A Novel PSO-Based Adaptive Filter Structure with Switching Selection Criteria for Active Noise Control

Abstract

In recent years, active noise control (ANC) systems have been widely used in advanced electronic appliances. Nowadays, several authors use gradient-optimization algorithms since they can be easily implemented in these devices. However, these algorithms need to estimate the secondary path in advance. As consequence, this factor can limit its use in real-ANC applications since the secondary path can undergo significant variations over time. To solve this problem, we propose an ANC system with switching filter selection based on particle swarm optimization (PSO) algorithms. Specifically, we use two sets of populations of particles with different acceleration coefficients and inertia weights to create an advanced structure in which the first PSO algorithm guarantees a high convergence speed while the use of the second PSO algorithm allows to achieve a high-level noise reduction. The results demonstrate that the proposed algorithm exhibits better convergence properties compared with previously reported solutions.

1. Introduction

Nowadays, environmental noise represents a worldwide health problem. Specifically, this noise is increased significantly due to the use of industrial equipment (motors, fans, extractors, transformers), means of transport (automotive vehicles, airplanes, etc.), among others. As consequence, the environmental noise critically affects the acoustic environment of houses, buildings, hospitals, offices, schools. During the last decades, ANC systems have emerged as a possible solution to solve the problem of environmental noise [1,2]. In general, ANC is conceived as a technique that employs adaptive filters to generate an anti-noise signal, which attenuates the noise signal by means of the principle of superposition, as shown in Figure 1. In conventional ANC systems, several authors have used gradient-optimization filtered-X least mean square (FXLMS) algorithms since these algorithms exhibit low computational complexity [3]. As a consequence, these algorithms are easy to implement in embedded devices [4,5,6,7]. However, these algorithms exhibit a low convergence speed, which reduces the performance of the ANC systems in terms of noise cancellation. To face these problems, several authors have proposed advanced convex combination filtering structures [8,9,10,11,12,13]. In these structures, two filters with complementary capabilities are used to improve the overall filter performance in comparison when a single filter is used. Commonly, in these combinations, the authors use a first filter with a large step-size to offer fast convergence speed. To guarantee low steady-state mean square error (MSE) a second filter with small step-size is used. However, these structures present a high computational cost since two filters are used at the same time. On the other hand, the performance of these algorithms in real acoustic environments can be affected critically due to two factors: (1) estimation of the secondary path; (2) time-variations of the secondary path. For example, if the estimated model of the secondary path is not accurate, the ANC systems presents instability [14,15,16,17]. Therefore, this aspect has a big impact, especially in practical situations, since some studies have demonstrated that secondary path can change over the time. On the other hand, the adaptability of the most commonly used off-line estimation methods is null when variations of the secondary path occur.

Some studies have demonstrated that a genetic algorithm (GA) can be applied to ANC systems [18,19,20,21,22]. However, it exhibits slow convergence and it requires many operations [23]. Recently, several authors have proposed the use of bio-inspired algorithms to overcome these problems. Specifically, particle swarm optimization (PSO) algorithm has demonstrated its highly computational capabilities for online ANC adaptation. Specifically, the existing PSO-based ANC systems are mainly composed of P adaptive filters to minimize the noise signal by generating an antinoise signal, as shown in Figure 2. In a PSO-based ANC system, the noise source, which travels via primary path, is obtained by the reference microphone. Once the source is captured, this signal is sent to one of the P adaptive filters [23,24]. After performing the adaptive filtering process, the output is sent to the loudspeaker to generate the antinoise signal. In addition, the ANC system employs the error microphone to sense the residual noise. Here, this noise is processed by P adaptive filters to adjust their weights at each generation, where a generation is defined as a time to perform swarming operations.

As mentioned above, diverse PSO-based ANC systems have been proposed to improve the noise cancelation. For example, in [23] presented a conditional reinitialization of the particle vectors of the PSO algorithm to process time-varying plants of the ANC systems. Specifically, the authors intend to improve the convergence properties, specially when abrupt changes in primary and/or secondary paths occurs since the conventional PSO is ineffective to regain convergence under these conditions. Another work was presented in [25], in which a functional-link-artificial-neural-network-based (FLANN) multichannel nonlinear ANC system is trained by a PSO algorithm. In [26], the authors evaluated the performance of an ANC system by using the PSO algorithm and Wilcoxon norm. In [27], presented an adaptive narrowband ANC system based on PSO algorithm. Specifically, their approach avoids the use of a synchronization signal. In [28], the PSO algorithm is used to attenuate engine noise inside vehicle enclosures. Therefore, these approaches intend to solve problems related to time-variations of the path, multi-channel ANC systems, nonlinear ANC systems. Recently, in [24] presented a new variant of the quantum-behaved PSO algorithm. This approach intends to decrease its computational complexity and improve the noise reduction level. However, this approach achieves a slight improvement in terms of noise reduction level compared with existing approaches. It should be noted that the noise reduction level is one of the most important factor to be considered since it determines the efficiency of noise cancellation. Therefore, a big challenging task is the development of an efficient PSO-based ANC system which can efficiently reduce the noise and it can be seen as a prominent solution in real-world ANC applications.

2. Proposed PSO-Based ANC System with Switching Filter Selection

Inspired by convex combination structures based on gradient descent algorithms [8,9,10,11,12,13], we present a new variant of the PSO-based ANC structure. Specifically, we introduce for the first time an ANC structure, in which two PSO algorithms are used, to significantly improve the convergence properties. As a consequence, the proposed structure can be used in real acoustic environments. Here, each PSO algorithm has been parameterized under different acceleration coefficients and inertia weights to create a global filter whose performance is better than the PSO-based ANC systems. During the last years, several studies have demonstrated that the convergence behavior of PSO algorithm is determined by the values of the inertia weight and the acceleration coefficients [29,30,31]. Although PSO algorithm is considered to be robust in many applications, it fails to find a balance between exploration and exploitation processes [32].

As can be observed from Figure 3, we use two PSO algorithms to create a new PSO-based ANC system. The first $PS{O}_1$ algorithm is dedicated to perform an efficient exploration process by providing a high convergence speed, while the second $PS{O}_2$ algorithm is used to efficiently perform the exploitation process by offering a low steady-state MSE. To control the selection between these two algorithms, we employ a selection criteria based on the evaluation of past samples of the error signal $\mathbf{e}\left(n\right)$. In this way, we evaluate the degree of change of the error signal over time. The use of these strategies have allowed us to create a PSO-based adaptive filter with switching filter selection to improve convergence properties compared with existing approaches.

Figure 4 shows the control process to perform the proposed PSO-based ANC system by executing the following steps:

• Specify the control parameters. Here, the PSO-based ANC system is composed of j population matrices $\mathbf{W}$, where each matrix $\mathbf{W}$ is composed of P adaptive filters and each particle represents an adaptive filter, as shown in Equation (1). Therefore, the order N of each adaptive filter defines the dimension of each particle. The whole population is defined as follows:

  $${\mathbf{W}}_j=\left[\begin{array}{cccc}{w}_1^1& w_1^2& \cdots & w_1^P\\ w_2^1& w_2^2& \cdots & w_2^P\\ ⋮& ⋮& \ddots & ⋮\\ w_N^1& w_N^2& \cdots & w_N^P\end{array}\right]$$

  (1)

  where $j\in \{1,2\}$ is the parameter to select between $PS{O}_1$ and $PS{O}_2$.
• Create an initial population. At the first iteration $n=1$, the position ${\mathbf{w}}_i\left(n\right)$ of each particle is initialized for both populations, where $i=1,2,\cdots ,P$.

  $${\mathbf{w}}_i\left(n\right)=(ub-lb)·\mathbf{r}+lb$$

  (2)

  where $\mathbf{r}$ is a gaussian process of length $(1,N)$ with zero mean and variance = 0.25. The values $lb=-1$ is a lower bound and $ub=1$ is a upper bound, since the impulse response of the primary path is normalized.
• Perform signal filtering and signal counteraction. The signal $\mathbf{x}\left(n\right)$ travels via primary path to generate the signal $\mathbf{d}\left(n\right)$. In addition, the signal $\mathbf{x}\left(n\right)$ is processed by the PSO algorithm to update its coefficients. Here, each particle is used to compute a single filter at any generation time n. Once the coefficients are updated, signal $\mathbf{y}\left(n\right)$ is produced and it travels via the secondary path $\mathbf{s}\left(z\right)$ to generate the anti-noise signal $\widehat{\mathbf{y}}\left(n\right)$. On the other hand, the calculation of signal counteraction or also called residual noise $\mathbf{e}\left(n\right)$ is given by

  $${\mathbf{e}}_i\left(n\right)=\mathbf{d}\left(n\right)+{\widehat{\mathbf{y}}}_i\left(n\right)$$

  (3)
• Evaluate fitness function. The mean squared error (MSE) of each P error signal, which represents the fitness function of each adaptive filter, is used by two PSO algorithms to determine the best position. Here, the fitness value $f_i$ of the position $\mathbf{w}$ can be obtained as follows:

  $$f_i\left(n\right)=\frac{1}{N}\sum _{k=1}^T{e}_i^2\left(k\right)$$

  (4)

  where T depicts the total number of iterations
• Select the best performance. As can be observed from Figure 3, the proposed ANC system is composed of two PSO algorithms ($PS{O}_1$ and $PS{O}_2$). Here, each particle of the PSO algorithms has a position $\mathbf{w}$ and a velocity $\mathbf{v}$. Equations (5) and (6) are used to update these parameters as follows:

  $${\mathbf{v}}_{j,i}\left(n\right)={\varphi }_j·{\mathbf{v}}_{j,i}(n-1)+c_1·r_1[{\mathbf{w}}_{pbes{t}_{j,i}}-{\mathbf{w}}_{j,i}\left(n\right)]+c_2·r_2[{\mathbf{w}}_{gbest}-{\mathbf{w}}_{j,i}\left(n\right)]$$

  (5)

  $${\mathbf{w}}_{j,i}\left(n\right)={\mathbf{w}}_{j,i}(n-1)+{\mathbf{v}}_{j,i}\left(n\right)$$

  (6)

  where $c_1$ and $c_2$ are the acceleration coefficients, $r_1$ and $r_2$ are the vectors of random numbers of length N, ${\mathbf{w}}_{pbest}$ and ${\mathbf{w}}_{gbes{t}_{j,i}}$ are the personal best position and global best position, respectively, and ${\varphi }_j$ is the inertia weight. In addition, $r_1$ and $r_2$ are defined in the interval [0, 1] and the inertia weight can be obtained as follows:

  $${\varphi }_j={\varphi }_{j_{max}}-\frac{t·({\varphi }_{j_{max}}-{\varphi }_{j_{min}})}{T}$$

  (7)
  To select between two PSO algorithms ($PS{O}_1$ and $PS{O}_2$), we propose a selection criteria based on the evaluation of past samples of the error signal to verify significant changes in the MSE level. Equations (8) and (9) are used to evaluate the past samples of the error as follows:

  $${\gamma }_1=|e_{gbest}^2(n-1)-e_{gbest}^2(n-2)|$$

  (8)

  $${\gamma }_2=|e_{gbest}^2\left(n\right)-e_{gbest}^2(n-1)|$$

  (9)
  Hence, the updates of the position and velocity of each PSO algorithm are ruled by the following equations:

  $${\mathbf{w}}_{1,i}\left(n\right)=\left\{\begin{array}{cc}{\mathbf{w}}_{1,i}(n-1)+{\mathbf{v}}_{1,i}\left(n\right),\hfill & \mathrm{if}{\gamma }_1\ge k_1\mathrm{and}{\gamma }_2\le k_2\hfill \\ {\mathbf{w}}_{2,i}\left(n\right),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

  (10)

  $${\mathbf{v}}_{1,i}\left(n\right)={\varphi }_1·{\mathbf{v}}_{1,i}(n-1)+c_1·r_1[{\mathbf{w}}_{pbes{t}_{1,i}}-{\mathbf{w}}_{1,i}\left(n\right)]+c_2·r_2[{\mathbf{w}}_{gbest}-{\mathbf{w}}_{1,i}\left(n\right)]$$

  (11)

  $${\mathbf{w}}_{2,i}\left(n\right)=\left\{\begin{array}{cc}{\mathbf{w}}_{1,i}\left(n\right),\hfill & \mathrm{if}{\gamma }_1\ge k_1\mathrm{and}{\gamma }_2\le k_2\hfill \\ {\mathbf{w}}_{2,i}(n-1)+{\mathbf{v}}_{2,i}\left(n\right),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

  (12)

  $${\mathbf{v}}_{2,i}\left(n\right)={\varphi }_2·{\mathbf{v}}_{2,i}(n-1)+c_3·r_1[{\mathbf{w}}_{pbes{t}_{j,i}}-{\mathbf{w}}_{2,i}\left(n\right)]+c_4·r_2[{\mathbf{w}}_{gbest}-{\mathbf{w}}_{2,i}\left(n\right)]$$

  (13)

  where $k_1<k_2$, $k_1$ is a small number close to zero, and $k_2$ is chosen by trial and error [23]. In general, Equations (10) and (12) indicate when the PSO algorithms are updated. If the degree of change of the MSE level is larger than $k_1$, the $PS{O}_1$ algorithm must be updated and transfer its coefficients to the $PS{O}_2$ allowing the system to update only one algorithm at a time. On the other hand, if according to Equation (12), $PS{O}_2$ must be updated, the new coefficients are transferred to $PS{O}_1$. With the proposed scheme, a fast convergence speed using $PS{O}_1$ algorithm is attained, and the MSE level is reduced by employing $PS{O}_2$ algorithm.
• Update the personal and global best position. In this step, two variables are selected, the first one is called personal best position, ${\mathbf{w}}_{pbes{t}_{j,i}}\left(n\right)$. To find the value of ${\mathbf{w}}_{pbes{t}_{j,i}}\left(n\right)$, we perform a comparison between the current value of $f_i\left(n\right)$ and the value of $f\left[{\mathbf{w}}_{pbes{t}_{j,i}}(n-1)\right]$ as follows:

  $${\mathbf{w}}_{pbes{t}_{j,i}}\left(n\right)=\left\{\begin{array}{cc}{\mathbf{w}}_{j,i}\left(n\right),\hfill & \mathrm{if}{f}_i\left[{\mathbf{w}}_{j,i}\left(n\right)\right]<f\left[{\mathbf{w}}_{pbes{t}_{j,i}}(n-1)\right]\hfill \\ {\mathbf{w}}_{pbes{t}_{j,i}}(n-1),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

  (14)
  To compute Equation (14) at the first generation, the ${\mathbf{w}}_i\left(1\right)$ is defined as ${\mathbf{w}}_{pbes{t}_i}\left(1\right)$. The second variable is denominated as global best position, ${\mathbf{w}}_{gbest}$. To calculate this variable, we compare the result of $f\left[{\mathbf{w}}_{pbes{t}_{min}}\left(n\right)\right]$ with the evaluation of the best global position $f\left[{\mathbf{w}}_{gbest}(n-1)\right]$, where ${\mathbf{w}}_{pbes{t}_{min}}\left(n\right)={\mathbf{w}}_{pbes{t}_g}\left(n\right)$, $g=argmi{n}_{1\le i\le P}\left\{{\mathbf{w}}_{pbes{t}_{j,i}}\left(n\right)\right\}$. The computation of the global best position, ${\mathbf{w}}_{gbest}$ is given by

  $${\mathbf{w}}_{gbest}\left(n\right)=\left\{\begin{array}{cc}{\mathbf{w}}_{pbes{t}_{min}}\left(n\right),\hfill & \mathrm{if}f\left[{\mathbf{w}}_{pbes{t}_{min}}\left(n\right)\right]<f\left[{\mathbf{w}}_{gbest}(n-1)\right]\hfill \\ {\mathbf{w}}_{gbest}(n-1),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

  (15)
• Update population. In this step, we use Equations (5) and (6) to update the velocity and position of each particle, respectively.

In general, $PS{O}_1$ algorithm is responsible of performing the exploration task. Once the adaptive filter reaches the steady-state, the $PS{O}_2$ algorithm executes the exploitation tasks to achieve a high noise reduction.

3. Simulation Results

Here, we simulate the proposed PSO-based ANC system (single channel) with switching filter selection in MATLAB to demonstrate its effectiveness in terms of convergence properties. Specifically, we develop a set of simulation experiments under the following conditions:

1.

We model the primary and secondary paths as finite impulse response (FIR) filters using 256 and 128 coefficients [33], respectively.

2.

The MSE was evaluated at error microphones.

3.

We consider a white Gaussian noise signal with a zero mean as input signal $x\left(n\right)$.

4.

Table 1. Acceleration coefficients and inertia weights used in all experiments.

${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{\varphi }}_{1_{\mathit{max}}}$	${\mathit{\varphi }}_{1_{\mathit{min}}}$	${\mathit{\varphi }}_{2_{\mathit{max}}}$	${\mathit{\varphi }}_{2_{\mathit{min}}}$
1.0599	1.0599	1.0993	1.0993	0.79	0.0009	0.9449	0.01649

shows the selected acceleration coefficients and inertia weights to produce the best convergence properties. As is well known, acceleration coefficients are static. Therefore, the optimal values need to be found empirically. In general, the use of low values for $c_1$ and $c_2$ produces smooth particle trajectories [34]. Here, the proper selection of these values have allowed the particles to roam close from good regions which produces more acceleration. Based on [34], we select of the inertia weights (${\varphi }_{max}$) and values for $c_1$ and $c_2$, as follows:

$${\varphi }_{max}>{\displaystyle \frac{1}{2}}(c_1+c_2)-1$$

(16)

According to previous equation, if the condition is not satisfied, the PSO algorithm may diverge or could present cyclic behavior.

To validate the performance of the proposed algorithm, we made a comparison between this bio-inspired algorithm and the conventional FXLMS algorithm. In addition, we carried out several experiments by varying the filter’s order, number of particles and changing the secondary/primary path during the filtering process to validate the performance of the proposed PSO-based ANC system.

• Comparison between the proposed algorithm and the FXLMS algorithm. To evaluate the MSE level, we use a population of 100 particles. To verify the tracking capabilities of the algorithms, we induce an abrupt change in the middle of the generations (1500) by multiplying the secondary path by −1. The FXLMS step-size was adjusted by trial and error; the value producing the fastest convergence speed was selected.As can be observed from Figure 5, the proposed algorithm exhibits higher convergence speed when compared to the conventional FXLMS. Besides, the FXLMS requires an estimation of the secondary path to avoid instability and low accuracy. In contrast, this estimation is not required when the PSO algorithm is used.
• Effect of changing the order of adaptive filter. To evaluate the MSE level, we use a population of 100 particles and different adaptive filter’s length (N from 20 to 50), as shown in Figure 6. To verify the tracking capabilities of the proposed PSO-based ANC system, we induce an abrupt change in the middle of the generations (1500) by multiplying the secondary path by −1. As can be observed from Figure 6, the MSE level is high when low number of coefficients are used. Otherwise, the use of large number of coefficients improves the MSE level by paying a penalty in terms of convergence speed. In addition, the use of high order adaptive filters demands a high computational cost by improving significantly the MSE level, where this parameter is crucial in practical ANC systems since it determines the level of noise reduction.

• Effect of changing the number of particles of the PSO algorithm. According to previous results, we obtained the higher level reduction when 50 filter coefficients are used. When selecting a number higher than 50, the performance of the algorithm does not change and remains the same. Therefore, in this experiment, we use this number of filter coefficients and vary the number of particles to verify how the performance of the proposed PSO-based ANC system is affected. As can be observed from Figure 7, the number of particles is an important factor to be considered, especially when high level of noise reduction and fast convergence speed are required. Hence, the use of a large number of particles and high order filters guarantees good convergence properties at the cost of increasing the computational complexity.

• Comparison between the proposed PSO-based ANC system with switching filter selection and existing approaches. In addition to previous simulations, we simulate the proposed PSO-based ANC system and existing approaches [23,24] to make a consistent comparison between them in terms of the MSE level.

  Table 2. Acceleration coefficients and inertia weights of the PSO and QPSO algorithms.

  Approach	Algorithm	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{\varphi }}_{\mathit{min}}$	${\mathit{\varphi }}_{\mathit{max}}$
  Rout et al. [23]	PSO	1	1	0.0009	0.79
  Yu et al. [24]	QPSO	1.5	1.5	0.5	1

  shows the fixed parameters of the existing approaches, such as acceleration coefficients and inertia weights, which were selected to provide the best performance of these approaches. Here, the primary and secondary path of the proposed PSO-based ANC system and existing approaches were multiplied by −1 at generation 750 to test their tracking behavior, as shown in Figure 8 and Figure 9, respectively. As can be observed, the proposed PSO-based ANC system achieves a higher MSE level when compared with existing approaches. It should be noted that the most recent approach based on quantum-behaved (QPSO-based ANC system) [24] achieves a slight improvement of the MSE level compared with the PSO-based ANC system [23] at the cost of increasing its computational complexity, as shown in

  Table 3. Comparison between the proposed PSO-based ANC system and existing approaches [23,24,35] in terms of computational complexity.

  Approach	Algorithm	Additions	Multiplications	Logarithms
  Adapted from Burgess et al. [35]	FXLMS	$2N+M$−1	$N+M+1$	0
  Adapted from Rout et al. [23]	PSO	$5NP+2$	$5NP+2$	0
  Adapted from Yu et al. [24]	QPSO	$5NP+2P+3$	$4NP+3$	N
  This work	PSO	$5NP+7$	$5NP+5$	0

  . As can be observed, the proposed PSO-based ANC system with switching filter selection reaches similar convergence speed compared with the QPSO-based ANC system. However, the proposed approach exhibits better MSE-level by expending lower computational cost. On the other hand, we include the FXLMS algorithm in the comparison. Obviously, the FXLMS algorithm exhibits lower computational cost compared with the existing bio-inspired approaches. Nonetheless, the FXLMS algorithm presents the lowest convergence speed.

• Power Spectrum analysis. Apart from previous results, we obtained the residual noise at the error microphone by using two different reference signals. Figure 10 depicts the residual noise and the power spectrum when a multi-tonal signal is used as undesired-noise. As can be observed from this figure, if the adaptation process occurs, the noise at the error microphone decreases.
  Figure 11 depicts the residual noise and the power spectrum when a broadband noise is used as undesired-noise. As can be observed from Figure 11, the proposed algorithm achieves an average noise reduction of 11.54 dB by using broadband noise as a reference signal.

4. Conclusions

In this brief, we introduce for the first time a PSO-based ANC structure, in which two PSO algorithms are used. Here, these algorithms are initialized with different parameters and involves a selection criteria to create an ANC filtering structure to significantly improve the convergence properties in comparison with existing approaches. Therefore, we present two contributions:

1.

We take advantage of the complementary capabilities of two PSO algorithms to improve the MSE level.

2.

We propose a selection criteria to determine which PSO algorihm is used. Therefore, both algorithms are not executed at the same time during the whole adaptive processing. In this way, we intend to decrease the computational cost of the proposed ANC system since two algorithms are used. However, the proposed filtering scheme has a significantly improved steady-state MSE level.

Our experiments have demonstrated that the proposed PSO-based ANC filtering structure can regain convergence when abruptly changes occurs in the primary and secondary paths. Here, we use large paths, in contrast to previous studies [23,24], to recreate real acoustic environments which will potentially lead to the development of practical ANC applications in near future [36].