# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 2 Fractional Lower-Order Power-Based Algorithms
### Other Algorithms  
\(p\)-power algorithms need to satisfy \(p<\alpha\) to prevent instability in ANC systems. To ensure stability in all situations, the choice of \(p=1\) has been proposed, leading to the sign algorithm (SA). The SA algorithm, in comparison to the FxLMP algorithm, does not necessitate prior knowledge of the distinctive parameter \(\alpha\) of impulsive noise, leading to a reduction in processing demands [12; 13]. However, traditional SA trades convergence speed for robustness. To address this, [PERSON] introduced the affine projection sign algorithm, commonly known as APSA, which minimizes the posterior error \(l_{1}\) moment as the optimization goal [14]. By introducing the error sign vector, APSA eliminates the need to invert matrices during weight coefficient updates, significantly reducing algorithmic computational complexity. Simultaneously, the introduction of the error \(l_{1}\) moment enhances the algorithm's resilience to impulsive noise. The algorithm exhibits excellent resistance to impulse noise interference, enhancing the convergence speed under colored input signals. [PERSON] and [PERSON] simplified the formula for the proportionality factor in the proportionate affine projection sign algorithm, commonly known as PAPSA, improving the algorithm's computational efficiency and impulsive noise resilience. The performance of noise reduction in sparse systems is improved [15]. [PERSON] improved the efficiency and impulsive noise resilience of the proportionate affine projection sign algorithm (PAPSA) by introducing the Gaussian entropy as a substitute for the error \(l_{0}\) moment in the proportionality matrix function along with its approximate formula [16]. Additionally, [PERSON] introduced the error sign function and reference signal into the SA step size, allowing for a larger step size during the initial working phase for rapid convergence and a smaller step size with minimal errors to achieve significant steady-state error reduction. The algorithm demonstrates fast convergence under colored input signals, where the smoothing factor parameter satisfies \(0<\beta<1\)[17]. [PERSON] implemented an enhanced Gaussian kernel formula, replacing the traditional variable step size formula. \(\beta\) represents a smooth factor [18]. [PERSON] expedites the convergence of the APSA algorithm through adaptive step size adjustment [19]. In this context, the parameter \(\delta\) is typically set to 0, and \(\alpha\) is defined as 1-M/5L. This ensures stability when outliers are present. This formula adjusts the step size to be larger when the error is significant, enhancing convergence speed, and smaller when the error is minor, improving steady-state error. [PERSON] and [PERSON] enhanced the convergence speed of the sign algorithm (SA) by introducing a natural logarithmic momentum term related to the absolute value of weight coefficients in the weight coefficient update formula [20]. This algorithm enhances system performance in sparse environments, where \(\varepsilon\) is a non-zero decimal number. [PERSON] proposed a series of dual sign algorithms (DSA) and introduced compensation terms related to the \(l_{1}\) and \(l_{0}\) moments of weight coefficients in the weight coefficient update formula to accelerate the convergence process in sparse environments [21]. In addition, there are mixed-moment algorithms and continuous \(p\) power algorithms. In 1997, [PERSON] and [PERSON] proposed a method called the mixed norm (MN) algorithm, which forms a new cost function by convexly combining the \(l_{2}\) and \(l_{1}\) norms of the error [22]. [PERSON] and [PERSON] improved the convergence speed and steady-state performance of the MN algorithm by adapting the fixed mixing parameter in the MN algorithm based on the error and normalizing the step parameter in the weight coefficient update formula [23]. [PERSON] and [PERSON] summarized the general form of the MN algorithm, proposing the filtered-x general mixed norm (FxGMN) algorithm [24].