# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 3 Non-Linear Transformation-Based Algorithms  
### Threshold Method  
Grounded in the error \(l_{2}\) moment criterion, the FxLMS algorithm experiences significant performance degradation when encountering impulsive noise interference. The simplest and most direct method to mitigate impulsive noise interference is to set thresholds for both the error signal and reference signal. Sun introduced the threshold FxLMS (ThFxLMS) algorithm, which ignores signals with values exceeding a predefined threshold parameter [28]. In this approach, the corresponding filter coefficients remain unchanged,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Cost Function** & **\(\varphi\)(\(e\)(\(n\)))** & **References** \\ \hline \(J(n)=E\left[\left|e(n)\right|\right]\) & \(\operatorname*{sign}(\varepsilon(n))\) & [12; 13;  
ensuring the stability of the ANC system. However, in specific scenarios, this method may lead to frequent coefficient freezing, resulting in a slower convergence speed. In 2009, [PERSON] further improved the algorithm based on the ThFxLMS algorithm [29]. In addition to correcting the reference signal, constraints were applied to the error signal, replacing outliers with critical values. [PERSON] proposed a method using box plot (BP) sliding windows for online threshold estimation [30]. This method does not require prior knowledge of the impulsive noise characteristic parameter \(\alpha\), and the threshold exhibits time-varying characteristics. Non-linear threshold transformations applied to the reference signal can be utilized not only in FxLMS but also in other algorithms. [PERSON] enhanced the algorithm's control performance against impulsive noise by applying a threshold non-linear transformation to both the error and reference signals and adopting a variable step size approach [31]. In 2007, [PERSON] introduced the FxLMM algorithm, utilizing a non-linear transformation of the error signal with the Hampel function. [PERSON] combining the Huber M-estimation method with the LMS algorithm and presented a simplified FxLMM algorithm [32]. [PERSON] and [PERSON] utilized the Fair function to transform the error signal, facilitating the setting of thresholds for error signals [33]. [PERSON] introduced a hyperbolic tangent threshold transformation for both the reference and error signals, replacing the manual threshold-setting approach [34]. [PERSON] improved the FLMM's performance by substituting a more precise method for estimating the proportion of the distribution in the reference noise with a higher degree of accuracy [35]. The threshold of the algorithm in this classification is obtained through parameter estimation by the formula. Table 4 summarizes the threshold formulas and corresponding weight coefficient update formulas for threshold-based algorithms. The characteristics of Threshold-Based Algorithms can be seen in Table 4. The fixed threshold method has low computational complexity, but its performance depends on the selection of threshold parameters. Foreknowledge of the distinctive parameter \(\alpha\) of abrupt noise is required for the selection of threshold parameters, and adjustments to this parameter are not feasible when noise conditions change. On the other hand, online