Filtered-x Set Membership Algorithm With Time-Varying Error Bound for Nonlinear Active Noise Control

In order to reduce the computational complexity for nonlinear Active Noise Control (ANC) systems, we propose a filtered-x set membership with a time-varying error bound (Fx-SM-VEB) algorithm based on the adaptive approximation principle and data-selective update strategy. By introducing the time-varying error bound (VEB) to replace a pre-specified threshold when nonlinearity exists in the components of the ANC system, the performance of the proposed algorithm is significantly improved. Moreover, the VEB is also easily expanded against the impulsive noise that is normally encountered in actual ANC systems. Based on the impulsive-free estimation, we develop a robust Fx-SM-VEB (RFx-SM-VEB) algorithm for the ANC system which is corrupted by impulsive noise in the reference input signal. Besides, we provide analyses of the steady-state behavior, stability conditions, and computational complexity of the proposed algorithms. Many simulation results in different scenarios of the ANC system have shown that the proposed algorithms are efficient under the nonlinear environment and impulsive noise.


I. INTRODUCTION
For the purpose of noise reduction at low frequencies (f <= 500 Hz), the ANC system has shown many outstanding advantages such as low cost, and simple implementation [1], [2], [3]. Its principle is based on the superposition of two sound wave sources, the primary path (unwanted noise source) and the secondary path (anti-noise source) to suppress unwanted noise. The noise cancellation quality of the ANC system depends not only on the correct coordination between the audio domain and the electrical domain, but also on the problem of reducing nonlinear distortion occurring in the primary path, secondary path, and noise source [4], [5]. Many studies have shown that the ANC system using a linear controller suffers from performance degradation because it does not take into account the influence of nonlinear distortion [4], [5], [6], [7], [8].
The associate editor coordinating the review of this manuscript and approving it for publication was Prakasam Periasamy .
The strategy of updating the filter weights of the set-membership algorithm is to update only when the magnitude of the priori error exceeds a predefined threshold [17]. Based on this technique, data with no information or insignificant amount of information will be ignored, resulting in reduced update costs for redundant data. The set membership algorithm has been developed for many applications to reduce computational complexity [18], [19], [20], [21], [22], [23], [24]. In work [18], a nonlinear adaptive filter based on the VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ kernel set membership method was introduced for the purpose of reducing complexity and increasing tracking ability. Many improvements to the set membership NLMS algorithm for the acoustic echo and sparse channel estimation problem have been proposed [19], [20], [21], [22]. To reduce power consumption in wireless sensor networks, solutions based on SM filtering strategy have also received the attention of many researchers [23], [24]. Much literature [25], [26], [27] dealing with identification systems in the set-membership framework exhibits robust performance in the impulsive noise environment. In particular, the filtered-x set-membership affine projection (Fx-SM-AP) algorithms have also been applied in the linear active noise control system to reduce the computation for the Fx-AP algorithm [28], [29].
In order to improve the efficiency of nonlinear ANC systems, in this study, we propose the filtered-x set membership with time-varying error bound (Fx-SM-VEB) and robust Fx-SM-VEB (RFx-SM-VEB) algorithms. The proposed Fx-SM-VEB algorithm updates only a small part of the filter weights, thus it can significantly reduce the computational complexity. Furthermore, due to using a time-varying error bound (VEB), the proposed algorithm is more suitable for nonlinear ANC systems, resulting in performance improvement compared to using the predetermined threshold. Besides, by calculating the threshold parameter through an impulsive-free estimation, the time-varying error bound can be easily applied to ANC systems containing impulsive noise. In practice, impulsive noise which often occurs with low probability, but large amplitude is a challenging problem for the stability of adaptive algorithms [15]. The proposed RFx-SM-VEB algorithm can against the impulse noise because the error bound is robust to outliers and achieves a good convergence performance resulting from a time-varying error bound. Many simulation studies have been conducted to evaluate the performance of the proposed algorithms in different scenarios of ANC systems.
The remainder of the paper is structured as follows. Section II provides a summary of the ANC system based on the LIP nonlinear expansion function. Section III proposes the set membership filter-based algorithms for the nonlinear ANC system. Sections IV and V give the analysis and the computational complexity of the algorithm, respectively. Sections VI discusses simulation results. Section VII concludes our work.

II. BRIEF OF NONLINEAR ANC SYSTEM
Many linear-in-the-parameters (LIP) nonlinear filters (such as Volterra, FLANN, EMF, etc.) have been used to replace neural networks in ANC systems because of their simple structure and low complexity. Many researchers [4], [5], [9], [10], [11], [12], [13], [14], [15], [16] have also shown the effectiveness of the LIP filters for reducing the nonlinear distortion that exists in the primary path, secondary path, or reference noise. Fig. 1 illustrates a nonlinear ANC system based on LIP controller using Fx-LMS algorithm [1], [2], where X(n) is the reference noise source; y(n) is the output of the controller; P(z) is the transfer function of the primary path from the reference source to the error sensor; S(z) is the transfer function of the secondary path from the controller output to the error sensor;Ŝ(z) is an estimate of the secondary path; d(n) is the main path noise measured at the noise cancellation point;d(n) is the estimate of d(n); and W (n) is the filter coefficient of the controller.
For expansion function using a truncated Volterra series, linearly independent functions can be represented as [4], ϕ k {X(n)} = x(n − n k,1 )x(n − n k,2 ) . . . x(n − n k,P k ), where n k,1 , n k,2 , . . . , n k,P denote an arbitrary integer (0, 1, 2, . . .) representing delay (note that the maximum value of the delay is N , which is the memory size of the input vector X(n)), and P is the order of the nonlinear expansion function. As shown in [4], the memory length of the expansion function based on the truncated Volterra series is defined as

III. THE ALGORITHM BASED ON SET MEMBERSHIP FILTERING (SMF) FOR NONLINEAR ANC SYSTEM
Many works in [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], and [29] have pointed out the effectiveness of adaptive algorithms based on the SMF technique in reducing computational complexity and the effect of impulse noise. However, only a few works [28], [29] focus on solving for the ANC framework, and they also only apply to linear systems. To overcome the problem of nonlinearity in ANC systems, LIP controllers (Volterra, FLANN. . . ) have been used. But this remarkably increases the computational complexity. In this section, we propose filtered-x SM with a time-varying error bound (Fx-SM-VEB) algorithm for the nonlinear ANC system to reduce the computational burden. In addition, we introduce a robust error bound that uses an impulsive-free estimation, to develop a robust Fx-SM-VEB (RFx-SM-VEB) algorithm for nonlinear ANC systems containing impulsive noise.

A. PROPOSED FX-SM-VEB ALGORITHM
Based on the least-perturbation property, the constrained cost function of the algorithm is represented as follows: where ξ denotes the pre-specified error bound and must be chosen appropriately. The constraint problem can be solved by using the Lagrange multipliers method, and an unconstrained cost function is given by where λ is a Lagrange multiplier. Taking the gradient with respect to W (n + 1), then setting it equals to zero, we obtain To achieve a structure for both nonlinear and linear secondary paths, the virtual secondary path filter [8] is used bỹ , · · · , ∂d(n) ∂y(n − K ) ] T , (11) where K is the memory size of the secondary path. Note that the term ∂d(n) ∂W (n+1) in (10) can be written as Assuming that the step size is small (i.e., the weight taps are slowly varying), applying the approximation principle, we get ∂y(n − k) where S(n) is the expanded version of the external input signal X(n) based on the LIP nonlinear functions (FLANN, Volterra,. . .).
Substituting (13), (12) into (10), and combining with (11), we have It is worth noting that the term K k=0 h(n, k)S(n − k) is the result of filtering the extended signal S(n) through the virtual secondary path filter. Thus, we can define the filtered signal of S(n) as where the asterisk * indicates convolution. Combining (15) and (14), we get Taking the gradient with respect to λ, then setting it equals to zero, we obtain To compute (17), we can rewrite the estimate of d(n) aŝ Assuming the step size is small, W (n+1) can be considered as very slowly varying. Thus, (18) can be expressed aŝ Substituting (15) and (16) into (19), we havê Substituting (20) in to (17), we get Note that and therefore, we have Combining (16) and (23), we get the equation to update the weights as where is a time-varying step-size. At the time instant n, if |e(n)| is larger than a pre-specified error bound ξ , the equation (24) needs updating the filtering weights. Otherwise, there is no need to update. In other words, the algorithm used a data-selection update mechanism to reduce the computational complexity. We hereafter call this algorithm the filtered-x set membership (Fx-SM) algorithm. The algorithm achieves good convergence performance and low computational complexity when a pre-specified error bound ξ is chosen appropriately. However, this is difficult to achieve since we lack understanding of the environment of the nonlinearity that exist in the primary path and/or secondary path, the reference input signal. In order to tackle this disadvantage, a time-varying error bound (VEB) has been developed, which can automatically adjust the error bound in each iteration during the update process.
Referenced by works [25], [30], a time-varying error bound can be represented as follows: where ξ = τ δ 2 v is a pre-specified error bound, δ 2 υ is the variance of observation noise υ(n), ρ is usually choose 0 < ρ 1 to achieve low steady-state misalignments, θ(n) = δ(n) is a threshold parameter, with being a scalar depending on the nonlinear model of the system; δ(n) calculated through a residual error estimate as follows: where 0 < ω 1. Therefore, when |e(n)| > θ(n) then the error bound ξ (n) changes according to residual error, otherwise it is equal to a pre-specified error bound. To increase convergence, it is common to set the initial δ 2 (0) to large (i.e., |e(n)| will be less than θ(n) during transient states), and thus the algorithm will work with a pre-specified error bound ξ (n) = ξ . Fig. 2 illustrates the nonlinear ANC system based on the proposed algorithm.
if |e(n)| > θ(n) then 9: ξ (n) = |e(n)| − ρθ(n) 10: else 11: end if 13: if |e(n)| > ξ (n)) then 14: A serious challenge is that the adaptive Fx-SM-VEB algorithm becomes unstable when impulsive noise is present in the reference input signal. To address this, we extend the residual noise estimate δ 2 (n) in (27) to a robust estimate for impulse noise as with φ(n) = [e 2 (n), e 2 (n − 1), . . . , e 2 (n − + 1)] T , and is length of the estimation window. It is easy to see that the estimate δ 2 (n) is robust for outliers, (i.e., when the impulsive noise-corrupted error signals, then θ(n) = δ(n) would not increase). Hence, if the ANC system contains impulse noise, it is certain that |e(n)| > θ (n), and from (26) we use the time-varying error bound ξ (n) = |e(n)| − ρθ(n). However, due to the impulse noise is very large, ρθ(n) |e(n)|, and thus we can deduce the error bound ξ (n) ≈ |e(n)|. In this case, the step-size in (25) equals to zero, resulting in the RFx-SM-VEB algorithm stopping the update process. This also means that the RFx-SM-VEB algorithm will not be unstable in the presence of the impulse noise.

IV. ANALYSIS OF THE ALGORITHM A. THE STEADY-STATE BEHAVIOR ANALYSIS
In this section, analysis of the algorithm based on the SMF technique for the ANC system have been presented.
From (22) we can represent the residual error in the following form: where S f (n) is the filtered version of the extended signal S(n) via the secondary path. We can define the estimation error vector as W (n) = W 0 − W (n), where W 0 is the weight vector that needs to be estimated, then the primary signal d(n) at the noise cancellation point can be represented as where υ(n) denotes noise measured at the noise cancellation point, which is assumed to be a zero-mean Gaussian process and have variance δ 2 υ . Substituting (30) into (29), we get Thus, we can represent the output MSE of e(n) as follows: From (32), the excess MSE can be deduced as (33) On the other hand, the weight update equation of the SMF-based system can be rewritten by if |e(n)| > θ(n) then 10: ξ (n) = |e(n)| − ρθ(n) 11: else 12: ξ (n) = τ δ 2 v 13: end if 14: if |e(n)| > ξ (n)) then 15: 16: else 17: and δ 2 s is the variance of the filtered input signal through the secondary path S f (n).
Therefore, the updated equation of the estimation error W (n) is given by Substituting (36) into (33), the output excess MSE can be expressed by |e 0 (n)| e(n)S f (n). By using calculation given in Appendix, we derive the output excess MSE in (37) as Let P u be the probability of updating in each iteration, E(|) the conditional expected value, we can write where A = E( 1 |e(n)| |e(n)| > ξ (n)), B = E(e(n) |e(n)| > ξ (n)), and C = E(e 2 (n) |e(n)| ξ (n)). Substituting (39-41) in (38) we have When the algorithm reaches the steady state (i.e., ξ ex (n + 1) → ξ ex (n), the output excess MSE becomes (43), as shown at the bottom of the next page.

B. THE CONVERGENCE AND STABILITY CONDITIONS ANALYSIS
To establish the convergent behavior of the proposed algorithms, we conduct the behavior analysis of the weight deviation vector ( W (n) = W 0 -W (n)). The weighted deviation vector update equation for algorithms (24) and (25) can be expressed as, Substituting (31) into (44), the above expression can be rewritten as, where I is the 1 by M identity vector.
The expected coefficient vector error is given by, where R is the correlation matrix of the signal S f (n). Equation (46) can be expressed as, Based on the method of analyzing the convergence behavior of the NLMS algorithm [31], we deduce the stability condition of the proposed algorithm as, On the other hand, from (25) we have 0 < ψ(n) < 1. Therefore, it can be concluded that the proposed algorithms are always stable. Table 1 shows the comparison of computational complexity in each iteration in terms of multiplication and addition for Fx-LMS, Fx-SM, and proposed Fx-SM-VEB algorithms. For the proposed Fx-SM-VEB algorithm, the basic operations are analyzed specifically as follows:

V. ANALYSIS OF COMPUTATIONAL COMPLEXITY
• Computing the filter output in the equation (2)  • Computing the term S f (n) 2 2 in the update equation (24) requires one multiplication and two additions. (Refer to [32], we can decompose S f (n) 2 2 into N ). Notice that the terms S f (n − 1) 2 2 and S 2 f (n − N ) have been computed from earlier iterations, and therefore we only need to compute for term S 2 f (n)). • Computing the time-varying error bound ξ (n) in equations (26,27) requires 4 multiplications, 2 additions, and 1 comparison.
• Computing the weight update W (n) in the equation (24) requires P u M +1 multiplication and P u (M −1) addition, and 1 comparison, where P u =UR/100 is the probability of updating the filter weight in each iteration, and UR is the update rate which is defined by the percentage of update numbers of the equation (24) per the total iterations. • Computing the filtering of the extended signal S(n) via the virtual secondary path (15) requires KL multiplication and K (L − 1) addition, where K is the memory length of S(z) and L is the number of channels of extended signal. Note that the extended signal S(n) can be analyzed in a multichannel fashion to exploit the shift structure in each channel. Instead of having to compute for all the operations of the extended signal filtering, we only need to compute the first term in each channel and then use the shift property to derive the remaining terms.

VI. SIMULATION
In this section, we apply the proposed algorithms for two nonlinear controllers commonly used in ANC systems, namely FLANN and Volterra, to evaluate the effectiveness in terms of reducing computational complexity and impulsive noise. First, we illustrate the comparison between the conventional Fx-LMS algorithm, Fx-SM algorithm, and the proposed Fx-SM-VEB algorithm for nonlinear controllers in the Gaussian noise environment. Then, we compare the proposed RFx-SM-VEB algorithm with previous robust algorithms (such as Fx-LogLMS [33], RFx-LMS [34], and Fx-LogLMP [35]) in an impulsive noise environment.
To compare the performance in the Gaussian noise environment, we use the normalized mean-square error (NMSE) defined by where δ 2 dn is the variance of the primary path noise and e 2 (n) is the residual noise power.
In all experiments, the controller parameters have chosen as follows: (i) For Volterra, memory size N = 10 and order of P = 2; (ii) For FLANN, memory size N = 10 and order of P = 3; (iii) Signal-to-noise ratio SNR = 30dB; and (iv) Learning curves were plotted after 100 independent runs to better discern the behavior of the NMSE or ANR.

A. PERFORMANCE COMPARISON IN THE GAUSSIAN NOISE ENVIRONMENT
Experiment 1: In this experiment, we assume the scenario of nonlinearity in the ANC system as referenced in [8]. Here, the nonlinear models of the primary path and the secondary path are expressed, respectively, by andd (n) = y(n) + 0.35y(n − 1) + 0.09y(n − 2) − 0.5y(n)y(n − 1) + 0.4y(n)y(n − 2). (52) The reference input signal is a white Gaussian noise process.

1) + CHOICE OF PARAMETERS
To determine the suitable parameters for the proposed algorithm, we conducted many experiments with different values for each parameter. The values of the parameters are chosen based on a trade-off between the NMSE performance and computational complexity (calculated through the update rate UR). In this section, we only illustrate the experiments for choosing the and τ parameters for the Fx-SM-VEB algorithm using the Volterra controller.
To determine the parameter , we first keep the τ fixed, then change the in turn to get the best compromise NMSE and UR values (and vice versa for parameter τ ). Fig. 3a illustrates the NMSE performance and the UR with different values. Fig. 3b illustrates the NMSE performance and the UR with different τ values. From Fig. 3a and 3b, it is easy to see that the parameters = 1.98 and τ = 11 achieves a good compromise between the NMSE performance and the UR update rate.
From Table. 2 and Fig. 4, it is easy to see that the proposed Fx-SM-VEB algorithm reduces the computational complexity significantly compared to the Fx-LMS algorithm while maintaining the equivalent noise cancellation performance. The Fx-SM algorithm with a pre-specified error bound is reduced performance compared to the Fx-LMS and proposed Fx-SM-VEB algorithms. This is probably due to the presence of nonlinearity in the primary path and secondary path, making the Fx-SM algorithm difficult to achieve a suitable pre-specified error bound. Experiment 2: In this experiment, we keep the primary path model as in Experiment 1, but the secondary path is approximated by the Hammerstein [10] nonlinear model as follows: (53) 90086 VOLUME 10, 2022 The reference noise source is the color noise generated by filtering the white Gaussian noise v(n) through the ARMA (Autoregressive-moving-average) model as   systems using the proposed Fx-SM-VEB algorithm can reduce the number of operations to update the weights by 78% and 75%, respectively. The Fx-SM algorithm using prespecified error bound suffers from attenuation in nonlinear environments, in particular, the noise cancellation performance is attenuated by 2.5 dB for the FLANN controller, and 2 dB for the Volterra controller.

Experiment 3:
In this experiment, we use the reference input as an actual noise generated by the fan. This noise source is recorded at a frequency of 44100 Hz and sampled at 16 bits per sample. The length of the reference signal is equal to 30 000 samples. In order to evaluate the performance of the proposed RFx-SM-VEB algorithm, we add impulsive noise to the reference input signal at iterations 1000, 7000, 15000, and 25000. The impulsive noise is modeled by (n) = σ (n) (n) [27], where σ (n) is a Bernoulli random sequence, independent and identically distributed (i.i.d) and σ (n) has an instantaneous value at time n being either zero or one (occurrence probability (P(σ (n) = 1) = )); and (n) is an i.i.d zero-mean Gaussian sequence, with variance δ 2 = ϒδ 2 s , here ϒ 1, δ 2 s is the variance of the reference input signal. Fig. 6 illustrates the reference input signal corrupted by impulsive noise (n), with ϒ = 10000 and = 0.8. Fig. 7 depicts the ANR learning curves of the algorithms on the FLANN (7a), and Volterra (7b) controllers. It is found that the proposed RFx-SM-VEB algorithm is stable and achieves better convergence performance than the Fx-LMS, and Fx-LogLMS algorithms for both controllers. The Fx-LMS algorithm suffer from diverges, while the RFx-LMS is stable but achieves worse performance than proposed RFx-SM-VEB when applied to the Volterra controller.
Experiment 4: Here we use the input reference noise as an impulse noise [36] modeled by a standard symmetric α-stable distribution as where parameter α denotes a characteristics exponent, 0 < α < 2. When α is close to 2, we can consider this distribution as a Gaussian function. For a small value of α, this distribution can be viewed as a function of strong impulse noise. The nonlinearity of the primary path noise at the canceling point is modeled by [15], where v(n) = x(n) * p(n), with p(n) is the impulse response of the function P(n) = z −3 − 0.3z −4 + 0.2z −5 . The secondary path is assumed to be a minimum phase transfer function S(z) = z −2 + 1.5z −3 − z −4 . Four scenarios corresponding to the impulsive noise reference input have been demonstrated in this experiment: scenario 1: α = 1.7; scenario 2: α = 1.8; scenario 3: α = 1.9; scenario 4: α = 2.0.   Fig. 9 depict the learning curve of the Fx-LMS, Fx-LogLMS, Fx-LogLMP, RFx-LMS, and proposed RFx-SM-VEB algorithms for the Volterra and FLANN based-ANC systems, respectively. As we can see, the Fx-LMS algorithm diverges in the cases α = 1.7, α = 1.8, and α = 1.9 which is obvious since this algorithm is only for assuming the reference noise is a Gaussian process. For the FLANN based-ANC system, the proposed RFx-SM-VEB algorithm is equivalent to RFx-LMS and Fx-LogLMP algorithms, and outperforms the Fx-LogLMS algorithm. Notice that the Fx-LogLMP algorithm needs to know the a priori information of the reference noise. For the Volterra-based ANC system, the proposed RFx-SM-VEB algorithm outperforms all the above algorithms. It is easy to see that this superiority is more obvious in the case of strong impulsive noise.

VII. CONCLUSION
In this paper, we proposed two Fx-SM-VEB and RFx-SM-VEB algorithms for nonlinear ANC systems. The Fx-SM-VEB algorithm greatly reduces the computational complexity and avoids the difficult problem of choosing a predetermined threshold thanks to the data-selection update strategy and the time-varying error bound. Our study also shows that the algorithm based on the SMF technique using the time-varying bound outperforms that of using a prespecified bound, especially in a nonlinear environment. The proposed RFx-SM-VEB algorithm can combat the impulse noise that occurs in a nonlinear ANC system, which is implemented by introducing an impulsive-free estimation into the time-varying error bound to ensure the stability of the algorithm. Analysis of the computational complexity and the steady-state output MSE has also been derived. Comparison results between the Fx-LMS, Fx-SM, and proposed Fx-SM-VEB algorithms in the Gaussian environment; as well as between the Fx-LogLMS, Fx-LogLMP, RFx-LMS, and proposed RFx-SM-VEB algorithms in the impulsive noise environment have demonstrated the efficiency of the proposed algorithms for nonlinear ANC system. In addition, we believe that the proposed algorithms will be highly efficient when applied to the multichannel ANC system. A study on developing these algorithms for the multichannel ANC system will be conducted in the near future.
For calculating β 3 , we have: (66) degree in signal and information processing from Southwest Jiaotong University, Chengdu, China, in 2019. He is currently a Lecturer with the School of Engineering and Technology, Vinh University, Vietnam. His current research interests include adaptive signal processing, active noise control, and control systems.
HOANG HUU VIET received the B.S. degree in mathematics from Vinh University, Nghe An, Vietnam, in 1994, the B.S. and M.S. degrees in computer science from the Hanoi University of Technology, Hanoi, Vietnam, in 1998 and 2002, respectively, and the Ph.D. degree in computer engineering from Kyung Hee University, Republic of Korea, in August 2013. He is currently an Associate Professor with the School of Engineering and Technology, Vinh University. His current research interests include artificial intelligence, reinforcement learning, and adaptive signal processing. VOLUME 10, 2022