## 3 The Proposed Optimal Power Allocation Strategy
### Illumination and Communication Constraints  
**(1) \(IIuminance\) Constraint**  
In order to achieve the desired brightness levels for users, illuminance constraint should be satisfied [21]. The multi-color LED transmitter should maintain constant brightness level where the mixed white light should be unflickering, so the illuminance constraint can be expressed as  
\[\mathbf{1}^{T}\mathbf{\Phi}=\sum_{i=1}^{N}\Phi_{i}=P_{t}, \tag{9}\]  
where \(\mathbf{\Phi}=[\Phi_{1},\Phi_{2},\cdots,\Phi_{N}]^{T}\) is the luminous flux vector, \(\Phi_{i}\) is the luminous flux for the \(i\)th color LED chip and \(P_{t}\) denotes the total luminous flux for the multi-color LED.  
**(2) \(Quadrangle\)** \(Constration\)  
Chromaticity is the basic characteristic of the color perceived by human eyes. In CIE 1931 color space chromaticity diagram, the chromaticity of color can be represented by the coordinate point \((x,y)\), as shown in Figure 3. Accordingly, we assume the the corresponding chromaticity coordinate is \((x_{i},y_{i})\) for the \(i\)th color LED chip. Based on the Grassmann's laws, the desired chromaticity coordinate of the mixed light \(\mathbf{p}=(\hat{x},\hat{y})\) can be calculated by [22]  
\[\hat{x}=\frac{\mathbf{a}^{T}\mathbf{\Phi}}{\mathbf{b}^{T}\mathbf{ \Phi}}=\frac{\sum_{i=1}^{N}\frac{x_{i}}{y_{i}}\Phi_{i}}{\sum_{i=1}^{N}\frac{1 }{y_{i}}\Phi_{i}}, \tag{10}\] \[\hat{y}=\frac{\mathbf{1}^{T}\mathbf{\Phi}}{\mathbf{b}^{T}\mathbf{ \Phi}}=\frac{\sum_{i=1}^{N}\Phi_{i}}{\sum_{i=1}^{N}\frac{1}{y_{i}}\Phi_{i}},\]  
where \(\mathbf{a}=[\frac{x_{1}}{y_{1}},\frac{x_{2}}{y_{2}},\cdots,\frac{x_{N}}{y_{N} }]^{T}\) and \(\mathbf{b}=[\frac{1}{y_{1}},\frac{1}{y_{2}},\cdots,\frac{1}{y_{N}}]^{T}\) are the constant coefficients vectors.  
Since human eyes have limitation on color discrimination, small chromaticity change for the white light can be tolerated. According to Ref. [14], quadrangle chromaticity tolerance can be utilized as a statistical measurement to distinguish chromaticity difference between two colors in CIE 1931 chromaticity diagram, which is recommended for LED products. If the chromaticity point moves within the quadrangle range, human eyes almost could not notice the light color variation. Compared with the fixed chromaticity point, quadrangle chromaticity constraint can provide more freedom to improve the system performance. The chromaticity tolerance with quadrangle range for the target white color under different CCTs is shown in Figure 4, and the corresponding chromaticity center points are listed in Table 1 where A, B, C and D represent the four corners of the quadrangle. It should be noted that the proportion of the blue and green component becomes higher with the increase of CCT values. Accordingly, the quadrangle chromaticity constraint can be calculated as  
\[\left\{\begin{aligned} & k_{1}x-y+l_{1}\leq 0,\\ & k_{2}x-y+l_{2}\leq 0,\\ &-k_{3}x+y-l_{3}\leq 0,\\ &-k_{4}x+y-l_{4}\leq 0,\\ \end{aligned}\right. \tag{11}\]  
where \(k_{i}\) and \(l_{i}\) denote the coefficients determined by the quadrangle range in the chromaticity diagram. We set \(\mathbf{K}=\begin{bmatrix}k_{1}&k_{2}&-k_{3}&-k_{4}\\ -1&-1&1&1\end{bmatrix}^{T}\) and \(\mathbf{L}=[l_{1},l_{2},-l_{3},-l_{4}]^{T}\) here, so the quadrangle chromaticity constraint can be rewritten as  
\[\mathbf{Kp}+\mathbf{L}\leq\mathbf{0}. \tag{12}\]  
Figure 3: CIE 1931 color space chromaticity diagram.