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Abstract

In the conventional hybrid active noise control (CHANC) algorithm, the filtered-x least mean square (FXLMS) algorithm is used in both feedforward and feedback structures. However, the FXLMS algorithm does not process the reference signal and the error signal before updating the weight vector, which leads to the inadequate robustness of the CHANC algorithm in the impulsive noise environment. To solve this problem, the maximum versoria criterion is incorporated into the HANC (MVC-HANC) algorithm in this paper. Firstly, the MVC-HANC algorithm employs a novel nonlinear function to compress the error signal. Secondly, the proposed algorithm uses the modified sigmoid function to constrain the reference signal. To further improve the noise attenuation performance, the MVC-HANC algorithm uses a novel exponential function to adjust the step-size adaptively. Simulation and experiment results demonstrate that the proposed algorithm has the better attenuation performance than the conventional HANC algorithm.

Keywords

hybrid active noise control FXLMS algorithm MVC-HANC algorithm impulsive noise environment

Introduction

The noise emitted by the industrial products seriously affects people’s normal life. In the past, the passive noise control (PNC) technology has been widely used and achieved good noise reduction performance.¹ However, the PNC technology has significant limitations when dealing with low-frequency noise. Active noise control (ANC) technology has good noise reduction ability for the low-frequency noise.² Most of the ANC algorithms used nowadays are adaptive algorithms, and non-adaptive algorithms are rarely used anymore. The ANC algorithms used in this paper are the adaptive algorithms.

The principle of the ANC system is that the loudspeaker emits a secondary sound; the phase of this secondary sound is opposite to the primary noise, so as to achieve the purpose of noise reduction.³ The ANC algorithm can be divided into two structures, one is the feedforward structure and the other is the feedback structure. In recent years, many scholars combined the feedforward structure with the feedback structure to form a hybrid system. We refer to this hybrid system as a conventional hybrid ANC (CHANC) algorithm.

Based on the CHANC algorithm, Akhtar MT developed a new HANC algorithm, which we call AMHANC algorithm in this paper.⁴ The AMHANC algorithm uses the third filter to improve the attenuation performance. Based on the AMHANC algorithm, Trideba proposed a new HANC (T-HANC) algorithm, the T-HANC adopts the joint-optimized normalized least mean square (JO-NLMS) algorithm to update the weight vector of the third filter.⁵ Mokhtarpour proposed a self-tuning HANC algorithm, this algorithm can adaptively adjust the step-size.⁶ Jiang proposed a novel adaptive step-size HANC algorithm, the parameters in this algorithm can be adjusted by the intelligent algorithm.⁷

In the CHANC algorithm, the filtered-x least mean square (FXLMS) algorithm is usually applied to the feedforward and feedback structures.⁸ When it deals with the impulsive noise, the attenuation performance have great limitations.⁹ Many scholars have developed other ANC algorithms to improve the robustness of the FXLMS algorithm.

The error signal is often compressed before the weight vector is updated. The filtered-x least mean M-estimate (FXLMM) algorithm has tried to compress the error signal.¹⁰ The filtered-x logarithmic error LMS (FXlogLMS) algorithm uses the logarithmic function to compress the error signal.¹¹ The general step-size normalized FXLMS (FXgsnLMS) algorithm adopts a smooth compression curve to compress the error signal.¹² The filtered-x least mean p-power (FXLMP) algorithm and the filtered-x least mean absolute deviation (FXLMAD) algorithm form a new error signal processing method by modifying the cost function.^13,14 The filtered-x maximum versoria criterion (FXMVC) algorithm proposes a new error signal compression function.^15,16 The algorithm proposed in this paper is based on this algorithm.

Constraining the reference signal can significantly improve the robustness of the algorithm. The threshold based FXLMS (Th-FXLMS) algorithm constrains the reference signal, the noise attenuation performance is improved.¹⁷ The weight-constrained FXLMS (CFXLMS) algorithm constrains the weight vector, which can achieve the same noise attenuation effect.¹⁸

Many variable step-size functions are proposed, which can improve the attenuation performance. The normalized step-size FXLMS (NSS-FXLMS) algorithm proposes a normalized step-size method.¹⁹ Based on the NSS-FXLMS algorithm, the convex-combined step-size (CCSS) based filtered-x generalized normalized LMP (CCSS-FXGNLMP) algorithm considers the energy of the reference signal and the error signal at the same time, and proposes a new variable step-size strategy.²⁰ The filtered-x maximum correntropy criterion (FXMCC) algorithm and the filtered-x least mean kurtosis (FXLMK) algorithm implement the variable step-size strategy by modifying the cost function.^21,22 Other variable step-size strategies can also achieve the effect of improving noise attenuation performance.^23,24 Based on the summary of the variable step-size strategies, a novel variable step-size function is proposed in this paper.

The motivation for this paper is that the CHANC algorithm does not achieve satisfactory result in attenuating impulsive noise. To solve the problem, the MVC-HANC algorithm is proposed in this paper. The proposed algorithm uses a novel function to compress the error signal. Meanwhile, the MVC-HANC algorithm adopts a modified sigmoid function to constrain the reference signal. To further improve the noise attenuation performance, an adaptive step-size function is applied to the algorithm. The main contributions of the proposed algorithm can be summarized as follows:

(1) The MVC-HANC algorithm uses a novel nonlinear function to compress the error signal, the compression strength of the error signal can be adjusted.

(2) The proposed algorithm adopts a modified sigmoid function to constrain the reference signal.

(3) The proposed algorithm uses an adaptive step-size function to adjust the step-size adaptively.

Conventional HANC algorithm

The conventional HANC (CHANC) algorithm is a classic hybrid algorithm. Figure 1 shows the block diagram of the CHANC algorithm. In Figure 1, x₁(n) and x₂(n) represent the reference noises in the feedforward and feedback structures, respectively. P(z) is the transfer function of the primary path, its transfer path is the distance between the primary sound source and the error microphone. d(n) is the primary noise collected by the error microphone. W₁(z) and W₂(z) represent the feedforward and feedback adaptive filters, respectively. y(n) is the output signal of the whole algorithm. y_s(n) is the output signal y(n) collected by the error microphone. e(n) is the error signal, it is obtained by the mutual cancellation of d(n) and y_s(n). S(z) is the transfer function of secondary path, $\hat{S} (z)$ is the estimation of the secondary path. ${\hat{x}}_{1} (n)$ is the reference noise x₁(n) filtered by the $\hat{S} (z)$ . ${\hat{x}}_{2} (n)$ is the reference noise x₂(n) filtered by the $\hat{S} (z)$ . LMS represents the weight update algorithm.

Figure 1.

Block diagram of the conventional HANC algorithm.

In the CHANC algorithm, the output signals of the feedforward and feedback structures can be written as

y_{1} (n) = w_{1}^{T} x_{1} (n)

(1)

y_{2} (n) = w_{2}^{T} x_{2} (n)

(2)In equation (1), x ₁(n) is the reference noise of the feedforward structure and its memory length is L, the x ₁(n) can be expressed as

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

. w ₁(n) is the weight vector of the feedforward structure and its length is L, the expression of the w ₁(n) can be written as

w_{1} (n) = {[w_{1,0} (n), w_{1,1} (n), \dots, w_{1, L - 1} (n)]}^{T}

. In equation (2), x ₂(n) is the reference noise of the feedback structure, the memory length is L, and the x ₂(n) can be expressed as

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

. w ₂(n) is the weight vector of the feedback structure, the length of the w ₂(n) is L, the expression of the w ₂(n) can be written as

w_{2} (n) = {[w_{2,0} (n), w_{2,1} (n), \dots, w_{2, L - 1} (n)]}^{T}

The output of the whole algorithm can be written as

y (n) = y_{1} (n) + y_{2} (n)

(3)In equation (3), y₁(n) and y₂(n) are the outputs of the feedforward and feedback structures, respectively.

The expression of the error signal e(n) can be written as

e (n) = d (n) - y_{s} (n) + v (n)

(4)In equation (4), v(n) is the uncorrelated signal. y_s(n)=s(n) * y(n), s(n) is the impulse response of the S(z). * is the linear convolution.

The expression of x₂(n) can be written as

x_{2} (n) = e (n) + {\hat{y}}_{s} (n) = e (n) + \hat{s} (n) * y (n)

(5)In equation (5), the

\hat{s} (n)

is the impulse response of the

\hat{S} (z)

The update formula of the w ₁(n) and w ₂(n) can be written as

w_{1} (n + 1) = w_{1} (n) + μ_{1} e (n) {\hat{x}}_{1} (n)

(6)

w_{2} (n + 1) = w_{2} (n) + μ_{2} e (n) {\hat{x}}_{2} (n)

(7)In equations (6) and (7), μ₁ and μ₂ are the step-sizes in the feedforward and feedback structures, the e(n) is the error signal, the expressions of the

{\hat{x}}_{1} (n)

and

{\hat{x}}_{2} (n)

can be written as

{\hat{x}}_{1} (n) = \hat{s} (n) * x_{1} (n)

{\hat{x}}_{2} (n) = \hat{s} (n) * x_{2} (n)

As shown in equations (6) and (7), the values of e(n) and $\hat{x} (n)$ become sharply larger when the impulsive signals are encountered during the update of the weight vector w (n). Then, the value of w (n+1) will also become sharply larger, which means that w (n+1) will diverge. When calculating the w (n+1), we need to use w (n). Once the weight vector diverges, it is difficult to converge again. Therefore, we need to constrain the e(n) and x(n).

The process of the CHANC algorithm is shown in Table 1.

Table 1.

Summary of the conventional HANC algorithm.

Initialization	w ₁(0) = 0, w ₂(0) = 0,μ₁ = 0,μ₂ = 0
Parameters	μ₁,μ₂
Adaptive process	For n = 0,1,2⋯
—	$y_{1} (n) = w_{1}^{T} x_{1} (n)$
—	$y_{2} (n) = w_{2}^{T} x_{2} (n)$
—	y(n) = y₁(n)+y₂(n)
—	y_s(n) = s(n) * y(n)
—	e(n) = d(n) − y_s(n)+v(n)
—	$x_{2} (n) = e (n) + {\hat{y}}_{s} (n) = e (n) + \hat{s} (n) * y (n)$
—	${\hat{x}}_{1} (n) = \hat{s} (n) * x_{1} (n)$
—	${\hat{x}}_{2} (n) = \hat{s} (n) * x_{2} (n)$
—	$w_{1} (n + 1) = w_{1} (n) + μ_{1} e (n) {\hat{x}}_{1} (n)$
—	$w_{2} (n + 1) = w_{2} (n) + μ_{2} e (n) {\hat{x}}_{2} (n)$

The proposed MVC-HANC algorithm

FXMVC algorithm

The cost function of the FXMVC algorithm can be written as

J (n) = E {\frac{1}{1 + k {| e (n) |}^{p}}}

(8)In equation (8), k = (2a)^−p, a is the radius of the versoria, p is the shape parameter. The gradient function of J(n) can be expressed as

\nabla J (n) = \frac{\partial J (n)}{\partial w (n)}

= \frac{\partial (\frac{1}{1 + k {| e (n) |}^{p}})}{\partial w (n)}

= - k p \frac{1}{{(1 + k {| e (n) |}^{p})}^{2}} {| e (n) |}^{p - 1} s i g n (e (n)) \frac{\partial e (n)}{\partial w (n)}

= k p \frac{1}{{(1 + k {| e (n) |}^{p})}^{2}} {| e (n) |}^{p - 1} s i g n (e (n)) \hat{x} (n)

(9)

Then, the weight vector update equation of the FXMVC algorithm can be written as

w (n + 1) = w (n) + μ k p \frac{1}{{(1 + k {| e (n) |}^{p})}^{2}} {| e (n) |}^{p - 1} s i g n (e (n)) \hat{x} (n)

(10)In equation (10), we set p = 2 in this paper, then the equation (10) can be rewritten as

w (n + 1) = w (n) + 2 μ k \frac{| e (n) |}{{(1 + k {| e (n) |}^{2})}^{2}} s i g n (e (n)) \hat{x} (n)

(11)In this paper, we define

ψ (e (n)) = 2 k \frac{| e (n) |}{{(1 + k {| e (n) |}^{2})}^{2}} s i g n (e (n))

(12)In equation (12), the ψ(e(n)) is the compression function, the k is defined as the compression factor in this paper. As shown in equation (12), the compression function ψ(e(n)) can effectively achieve the compression of the error signal, and we can change the compression degree by adjusting the coefficients. In this paper, we obtain a compression function by fixing p = 2 and varying the value of k. Figure 2 shows the characteristic curves of the compression function ψ(e(n)). As shown in Figure 2, this setup makes it possible to obtain a set of symmetric compression curves. It is obvious that this compression function compresses the error signal e(n) with the same degree regardless of whether e(n) is positive or negative. The value of k has a great influence on the compression degree of the function ψ(e(n)).

Figure 2.

The characteristic curves of the function ψ(e(n)) with different factor k.

Constraint function

To further improve the noise attenuation performance, we adopt a modified sigmoid function to constrain the reference signal. Equation (13) shows the modified sigmoid function.

h (x (n)) = α (\frac{1}{1 + e^{- β x (n)}} - \frac{1}{2})

(13)In equation (13), the x(n) is the reference noise, the parameter α determines the threshold of the constraint function, the parameter β determines the constraint strength. As shown in equation (13), the constraint function h(x(n)) can effectively achieve the constraint of the reference signal. We can change the constraint degree by adjusting the coefficients. Figure 3 shows the characteristic curves of the constraint function h(x(n)) with different parameter β, the parameter α is set as 1.

Figure 3.

The characteristic curves of the function h(x(n)) with different parameter β.

Adaptive step-size function

When the ANC algorithm encounters impulsive noise, the algorithm is prone to divergence. In order to prevent divergence, a smaller step-size is generally set for the ANC algorithm. However, when the step-size is small, the convergence rate of the ANC algorithm will decrease. Therefore, we hope that when the impulsive noise appears, the step-size will become smaller to prevent the algorithm from diverging, and when there is no impulsive noise, the step-size becomes larger to improve the convergence rate. Based on this idea, we propose a novel variable step-size equation. Equation (14) shows the variable step-size equation.

μ (n) = \tilde{μ} e^{- (a {‖ {\hat{x}}^{'} (n) ‖}^{2} + b {‖ e (n) ‖}^{2})}

(14)In equation (14),

\tilde{μ}

is the basic step-size, a and b are the parameters, ‖∙‖² is the Euclidean norm,

{\hat{x}}^{'} (n) = \hat{s} (n) * x^{'} (n)

, x ^′(n) is the reference signal compressed by function h(x(n)), x ^′(n)=[h(x(n)) h(x(n−1))⋯h(x(n−L+1))]^T.

As shown in equation (14), when the reference signal x(n) and the error signal e(n) become larger, the step-size μ(n) becomes smaller. According to equations (6) and (7), this variable step-size formula can avoid the weight vector divergence during the weight vector update. Furthermore, the variable step-size formula can be changed by adjusting the parameters.

MVC-HANC algorithm

Figure 4 shows the block diagram of the MVC-HANC algorithm, the calculation process of the proposed algorithm is as follows.

Figure 4.

Block diagram of the MVC-HANC algorithm.

The output signals of the feedforward and the feedback structures can be written as

y_{1} (n) = w_{1}^{T} x_{1} (n)

(15)

y_{2} (n) = w_{2}^{T} x_{2} (n)

(16)

The output signal of the whole algorithm is

y (n) = y_{1} (n) + y_{2} (n)

(17)

The error signal can be written as

e (n) = d (n) - y_{s} (n) + v (n) = d (n) - s (n) * y (n) + v (n)

(18)

We need to calculate the reference signal of the feedback structure

x_{2} (n) = e (n) + {\hat{y}}_{s} (n) = e (n) + \hat{s} (n) * y (n)

(19)Then, we need to calculate the update equation of the weight vector. Firstly, the error signal e(n) is compressed by formula ψ(∙)

ψ_{1} (e (n)) = 2 k_{1} \frac{| e (n) |}{{(1 + k_{1} {| e (n) |}^{2})}^{2}} s i g n (e (n))

(20)

ψ_{2} (e (n)) = 2 k_{2} \frac{| e (n) |}{{(1 + k_{2} {| e (n) |}^{2})}^{2}} s i g n (e (n))

(21)In equations (20) and (21), the ψ₁(e(n)) and ψ₂(e(n)) are the compression function of the feedforward and feedback structures, respectively.

Secondly, we constrain the reference signal by the formula h(∙)

x_{1}^{'} (n) = h (x_{1} (n)) = α_{1} (\frac{1}{1 + e^{- β_{1} x_{1} (n)}} - \frac{1}{2})

(22)

x_{2}^{'} (n) = h (x_{2} (n)) = α_{2} (\frac{1}{1 + e^{- β_{2} x_{2} (n)}} - \frac{1}{2})

(23)

Thirdly, we propose the adaptive step-size formula

μ_{1} (n) = {\tilde{μ}}_{1} e^{- (a_{1} {‖ {\hat{x}}_{1}^{'} (n) ‖}^{2} + b_{1} {‖ ψ_{1} (e (n)) ‖}^{2})}

(24)

μ_{2} (n) = {\tilde{μ}}_{2} e^{- (a_{2} {‖ {\hat{x}}_{2}^{'} (n) ‖}^{2} + b_{2} {‖ ψ_{2} (e (n)) ‖}^{2})}

(25)In equations (24) and (25),

{\hat{x}}_{1}^{'} (n) = \hat{s} (n) * x_{1}^{'} (n)

x_{1}^{'} (n) = {[h (x_{1} (n)) h (x_{1} (n - 1)) \dots h (x_{1} (n - L + 1))]}^{T}

{\hat{x}}_{2}^{'} (n) = \hat{s} (n) * x_{2}^{'} (n)

x_{2}^{'} (n) = {[h (x_{2} (n)) h (x_{2} (n - 1)) \dots h (x_{2} (n - L + 1))]}^{T}

Finally, the update equation of the weight vector can be written as

w_{1} (n + 1) = w_{1} (n) + μ_{1} (n) ψ_{1} (e (n)) {\hat{x}}_{1} (n)

(26)

w_{2} (n + 1) = w_{2} (n) + μ_{2} (n) ψ_{2} (e (n)) {\hat{x}}_{2} (n)

(27)

The process of the MVC-HANC algorithm is shown in Table 2.

Table 2.

Summary of the MVC-HANC algorithm.

Initialization	w ₁(0) = 0, w ₂(0) = 0,μ₁(n) = 0,μ₂(n) = 0
Parameters	${\tilde{μ}}_{1}, {\tilde{μ}}_{2}, k_{1}, k_{2}, α_{1}, α_{2}, β_{1}, β_{2}, a_{1}, a_{2}, b_{1}, b_{2}$
Adaptive process	For n = 0,1,2⋯
—	$y_{1} (n) = w_{1}^{T} x_{1} (n)$
—	$y_{2} (n) = w_{2}^{T} x_{2} (n)$
—	y(n) = y₁(n) + y₂(n)
—	y_s(n) = s(n) * y(n)
—	e(n) = d(n) − y_s(n) + v(n)
—	$x_{2} (n) = e (n) + {\hat{y}}_{s} (n) = e (n) + \hat{s} (n) * y (n)$
—	$ψ_{1} (e (n)) = 2 k_{1} \frac{\| e (n) \|}{{(1 + k_{1} {\| e (n) \|}^{2})}^{2}} s i g n (e (n))$
—	$ψ_{2} (e (n)) = 2 k_{2} \frac{\| e (n) \|}{{(1 + k_{2} {\| e (n) \|}^{2})}^{2}} s i g n (e (n))$
—	$x_{1}^{'} (n) = h (x_{1} (n)) = α_{1} (\frac{1}{1 + e^{- β_{1} x_{1} (n)}} - \frac{1}{2})$
—	$x_{2}^{'} (n) = h (x_{2} (n)) = α_{2} (\frac{1}{1 + e^{- β_{2} x_{2} (n)}} - \frac{1}{2})$
—	${\hat{x}}_{1}^{'} (n) = \hat{s} (n) * x_{1}^{'} (n)$
—	${\hat{x}}_{2}^{'} (n) = \hat{s} (n) * x_{2}^{'} (n)$
—	$μ_{1} (n) = {\tilde{μ}}_{1} e^{- (a_{1} {‖ {\hat{x}}_{1}^{'} (n) ‖}^{2} + b_{1} {‖ ψ_{1} (e (n)) ‖}^{2})}$
—	$μ_{2} (n) = {\tilde{μ}}_{2} e^{- (a_{2} {‖ {\hat{x}}_{2}^{'} (n) ‖}^{2} + b_{2} {‖ ψ_{2} (e (n)) ‖}^{2})}$
—	$w_{1} (n + 1) = w_{1} (n) + μ_{1} (n) ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n)$
—	$w_{2} (n + 1) = w_{2} (n) + μ_{2} (n) ψ_{2} (e (n)) {\hat{x}}_{2}^{'} (n)$

Stability analysis

The step-sizes in the feedforward and feedback structures of the MVC-HANC algorithm need to be controlled within a reasonable range. We first focus on the step-size range in the feedforward structure. Define w _opt as the optimal weight vector. The relationship between the weight vector at the current moment and the optimal weight vector is ϵ (n) = w _opt− w ₁(n). Define $d (n) = w_{o p t}^{T} * {\hat{x}}_{1} (n)$ . Then, we can get the following equation

ϵ (n + 1) = ϵ (n) - μ_{1} (n) ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n)

(28)

Define the mathematical expectation of ‖ ϵ (n)‖² as

φ (n) = E {{‖ ϵ (n) ‖}^{2}}

(29)

Then,

φ (n + 1) = E {{‖ ϵ (n + 1) ‖}^{2}} = E {{‖ ϵ (n) - μ_{1} (n) ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} = {‖ ϵ (n) ‖}^{2} + μ_{1}^{2} (n) E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} - 2 μ_{1} (n) E {ψ_{1} (e (n)) ϵ^{T} (n) {\hat{x}}_{1}^{'} (n)}

(30)

The difference between φ(n+1) and φ(n) is

φ (n + 1) - φ (n) = μ_{1}^{2} (n) E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} - 2 μ_{1} (n) E {ψ_{1} (e (n)) ϵ^{T} (n) {\hat{x}}_{1}^{'} (n)}

(31)Here,

ϵ^{T} (n) {\hat{x}}_{1}^{'} (n) = {[w_{o p t} - w_{1} (n)]}^{T} {\hat{x}}_{1}^{'} (n) \approx e (n)

(32)

The equation (31) can be rewritten as

ρ (n) = φ (n + 1) - φ (n) = μ_{1}^{2} (n) E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} - 2 μ_{1} (n) E {ψ_{1} (e (n)) e (n)}

(33)In order to ensure the convergence of the algorithm, the condition ρ(n) < 0 must be satisfied, then

μ_{1}^{2} (n) E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} - 2 μ_{1} (n) E {ψ_{1} (e (n)) e (n)} < 0

(34)

It means

0 < μ_{1} (n) < \frac{2 E {ψ_{1} (e (n)) e (n)}}{E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}}}

(35)

Combining equation (24) and equation (35), we can get

0 < {\tilde{μ}}_{1} < \frac{2 E {ψ_{1} (e (n)) e (n)}}{E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} e^{- (a_{1} {‖ {\hat{x}}_{1}^{'} (n) ‖}^{2} + b_{1} {‖ ψ_{1} (e (n)) ‖}^{2})}}

(36)

The equation (36) shows the boundary of the basic step-size ${\tilde{μ}}_{1}$ in the feedforward structure. In the conventional HANC algorithm, the convergence coefficient μ₁ in the feedforward structure is bounded by

0 < μ_{1} < \frac{2 E {{e (n)}^{2}}}{E {{e (n)}^{2} {‖ {\hat{x}}_{1} (n) ‖}^{2}}}

(37)Obviously, the ψ₁(e(n)) and the

{\hat{x}}_{1}^{'} (n)

represent the error signal e(n) and reference signal x(n) are constrained. Thus,

0 < \frac{2 E {{e (n)}^{2}}}{E {{e (n)}^{2} {‖ {\hat{x}}_{1} (n) ‖}^{2}}} < \frac{2 E {ψ_{1} (e (n)) e (n)}}{E {{‖ ψ_{1} (e (n)) {\hat{x}}_{1}^{'} (n) ‖}^{2}} e^{- (a_{1} {‖ {\hat{x}}_{1}^{'} (n) ‖}^{2} + b_{1} {‖ ψ_{1} (e (n)) ‖}^{2})}}

(38)In other words, the range of

{\tilde{μ}}_{1}

is larger than the range of μ₁.

Similarly, we can get the boundary of the basic step-size ${\tilde{μ}}_{2}$

0 < {\tilde{μ}}_{2} < \frac{2 E {ψ_{2} (e (n)) e (n)}}{E {{‖ ψ_{2} (e (n)) {\hat{x}}_{2}^{'} (n) ‖}^{2}} e^{- (a_{2} {‖ {\hat{x}}_{2}^{'} (n) ‖}^{2} + b_{2} {‖ ψ_{2} (e (n)) ‖}^{2})}}

(39)For the same reason, the range of

{\tilde{μ}}_{2}

is larger than the range of μ₂ in feedback structure of the CHANC algorithm.

Complexity analysis

Computational complexity is an important aspect to evaluate ANC algorithm. Table 3 shows the summary of the computational complexity analysis. In Table 3, L₁ represents the length of the weight vector in the feedforward structure, L₂ represents the length of the weight vector in the feedback structure, L₃ represents the length of the weight vector in the third filter, M represents the length of the secondary path, Abs(∙) represents the absolute value operation, Sign(∙) represents the signum function, Exp(∙) represents the exponential operation.

Table 3.

Summary of the computational complexity analysis.

System	*/÷	+/−	Other operations
FXLMS	2L₁ + 2M + 1	2L₁ + 2M − 2	$-$
CHANC	2L₁ + 2L₂ + 3M + 2	2L₁ + 2L₂ + 3M − 3	$-$
AMHANC	2L₁ + 2L₂ + 2L₃ + 3M + 3	2L₁ + 2L₂ + 2L₃ + 3M − 3	$-$
MVC-HANC	3L₁ + 3L₂ + 3M + 32	3L₁ + 3L₂ + 3M + 5	Abs(∙):1,Sign(∙):1,Exp(∙):4

As shown in Table 3, the FXLMS algorithm has the lowest computational complexity, but its noise attenuation performance is worse compared with other algorithms mentioned in this paper. In the subsequent simulations and experiments, it can be seen that the FXLMS algorithm provides the worse attenuation performance, and provides the worst attenuation performance in Experiment B and Experiment C. The MVC-HANC algorithm proposed in this paper has the highest computational complexity. The MVC-HANC algorithm is more than FXLMS algorithm L₁+3L₂+M+31 multiplications and L₁+3L₂+M+7 additions. The attenuation performance of the MVC-HANC algorithm is the best among these four algorithms. Therefore, its computational complexity increase is acceptable compared to its noise attenuation performance. If the number of channels is drastically increased, the MVC-HANC algorithm still has the highest computational complexity and the best noise attenuation performance when using the same number of channels.

Simulation results

In this section, the computer simulations are conducted. The physical system represented in the simulation results is the room. The data of acoustic paths need to be obtained in the simulation.

The primary path P(z) and the secondary path S(z) are modeled as FIR filters, the length of the filters is 128. The estimated secondary path $\hat{S} (z)$ is considered the same as secondary path S(z). The frequency responses of these two acoustic paths are shown in Figure 5. The impulse responses of the primary and secondary paths are obtained by identifying the acoustic paths on the test rig. The test rig is described in detail in the section of Experiment results. Obviously, the impulse responses of the primary and secondary paths are obtained from experiments.

Figure 5.

Frequency response of the primary path and the secondary path: (a) magnitude response and (b) phase response.

The averaged noise reduction (ANR) is used to validate the attenuation performance of the ANC algorithms. The calculation formula of the ANR is

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

(40)In equation (40), the A_e(n) and A_d(n) can be written as

{\begin{array}{c} A_{e} (n) = η A_{e} (n - 1) + (1 - η) | e (n) | \\ A_{d} (n) = η A_{d} (n - 1) + (1 - η) | d (n) | \end{array}

(41)In equation (41), A_e(0) = 0, A_d(0) = 0. As shown in equation (41), A_e(n−1) and A_d(n−1) represent the A_e(n) and A_d(n) at the previous moment, respectively. The larger the value of η takes, the larger the proportion of A_e(n−1) and A_d(n−1) in the calculation of A_e(n) and A_d(n), the smoother the ANR curve is. η determines the smoothness of the ANR curve. According to the references, the range of η is [0.9990,0.9999].^25,26 To better present the noise attenuation performance of the algorithms, we choose η = 0.9998 in this paper.

The reference noise is the impulsive noise generated by MATLAB. The impulsive noise conforms to the SαS distribution. In most cases, the α range of the impulsive noise is set as (1, 2). We choose three typical cases to calculate the simulations. Firstly, we set α = 1.6, this case represents the mild impulsive noise. Secondly, we set α = 1.4, this case represents the medium impulsive noise. Thirdly, we set α = 1.2, this case represents the heavy impulsive noise. The sampling frequency is set as 8000 Hz.

Simulation A

In Simulation A, the reference noise is the mild impulsive noise (α = 1.6). The uncorrelated noise v(n) is the Gauss white noise, and its bandwith range is [800, 900] Hz. Table 4 shows the step-sizes used in this case.

Table 4.

Summary of the step-sizes used in Simulation A.

Algorithms	Step-sizes
FXLMS	μ₁ = 3.5×10⁻⁵
CHANC	μ₁ = 3.5×10⁻⁵, μ₂ = 5×10⁻⁴
AMHANC	μ₁ = 3.5×10⁻⁵, μ₂ = 5×10⁻⁴, μ_h = 1.4×10⁻⁵
MVC-HANC	${\tilde{μ}}_{1} = 0.2$ , ${\tilde{μ}}_{2} = 0.9$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

In Table 4, μ₁ and μ₂ represent the step-sizes in the feedforward and feedback structures, respectively. μ_h represents the step-size in the third filter. As for the MVC-HANC algorithm, ${\tilde{μ}}_{1}$ and ${\tilde{μ}}_{2}$ represent the step-sizes in the feedforward and feedback structures, respectively. k₁,k₂,α₁,α₂,β₁,β₂,a₁,a₂,b₁,b₂ are the parameters in the MVC-HANC algorithm. The selected step-sizes and parameters are the optimal values obtained through repeated simulations.

Figure 6 shows the ANR results. As shown in Figure 6, the AMHANC algorithm has the worst convergence rate, the convergence rate of the FXLMS algorithm is better than the AMHANC algorithm, the CHANC algorithm has better convergence rate than the FXLMS algorithm, and the MVC-HANC algorithm proposed in this paper has the best convergence rate. The CHANC algorithm has the worst steady-state performance, the ANR result of the CHANC algorithm is −2.7 dB at iterations 8.0×10⁴. The MVC-HANC algorithm has the best steady-state performance, the ANR result of this algorithm is −3.5 dB in the final stage of the iterations.

Figure 6.

ANR curves for the Simulation A.

The results show a 0.8 dB improvement for steady-state performance of the MVC-HANC algorithm compared to the CHANC algorithm. The improvement for steady-state performance is acceptable. Firstly, the references can demonstrate that the algorithms proposed by other scholars also have limited improvements for the impulsive noise.^27–29 Secondly, in order to ensure that the ANC algorithms converge without divergence, the ANC algorithms have the smaller step-size setting when attenuating impulsive noise compared to attenuating steady-state noise, Thirdly, the appearance of impulsive noise is random; therefore, the algorithms tend to have better noise attenuation performance during the iterative process compared to the performance in the final stage of the iterations. We need to examine the whole iteration of the ANC algorithm to evaluate its noise attenuation performance.

Simulation B

In Simulation B, the reference noise is the medium impulsive noise (α = 1.4). The uncorrelated noise v(n) is the same as the uncorrelated noise in Simulation A. The step-sizes used in this case are shown in Table 5.

Table 5.

Summary of the step-sizes used in Simulation B.

Algorithms	Step-sizes
FXLMS	μ₁ = 1×10⁻⁷
CHANC	μ₁ = 1×10⁻⁷, μ₂ = 1×10⁻⁶
AMHANC	μ₁ = 1×10⁻⁷, μ₂ = 1×10⁻⁶, μ_h = 2×10⁻⁷
MVC-HANC	${\tilde{μ}}_{1} = 0.2$ , ${\tilde{μ}}_{2} = 0.1$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

Figure 7 shows the ANR curves. As shown in Figure 7, the FXLMS algorithm, the CHANC algorithm, and the AMHANC algorithm have the same attenuation performance, the ANR curves of the three ANC algorithms are almost coincident, the ANR results of these three algorithms are −1.4 dB at iterations 8.0×10⁴. The attenuation performance of the MVC-HANC algorithm is best compared with other three algorithms, the ANR result of the MVC-HANC algorithm is −2.2 dB in the final stage of the iterations, it is 0.8 dB lower than other three algorithms. For the same reasons as in Simulation A, the ANR result is acceptable.

Figure 7.

ANR curves for the Simulation B.

Simulation C

In Simulation C, the reference noise is the heavy impulsive noise (α = 1.2). The uncorrelated noise is the same as the uncorrelated noise in Simulation A. Table 6 shows the step-sizes used in this case.

Table 6.

Summary of the step-sizes used in Simulation C.

Algorithms	Step-sizes
FXLMS	μ₁ = 4×10⁻⁸
CHANC	μ₁ = 4×10⁻⁸, μ₂ = 1×10⁻⁷
AMHANC	μ₁ = 4×10⁻⁸, μ₂ = 1×10⁻⁷, μ_h = 5×10⁻⁸
MVC-HANC	${\tilde{μ}}_{1} = 0.3$ , ${\tilde{μ}}_{2} = 0.1$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

Figure 8 shows the ANR curves. As shown in Figure 8, the FXLMS, CHANC, and AMHANC algorithms have almost the same attenuation performance, the ANR results are −1.2 dB at iterations 8.0×10⁴. The ANR result of the MVC-HANC algorithm is about −2.0 dB in the final stage of the iterations, it is 0.8 dB lower than the three conventional algorithms. For the same reasons as in Simulation A, the ANR result is acceptable.

Figure 8.

ANR curves for the Simulation C.

Experiment results

In this section, the primary impulsive noise was measured from a real vehicle going over a bump, the experimental results are based on the laboratory test rig setup. In the following, we will introduce the method of primary noise collection and the arrangement of the test rig.

To collect the primary noise signal, we conducted a vehicle experiment. The vehicle made from China, the test site is an asphalt pavement. When the noise signal was collected, the vehicle was passing through the speed bump at 25 km/h, 30 km/h, and 35 km/h. The sampling rate is 20,480 Hz. To improve the computational efficiency, we resampled the data with a new sampling rate of 8000 Hz. As shown in Figure 9(a), the microphone is placed at the driver’s left ear. As shown in Figure 9(b), the noise data is processed by the computer and the LMS SCADAS Mobile. Figure 10 shows time domain plots of the vehicle interior noise, Figure 11 shows the time-frequency domain plots of the vehicle interior noise.

Figure 9.

Noise data acquisition equipment: (a) microphone, (b) computer and LMS SCADAS Mobile.

Figure 10.

The time domain plots of the vehicle noise when the vehicle was passing through the speed bump at: (a) 25 km/h, (b) 30 km/h, (c) 35 km/h.

Figure 11.

The time-frequency domain plots of the vehicle noise when the vehicle was passing through the speed bump at: (a) 25 km/h, (b) 30 km/h, (c) 35 km/h.

Figure 12 shows the ANC experimental platform, the MicroAutoBox III is the main controller, the laptop is used to build the algorithm model and establish communication with the MicroAutoBox III. The acoustic devices are arranged through the test rig.

Figure 12.

The ANC experimental platform.

Figure 13 shows the mounting position of the acoustic devices. As shown in Figure 13, the test rig is equipped with three loudspeakers and three microphones, in this experiment, Loudspeaker 3 and Microphone 3 are not used. Loudspeaker 1 emits primary noise, the primary noise is the vehicle interior noise. Microphone 1 is used to collect the primary noise signal. Loudspeaker 2 emits the secondary sound, microphone 2 is used to collect the error signal. The MicroAutoBox III controls the Loudspeaker 1 and 2 to make the primary noise and the secondary sound. The sound signals from the MicroAutoBox III are amplified by the power amplifier and then passed to the loudspeakers. The sampling frequency is 8000 Hz. The secondary path is obtained by off-line identification. The frequency response of the secondary path is shown in Figure 5.

Figure 13.

The ANC test rig.

Experiment A

In this case, the reference noise is the vehicle interior noise. When the noise signal was collected, the vehicle was passing through the speed bump at a speed of 25 km/h. The uncorrelated noise v(n) is the Gauss white noise, and it has limited bandwidth of 800–900 Hz. The step-sizes used in this case are shown in Table 7.

Table 7.

Summary of the step-sizes used in Experiment A.

Algorithms	Step-sizes
FXLMS	μ₁ = 3×10⁻⁴
CHANC	μ₁ = 3×10⁻⁴, μ₂ = 1×10⁻⁴
AMHANC	μ₁ = 3×10⁻⁴, μ₂ = 1×10⁻⁴, μ_h = 1×10⁻⁴
MVC-HANC	${\tilde{μ}}_{1} = 0.5$ , ${\tilde{μ}}_{2} = 1 \times 10^{- 4}$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

Figure 14 shows the ANR results. As shown in Figure 14, the FXLMS, CHANC, and AMHANC algorithms have almost the same convergence rate. The convergence rate of the MVC-HANC algorithm is better than other three algorithms. The MVC-HANC algorithm proposed in this paper has the best steady-state performance, the ANR result is about −4.1 dB in the final stage of the iterations. For the same reasons as in Simulation A, the ANR result is acceptable.

Figure 14.

ANR curves for the Experiment A.

Figure 15 shows the comparison of the residual error signals when using different ANC algorithms to suppress the reference noise mentioned in Experiment A. As shown in Figure 15, after using the ANC algorithms, the residual error signals still exhibit the impulsive signal characteristic, but the significant noise attenuation effect of the ANC algorithm can be seen in the time domain.

Figure 15.

Comparison of the residual error signals using different ANC algorithms in Experiment A: (a) ANC off and FXLMS algorithm, (b) FXLMS algorithm and CHANC algorithm, (c) CHANC algorithm and AMHANC algorithm, (d) AMHANC algorithm and MVC-HANC algorithm.

Experiment B

In this case, the vehicle was passing through the speed bump at a speed of 30 km/h when the reference noise was collected. The uncorrelated noise v(n) is the same as the uncorrelated noise in Experiment A. Table 8 shows the step-sizes used in this case.

Table 8.

Summary of the step-sizes used in Experiment B.

Algorithms	Step-sizes
FXLMS	μ₁ = 2×10⁻⁴
CHANC	μ₁ = 2×10⁻⁴, μ₂ = 1×10⁻⁴
AMHANC	μ₁ = 2×10⁻⁴, μ₂ = 1×10⁻⁴, μ_h = 1×10⁻⁴
MVC-HANC	${\tilde{μ}}_{1} = 0.5$ , ${\tilde{μ}}_{2} = 1 \times 10^{- 4}$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

Figure 16 shows the ANR curves. As shown in Figure 16, the FXLMS algorithm has the worst convergence rate. The MVC-HANC algorithm has the best convergence rate. The MVC-HANC algorithm has the best steady-state performance, the ANR result is −4.1 dB in the final stage of the iterations. For the same reasons as in Simulation A, the ANR result is acceptable. Figure 17 shows the comparison of the residual error signals in Experiment B. As shown in Figure 17, the residual error signals still exhibit the significant impulsive signal characteristic.

Figure 16.

ANR curves for the Experiment B.

Figure 17.

Comparison of the residual error signals using different ANC algorithms in Experiment B: (a) ANC off and FXLMS algorithm, (b) FXLMS algorithm and CHANC algorithm, (c) CHANC algorithm and AMHANC algorithm, (d) AMHANC algorithm and MVC-HANC algorithm.

Experiment C

In this case, the vehicle was passing through the speed bump at a speed of 35 km/h when the reference noise signal was collected. The uncorrelated noise v(n) is the same as the uncorrelated noise in Experiment A. Table 9 shows the step-sizes used in this case.

Table 9.

Summary of the step-sizes used in Experiment C.

Algorithms	Step-sizes
FXLMS	μ₁ = 2×10⁻⁴
CHANC	μ₁ = 2×10⁻⁴, μ₂ = 1×10⁻⁴
AMHANC	μ₁ = 2×10⁻⁴, μ₂ = 1×10⁻⁴, μ_h = 1×10⁻⁴
MVC-HANC	${\tilde{μ}}_{1} = 0.5$ , ${\tilde{μ}}_{2} = 1 \times 10^{- 4}$ , k₁ = 1×10⁻³, a₁ = 1×10⁻², b₁ = 1.5, α₁ = 60, β₁ = 7.5×10⁻² k₂ = 1×10⁻³, a₂ = 1×10⁻², b₂ = 1.5, α₂ = 60, β₂ = 7.5×10⁻²

Figure 18 shows the ANR curves. As shown in Figure 18, the FXLMS, CHANC, and AMHANC algorithms have almost the same convergence rate. The MVC-HANC algorithm has the best convergence rate. The FXLMS algorithm has the worst steady-state performance, the ANR result is −2.7 dB at iterations 7.5×10⁴. The MVC-HANC algorithm has the best steady-state performance, the ANR result of this algorithm is −3.4 dB in the final stage of the iterations. For the same reasons as in Simulation A, the ANR result is acceptable. Figure 19 shows the comparison of the residual error signals in Experiment C.

Figure 18.

ANR curves for the Experiment C.

Figure 19.

Comparison of the residual error signals using different ANC algorithms in Experiment C: (a) ANC off and FXLMS algorithm, (b) FXLMS algorithm and CHANC algorithm, (c) CHANC algorithm and AMHANC algorithm, (d) AMHANC algorithm and MVC-HANC algorithm.

Conclusion

This paper proposes the MVC-HANC algorithm for the active noise control. In the section of Simulation results, the ANR results of MVC-HANC algorithm in the final stage of the iterations are −3.5 dB, −2.2 dB, and −2.0 dB, respectively. In the section of Experiment results, the ANR results of MVC-HANC algorithm in the final stage of the iterations are −4.1 dB, −4.1 dB, and −3.4 dB, respectively. The limitation of the MVC-HANC algorithm is that it has the high computational complexity. In future work, we need to reduce the computational complexity to ease the controller burden. We need to optimize the placement of acoustic devices to make the noise attenuation performance of ANC algorithm more obvious.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Chongqing, China (cstc2021jcyj-msxmX0152).

ORCID iD

Rui Zhang

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Maximum versoria criterion applied to hybrid active noise control algorithm for the impulsive noise

Abstract

Keywords

Introduction

Conventional HANC algorithm

The proposed MVC-HANC algorithm

FXMVC algorithm

Constraint function

Adaptive step-size function

MVC-HANC algorithm

Stability analysis

Complexity analysis

Simulation results

Simulation A

Simulation B

Simulation C

Experiment results

Experiment A

Experiment B

Experiment C

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References