## 2 System Model
\[\mathbf{h}_{\alpha,1} =\frac{\mathbf{h}_{\mathrm{t},\mathrm{L}_{\mathrm{c}},\mathrm{L}_{ \mathrm{c}}}+\mathbf{h}_{\mathrm{t},\mathrm{R}_{\mathrm{c}},\mathrm{R}_{ \mathrm{c}}}}{2},\quad\mathbf{h}_{\alpha,2}=\frac{\mathbf{h}_{\mathrm{t}, \mathrm{R}_{\mathrm{c}},\mathrm{L}_{\mathrm{c}}}-\mathbf{h}_{\mathrm{t}, \mathrm{R}_{\mathrm{c}},\mathrm{R}_{\mathrm{c}}}}{2}, \tag{9}\] \[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]0  
with the purpose of obtaining the complex valued forms [4]  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]1 \[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]2  
and express the output complex signal as  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]3  
where the superscripts \(H\) and \(*\) denote the transpose-conjugate and conjugate operators, respectively, and  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]4 \[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]5  
Consequently, the complex vectors from (11) and (12) can be concatenated to obtain the \(2L\times 1\) complex valued vector comprising the information corresponding to the coefficients of the four unknown impulse responses  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]6  
and a corresponding \(2L\times 1\) input vector can be defined as  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]7  
Using (16) and (17), a more compact expression can be written for the complex echo signal as  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]8  
Finally, by also expressing the complex valued noise \(w(n)\) as  
\[\mathbf{x}_{\rm R_{c}}(n)=[x_{\rm R_{c}}(n)\quad x_{\rm R_{c}}(n-1)\quad \cdots\quad x_{\rm R_{c}}(n-L+1)]^{T}. \tag{2}\]9  
the microphone signals corresponding to the _left_ and _right_ channels can be written as:  
\[y_{\rm L_{c}}(n)=\mathbf{h}_{\rm t,L_{c}L_{c}}^{T}\mathbf{x}_{\rm L_{c}}(n)+ \mathbf{h}_{\rm t,R_{c}L_{c}}^{T}\mathbf{x}_{\rm R_{c}}(n), \tag{3}\]0  
The error signal can now be expressed as  
\[y_{\rm L_{c}}(n)=\mathbf{h}_{\rm t,L_{c}L_{c}}^{T}\mathbf{x}_{\rm L_{c}}(n)+ \mathbf{h}_{\rm t,R_{c}L_{c}}^{T}\mathbf{x}_{\rm R_{c}}(n), \tag{3}\]1  
where \(\widetilde{y}(n)\) is the estimate of the complex valued system's output computed using the adaptive filter's set of coefficients \(\widetilde{\mathbf{h}}(n)\) (which approximates \(\widetilde{\mathbf{h}}_{\mathrm{t}}\)). The described framework improves the handling of SAEC scenarios using adaptive algorithms. Moreover, it favors the development of corresponding new features by recasting four unknown system identification problems into a single one. The two-input/two-output system with real random variables is viewed as a single-input/single-output system with CRVs.