## 1 Introduction
Here, IIR-based structures have inherent stability problems, RLS-based ANCS exhibit numerical instability issues, and the other approaches mentioned above increase the computational complexity. These reasons make FxLMS still a good choice for ANC applications. However, the FxLMS algorithm does not appear as a viable choice for ANC of noise generated from non-Gaussian stable sources of an impulsive nature. In many practical situations, viz., office, infant incubators, punching and cutting machines in industrial setups, traffic noise, gun shots and explosions, and noise in an MRI room, the target noise is non-Gaussian [20; 21; 22; 23]. Such a noise signal exhibits an impulsive nature and hence can be better modeled by heavy tailed non-Gaussian (stable) distribution than Gaussian distribution. In order to address poor performance of the FxLMS algorithm for impulsive ANC systems, researchers have proposed many variants of the FxLMS algorithm, viz., [PERSON]'s algorithm [24], the thresholding FxLMS (Th-FxLMS) algorithm [25], the improved normalized step-size-based FxLMS (INSS-FxLMS) algorithm [26], the Fx logarithmic LMS (FxlogLMS) algorithm [27], and the Fx least mean M-estimate (FxLMM) algorithm [21; 28; 29]. Here, the INSS-FxLMS algorithm may exhibit poor performance for strongly impulsive sources, the FxlogLMS algorithm suffers from a problem of entering a dead zone during adaptation (due to characteristics of log operation), and the rest of the algorithms require estimating appropriate thresholding parameters such that reference and/or error signals are thresholded before being used in the update process. Obtaining appropriate threshold parameter values may pose a great challenge, especially if a suitable threshold needs to be estimated during the online operation of impulsive ANC systems. An excellent review of threshold adaptive ANC algorithms for impulsive sources is found in [30]. In order to take into account the non-Gaussian nature of impulsive sources [31], the filtered-x least mean p-power (FxLMP) algorithm minimizes \(p\)-power of the error signal instead of the mean squared value [32]. Here, \(p\) is the fractional low-order moment which appears to be a robust choice compared with the second order moment considered in the derivation of the classical FxLMS algorithm [32]. The FxLMP algorithm does outperform the FxLMS algorithm for impulsive ANC systems; however, the convergence speed of the FxLMP algorithm is very slow, especially for strongly impulsive sources. Great effort has been devoted to improving the performance of the classical FxLMP algorithm, and many variants have been proposed [33; 34; 35; 36]. It is worth mentioning that a suitable value of the parameter \(p\) must be found in relation to the characteristic of the noise source, which may be quite challenging in many situations. For several other robust cost-function-based algorithms, for example the least-mean-kurtosis-based adaptive algorithm, the interested reader is referred to [37]. This paper develops a robust adaptive algorithm (and its variant) for impulsive ANC systems. The key features of the developed algorithm(s) are summarized as follows:  
* It is well-known that the RLS-based adaptation exhibits a far better convergence speed compared with the stochastic gradient-based adaptation (as in the LMS algorithm), where the price to be paid is increased computational complexity. The computational complexity may not be an issue with the recent advancements in semiconductor technology; therefore, in order to realize a fast convergence speed, the proposed algorithm is derived in the framework of an RLS-based adaptation. * Instead of FxLMS-based ANC, the'modified Fx' (Mfx) LMS (MfxLMS)-based ANC structure [38] is used to implement the proposed algorithm. The MfxLMS ANC uses two copies of the ANC filter so that generation of the noise control signal and adaptation of coefficients of the ANC filter are treated as separate processes.