Design of delayless multi-sampled subband functional link neural network with application to active noise control

doi:10.1016/j.sigpro.2022.108757

Signal Processing

Volume 202, January 2023, 108757

https://doi.org/10.1016/j.sigpro.2022.108757 Get rights and content

Highlights

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The problem of designing a delayless multi-sampled multiband-structured subband functional link neural network (DMSFLNN) structure is addressed.
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This paper firstly proposes the DMSFLNN, which employs the multi-sampled multiband-structured subband adaptive filter (MSAF) scheme to accelerate the convergence speed of the trigonometric FLNN (TFLNN).
•
For the proposed algorithmic formulation over the DMSFLNN, the mean, mean-square stabilities and optimal step size are derived.
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Based on the DMSFLNN structure, the delayless multi-sampled multiband-structured filtered-s normalized LMS (NLMS) algorithm, called DMSFsNLMS, is devised to handle nonlinear active noise control (NANC) systems.
•
Monte-Carlo simulation results have been provided to demonstrate the effectiveness of the DMSFLNN filter and theoretical results.

Abstract

To accelerate the convergence speed of the functional link neural network (FLNN) particularly for colored input signals, this paper proposes a delayless multi-sampled multiband-structured subband FLNN (DMSFLNN) structure. Then, to update the weights of the DMSFLNN, a normalized subband adaptive algorithm is devised. Next, the stability conditions, optimal step size and computational complexity are investigated. Moreover, the proposed method is applied to the nonlinear active noise control, obtaining the delayless multi-sampled multiband-structured filtered-s normalized least mean square (DMSFsNLMS) algorithm. Finally, simulation results demonstrate that the proposed method improves the convergence speed of the FLNN.

Introduction

Neural networks (NNs), such as radial basis function NN, echo state network, and reaction-diffusion NN, have had a great deal of success in the study of many complex applications [1], [2], [3], [4], [5], for example, nonlinear system identification, communication channel equalization, and image encryption. Recently, functional link NNs (FLNNs) as a single-layer feedforward neural network with low computational complexity have attracted considerable interest in active noise control (ANC) [6], [7], [8], digital predistortion [9], and many other applications. According to different ways of functional expansion, the commonly used FLNN includes trigonometric-FLNN (TFLNN) [10], Legendre-FLNN (LFLNN) [11], Chebyshev-FLNN (CFLNN) [12] and so on.

In the past decades, to enhance the nonlinear identification capability of the FLNN, several different filter structures and methods have been proposed, such as pipelined filter [13], sparse representations [9], set-membership method [14] and collaborative TFLNNs [15], [16]. By means of an exponential sinusoid, an adaptive exponential TFLNN (AETFLNN) was reported in [8]. Using the infinite impulse response (IIR) scheme, a recursive adaptive sparse exponential TFLNN (RASETFLNN) was proposed in [17]. Furthermore, a generalized exponential TFLNN structure was developed in [18]. Recently, many improved variants of the FLNNs were proposed in the literature [19], [20], [21]. For example, a novel CFLNN based on correntropy was introduced in [19], which can achieve excellent performance in impulsive noise environments. However, because all these FLNNs expand the original input signal to a higher dimensional expansion space, it will bring an expanded input covariance matrix with larger eigenvalue spread. Its disadvantage is that using the well-known least mean square (LMS) strategy will lead to a slow rate of convergence.

It is known that the subband adaptive filter (SAF) structures increase the algorithmic convergence speed when the input signal involves the covariance matrix with large eigenvalue spread [22]. For the linear SAF, several subband structures had been designed, including conventional SAF [23], [24], [25], [26] and multiband-structured SAF (MSAF) mechanism [27], [28]. Compared with the conventional SAF method, the MSAF mechanism updates the fullband taps based on the subband error signals. Recently, several modified MSAFs have been presented, such as variable step size MSAF [29], [30], robust MSAF [31], [32], multi-sampled MSAF [33], etc. To increase the nonlinear processing capability of subband filtering, subband second-order Volterra filter (SBVF) and its proportionate version were developed [34], [35]. Because of using the conventional SAF structure, the SBVF updates the taps of each subfilter in the first- and second-order subband channels independently.

By generating an anti-noise with equal amplitude and opposite phase, the ANC system can cancel the undesired noise signal, such as in the headset, car and infant incubators. The linear subband ANC systems were developed in [36], [37], [38]. However, in real-time implementation, significant nonlinearities are frequently generated by the loudspeaker of the ANC system. The performance of linear ANC systems deteriorates when applied for such nonlinear ANC (NANC) scenario [6], [39]. To overcome this situation, the FLNN-based ANC has been increasingly employed, such as [8], [40]. Unfortunately, these FLNN-based ANC systems using the LMS method will be subjected to slow convergence speed for correlated noise.

In order to speed up the convergence of the TFLNN, in this work we will design a delayless multi-sampled multiband-structured subband FLNN (DMSFLNN) structure. The key ingredient is to employ the multi-sampled multiband-structured SAF scheme proposed in [33] in the TFLNN. In the meantime, to avoid the delay caused by subband filters, a delayless channel is introduced, which copies the multi-sampled MSAF taps. With the use of the principle of minimum disturbance over the DMSFLNN, an effective normalized subband adaptive algorithm is designed. In order to study the performance, stability conditions, optimal step size and computational complexity of the proposed algorithm are deduced. Additionally, for application to the NANC, a delayless multi-sampled multiband-structured filtered-s normalized LMS (NLMS) algorithm, called DMSFsNLMS, is designed. To assess the performance of the DMSFLNN, extensive simulation experiments are conducted in nonlinear system identification and NANC applications. Finally, to clarify the contribution in this work, the main contributions have been summarized as follows:

i)
This paper firstly proposes the DMSFLNN, which employs the multi-sampled MSAF scheme to accelerate the convergence speed of the TFLNN.
ii)
For the proposed algorithmic formulation over the DMSFLNN, the mean, mean-square stabilities and optimal step size are derived.
iii)
Based on the DMSFLNN structure, the DMSFsNLMS algorithm is devised to handle NANC systems.

The remainder of the paper is arranged as follows. The TFLNN is briefly reviewed in Section 2. The DMSFLNN is presented in Section 3. Section 4 provides an analysis of the stability conditions, optimal step size and computational complexity. Section 5 develops the DMSFLNN-based NANC algorithm. Section 6 provides Monte-Carlo simulations and the conclusion is drawn in Section 7. The symbols and abbreviations are listed as follows:

Section snippets

Overview of traditional TFLNN

Consider a nonlinear system identification problem. At time instant $k$ , the observed output signal $d (k)$ and known input signal $x (k)$ are related by $d (k) = f (x (k)) + υ (k)$ where $υ (k)$ denotes the additive background noise, $f (\cdot)$ is an unknown nonlinear system, and $x (k) = {[x (k), x (k - 1), \dots, x (k - D + 1)]}^{T}$ with $D$ being the delayed number.

To model and approximate the nonlinear system $f (\cdot)$ , the signal $x (k)$ in the TFLNN is expanded to a higher dimensional vector $s (k)$ : $\begin{matrix} s (k) & = [1, x (k), x (k - 1), \dots, x (k - D + 1), \\ \sin (π x (k)), \cos (π x (k)), \dots, \end{matrix}$

Proposed DMSFLNN and its weight update

In this section, to speed up the convergence of the TFLNN, we aim to design the DMSFLNN filter, which integrates the multi-sampled multiband-structured subband method into the TFLNN. In the DMSFLNN, a delayless channel is also introduced to avoid the delay caused by subband filters. Moreover, we provide an algorithmic formulation over the DMSFLNN.

Analysis of stability conditions, optimal step size and computational complexity over the DMSFLNN

In the section, we will analyze the stability conditions, optimal $μ$ and computational complexity for the proposed subband adaptive algorithm (11).

To begin with, we introduce a model as follows $d (k) = s^{T} (k) w_{opt} + υ (k)$ where $w_{opt}$ denotes the deterministic coefficients vector of the linear MSE estimator of $d (k)$ based on the trigonometric extended input vector $s (k)$ . Some commonly used assumptions on the signal $s (k)$ and the estimation error $υ (k)$ are introduced as below [33].

Assumption 1

The input sequence ${x (k)}$ is a

DMSFLNN-based ANC application

In this section, the DMSFLNN-based NANC system is designed, as depicted in Fig. 2. In this figure, $P (z)$ and $S (z)$ denote the transfer functions of the primary path and secondary path, respectively, $x (k)$ is the reference noise measured by a reference microphone, $d (n)$ is the primary noise to be cancelled, $\hat{d} (k)$ is the output signal of the secondary path at the cancellation point, $\hat{S} (z)$ is an estimate of $S (z)$ , and $\hat{w} (k)$ denotes the weight of the adaptive controller.

At time $k$ , the error signal is

Monte-Carlo simulation study

In this section, we provide Monte-Carlo simulation results to validate the usefulness of the DMSFLNN. The cosine-modulated filter banks are adopted in the subband structure[33], where the lengths of the prototype filter are $M = 64$ , 128, and 256 for $N = 8, 16, 32$ , respectively. The initial weight vector is a zero vector. All curves are generated by averaging the results of 100 Monte-Carlo runs.

Conclusion

In this paper, the DMSFLNN filter has been designed. By employing the multi-sampled MSAF method, the DMSFLNN filter can increase the convergence speed of the TFLNN in the case of colored input signals. The principle of minimum disturbance has been used over the DMSFLNN and then a subband algorithm has been constructed. The analysis of stability conditions, optimal step size and computational complexity has been undertaken for the proposed DMSFLNN filter. Moreover, the DMSFLNN-based NANC

Clarification of Relationship to Conference Paper

A part of the initial work in this SIGPRO submission was presented at the 52nd IEEE International Symposium on Circuits and Systems (ISCAS’2019) in Sapporo, Hokkaido, Japan in May 2019, which is also cited as Reference [1] in this SIGPRO submission.

In this SIGPRO submission, we have significantly expanded the ISCAS’2019 conference paper by including a lot of substantive new material. In more details, in this SIGPRO submission,

•
we show the mean stability and mean-square stability of the proposed

CRediT authorship contribution statement

Sheng Zhang: Conceptualization, Formal analysis, Methodology, Writing – original draft. Wei Xing Zheng: Methodology, Writing – review & editing. Hongyu Han: Validation, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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For the correlated input, the Volterra kernel-based least mean-square (LMS) algorithm in the time-domain exhibits a slow learning rate caused by the large eigenvalue spread of the input covariance matrix. To tackle such an issue, this paper develops a novel frequency-domain Volterra kernel-based filter, resulting in the periodic update constrained frequency-domain second-order Volterra normalized LMS (named as P-CFDSOV-NLMS1) algorithm. Subsequently, by using one- and two-dimensional discrete Fourier transforms (DFTs) simultaneously, another frequency-domain implementation and corresponding P-CFDSOV-NLMS2 algorithm are constructed. In contrast, the P-CFDSOV-NLMS1 scheme only requires one-dimensional DFT operations and takes advantage of the joint information between the block input vectors. Then, the mean and mean-square convergence behaviors of the P-CFDSOV-NLMS1 algorithm are investigated. Furthermore, the designed frequency-domain method is extended to three different widely complex-valued Volterra kernel-based models. Finally, computer simulations reveal that the suggested algorithms outperform the previously reported frequency-domain techniques in terms of convergence speed and tracking ability.
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Road noise is one of the dominant noise sources in pure electric vehicles. Active control for road noise is an attractive solution since this technique is suitable for controlling low-frequency noise. Feed-forward active road noise control (ARNC) systems utilize lots of reference signals and corresponding adaptive filters, and the control performance usually improves as the number of reference signals increases. To reduce the computational cost of ARNC systems so that more reference signals can be processed, we introduce the delayless subband structure into ARNC system and a delayless partial update subband with weighted reference signal normalized filtered-x least mean square (DPUS-WRS-NFxLMS) algorithm is proposed in this paper. Based on the traditional delayless subband algorithm, a partial subband update strategy is proposed to reduce the computational cost. Furthermore, the weighted reference signal is introduced for each subband filter. In this way, a modified vector is designed to assign different weights to different reference signal samples so that the convergence performance can be improved and the round-off error can be reduced. In addition, an adjustment strategy for the step size and reference signal weights is designed to make the algorithm have a better balance between the convergence rate and steady-state behavior. Finally, a purely time-delayed secondary path is constructed as the estimated secondary path to avoid the negative impact of the magnitude-frequency characteristics of the actual secondary path. The simulation results prove that the proposed algorithm achieves better control results than the traditional algorithms with lower computational complexity.
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^☆: This work was supported in part by the National Natural Science Foundation of China under Grants 61801401 and 61701331. A part of the initial work in this paper was presented at the 52nd IEEE International Symposium on Circuits and Systems (ISCAS’2019), Sapporo, Hokkaido, Japan, May 2019 [1].

View full text

Design of delayless multi-sampled subband functional link neural network with application to active noise control☆

Highlights

Abstract

Introduction

Section snippets

Overview of traditional TFLNN

Proposed DMSFLNN and its weight update

Analysis of stability conditions, optimal step size and computational complexity over the DMSFLNN

DMSFLNN-based ANC application

Monte-Carlo simulation study

Conclusion

Clarification of Relationship to Conference Paper

CRediT authorship contribution statement

Declaration of Competing Interest

Signal Process.

Signal Process.

Signal Process.

Signal Process.

Neural Netw.

Signal Process.

Appl. Acoust.

Digit. Signal Process.

Signal Process.

IEEE Trans. Audio Speech Lang. Process.

Syst. Signal Process.

Subband adaptive filtering algorithm over functional link neural network

Proc. 52nd IEEE Int. Symp. Circuits Syst.

Polynomial Signal Processing

Impulsive synchronization of reaction diffusion neural networks with mixed delays and its application to image encryption

IEEE Trans. Neural Netw. Learn. Syst.

Frequency domain exponential functional link network filter: design and implementation

Signal Process.

Design of adaptive exponential functional link network-based nonlinear filters

IEEE Trans. Circuits Syst. I Reg. Pap.

Adaptive Pattern Recognition and Neural Networks

Identification of nonlinear dynamic systems using functional link artificial neural networks

IEEE Trans. Syst. Man Cybern. B Cybern.

Nonlinear dynamic system identification using Chebyshev functional link artificial neural networks

IEEE Trans. Syst. Man Cybern. B Cybern.

Functional link adaptive filters for nonlinear acoustic echo cancellation

IEEE Trans. Audio Speech Lang. Process.