## 3 System Design
### Illumination and Communication Constraints  
(1) Chromaticity diagram  
Chromaticity is the basic characteristic of the color perceived by human eyes. Based on CIE 1931 [19], the XYZcolor system has been established to reference tristimulus value matrix \(\mathbf{C}=[X,Y,Z]^{T}\) corresponding to the three kinds of cones of the human eyes, which can be modeled as:  
\[ \begin{split} X&=\int S(\lambda)\varphi_{1}( \lambda)d\lambda,\\ Y&=\int S(\lambda)\varphi_{2}(\lambda)d\lambda, \\ Z&=\int S(\lambda)\varphi_{3}(\lambda)d\lambda, \end{split}  \tag{9}\]  
where the color matching functions \(\varphi_{1}\), \(\varphi_{2}\), and \(\varphi_{3}\) denote the spectral responses of the three kinds of cones. The SPD of the multi-color LED is given by \(S(\lambda)=\sum_{i=1}^{N}\tilde{P}_{i}S_{i}(\lambda)\), where \(S_{i}(\lambda)\) denotes the SPD of the \(i^{\text{th}}\) color LED chip. Accordingly, the tristimulus values \(XYZ\) can be respectively expressed as:  
\[ \begin{split} X&=\int\sum_{i=1}^{N}\tilde{P}_{i}S_ {i}(\lambda)\varphi_{1}(\lambda)d\lambda\\ &=\sum_{i=1}^{N}f_{i1}\tilde{P}_{i}=\mathbf{\ell}_{1}^{T}\tilde{\mathbf{ \rho}},\end{split}  \tag{10}\]  
Figure 3: The model of the multi-color-based VLC system. and:  
\[ \begin{split} Y&=\sum_{i=1}^{N}f_{i2}\bar{P}_{i}=\mathbf{ f}_{2}^{T}\bar{\mathbf{P}},\\ Z&=\sum_{i=1}^{N}f_{3}\bar{P}_{i}=\mathbf{f}_{3}^{T} \mathbf{P},\end{split}  \tag{11}\]  
where \(\mathbf{f}_{j}=[f_{1j},f_{2j},\cdots,f_{Nj}]^{T}\) with \(f_{ij}=\int S_{i}(\lambda)\varphi_{j}(\lambda)d\lambda\) for \(i\in\{1,2,\cdots,N\}\), \(j\in\{1,2,3\}\), and the matrix \(\mathbf{F}\) can be defined as \(\mathbf{F}=[\mathbf{f}_{1},\mathbf{f}_{2},\mathbf{f}_{3}]_{3\times N}^{T}\). Thus, we can get the relation between the average transmitted optical power and the tristimulus values as:  
\[\mathbf{C}=\mathbf{F}\bar{\mathbf{P}}. \tag{12}\]  
The human eye color vision is the projection of \((X,Y,Z)\) onto the plane of \(x+y+z=1\) in the \(x-y-z\) space, given by \((x,y,z)\) as follows [11]:  
\[ \begin{split} x&=\frac{X}{X+Y+Z},\\ y&=\frac{Y}{X+Y+Z},\\ z&=\frac{Z}{X+Y+Z}.\end{split}  \tag{13}\]  
Based on the above analysis, we define \(\mathbf{c}=[x,y]^{T}\), and we can have the relation between the chromaticity coordinate of mixed color \((x,y)\) in the CIE 1931 chromaticity diagram and the average transmitted optical power as:  
\[\mathbf{c}=\mathbf{G}\bar{\mathbf{F}}\bar{\mathbf{P}}, \tag{14}\]  
where \(\mathbf{G}= \begin{bmatrix}\kappa&0&0\\ 0&\kappa&0\end{bmatrix} \) with \(\kappa=\frac{1}{X+Y+Z}=\frac{1}{\mathbf{1}^{T}(\mathbf{F}\bar{\mathbf{P}})}\). (2) Quadrangle constraint:  
According to the CIE 1931 chromaticity diagram, each coordinate represents a certain chromaticity. It is worth noting that the chromaticity change cannot be perceived in a certain range due to the limitation of human eyes. To specify the range of chromaticity recommended for LED products, ANSI C78.377-2015 [15] has been established to modify the chromaticity tolerances in the quadrangle range instead of a fixed point in the CIE 1931 chromaticity diagram, which provides more degrees of freedom to improve the performance of the system. Table 1 lists the \((x,y)\) chromaticity coordinates of the center points and the four corners of each quadrangle range constraint for the target white color at different CCTs, which is shown in Figure 4. Therefore, the mixed white light chromaticity constraint in a specific quadrangle range can be expressed as:  
\[ \begin{cases}y\geq k_{1}x+l_{1},\\ y\geq k_{2}x+l_{2},\\ y\leq k_{3}x+l_{3},\\ y\leq k_{4}x+l_{4},\end{cases}  \tag{15}\]  
where \(k_{i}\) and \(l_{i}\) are the coefficients determined by the quadrangle range in the chromaticity diagram with \(i=1,2,3,4\). In order to make the consistent form, Equation (15) can be rewritten as:\[\left\{ \begin{array}{c}k_{1}x-y+l_{1}\leq 0,\\ k_{2}x-y+l_{2}\leq 0,\\ -k_{3}x+y-l_{3}\leq 0,\\ -k_{4}x+y-l_{4}\leq 0.\end{array} \right.