# Effective Receiver Design for MIMO Visible Light Communication with Quadrichromatic LEDs
## 3 Proposed Receiver Design  
### Problem Formulation  
Denote \(\mathbf{t}_{j}\), \(\mathbf{u}_{i}\), and \(\mathbf{v}_{ji}\) as the normal vector of the \(j\)-th QLED, the normal vector of the \(i\)-th receiver, and the vector between positions of the \(j\)-th QLED and the \(i\)-th receiver as in Figure 2. Without loss of generality, we normalize the normal vector lengths to 1, that is, \(\|\mathbf{t}_{j}\|=1\), \(\|\mathbf{u}_{i}\|=1\). Consequently, we have \(\cos\phi_{ij}=\frac{\mathbf{t}_{j}^{\mathrm{T}}\mathbf{v}_{ji}}{\|\mathbf{v}_{ ji}\|}\), \(\cos\theta_{ij}=\frac{-\mathbf{v}_{ji}^{\mathrm{T}}\mathbf{u}_{i}}{\|\mathbf{v}_{ ji}\|}\). Therefore, as described in [10], in the condition of \(0<\theta_{ij}\leqslant\theta_{\mathrm{f}ov}\), the channel coefficient in (3) can be rewritten as  
\[h_{ij}=\frac{(\gamma+1)\,A_{r}\delta}{2\pi d_{ij}^{2}}\left(\frac{\mathbf{t}_{ j}^{\mathrm{T}}\mathbf{v}_{ji}}{\|\mathbf{v}_{ji}\|}\right)^{\gamma}\frac{-\mathbf{v}_{ ji}^{\mathrm{T}}\mathbf{u}_{i}}{\|\mathbf{v}_{ji}\|}. \tag{4}\]  
Consequently, \(h_{ij}\) can be expressed as  
\[h_{ij}=q_{ij}p_{ij}, \tag{5}\]where \(q_{ij}=\frac{-(\gamma+1)A_{r}\delta}{2\pi d_{ij}^{2}\|{\bf v}_{ji}\|}\left(\frac{{ \bf t}_{ij}^{T}{\bf v}_{ji}}{\|{\bf v}_{ji}\|}\right)^{\gamma}\) and \(p_{ji}={\bf v}_{ji}^{T}{\bf u}_{i}\). Theoretically, the vector \({\bf u}_{i}\) will directly affect the value of the channel coefficient \(h_{ij}\). Moreover, to better represent the received signal according to the orientation vectors of all receivers, we have  
\[{\bf r}_{m}={\bf H}{\bf x}_{m}+{\bf n}=\left({\bf x}_{m}^{T}\otimes{\bf I}_{4N _{r}}\right)\,{\rm vec}\left({\bf H}\right)+{\bf n}. \tag{6}\]  
With the results from Equations (2) and (5), we can write \({\rm vec}\left({\bf H}\right)={\rm vec}\left({\bf\hat{H}}\right)\circ{\rm vec }\left({\bf W}\right)={\rm vec}\left({\bf P}\right)\circ{\rm vec}\left({\bf Q} \right)\circ{\rm vec}\left({\bf W}\right)\) and so  
\[{\bf r}_{m}=\left({\bf x}_{m}^{T}\otimes{\bf I}_{4N_{r}}\right)\left[{\rm vec }\left({\bf P}\right)\circ{\rm vec}\left({\bf Q}\right)\circ{\rm vec}\left({ \bf W}\right)\right]+{\bf n}, \tag{7}\]  
where \(q_{ij}\) corresponds to the element in the \(i\)-th row and \(j\)-th column of the \(4N_{t}\times 4N_{r}\) matrix \({\bf Q}\). Similarly, \(p_{ij}\) corresponds to the element in the \(i\)-th row and the \(j\)-th column of the matrix \({\bf P}\). Moreover, since \(p_{ij}={\bf v}_{ji}^{T}{\bf u}_{i}\), we can express  
\[{\rm vec}\left({\bf P}\right)={\bf K}{\bf U}, \tag{8}\]  
where \({\bf U}=\left[{\bf u}_{1}^{T},{\bf u}_{2}^{T},...,{\bf u}_{N_{r}}^{T}\right]^{T}\). In addition, \({\bf K}={\bf 1}_{4\times 1}\otimes{\bf G}\) where \({\bf G}=\left[ \begin{matrix}{\bf G}_{1}&{\bf 0}&{\bf 0}\\ {\bf 0}&\ddots&{\bf 0}\\ {\bf 0}&{\bf 0}&{\bf G}_{N_{r}}\end{matrix} \right]\) and \({\bf G}_{i}={\bf 1}_{4\times 1}\otimes\left[ \begin{matrix}{\bf v}_{i1}^{T}\\ \vdots\\ {\bf v}_{iN_{t}}^{T}\end{matrix} \right]\). Note that \({\rm vec}\left({\bf Q}\right)\), \({\rm vec}\left({\bf W}\right)\) are the vectors of size \(16N_{t}N_{r}\times 1\); \({\bf K}\) and \({\bf U}\) are of sizes \(16N_{t}N_{r}\times 12\) and \(12\times 1\), respectively. Finally, we have  
\[ \begin{split}{\bf r}_{m}&=\left({\bf x}_{m}^{T}\otimes{\bf I }_{4N_{r}}\right)\left[\left({\bf K}{\bf U}\right)\circ{\rm vec}\left({\bf Q} \right)\circ{\rm vec}\left({\bf W}\right)\right]+{\bf n}\\ &=\left({\bf x}_{m}^{T}\otimes{\bf I}_{4N_{r}}\right){\bf C}{\bf U }+{\bf n}={\bf B}_{m}{\bf U}+{\bf n},\end{split}  \tag{9}\]  
where \({\bf C}\) is a \(16N_{t}N_{r}\times 12\) matrix with each row \({\bf c}_{z}\) computed by \({\bf c}_{z}=q_{z}w_{z}{\bf k}_{z}\) and \({\bf B}_{m}=\left({\bf x}_{m}^{T}\otimes{\bf I}_{4N_{r}}\right){\bf C}\). In here, \({\bf k}_{z}\) is the \(z\)-th row of matrix \({\bf K}\), \(q_{z}\) and \(w_{z}\) are the corresponding \(z\)-th elements of \({\rm vec}\left({\bf Q}\right)\) and \({\rm vec}\left({\bf W}\right)\), respectively.