ReviewA survey on active noise control in the past decade–Part II: Nonlinear systems☆
Introduction
The celebrated filtered-x least mean square (FxLMS) algorithm and its variants have been successfully applied in active noise control (ANC) systems. However, in practical situations, the primary path and secondary path may be nonlinear [1]. Moreover, the reference noise arises from dynamic systems and as such, the noise may be a nonlinear and deterministic or stochastic, colored, and non-Gaussian signal [2]. In these cases, linear ANC techniques cannot fully utilize the coherence in the noise and achieve suboptimal performance.
To address these limitations, several nonlinear ANC (NLANC) algorithms were proposed, yielding a more stable performance and lower noise level [3], [12], [21]. It should be emphasized that some NLANC methods mentioned in this paper first appeared before 2009. The research on NLANC dates back to 1995, 59 years after the concept of ANC was proposed by Lueg [22]. In this year, Snyder and Tanaka developed ANN for NLANC problem, and so far a large number of NLANC algorithms have been proposed. The milestones in the progress of NLANC before 2009 are presented in Table 1. Survey papers on NLANC techniques have been reported [2]. However, these surveys do not cover the literature since 2013. To complete the review of NLANC techniques and include the advances, in this paper we review the research progress of NLANC models in recent years.
The Volterra filter is gaining importance in NLANC. According to the Stone-Weierstrass approximation theorem, the Volterra filter is a ‘universal approximator’, i.e., the filter can arbitrarily approximate discrete-time with fading memory, time-invariant, continuous, finite-memory, nonlinear system [23]. The Volterra filter belongs to linear-in-the-parameters (LIP) filters, where the output depends linearly on the parameters of the filter itself. Note that this type of filter becomes computationally expensive when a large number of coefficients are required [23]. To solve this problem, the second-order Volterra (SOV) series and the third-order Volterra (TOV) series are often used. For example, in [24], [25], the Volterra filtered-x-based algorithms that use of SOV were developed for active impulsive noise control (AINC) problem.
The NLANC based on an artificial neural network (ANN) method was first studied, which provides the capacity to maintain causality within the control scheme. From 1997 to 2008, many papers discussed the usage of radial basis function (RBF) networks, recurrent neural networks (RNNs), and fuzzy neural networks for NLANC (see Table 1). In the last decade, some attempts were made by using an ANN as the controller [26], [27], [28]. A neural controller equipped with the filtered-u least mean square (FuLMS) algorithm with correction terms momentum was proposed in [29]. The correction terms momentum can efficiently prevent the system from unstable poles and can enhance the performance as compared with the conventional leaky FuLMS algorithm. Regrettably, the above mentioned ANNs have heavy computational complexity which may hinder their practical applications [2].
The functional link ANN (FLANN) exploits the single layer of ANN to obtain reliable estimation performance with acceptable error level [30]. The FLANN filter is also a type of LIP filter, which shares a unified implementation with the Volterra filter. Moreover, the linear adaptive filtering algorithms can easily extend to this framework with favorable implications on the algorithm stability and computational complexity [31]. Over the past decade, a number of filtered-s least mean square (FsLMS) algorithms were proposed by employing FLANN, such as fast FsLMS [30], robust FsLMS (RFsLMS) [32]. Most of these algorithms use trigonometric functional expansion as the basis function, and a small number of algorithms that consider using Chebyshev [33], Fourier [34], and Legendre expansions [35] were also developed.
To compensate for the strong nonlinearity of the system with saturated signals, several bilinear algorithms with reduced computational complexity were proposed, which employ the cross-terms based on both feedback and feedforward polynomials and they can model NLANC systems with shorter filter length [36], [37]. In [38], a new bilinear filter was proposed, which uses FLANN to expand the input vector and then adapts coefficients via bilinear filter to obtain excellent nonlinear modeling capability.
In this context, the particle swarm optimization (PSO) algorithm is an effective biologically inspired meta-heuristic algorithm, which performs by a competition and cooperation mechanism between the particle swarm and thus it retains the global search strategy of population [39]. Such algorithm has been extensively studied in ANC technique, especially for nonlinear system [40]. The most obvious advantage of the PSO-ANC algorithm is that it can reduce noise without online estimation of the secondary path [40], [41]. In addition, other heuristic ANC algorithms were also developed, see [42] and references therein.
Since 2009, several attractive methods were proposed for NLANC. For performance improvement, the kernel adaptive filter (KAF) algorithm and spline adaptive filter were introduced [43], [44]. The KAF recasts the input data into a high-dimensional feature space via a reproducing kernel Hilbert space (RKHS) and as such, the linear adaptive filter is applied in the feature space [43]. In contrast, the spline adaptive filter establishes a nonlinear mathematics model by an adaptive look-up table (LUT) in which the control points are interpolated by a local low-order polynomial spline curve [44]. An important merit of the spline adaptive filter is that its computational complexity is much lower as compared to other traditional NLANC algorithms [45]. Motivated by distributed algorithms, some nonlinear distributed algorithms were proposed by integrating FLANN into distributed ANC algorithms, which is important for ensuring that the wireless acoustic sensor networks (WASNs) can combat nonlinear distortions [46].
Initial studies have shown that the KAF is limited by high computational complexity. Specifically, the input data of KAF must be a dictionary, where every new data input that arrives is used to calculate the filter output [47]. To overcome this limitation, the quantized scheme [48] and the set-membership scheme [49] can be taken into account to start new research on the field of NLANC. For example, the adaptive step size scheme and data-selective update strategy proposed in [49] can be directly combined with the KAF in NLANC. On the other front, Internet of Things (IoT) has been shown to be compatible with communication and signal processing techniques for creating a smart world. In [50], the concept of ANC was introduced to IoT, generating MUTE leverages, for environmental noise cancellation. In [51], the concept system of ANC-IoT is tested. All these works have demonstrated the feasibility and great potential of the combination of the IoT and ANC.
The goal of this survey is to provide a comprehensive overview of the latest NLANC research results and innovations. We also discuss the applications of ANC techniques that have been introduced in many fields, such as functional magnetic resonance imaging (fMRI) acoustic noise control [52] and transformer noise control [53]. Moreover, we present some remarks on future research directions for ANC. The papers that have been published since 2009 are considered in this survey. In particular, the main problems that the NLANC algorithms should address are presented in Section 2. The modeling method of NLANC is discussed in Section 3, where Volterra ANC algorithms, FLANN-based algorithms, and bilinear ANC algorithms are used as examples. Section 4 focuses on evolutionary-computing-based ANC algorithms. In Section 5, some novel NLANC methods emerging in the past decade are investigated. Some important implementation issues and future research challenges of ANC are discussed in Sections 6 and 7, respectively. Finally, conclusions are provided in Section 8. All the brackets follow the order .
Section snippets
Problem formulation
In practical situations, three types of nonlinearities may exist in ANC systems. First of all, the primary noise at the cancellation point may exhibit nonlinear effects. For instance, when the primary noise propagates in a duct with very high sound pressure, the transfer function of can be well modeled by nonlinear function [33], [54].
On the other hand, the secondary plant models the signal converters (A/D and D/A), power amplifiers, and transducers (actuators or speakers). The
NLANC algorithms
The modeling accuracy of NLANC systems is crucial for meeting the performance requirements of noise reduction level. This section presents different approaches for the design of NLANC.
Heuristic-based ANC algorithms
Both linear and nonlinear ANC tasks can be solved by transforming into a global optimization problem. Especially for NLANC systems, such problems are well-suitable for considering global optimization problems, because the nonlinear secondary path may cause the conventional adaptive filter to suffer from the local minima trapping phenomenon in non-convex or non-deterministic polynomial (NP)-hard problems, and it turns out the degraded reduction performance. The first heuristic algorithm used to
Novel NLANC methods emerging in past decade
In this section, we present some novel NLANC approaches, such as spline ANC algorithms, KAF ANC algorithms, and nonlinear distributed ANC algorithms, proposed in the last ten years.
Recent implementations and applications of ANC
In this section, the application development of ANC in the past decade is discussed. Before we start, it should be highlighted that many applications have been developed before last decade (see time line of ANC applications in Table 5). These works have laid the foundation for the implementations and applications of ANC in the past ten years.
Future research challenges
Some future challenges of ANC have been discussed in [56], [197]. In this section, we present a number of promising directions for future ANC research from other perspectives.
Conclusion
In this survey paper, we have discussed state-of-the-art methods related to NLANC technique, including FsLMS-based algorithms, distributed extension and selected applications. We have also summarized FLANN approaches as well as the Chebyshev, EMFN, and LN filters. The spline adaptive filter, kernel adaptive filter, and nonlinear distributed ANC algorithms are the new nonlinear modeling approaches considered for NLANC. Due to technical limitations, these open areas need further investigations of
Declaration of Competing Interest
The authors declare no conflict of interest.
Acknowledgment
The authors would like to thank the associate editor and the anonymous referees for their valuable comments.
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The work is supported by the National Science Foundation of P.R. China under Grant 61901285, 61901400, and 61701327, Sichuan Science and Technology Fund under Grant 20YYJC3709, China Postdoctoral Science Foundation under Grant 2020T130453, and Sichuan University Postdoctoral Interdisciplinary Fund.