# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 3 Non-Linear Transformation-Based Algorithms
### Logarithmic Transformation and Trigonometric Transformation  
For impulsive noise signals following an \(\alpha\)-stable distribution, the signal becomes more Gaussian-like after undergoing a logarithmic transformation. Consequently, [PERSON] introduced the FxlogLMS algorithm, or the filtered-x logarithmic transformation least Mean square algorithm. This algorithm utilizes the error signal after logarithmic transformation as the objective function [36]. This algorithm exhibits stability without the need for prior knowledge of the impulsive noise characteristic parameter \(\alpha\) or the estimation of threshold parameters, making it more robust compared to the FxLMP method. However, the FxlogLMS algorithm introduces additional computational complexity due to the logarithmic calculation involved in updating the weight coefficients. Additionally, careful consideration is required for cases where l exceeds the amplitude of the error signal. [PERSON] and [PERSON] introduced an algorithm that utilizes the logarithmic transformation of the squared error signal as the objective function, specifically designed for impulsive noise elimination [37]. Furthermore, [PERSON] introduced a convex combination approach, incorporating the logarithmic conversion of the error signal \(\alpha\) (\(\alpha\) = 1.2) moments [38]. The algorithm corresponding to the \(l_{2}\) moment is referred to as the least mean logarithmic square (LMLS) algorithm, while the \(l_{1}\) moment corresponds to the least logarithmic absolute difference (LLAD) algorithm. Both algorithms exhibit rapid convergence and resilience against impulsive noise interference, although careful parameter selection is essential for their combination. LU introduced a recursive error fractional moment logarithmic transformation algorithm that combines logarithmic transformation with fractional lower-order moment methods [39]. The recursive algorithm enhances the convergence speed of the algorithm but also increases computational complexity. The Lorenz norm, another form of logarithmic transformation, is utilized in algorithms presented by [PERSON] and [PERSON] including the Lorenz norm algorithm based on the error signal, the normalized Lorenz norm algorithm and the momentum Lorenz norm algorithm [40]. These algorithms demonstrate excellent steady-state performance and convergence speeds. In addition to logarithmic transformations, trigonometric function transformations can enhance the algorithm's robustness against impulse noise. [PERSON] utilized the logarithmic hyperbolic cosine transformation of the error signal as the objective function in the filtered-x least cosine hyperbolic (FxLCH) algorithm and proposed the simultaneous use of variable step sizes and threshold nonlinear transformation methods for active elimination of impulse noise [41]. [PERSON] employed the logarithmic hyperbolic cosine transformation of the error signal as the cost function, enhancing the convergence speed of the algorithm through a variable step size form [42]. [PERSON] used the logarithmic hyperbolic cosine transformation of the error signal as the cost function [43]. Since the weight coefficient update formula of this type of algorithm requires calculating the exponential form of the error, the authors proposed two approximation equations, reducing computational complexity. [PERSON] adopted the logarithmic hyperbolic cosine transformation of the error signal as the cost function and improved the algorithm's convergence speed through an accelerated gradient method [44]. [PERSON] utilized the hyperbolic tangent function of the error as the cost function [45]. The parameter \(\beta\) needs adjustment based on the signal impulses and the algorithm. The hyperbolic tangent function adjusts the step size based on the amplitude variation of the error signal in weight coefficient updates. Under impulse noise conditions, the algorithm adapts the stride to bolster system robustness. Conversely, with a decrease in the error signal, the stride is increased to ensure the system convergence rate.