# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 3 Non-Linear Transformation-Based Algorithms
### Logarithmic Transformation and Trigonometric Transformation
Integrating the hyperbolic secant function and the mean difference function, [PERSON] formulated the cost function [46]. This method, with a clever choice of combination parameters, allows the step scaling factor to play a more excellent role. When the system has not encounteredimpulse noise and the error signal amplitude is large, the step size is not scaled to accelerate convergence. When the error is small, the step size is scaled to enhance steady-state performance. When encountering impulse noise interference, the step size is scaled to strengthen system stability. In [PERSON]'s proposed algorithm, the random Fourier feature (RFF) technique and CG approach are employed to address the slow convergence issues posed by the generalized Hough transform (GHT) criterion [47]. [PERSON] proposed an algorithm based on a random Fourier filter to handle impulse noise. The algorithm's convergence capability is confirmed through numerical simulations under different conditions [48]. [PERSON] introduced the error arctangent as the cost function, presenting the filtered-x arctangent error least mean square (FxatanLMS) algorithm [49]. The study showed outstanding performance of the normalized error arctangent algorithm in handling narrowband, wideband, and impulse noise. [PERSON] proposed a constrained least variance algorithm, demonstrating superior performance in impulse noise environments compared to other algorithms. The Incosh cost function is the natural logarithm of the hyperbolic cosine function, representing a combination of mean square error and mean absolute error criteria [50]. In this paper, setting \(\beta\) to 0.8 achieves the best convergence performance under pulse noise. [PERSON] introduced a new robust norm based on the exponential hyperbolic cosine function (EHCF) and developed corresponding adaptive filters based on EHCF [51]. [PERSON] proposed two adaptive filters based on hyperbolic sine functions that are suitable for impulse noise environments [52]. The study explored learning rate-based stability conditions and steady-state analysis. Additionally, a variable scale parameter format was proposed to eliminate the trade-off between convergence speed and steady-state excess mean squared error. [PERSON] proposed two adaptations based on the hyperbolic secant function (HSF). In the presence of pulse interference, both algorithms exhibit superior performance compared to existing algorithms [53]. Table 5 summarizes the corresponding weight coefficient-updating formulas for algorithms based on logarithmic and trigonometric transformations. The FxlogLMS algorithm involves logarithmic calculations during the filter weight coefficient updating process, significantly increasing computational complexity. However, other logarithmic transformation weight coefficient updating formulas, excluding FxlogLMS, do not require logarithmic calculations. Logarithmic hyperbolic cosine trans  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost Function** & \(\mathbf{\varphi(e(n))}\) & **References** \\ \hline \(J(n)=E\left[\log^{2}\lvert e(n)\rvert\right]\) & \(\frac{\log(\lvert e(n)\rvert)\text{sign}(e(n))}{\lvert e(n)\rvert}\) & [36] \\ \(J(n)=E\left[\log[1+\frac{  
formation algorithms essentially limit the amplitude of the error signal. Other types of logarithmic transformations are similar to step size normalization algorithms. These algorithms exhibit slow convergence rates in the presence of Gaussian noise. Three trigonometric functions that are suitable for nonlinear transformation of the error signal are the hyperbolic tangent, hyperbolic secant and arctangent. This type of transformation is a relatively recent introduction as a nonlinear cost function, demonstrating superior performance. However, the weight coefficient updating formulas involve complex trigonometric functions, especially hyperbolic functions, thereby increasing algorithmic complexity.