# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 2 Fractional Lower-Order Power-Based Algorithms  
### p-Power Algorithms  
From the above, it is evident that when the impulse noise signal follows an \(\alpha\)-stable distribution, the \(l_{2}\) norm of the \(e(n)\) does not exist, but the \(p\) power of the \(e(n)\) is bounded. In lieu of minimizing the variance, as in standard algorithms, [PERSON] et al. proposed minimizing the existing fractional lower-order moments (\(p<\alpha\)) [3]. Here, \(p\) represents the order less than the dispersion \(\alpha\). They introduced the filter-x least mean \(p\)-power (FxLMP) algorithm, where the cost function aims to minimize the \(p\)-power of the \(e(n)\). [PERSON] and colleagues studied the impact of parameter \(p\) on the algorithm's performance and concluded that the closer \(p\) is to the characteristic parameter of impulsive noise, the better the algorithm performs. In fact, when setting the parameter \(p\) in the FxLMP algorithm to a fixed value, the algorithm degrades into different algorithms such as LMAD, LMAT and LMF [4; 5]. Therefore, the FxLMP algorithm can be considered a general algorithm for this type. Subsequently, many researchers improved the FxLMP algorithm based on the study by [PERSON]. [PERSON] and [PERSON], based on the FxLMP algorithm, applied threshold nonlinear transformations to both the reference and error signals. This further improved the robustness of the FxLMP method without increasing computational complexity [6]. However, the stability of the FxLMP algorithm is contingent on the parameter \(p\). The algorithm exhibits better stability when \(p\) is less than the characteristic parameter \(\alpha\) of impulsive noise. Consequently, it is necessary to estimate the characteristic parameter of impulsive noise in advance when applying the FxLMP method. [PERSON] and [PERSON] delved into the dependency of the FxLMP algorithm on the characteristic parameter \(\alpha\) of impulsive noise [7]. They proposed a sliding window method for online estimation of the characteristic parameter \(\alpha\). This method circumvents the need for prior knowledge of the characteristic parameter \(\alpha\) in \(p\) power algorithms but comes at the cost of increased algorithm complexity. In addition, another drawback of the FxLMP series algorithms is the increased computational complexity and slow convergence speed of ANC systems due to the computation of fractional lower-order powers. To address this, [PERSON] and [PERSON], building upon the  
Figure 3: \(\alpha\)-stable distributed noise signals of different intensities (**a**) \(\alpha=1.8\); (**b**) \(\alpha=1.6\); (**c**) \(\alpha=1.3\). FxLMP algorithm, introduced two modified algorithms: the modified FxLMP (MFxLMP), which incorporates a threshold method, and the filtered-x modified normalized least mean \(p\) power, which utilizes a normalization step size method [8]. The filter-x affine projection (FxAP) algorithm was introduced by [PERSON] to address convergence speed in correlated input environments. Subsequently, [PERSON] and [PERSON] combined the affine projection algorithm with the normalized \(p\) power algorithm (FxLMLP) [9]. This integration leveraged the characteristics of the projection algorithm to improve the convergence speed of the algorithm. The results indicate that the algorithm maintains good convergence when transitioning from low-intensity impulse noise (\(\alpha=1.7\)) to high-intensity impulse noise (\(\alpha=1.1\)) and maintains insensitivity to changes in the noise source. In contrast, the FxLMP algorithm becomes unstable. Additionally, the introduction of the projection algorithm results in an increase in computational complexity. [PERSON], building upon the FxLMP, FxLMP and filtered-x generalized normalized LMP (FxGNLMP) algorithms, proposed the convex-combined step size FxLMP algorithm, which enhanced the convergence speed and steady-state performance of active impulsive noise control algorithms [10]. Nevertheless, the convex combination algorithm increased computational complexity due to the parallel computation of two filters.