## 3 Proposed Algorithm
### Developing the Convex Combined Step Size Approach  
It is no surprise that, like any FSS-based adaptive algorithm, the proposed FSS-MFxRRLS algorithm would also exhibit a trade-off situaiton for selecting the FSS: a large step size gives fast convergence but poor steady-state performance, and a small value would slow down the convergence speed while improving the performance at the steady state. Such a trade-off situation can be overcome by considering a variable step size (VSS) procedure which would automatically 'tune' the step size starting from a large value at the start to a small value as time progresses [45; 46; 47; 48]. This would help in achieving both a fast initial convergence as well as a good steady-state performance. Recently, employing a convex combination of two adaptive filters has gained some popularity [49], where one adaptive filter is adapted using a large step size and the other using a small one. Their outputs are combined using a (convex) mixing parameter such that the overall output is decided by a fast converging adaptive filter (using a large step size) during the transient state and by a slow converging adaptive filter (using a small step size) at the steady state [50; 51]. Since both adaptive filters are adapted simultaneously, it is obvious that this approach wound result in an increased computational complexity. Figure 2: Block diagram of ANC structure employing proposed RLS-based adaptive algorithms. As in the previous work [39; 40] for FxLMS-based ANC, a CCSS is developed for the FxRLS-based ANC algorithm described above. It is worth mentioning that the CCSS approach borrows the concepts of the convex combination of adaptive filters, yet uses only ONE adaptive filter to perform the adaptation [52; 53] as explained below. The key idea is to develop a time-varying step size, as in the classical VSS-based algorithms, however by employing the concepts of a convex combination of adaptive filters. With reference to Figure 2 for the proposed method, it is 'assumed' that two copies of \(W(z)\) (adaptive one) are available to be updated according to (28) as  
\[\mathbf{w}_{1}(n) =\mathbf{w}_{1}(n-1)+\mu_{1}g(n)\mathbf{k}_{\text{m}}(n), \tag{31}\] \[\mathbf{w}_{2}(n) =\mathbf{w}_{2}(n-1)+\mu_{2}g(n)\mathbf{k}_{\text{m}}(n). \tag{32}\]  
By selecting \(\mu_{1}>\mu_{2}\), it is understood that \(W_{1}(z)\) shows fast convergence speed and \(W_{2}(z)\) would exhibit good performance at the steady-state. In order to combine these two desired properties, the adaptive ANC filter \(W(z)\) can have a coefficient vector obtained via the convex combination of coefficient vectors given in (31) and (32) for \(W_{1}(z)\) and \(W_{2}(z)\), respectively,  
\[\mathbf{w}(n)=\gamma(n)\mathbf{w}_{1}(n)+(1-\gamma(n))\mathbf{w}_{2}(n), \tag{33}\]  
where \(\gamma(n)\) is a mixing parameter. It is a time-varying parameter (as explained later) whose value must be selected in the range \(0\leq\gamma(n)\leq 1\). From (31)-(33), we obtain the following expression  
\[\mathbf{w}(n)=\mathbf{w}(n-1)+\mu(n)g(n)\mathbf{k}_{\text{m}}(n), \tag{34}\]  
for updating the coefficient vector of the adaptive ANC filter \(W(z)\). In (34), \(\mathbf{w}(n-1)\) is the convex combination of previous values of the coefficient vectors of \(W_{1}(z)\) and \(W_{2}(z)\) and can be expressed as  
\[\mathbf{w}(n-1)=\gamma(n)\mathbf{w}_{1}(n-1)+(1-\gamma)\mathbf{w}_{2}(n-1), \tag{35}\]  
and \(\mu(n)\) is the CCSS which combines \(\mu_{1}\) and \(\mu_{2}\) via mixing parameter \(\gamma(n)\) as  
\[\mu(n)=\gamma(n)\mu_{1}+(1-\gamma(n))\mu_{2}. \tag{36}\]  
From (34) and (36), it is noticed that \(\gamma(n)=1\) (\(\gamma(n)=0\)) effectively means as if \(W(z)\) is adapted using a large (small)-valued step size \(\mu_{1}\) (\(\mu_{2}\)). As in any VSS-based algorithm, we would like to have \(\mu(n)\approx\mu_{1}\) at \(n=0\) to achieve fast convergence at the start-up and \(\mu(n)\approx\mu_{2}\) to obtain improved steady-state performance when \(n\rightarrow\infty\).