and the error microphone, $\hat{S} (z)$ represents the approximate estimate of $S (z)$ , typically determined using either online or offline algorithms. $x (n)$ denotes the reference signal, $x_{f} (n)$ represents the signal filtered by a nonlinear function and $\hat{S} (z)$ , and $d (n)$ is the interference signal at the error microphone’s position. $W (z)$ denotes the controller composed of a FIR filter, and $y (n)$ refers to the signal produced by the controller after processing, aimed at canceling the noise. The signal $y_{f} (n)$ results from $y (n)$ passing through the secondary path, and $e (n)$ is the outcome of superimposing $d (n)$ and $y_{f} (n)$ at the position of the error microphone. The control system utilizes the secondary loudspeaker to propagate the cancellation signal $y (n)$ , the error signal $e (n)$ is collected to update the control filter coefficients, and the corresponding computation within the filter are performed via adaptive algorithms.

Figure 2.

Block diagram of the MRFxatanLMS algorithm.

To enhance the stability of the ANC system in controlling impulsive noise, the reference signal $x (n)$ is performed a nonlinearly transformation through an improved arctangent function, generating a modified reference signal $x^{'} (n)$ , as depicted in the following equation.

x^{'} (n) = f [x (n)] = \frac{1}{a^{2}} \arctan [a^{2} x (n)]

(1)

where a controls the signal range after nonlinear transformation. Figure 3 illustrates the curves of the nonlinear function

f [x (n)]

with various values of a, and it is evident that

f [x (n)]

regulates the amplitude of the reference signal

x (n)

within a reasonable range, thereby minimizing the impact of abnormally large values in the impulsive noise signal on the system’s stability.

Figure 3.

Curves of the nonlinear function f[x(n)] with various values of a.

Assuming the order of the control filter is L, define $x (n) = {[x (n), x (n - 1), x (n - 2), \dots, x (n - L + 1)]}^{T}$ , $w (n) = {[w_{0} (n), w_{1} (n), w_{2} (n), \dots, w_{L - 1} (n)]}^{T}$ , then

y (n) = w^{T} (n) x (n) = \sum_{l = 0}^{L - 1} w_{l} (n) x (n - l)

(2)

If the order of the secondary path $S (z)$ is M, i.e. $s (n) = {[s_{0} (n), s_{1} (n), s_{2} (n), \dots, s_{M - 1} (n)]}^{T}$ , then we have

y_{f} (n) = s^{T} (n) y (n) = \sum_{m = 0}^{M - 1} s_{m} (n) y (n - m)

(3)

For impulsive noise signals following an α-stable distribution, the expected value of the squared error signal is infinite. Consequently, the FxLMS algorithm based on the Minimum Mean Square Error (MMSE) criterion is ineffective in controlling impulsive noise. To address this issue, the cost function of the traditional MMSE criterion is embedded into the framework of the arctangent function, and a new cost function is defined as

J (n) = E [\frac{1}{b^{2}} \arctan [b^{2} e^{2} (n)]] \approx \frac{1}{b^{2}} \arctan [b^{2} e^{2} (n)]

(4)

where

e (n) = d (n) - y_{f} (n)

, b is a compression factor, and modifying its value allows for controlling the degree of compression applied to the error signal. Define

x_{f} (n) = {[x_{f} (n), x_{f} (n - 1), x_{f} (n - 2), \dots, x_{f} (n - L + 1)]}^{T}

, where

x_{f} (n) = \sum_{m = 0}^{M - 1} {\hat{s}}_{m} (n) x^{'} (n - m)

, substitute equation (3) into equation (4) and calculate the instantaneous gradient

\nabla J (n) = \frac{1}{b^{2}} \cdot \frac{\partial [\arctan [b^{2} e^{2} (n)]]}{\partial w (n)} = \frac{1}{b^{2}} \cdot \frac{\nabla [b^{2} e^{2} (n)]}{1 + b^{4} e^{4} (n)} = \frac{2 e (n) \nabla [e (n)]}{1 + b^{4} e^{4} (n)} \approx - \frac{2 e (n)}{1 + b^{4} e^{4} (n)} x_{f} (n)

(5)

Define a new robust function

φ [e (n)] = \frac{e (n)}{1 + b^{4} e^{4} (n)}

(6)

Then the update formula for the control filter of the MRFxatanLMS algorithm is as follows

w (n + 1) = w (n) - \frac{μ}{2} \nabla J (n) = w (n) + μ φ [e (n)] x_{f} (n)

(7)

In summary, the calculation process of the MRFxatanLMS algorithm is presented in the table below. (Table 1).

Table 1.

Calculation process of the MRFxatanLMS algorithm.

Initialization	w(0) = 0
Parameters	μ, a, b
Iterative process	Step 1 $y (n) = w^{T} (n) x (n)$ , $y_{f} (n) = s^{T} (n) y (n)$
	Step 2 $e (n) = d (n) - y_{f} (n)$
	Step 3 $x^{'} (n) = \frac{1}{a^{2}} \arctan [a^{2} x (n)]$ , $x_{f} {(n) = x}^{'} (n) * \hat{s} (n)$
	Step 4 $φ [e (n)] = \frac{e (n)}{1 + b^{4} e^{4} (n)}$
	Step 5 $w (n + 1) = w (n) + μ φ [e (n)] x_{f} (n)$

Performance analysis

Algorithm comparison

In the presence of impulsive noise, the convergence process is stabilized by applying a nonlinear transformation to

e (n)

through the robust function

φ [e (n)]

. Table 2 lists various expressions for

φ [e (n)]

in different algorithms. Notably, the MRFxatanLMS algorithm features a relatively simple expression for

φ [e (n)]

, which obviates exponential, logarithmic or trigonometric operations.

Table 2.

The various expressions for φ[e(n)] in different algorithms.

Algorithm	φ[e(n)]
FxLMP¹⁶	$\| e (n) \|^{p - 1} sgn [e (n)]$
FxlogLMS¹⁹	$sgn [e (n)] \cdot \log \| e (n) \| / \| e (n) \|, \| e (n) \| > 1$
FxsigLMS²¹	$sgn [e (n)] f (\| e (n) \|) [\frac{\exp (- \| e (n) \|)}{(1 + \exp {(- \| e (n) \|))}^{2}}], f (x) = \frac{1 - e^{- x}}{1 + e^{- x}}$
MCCFxLMS²⁵	$\exp (- \frac{e^{2} (n)}{2 σ^{2}}) e (n)$
MRFxatanLMS	$\frac{e (n)}{1 + b^{4} e^{4} (n)}$

Figure 4 depicts the curves of $φ [e (n)]$ for different algorithms. As illustrated in Figure 4, $φ [e (n)]$ can smoothly regulate the error signal across the full range of its variation. When impulsive noise is detected, the curves of $φ [e (n)]$ approach zero, thereby the filter coefficients are hardly updated to maintain system stability. The $φ [e (n)]$ of the FxlogLMS algorithm remains zero when $| e (n) | < 1$ , causing the filter coefficients to cease updating, which creates a dead zone in the algorithm. Several other algorithms have addressed the dead zone issue, but only the proposed algorithm can modify the curve by adjusting the compression factor b. An increase in b enhances the curve’s ability to suppress impulsive noise but slows the algorithm’s convergence rate. Conversely, a decrease in b accelerates convergence but diminishes the noise suppression capability. When b approaches 0, the curve of $φ [e (n)]$ becomes nearly linear, and the algorithm can hardly suppress impulsive noise, actually degenerating into the FxLMS algorithm. Consequently, the proposed algorithm can more effectively adapt to varying intensities of impulsive noise by adjusting the value of the parameter b.

Figure 4.

Curves with different nonlinear transformation function φ[e(n)].

Convergence analysis

Similar to the FxLMS algorithm, the MRFxatanLMS algorithm also requires setting the step size within an appropriate range to maintain system stability. In this section, we will analyze the convergence conditions that the algorithm’s step size must meet to ensure proper functioning.

Let $w_{o}$ represent the vector of optimal filter coefficients that satisfies the following relation

w_{o}^{T} x_{f} (n) \approx d (n)

(8)

The deviation vector $ε (n)$ is defined as shown in the following equation

ε (n) = w_{o} - w (n)

(9)

Then subtract $w_{o}$ from both sides of equation (7) to obtain the following equation

ε (n + 1) = ε (n) - μ φ [e (n)] x_{f} (n)

(10)

Convergence analysis involves applying the 2-norm and taking the expected value on both sides of equation (10)

E [‖ ε (n + 1) ‖^{2}] = E [‖ ε (n) - μ φ [e (n)] x_{f} (n) ‖^{2}] = E [‖ ε (n) ‖^{2}] + μ^{2} E [φ^{2} [e (n)] ‖ x_{f} (n) ‖^{2}] - 2 μ E [φ [e (n)] ε^{T} (n) x_{f} (n)]

(11)

Assuming the following equation is valid

ε^{T} (n) x_{f} (n) = {[w_{o} - w (n)]}^{T} x_{f} (n) \approx d (n) - y_{f} (n) = e (n)

(12)

Then equation (11) can be rewritten as follows

E [‖ ε (n + 1) ‖^{2}] = E [‖ ε (n) ‖^{2}] + μ^{2} E [φ^{2} [e (n)] ‖ x_{f} (n) ‖^{2}] - 2 μ E [φ [e (n)] e (n)]

(13)

To guarantee that the proposed algorithm converges within a finite number of iterations, it is necessary to meet the inequality specified in equation (14).

E [‖ ε (n + 1) ‖^{2}] < E [‖ ε (n) ‖^{2}]

(14)

The range of values for the step size μ, derived through further simplification and calculations, is as follows

0 < μ < \frac{2 E [φ [e (n)] e (n)]}{E [φ^{2} [e (n)] ‖ x_{f} (n) ‖^{2}]}

(15)

Complexity analysis

Additionally, computational complexity analysis is also crucial for assessing algorithm performance. Excessive computational complexity not only increases the computational burden of the hardware but also impacts the real-time performance of the noise reduction system. Assuming that L and M represent the lengths of the control filter coefficients and the secondary channel, respectively. Table 3 presents the computational complexities of various algorithms used in the simulation experiments, with each algorithm evaluated based on the computational cost per iteration.

Table 3.

Computational complexities of various algorithms.

Algorithm	Multiplication/Division operations	Addition/Subtraction operations	Others
FxLMP¹⁶	2L + 2M + 3	2L + 2M - 1	abs:1,sgn:1
FxLMP¹⁶	2L + 2M + 3	2L + 2M - 1	fractional moment:1
FxlogLMS¹⁹	2L + 2M + 3	2L + 2M - 2	if:1, abs:1, sgn:1, log:1
FxgsnLMS²⁷	2L + 2M + 9	2L + 2M - 2	exp:1, sqrt:1
MCCFxLMS²⁵	2L + 2M + 6	2L + 2M - 2	exp:1
MRFxatanLMS	2L + 2M + 11	2L + 2M - 1	atan:1

As shown in Table 3, the computational complexity of the MRFxatanLMS algorithm is comparable to the other algorithms, but it avoids complex exponential and logarithmic operations, which is more suitable for real-time hardware implementation.

Simulations

In this section, the performance of the proposed MRFxatanLMS algorithm is investigated by comparing it with several previously discussed algorithms. The primary path $P (z)$ is modeled using an FIR filter, and the secondary path $S (z)$ is similarly implemented with an FIR filter. Both $P (z)$ and $S (z)$ have a length of 128, and their data are acquired through secondary path identification experiments.²⁸ For simplicity, we assume that $\hat{S} (z)$ is approximately equal to $S (z)$ . Figure 5 displays the frequency responses of $P (z)$ and $S (z)$ . The control filter $w (n)$ is modeled as a 128th-order FIR filter.

Figure 5.

Frequency responses of p(z) and S(z).

The impulsive signals used in the simulation are shown in Figure 6, with α = 1.8, α = 1.6 and α = 1.3 corresponding to mild, moderate and heavy impulsive noise, respectively. Figure 7 displays the noise control efficacy of the proposed MRFxatanLMS algorithm across four varying noise intensities. The MRFxatanLMS algorithm demonstrates a consistent suppression effect in both Gaussian noise (α = 2) and impulsive noise (α = 1.8, α = 1.6, α = 1.3).

Figure 6.

Impulsive signals with different α.

Figure 7.

Noise control efficacy of the MRFxatanLMS algorithm. (a) Gaussian noise (α = 2) (b) mild impulsive noise (α = 1.8). (c) Moderate impulsive noise (α = 1.6) (d) heavy impulsive noise (α = 1.3).

To visually demonstrate the performance of the proposed algorithm, the averaged noise reduction (ANR) metric is introduced to assess its performance, defined as

A N R = 20 \log_{10} (\frac{A_{e} (n)}{A_{d} (n)})

(16)

A_{e} (n) = ξ A_{e} (n - 1) + (1 - ξ) | e (n) | A_{d} (n) = ξ A_{d} (n - 1) + (1 - ξ) | d (n) |

(17)

where

ξ

is the forgetting factor,

A_{e} (n)

and

A_{d} (n)

represent the estimates of

| e (n) |

and

| d (n) |

at each iteration. In simulation, we define that

A_{e} (0) = 0

A_{d} (0) = 0

, and

ξ = 0.999

. In this paper, 20 experiments were conducted for each algorithm and the average of their ANR values was calculated as the final result. The parameters used in the following cases are the optimal values derived from multiple experiments, as listed in Table 4.

Table 4.

Controlling parameters for several algorithms.

Algorithm	α = 1.8	α = 1.6	α = 1.3
FxLMP	μ = 3×10^-5	μ = 1×10^-5	μ = 3×10^-6
FxlogLMS	μ = 3×10^-4	μ = 1×10^-4	μ = 5×10^-5
FxgsnLMS	μ = 3×10^-4, δ = 2	μ = 2×10^-5, δ = 2	μ = 1×10^-6, δ = 2
MCCFxLMS	μ = 1×10^-4, δ = 2.5	μ = 6×10^-5, δ = 2.5	μ = 2×10^-5, δ = 3
MRFxatanLMS	μ = 7×10^-5 a = 0.4, b = 0.1	μ = 4×10^-5 a = 0.4, b = 0.15	μ = 2×10^-5 a = 0.4, b = 0.15

Firstly, the effectiveness of the MRFxatanLMS algorithm in a mild impulsive noise environment with α = 1.8 is verified against other algorithms. Figure 8 displays the noise reduction performance curves of different algorithms, with the number of iterations on the horizontal axis and ANR on the vertical axis. It is clear that all algorithms achieve stable convergence in a mild impulsive noise environment, successfully reducing noise below 17 dB. The FxlogLMS and MCCFxLMS algorithms exhibit the slowest convergence speeds, although the FxlogLMS algorithm achieves a lower residual error. The FxgsnLMS and FxLMP algorithms converge rapidly, but their residual errors are relatively high, especially the FxgsnLMS algorithm experiences significant fluctuations after convergence. An effective noise reduction method should exhibit a rapid convergence rate as well as minimal residual error. The MRFxatanLMS algorithm not only converges at a speed comparable to the FxgsnLMS and FxLMP algorithms but also achieves the lowest residual error.

Figure 8.

ANR curves for mild impulsive noise (α = 1.8).

Figure 9 demonstrates the performance of the MRFxatanLMS algorithm and other algorithms in a moderate impulsive noise environment. Compared to the previous case, although the convergence speed of several algorithms has slightly decreased, they still achieve convergence. In this scenario, the FxlogLMS, FxgsnLMS, and MCCFxLMS algorithms exhibit relatively slow convergence speeds, while the FxLMP algorithm converges faster than these three but experiences significant fluctuations after convergence. The MRFxatanLMS algorithm surpasses the others, delivering the fastest convergence speed and the lowest residual error.

Figure 9.

ANR curves for moderate impulsive noise (α = 1.6).

Similarly, Figure 10 illustrates the performance of several algorithms in a heavy impulsive noise environment. It is evident that the performance of the MCCFxLMS and FxgsnLMS algorithms deteriorates dramatically in this situation, as both algorithms not only converge very slowly but also exhibit high residual error. The FxlogLMS and FxLMP algorithms outperform the above two algorithms, but only about 15 dB of noise reduction was achieved after 2 × 10⁴ iterations. In contrast, the MRFxatanLMS algorithm is particularly outstanding in heavy impulsive noise environments, as it achieves a noise reduction of approximately 18 dB after 1.5 × 10⁴ iterations, maintaining stable performance in both convergence speed and residual noise.

Figure 10.

ANR curves for heavy impulsive noise (α = 1.3).

Finally, a simulation experiment was conducted using noise generated by a punching machine, instead of the simulated noise with an SαS distribution. The noise control performance of the MRFxatanLMS algorithm is demonstrated in Figure 11. As shown in Figure 11(a), punching noise exhibits strong impact characteristics, caused by the instantaneous collisions between the punch periodically and the workpiece or mold, resulting in short-term peak noise. After 1 × 10⁴ iterations with the proposed denoising algorithm, the residual noise was significantly reduced. Figure 11(b) presents the frequency spectrum corresponding to the signal in Figure 11(a). It shows that the frequency components of the punching noise are primarily concentrated in the low-frequency range, mainly between 200 Hz and 600 Hz. After processing through the noise reduction system, low-frequency noise in the 200-800 Hz range was notably suppressed, confirming the effectiveness of the MRFxatanLMS algorithm.

Figure 11.

Performance of MRFxatanLMS algorithm for punching machine noise. (a) Time domain (b) Frequency domain.

Conclusions

In this paper, the MRFxatanLMS algorithm is proposed to improve the robustness of the ANC system against impulsive noise. Firstly, an improved arctangent function is proposed to constrain abnormally large values in the reference signal, thereby suppressing impulsive noise. Secondly, a robust function is utilized to perform nonlinear transformation on the error signal, preventing the filter coefficient updating process from diverging in impulsive noise environments. Finally, the compression factor is introduced to adjust the compression degree of the error signal to balance the convergence speed and robustness of the algorithm. Additionally, the algorithm’s performance is thoroughly analyzed. Numerical simulations verify the effectiveness of the algorithm against impulsive noise, demonstrating its faster convergence speed and lower residual error compared with several other algorithms.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Weiping Liu

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