# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 1 Introduction  
Noise control encompasses both active noise control (ANC) approaches and passive noise control (PNC). Passive noise control primarily involves addressing noise through the manipulation of sound waves and acoustic isolation materials or structural design. This method is highly effective for controlling higher-frequency noise but has several drawbacks when it comes to low-frequency noise control. The characteristics of active noise reduction and passive noise reduction are outlined in Table 1.  
Therefore, in recent years, active noise control systems have gained increasing favor due to their compact size and excellent low-frequency noise reduction capabilities. They utilize active noise control algorithms to generate anti-noise signals, which are emitted through acoustic devices such as speakers to eliminate unwanted noise. The two core components of the ANC system are adaptive filters and control algorithms. The coefficients of the adaptive filter are iteratively updated according to the rules of the control algorithm. The filter, guided by the control algorithm, produces a secondary sound source that is equal in magnitude but opposite in phase to the primary noise, resulting in destructive interference to cancel out the primary noise. In 1933, German physicist [PERSON] first applied active noise cancellation to the treatment of noise at lower frequencies in pipelines, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Characteristics** & **Active Noise Reduction** & **Passive Noise Reduction** \\ \hline Real-time & Faster response and adaptation & Relatively slower response \\ Adaptability & Automatically adjusts parameters & Typically lacks adaptability \\ Effective Noise Reduction & Effectively suppresses various & Limited effectiveness in handling \\  & & types of noise & complex noise patterns \\ \multirow{2}{*}{Power Consumption} & \multirow{2}{*}{Requires additional power} & Relatively lower \\  & & power consumption \\ \multirow{2}{*}{Complexity} & \multirow{2}{*}{Algorithm and hardware} & Material or structure \\  & & Relatively simple implementation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristics of active noise reduction and passive noise reduction. implementing active noise control using the least mean square (LMS) algorithm [1]. In 1981, [PERSON] introduced the filtered-x least mean square (FxLMS) algorithm to address phase shift in the secondary path. Since then, FxLMS has become a classic algorithm in active control methods [2]. Figure 1 illustrates the block diagram of a feedforward ANC system. Where the reference signal is denoted as \(\mathbf{x}(n)=[\mathbf{x}(n),\,\mathbf{x}(n-1),\ldots,\,\mathbf{x}(n-L+1)]^{ \mathrm{T}}\), \(\mathbf{P}(\mathbf{z})\) signifies the primary path, and the secondary route is represented by \(\mathbf{S}(z)\). The adaptive filter is used for obtaining the estimation of the secondary path, denoted as \(\widehat{\mathbf{S}}(z)\), through offline or online modeling techniques. When the filter length is \(L\), the adaptive filter, denoted as \(\mathbf{W}(z)\), has a weight vector represented as \(\mathbf{w}(n)=[\mathbf{w}_{0}(n),\,\mathbf{w}_{1}(n),\,\ldots,\,\mathbf{w}_{L -1}(n)]^{\mathrm{T}}\). The represented output response of the filter is \(\mathbf{y}(n)\):  
\[\mathbf{y}(n)=\mathbf{w}^{\mathrm{T}}(n)\mathbf{x}(n) \tag{1}\]  
The expression for the error signal is as follows:  
\[e(n)=d(n)-\mathbf{y}_{\mathrm{f}}(n)=d(n)-\mathbf{y}(n)*\mathbf{s}(n) \tag{2}\]  
where \(\mathbf{s}(n)\) corresponds to the response characteristics of \(\mathbf{S}(z)\) and \({}^{*}\) signifies the discrete convolution operator. Represented by \(\mathbf{y}_{f}(n)\) and \(\mathbf{x}_{f}(n)\), these terms stand for the filtered reference and secondary signal, respectively.