## 3 Proposed Algorithm  
### Derivation of Robust FxRLS Algorithm  
It is important to mention that the classical FxLMS and FxRLS algorithms are, respectively, based on MSE cost function \(\xi_{\text{MSE}}(n)\) (4) and LS cost function \(\xi_{\text{LS}}(n)\) (6) which are not robust for noise sources of impulsive nature. In order to develop a robust objective function for the impulsive ANC systems, we consider the following function [43]  
\[f(n)=\mathbb{E}\bigg{\{}\frac{1}{1+\beta e^{2}(n)}\bigg{\}}, \tag{14}\]  
where \(\beta\) is a constant selected empirically. It is noticed that \(f(n)\to 0\) for impulsive sources and hence would be robust in mitigating the effect of impulsiveness. On the basis of function \(f(n)\) in (14), following an LS-based objective function is proposed to derive the RLS-based algorithms presented in this paper  
\[\xi_{\text{m}}(n)=\sum_{i=1}^{n}\lambda^{n-i}\bigg{\{}\frac{1}{1+\beta e^{2}(i )}\bigg{\}}, \tag{15}\]  
where the subscript m signifies'modified' objective function (in comparison with the standard LS cost function in (6)), and \(e(i)\) is the same as expressed in (7). In order to minimize the error signal \(e(n)\) so that reduced residual noise appears around the error microphone, the objective function \(\xi_{\text{m}}(n)\) in (15) must be maximized. This can be achieved by computing the gradient of \(\xi_{\text{m}}(n)\) and equating it to zero as  
\[\frac{\partial\xi_{\text{m}}(n)}{\partial\mathbf{w}(n)}=0, \tag{16}\]  
which results in the following equation  
\[\sum_{i=1}^{n}\lambda^{n-i}m(i)\mathbf{x}_{f}(i)e(i)=0, \tag{17}\]  
where the parameter \(m(i)\) is given as  
\[m(i)=\bigg{\{}\frac{1}{(1+\beta e^{2}(i))^{2}}\bigg{\}}. \tag{18}\]  
Substituting (7) in (17) and performing some straightforward computations, we obtain the following normal equations  
\[\mathbf{R}_{\text{m}}(n)\mathbf{w}(n)=\mathbf{p}_{\text{m}}(n)\quad\Rightarrow\quad\mathbf{ w}(n)=\mathbf{\Phi}_{\text{m}}(n)\mathbf{p}_{\text{m}}(n), \tag{19}\]  
where \(\mathbf{R}_{\text{m}}(n)\) is the (modified) weighted correlation matrix given as  
\[\mathbf{R}_{\text{m}}(n)=\sum_{i=1}^{n}\lambda^{n-i}m(i)\mathbf{x}_{f}(i)\mathbf{x}_{f}^{ T}(i), \tag{20}\]  
and \(\mathbf{p}_{\text{m}}(n)\) is the (modified) weighted cross-correlation vector being expressed as  
\[\mathbf{p}_{\text{m}}(n)=\sum_{i=1}^{n}\lambda^{n-i}m(i)\mathbf{x}_{f}(i)d(i), \tag{21}\]and \(\mathbf{\Phi}_{\rm m}(n)=\mathbf{R}_{\rm m}^{-1}(n)\) denotes the so-called inverse correlation matrix. Following the derivation of the classical RLS algorithm [42; 44], we obtain  
\[\mathbf{u}_{\rm m}(n)=\mathbf{\Phi}_{\rm m}(n-1)\mathbf{x}_{f}(n), \tag{22}\]  
\[\mathbf{k}_{\rm m}(n)=\frac{\mathbf{u}_{\rm m}(n)}{\frac{\lambda}{m(n)}+\mathbf{x}_{f}^{ \rm T}(n)\mathbf{u}_{\rm m}(n)}, \tag{23}\]  
\[\mathbf{\Phi}_{\rm m}(n)=\lambda^{-1}\mathbf{\Phi}_{\rm m}(n-1)-\lambda^{-1}\mathbf{k}_{ \rm m}(n)\mathbf{u}_{\rm m}^{\rm T}(n), \tag{24}\]  
\[\mathbf{w}(n)=\mathbf{w}(n-1)+\varepsilon(n)\mathbf{k}_{\rm m}(n), \tag{25}\]  
where \(\mathbf{k}_{\rm m}(n)\) is the'modified' Kalman gain vector which, in addition to (23), can be equivalently expressed as \(\mathbf{k}_{\rm m}(n)=m(n)\mathbf{\Phi}_{\rm m}(n)\mathbf{x}_{f}(n)\), \(\varepsilon(n)\) is the a priori estimation error as given in (13), and \(m(n)\) may be computed from (18). As noted from the above details, the derivation of the proposed algorithm is quite straightforward; however, there are many issues from the viewpoint of implementation for impulsive ANC systems and are tackled in the following subsections.