## 1 Introduction  
Active noise control (ANC) is a method for removing unwanted noise in a specific area and is based on controlling a second speaker to generate an interference wave. The first patent on ANC was granted to a German engineer in 1936 [1]. Since its first patent, ANC has continually received considerable interest, with numerous related studies conducted. To apply ANC under changing environmental conditions and noise characteristics, a feedforward method has been used to process signals quickly, with adaptive filters used to provide effective responses to changes. A widely used adaptive algorithm for ANC systems is the filtered least-mean-square method [2], derived by Window [3]. As shown in Figure 1, a microphone measures a reference noise \(\mathbf{x}(n)\), and a control speaker generates an interfering wave signal \(\mathbf{y}(n)\) through an adaptive filter \(\mathbf{W}(z)\). The noise and control signals are summed over each primary path \(\mathbf{P}(z)\) and secondary path \(\mathbf{S}(z)\). The reference noise passes through the primary path and becomes the output noise \(d(n)\). The noise residue is measured using an error microphone. The filtered least-mean-square (FxLMS) algorithm performs an adaptation process on the filter \(\mathbf{W}(z)\), generating \(\mathbf{y}(n)\) to minimize the noise residue \(e(n)\). The LMS algorithm is given as follows:\[\mathbf{w}(n+1)=\mathbf{w}(n)+\mu e(n)[\mathbf{x}\mathbf{s}(n)*\mathbf{x}(n)], \tag{1}\]  
where \(\mu\) is the adaptation step size parameter, \(\mathbf{w}(n)\) is the adaptive weight vector, \(\mathbf{x}(n)\) is the reference noise signal vector, and \(\mathbf{x}\mathbf{s}(n)\) is the filtered signal from the estimated secondary path filter \(\hat{S}(z)\)[2]. The FxLMS algorithm has been widely used owing to its advantages of low computational complexity, robust performance, and simplicity. The algorithm must converge within a short time to provide efficient noise reduction performance. Many FxLMS algorithms have been proposed with the aim of improving the available convergence conditions and speed [4; 5; 6]. Based on theoretical modeling, the necessary and sufficient condition for the convergence of the FxLMS has been derived. In practice, impulsive and multi-tonal noises (which present many difficulties when performing ANC with FxLMS) occur together with the stationary components. A novel secondary path modeling system targeting multiple tonal disturbances has been presented to solve these problems [7]. The ANC performance for impulsive noises has also been enhanced to improve the robustness of FxLMS algorithms [8]. To improve the convergence of ANC, researchers have proposed applying artificial intelligence using neural networks, which have been applied to non-linear ANC to improve their performance [9; 10; 11]. Recently, deep neural networks have become a popular field of study and their application range is becoming increasingly diverse. The increased interest in neural networks has led to enhancements in their computational speed, ease of use, and accuracy. Various neural networks have recently been applied in ANC. [PERSON] et al. used recurrent fuzzy neural networks to propose adaptive non-linear noise-control approaches [12]. Low-frequency noise generated from multi-source railway vehicles was cancelled through a convolution fuzzy neural network [13]. Through the use of a 10-layer extended convolutional neural network (CNN) on a field-programmable gate array, a real-time streaming feedforward ANC system for in-ear headphones was shown in a practical application scenario [14]. A hybrid selective fixed-filter active noise control (SFANC) and filtered-X normalized least-mean-square (FxLMS) approach was proposed to overcome the adaptive algorithm's slow convergence and provide a better noise reduction level using deep learning [15]. Another study presented a neural-based FxLMS with an error backpropagation algorithm to cancel non-linear broadband noise in an ANC system [16; 17].