## 3 Wl-Rls-Dcd  
For the RLS family of adaptive algorithms employed in SAEC scenarios with the WL framework, the vectors from (16) and (17) are interleaved instead of concatenated (as it was presented in the previous section). However, the new formulations do not affect the definition of the complex reference signal \(d(n)\)[4; 5]. Firstly, the vector comprising the input samples from (14) and (17) is redefined to reflect the new positioning of the complex valued samples and their transpose-conjugate counterparts. Consequently, we write:  
\[\widetilde{\mathbf{x}}(n)=[x(n)\ x^{*}(n)\ \ldots\ x(n-L+1)\ x^{*}(n-L+1)]^{T}. \tag{22}\]  
In a similar manner, the elements of (11) and (12) are also interleaved in order to express the new complex valued impulse response to be estimated [4; 5]:  
\[\widetilde{\mathbf{h}}_{\mathbf{t}}=\big{[}h_{\alpha,0}\ h_{\beta,0}\ \ldots\ h_{\alpha,L-1}\ h_{\beta,L-1}\big{]}^{T}, \tag{23}\]  
where \(h_{\alpha,l}\) and \(h_{\beta,l}\), with \(l=0,1,\ldots,L-1\), are the elements of the vectors \(\mathbf{h}_{\alpha}\) and \(\mathbf{h}_{\beta}\), respectively. When employing the least-squares approach to generate the complex valued \(2L\times 1\) estimate vector \(\widetilde{\mathbf{h}}(n)\) of \(\widetilde{\mathbf{h}}_{\mathbf{t}}\), the corresponding cost function is written using (22), (23), and the complex observation \(d(n)\) as:  
\[J\Big{[}\widetilde{\mathbf{h}}(n)\Big{]}=\sum_{i=1}^{n}\lambda^{n-i}\Big{[}d (i)-\widetilde{\mathbf{h}}^{H}(n)\widetilde{\mathbf{x}}(i)\Big{]}^{2}, \tag{24}\]  
where \(0<\lambda\leq 1\) denotes the forgetting factor, which controls the memory of the algorithm (i.e., values closer to one increase the adaptive filter's performance in steady state, while low values for \(\lambda\) improve the tracking capabilities of the system) [1; 4]. When performing the minimization of the error criterion, two new notations are introduced, i.e., the \(2L\times 2L\) estimate of the correlation matrix  
\[\mathbf{R}(n)=\sum_{i=1}^{n}\lambda^{n-i}\widetilde{\mathbf{x}}(i)\widetilde{ \mathbf{x}}^{H}(i)=\lambda\mathbf{R}(n-1)+\widetilde{\mathbf{x}}(n)\widetilde {\mathbf{x}}^{H}(n), \tag{25}\]  
and the \(2L\times 1\) cross-correlation vector between the complex input signal and the desired signal  
\[\mathbf{p}(n)=\sum_{i=1}^{n}\lambda^{n-i}\widetilde{\mathbf{x}}(i)d(i)= \lambda\mathbf{p}(n-1)+\widetilde{\mathbf{x}}(n)d(n). \tag{26}\]  
Thus, the solution for the normal set of equations  
\[\mathbf{R}(n)\widetilde{\mathbf{h}}(n)=\mathbf{p}(n), \tag{27}\]  
leads to the straight-forward expression for the adaptive filter's coefficients:  
\[\widetilde{\mathbf{h}}(n)=\mathbf{R}^{-1}(n)\mathbf{p}(n). \tag{28}\]  
The direct approach for computing \(\widetilde{\mathbf{h}}(n)\) has a complexity of \(\mathcal{O}(8L^{3})\), which is too costly for practical implementations, such as the ones for SAEC configurations. In acoustic applications, the number of filter coefficients \(2L\) can easily be in the range of hundreds or even thousands. Other classical solutions, such as the one based on Woodburry's identity (also known as the matrix inversion lemma [1; 4]), provide an alternative to (28) with a complexity \(\mathcal{O}(4L^{2})\). However, the required amount of arithmetic operations is still prohibitive for most computer chips and would considerably increase device prices, power consumption, etc. In this context, the combination between the exponentially weighted RLS adaptive algorithm and the DCD iterations was proposed as a computationally efficient method, with a complexity proportional to the adaptive filter's length multiplied by a small number (the actual details will be discussed in the next paragraphs). The WL-RLS-DCD is accepted as a reliable algorithm in multiple types of configurations for acoustic echo cancellation, including for stereophonic communication setups [5; 18; 21].