# Effective Receiver Design for MIMO Visible Light Communication with Quadrichromatic LEDs
## 3 Proposed Receiver Design
### Proposed Solution  
Based on the criterion in (10), the normal vectors optimization problem can be formulated as follows with additional constraints:  
\[\text{(P1):}\quad\max_{\mathbf{u}_{i}\theta_{ij}}\min_{\mathbf{x }_{m}\
eq\mathbf{x}_{n}\in\mathcal{S}}\lVert\mathbf{H}\left(\mathbf{x}_{m}- \mathbf{x}_{n}\right)\rVert^{2} \tag{11a}\] \[s.t.\quad 0<\theta_{ij}\leqslant\theta_{\text{fov}},\] (11b) \[\lVert\mathbf{u}_{i}\rVert^{2}=1, \tag{11c}\]  
where constraints \(\lVert\mathbf{u}_{i}\rVert^{2}=1\) to ensure the normalization assumption for \(i=1,2,...,N_{r}\) and lead to the constraint \(\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3\times 1}\right)\lVert\mathbf{U} \rVert^{2}=\mathbf{1}_{N_{r}\times 1}\). Moreover, the condition \(0<\theta_{ij}\leqslant\theta_{\text{fov}}\) aims to ensure that the QLEDs are inside the FOV of the PDs, and can be expressed as \(\cos\theta_{\text{fov}}\leqslant\mathbf{\hat{v}}_{ji}^{T}\mathbf{u}_{i}<1\) or \(\left(\cos\theta_{\text{fov}}\right)\mathbf{1}_{N_{r}N_{r}\times 1}\leqslant \mathbf{KU}<\mathbf{1}_{N_{r}N_{r}\times 1}\). Furthermore, the objective function (11a) can be rewritten as  
\[\lVert\mathbf{H}\left(\mathbf{x}_{m}-\mathbf{x}_{n}\right)\rVert^{2} =\lVert\left(\mathbf{B}_{m}-\mathbf{B}_{n}\right)\mathbf{U} \rVert^{2} \tag{12}\] \[=\mathbf{U}^{T}\left(\mathbf{B}_{m}-\mathbf{B}_{n}\right)^{T} \left(\mathbf{B}_{m}-\mathbf{B}_{n}\right)\mathbf{U}\] \[=\mathbf{U}^{T}\mathbf{S}_{mn}\mathbf{U},\]  
where \(\mathbf{S}_{mn}=\left(\mathbf{B}_{m}-\mathbf{B}_{n}\right)^{T}\left(\mathbf{B }_{m}-\mathbf{B}_{n}\right)\). Consequently, with the additional slack variable \(\alpha\), the optimization problem (P1) can be equivalently transformed into  
\[\text{(P2):}\quad\max_{\mathbf{u}_{i},...,\mathbf{u}_{N_{r}}}\alpha \tag{13a}\] \[s.t.\quad\mathbf{U}^{T}\mathbf{S}_{mn}\mathbf{U}\geqslant\alpha,\] (13b) \[\left(\cos\theta_{\text{fov}}\right)\mathbf{1}_{N_{r}N_{r}\times 1} \leqslant\mathbf{KU}<\mathbf{1}_{N_{r}N_{r}\times 1},\] (13c) \[\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3\times 1}\right) \lVert\mathbf{U}\rVert^{2}=\mathbf{1}_{N_{r}\times 1}. \tag{13d}\]  
The optimization problem (P2) is still nonconvex because of constraints (13b) and (13d); and the global solution for (P2) is an NP-hard problem. Constraint (13d) can be split into two sub-constraints  
\[s.t.\quad 0<\theta_{ij}\leqslant\theta_{\text{fov}},\]0 \[s.t.\quad 0<\theta_{ij}\leqslant\theta_{\text{fov}},\]1To deal with the nonconvex constraints (13b) and constraint (14b), by denoting \(\mathbf{U}^{(l)}\) as the value of \(\mathbf{U}\) at the \(l\)-th iteration and employing the first-order Taylor series expansion around \(\mathbf{U}^{(l)}\), the left-hand side of the inequality in (13b) and (14b) can be approximated by  
\[s.t.\quad 0<\theta_{ij}\leqslant\theta_{\text{fov}},\]2  
and  
\[ \begin{split}&\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3 \times 1}\right)\left\|\mathbf{U}\right\|^{2}\\ &\approx\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3\times 1} \right)\left[2\left(\mathbf{U}^{(l)}\right)^{T}\mathbf{U}-\left(\mathbf{U}^{(l )}\right)^{T}\mathbf{U}^{(l)}\right].\end{split}  \tag{16}\]  
Finally, by inserting (15) and (16), the optimization problem (P2) can be reformed as  
\[\text{(P3):}\quad\max_{\mathbf{U}}\alpha \tag{17a}\] \[s.t. \quad 2\left(\mathbf{U}^{(l)}\right)\mathbf{S}_{mn}\mathbf{U}^{T}- \left(\mathbf{U}^{(l)}\right)^{T}\mathbf{S}_{mn}\mathbf{U}^{(l)}\geqslant\alpha,\] (17b) \[\quad\left(\cos\theta_{\text{fov}}\right)\mathbf{1}_{N_{l}N_{r} \times 1}\leqslant\mathbf{K}\mathbf{U}<\mathbf{1}_{N_{l}N_{r}\times 1},\] (17c) \[\quad\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3\times 1} \right)\mathbf{U}^{T}\mathbf{U}\leqslant\mathbf{1}_{N_{r}\times 1},\] (17d) \[\quad\left(\mathbf{I}_{N_{r}}\otimes\mathbf{1}_{3\times 1} \right)\left[2\left(\mathbf{U}^{(l)}\right)^{T}\mathbf{U}-\left(\mathbf{U}^{(l )}\right)^{T}\mathbf{U}^{(l)}\right]\] \[\quad\geqslant\mathbf{1}_{N_{r}\times 1}. \tag{17e}\]  
Clearly, (P3) is a convex optimization problem that can be solved by the CVX package [11; 12].