## 3 System Design
### Illumination and Communication Constraints
\tag{16}\]  
To simplify the expression of the quadrangle constraint (16), we denote:  
\[\mathbf{Kc}+\mathbf{L}\leq\mathbf{0}, \tag{17}\]  
where \(\mathbf{K}= \begin{bmatrix}k_{1}&-1\\ k_{2}&-1\\ -k_{3}&1\\ -k_{4}&1\end{bmatrix} \) and \(\mathbf{L}=[l_{1},l_{2},-l_{3},-l_{4}]^{T}\). (3) Luminance constraint:  
Since color matching function \(\varphi_{2}\) is defined as the luminous efficiency function, the tristimulus value \(Y\) is roughly correlated with the luminous flux \(\Phi\) induced by human visual strength, which can be expressed as: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**CCT** & **Center Point** & \multicolumn{4}{c}{**(x,y) Tolerance Quadrangle**} \\ \hline  & & **A** & **B** & **C** & **D** \\ \hline
3000 K & (0.4339,0.4033) & (0.4562,0.4260) & (0.4303,0.4173) & (0.4150,0.3821) & (0.4373,0.3893) \\
3500 K & (0.4078,0.3930) & (0.4303,0.4173) & (0.4003,0.4035) & (0.3895,0.3709) & (0.4150,0.3821) \\
4000 K & (0.3818,0.3797) & (0.4003,0.4035) & (0.3737,0.3880) & (0.3671,0.3583) & (0.3895,0.3709) \\
4500 K & (0.3613,0.3670) & (0.3737,0.3882) & (0.3550,0.3754) & (0.3514,0.3482) & (0.3672,0.3585) \\
5000 K & (0.3446,0.3551) & (0.3550,0.3753) & (0.3375,0.3619) & (0.3366,0.3373) & (0.3514,0.3481) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coordinate of the four corners for the quadrangle constraint at different correlated color temperatures (CCTs). Figure 4: The chromaticity tolerance with the quadrangle range for LED products on the CIE 1931 chromaticity diagram. \[\Phi=K_{m}\int S(\lambda)\varphi_{2}(\lambda)d\lambda=K_{m}Y, \tag{18}\]  
where \(K_{m}\) is the conversion coefficient for the power to luminous flux (P/F). Based on Equation (11), we have the relation between the luminous flux and the average transmitted optical power as:  
\[\Phi=K_{m}\xi_{2}^{T}\tilde{\Phi}. \tag{19}\]  
We define \(\xi\) as the minimum luminance requirement, so the luminance constraint can be expressed as:  
\[\mathbf{f}_{2}^{T}\mathbf{P}\geq\frac{\xi}{K_{m}}. \tag{20}\]  
(4) Transmission power constraint:  
Unlike the conventional radio frequency (RF), the minimum value of the instantaneous transmission power \(\mathbf{P}\) should guarantee nonnegativity. As the modulated PAM signal is normalized in the range of \([-\Delta_{i},\Delta_{i}]\), the nonnegative constraint should satisfy:  
\[\mathbf{\Delta}\leq\mathbf{P}-\mathbf{P}^{min}, \tag{21}\]  
where \(\mathbf{\Delta}=[\Delta_{1},\Delta_{2}\cdots,\Delta_{N}]^{T}\) and \(\mathbf{P}^{min}=[P_{1}^{min},P_{2}^{min},\cdots,P_{N}^{min}]^{T}\) is the minimum level of the transmission power. Meanwhile, considering the nonlinearity for the LED, the transmitted signal may suffer clipping distortion once the power exceeds the maximum value, so the maximum limitation for the power is necessary. Thus, the maximum transmission power constraint is:  
\[\mathbf{\Delta}\leq\mathbf{P}^{max}-\mathbf{\tilde{P}}, \tag{22}\]  
where \(\mathbf{P}^{max}=[P_{1}^{max},P_{2}^{max},\cdots,P_{N}^{max}]^{T}\) is the maximum level of the transmission power. Accordingly, combining Constraints (21) and (22), we can have the transmission power constraint as:  
\[\mathbf{\Delta}\leq\min(\mathbf{\tilde{P}}-\mathbf{P}^{min},\mathbf{P}^{max}- \mathbf{\tilde{P}}). \tag{23}\]  
(5) Communication quality constraint:  
The VLC technique is the combination of communication and illumination, so the communication quality needs to be satisfied, where SINR is an important factor for improving the overall system performance [20]. Based on Equation (8), the SINR constraint at the \(i^{\text{th}}\) receiver can be represented as:  
\[ \begin{split} SINR_{i}&=\frac{\mathbb{E}(\rho^{2}t_ {i}^{2}h_{ii}^{2}p_{i}^{2})}{\sum_{j=1,j\
eq i}^{N}\mathbb{E}(\rho^{2}t_{ij}^{ 2}h_{ij}^{2}p_{j}^{2})+\sigma^{2}}\\ &=\frac{\rho^{2}t_{ii}^{2}h_{ii}^{2}\mathbb{E}(p_{i}^{2})}{\sum_{ j=1,j\
eq i}^{N}\rho^{2}t_{ij}^{2}h_{ij}^{2}\mathbb{E}(p_{j}^{2})+\sigma^{2}}, \end{split}  \tag{24}\]  
where \(\mathbb{E}(p_{i}^{2})=\frac{\Delta_{i}^{2}(M+1)}{3(M-1)}\), and \(M\) is the modulation order.