# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 2 Fractional Lower-Order Power-Based Algorithms
### p-Power Algorithms
[PERSON] developed a filter-logarithmic recursive least \(p\)-power algorithm, integrating logarithmic moments with \(p\)-power norms to enhance the noise reduction performance in the \(\alpha\) state of 1.35 [11]. In order to facilitate a more intuitive comparison of various update functions, the filter coefficient update formulas are standardized as follows:  
\[\mathrm{w}(n+1)=\mathrm{w}(n)+\mu\varphi(e(n))\mathrm{x}_{f}(n) \tag{7}\]  
The cost function and filter coefficient update formulas for the fractional lower-order power algorithms are shown in Table 2.  
The characteristics of \(p\)-power algorithms are summarized. These methods are intricately linked to impulsive noise, influencing the robustness of parameter \(p\). More specifically, the algorithm's convergence is contingent on the relationship between \(p\) and \(\alpha\) amidst impulsive noise interference, where effective convergence occurs when \(p<\alpha\). Whether estimating \(\alpha\) online or offline, additional steps and computational burden are introduced, contributing to increased algorithm complexity. The computation of the \(p\) power of the error signal further amplifies the algorithm's computational load. \begin{table}
\begin{tabular}{c c c} \hline
**Cost Function** & \(\varphi\)**(\(e(n)\))** & **References** \\ \hline \(J(n)=E\left[|e(n)|^{p}\right]\) & \(p|e(n)|^{p}\mathrm{sign}(e(n))\) & [3,4] \\ \(J(n)=E\left[|e^{\prime}(n)|^{p}\right]\) & \(\left\{\begin{array}{cc}e^{\prime}(n)=\left\{\begin{array}{cc}c_{1}&if\;e^{ \prime}(n)\geqslant c_{1}\\ c_{2}&if\;e^{\prime}(n)\leqslant c_{2}\\ c_{1}&\leqslant e^{\prime}(n)\leqslant c_{2}\end{array}\right.\end{array}\right.\) & [5] \\ \(c_{1}\) & \(if\;e^{\prime}(n)\leqslant c_{1}\) & \(if\;e^{\prime}(n)\leqslant c_{2}\) \\ \(c_{2}\) & \(if\;e^{\prime}(n)\leqslant c_{2}\) & \\ \(\chi_{f}(n)=\left\{\begin{array}{cc}c_{1}&if\;e^{\prime}(n)\leqslant c_{2}\\ c_{2}&if\;e^{\prime}(n)\leqslant c_{2}\\ c_{2}&if\;e^{\prime}(n)\leqslant c_{2}\end{array}\right.\) & \\ \(\chi_{f}(n)\) & \(c_{1}\leqslant\chi_{f}(n)\leqslant c_{2}\) & \\ \(p|e^{\prime}(n)|^{p}\mathrm{sign}(e(n)\chi_{f}^{\prime}(n))\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}p|e^{\prime}(n)|^{p-1}\mathrm{sign}(e(n)\chi_{f}^{ \prime}(n))\\ \end{array}\right.\) & \(\left\{\begin{array}{cc}\mathbf{K}(n)=\frac{\mathrm{sgn}\mathrm{Pr}(n-1) \mathrm{sgn}(n)}{\lambda+\mathrm{sgn}(n)\mathrm{sgn}(n)\mathrm{sgn}(n)}\\ \mathbf{P}(n)=\lambda^{-1}\mathbf{P}(n-1)-\lambda^{-1}\mathbf{K}(n)\mathbf{P}( n-1)\\ \mathbf{v}(n)=\frac{\mathrm{sgn}^{\prime-1}(1+|e(n)|)}{(1+|e(n)|)(1)+\tau}\end{array}\right.\) & [11] \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of adaptive algorithms in Section 2.1.