## 3 The Proposed Optimal Power Allocation Strategy
### Problem Formulation  
In this subsection, an optimal power allocation strategy is proposed to maximize the power efficiency for the multi-color LED based VLC systems. Considering the electrical power, the total power consumption is given by \(\sum_{i=1}^{N}(c_{i}\Phi_{i})^{2}\). Thus, combining the above-mentioned illumination and communication constraints, the main problem can be formulated as:  
\[\begin{split}\text{Obj. }&\min_{\mathbf{m},\mathbf{ \Phi}}\ \sum_{i=1}^{N}(c_{i}\Phi_{i})^{2}\\ \text{s.t. }&\ \mathbf{1}^{T}\mathbf{\Phi}=\sum_{i=1}^{N} \Phi_{i}=P_{t},\\ &\ \Firstly, we show the performance comparison between our proposed optimal power allocation strategy and uniform power allocation method for the CMIMO system model. The chromaticity coordinate for the uniform allocation scheme is \((\frac{1}{3},\frac{1}{3})\), which is contained in the quadrangle chromaticity range with CCT\(=5700\) K and the corresponding parameters have been listed in Table 1. Figure 5 shows the transmission power for different colors versus the total luminous flux ranging from 10 lm to 180 lm with interference coefficient \(\tau=0.05\). The simulation results indicate that the proposed optimal power allocation strategy obtains significant performance gains compared with the uniform power allocation method, reducing about 12.4% power consumption where the power efficiency improvement is independent of the luminous flux. Meanwhile, we can see that the transmission power of red, amber and green color chips keep medium levels, while the transmission power of the blue color chip keeps a low level. As for the transmission power efficiency for each color chip, compared with the uniform power allocation method, the red, amber and blue color chips can also achieve significant transmission power improvement, while the rate of transmission power for the green color chip increased. Since quadrangle chromaticity tolerance range has been modified to replace a fixed chromaticity coordinate in CIE 1931 chromaticity diagram, we can obtain more degrees of freedom to improve the system performance, which is consistent with the simulation results. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Type of white LED & RAGB LED \\ Number of LED chips \(N\) & 4 \\ Angle of irradiance \(\phi\) & \(30^{\circ}\) \\ Angle of incidence \(\psi\) & \(40^{\circ}\) \\ Half Power angle of LED \(\Phi_{1/2}\) & \(60^{\circ}\) \\ FOV \(\Psi_{C}\) & \(60^{\circ}\) \\ Receiver responsivity \(\mu\) & 0.4 A/W \\ Distance between LED and PD \(d_{ij}\) & 2 m \\ Effective detector area of PD & 1 cm\({}^{2}\) \\ Optical filter gain & 1 \\ Optical concentrator gain & 1.5 \\ Maximum forward current for each chip \(I_{max}\) & 0.7 A \\ SINR requirement \(\xi\) & 5 dB \\ Noise variance \(\sigma^{2}\) & \(10^{-14}\) W \\ Peak wavelength of each color & R 664 nm \\  & A 594 nm \\  & G 538 nm \\  & B 465 nm \\ Chromaticity point of each color & R (0.69406, 0.30257) \\  & A (0.59785, 0.39951) \\  & G (0.22965, 0.70992) \\  & B (0.12301, 0.09249) \\ Luminance flux-Forward current & R 0.021 A/lm \\ conversion coefficient of each color & A 0.014 A/lm \\  & G 0.005 A/lm \\  & B 0.015 A/lm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters utilized in the QLED VLC system. Next, the performance for the proposed optimal power allocation strategy under different CCT values is compared, where six CCT values are investigated with quadrangle chromaticity range based on Ref. [14]. As shown in Figure 6, the curves of total transmission power have the consistent variation tendency for different CCT values. With the increase of CCT values, higher power efficiency can be achieved. As illustrated in Section 3.1, the proportion of the blue and green component becomes higher at larger CCT values. Due to the higher forward current-luminance flux efficiency for blue and green component, the total transmission power can be reduced. Thus, the optimal power allocation under the required CCT values can be obtained for high-quality illumination.