## 1 Introduction
### Contributions  
We present a novel method to optimize the weighted sum rate of an MU-PhC-MIMO VLC system in the presence of Poisson shot noises. This is accomplished by utilizing a ZF-based approach to mitigate MUI and approximate the system's weighted sum-rate expression and then using a sequence of convexification processes to iteratively optimize the weighted sum rate of the system based on this approximation. This paper's primary contributions can be outlined as follows:  
* This paper derives the expression for the weighted sum rate of an MU-PhC-MIMO VLC system based on the definition of mutual information. Additionally, it provides an approximate expression for the proposed system's weighted sum rate under the minimized MUI, which is obtained utilizing a ZF approach.
* We propose a new optimization problem targeting the precoding matrix, aiming to maximize the weighted sum rate of the proposed MU-PhC-MIMO VLC system while minimizing MUI using the aforementioned ZF scheme.
* A novel sub-algorithm is developed to address the updated problem by employing variable substitution and successive convex approximation (SCA). After the analysis, this sub-algorithm is expected to converge to a robust solution that meets the Karush-Kuhn-Tucker (KKT) conditions. Afterward, by utilizing this sub-algorithm to traverse through all feasible scenarios, we can eventually obtain the optimal solution for the entire optimization problem.
* We also introduce a low-complexity alternative algorithm that achieves results close to those of the exhaustive algorithm. However, it significantly reduces discussions about possibilities, thereby reducing the algorithm's complexity from exponential to polynomial levels.  
Extensive simulations illustrate that the proposed algorithms can save the system 2.5 dB of transmit peak power compared to the classic ZF precoding scheme at low transmit power. At high transmit power, the sum rate of our system significantly surpasses that of the repetition coding scheme. Both of the proposed algorithms exhibit nearly identical performance. Furthermore, we show that increased active LEDs and lower DC bias power can further enhance system performance.  
The structure of the remainder of this paper is as follows. In Section 2, we introduce the proposed MU-PhC-MIMO VLC system and give the system's framework. In Section 3, we analyze the weighted sum rate of the MU-PhC-MIMO VLC system and provide an asymptotic approximation of the weighted sum rate based on the ZF scheme. We outline the problem under consideration and describe the proposed algorithms in Section 4. In Section 5, we present the outcomes of our simulations. Finally, we summarize this paper in Section 6.  
Notations: \(\mathcal{R}\) denotes the set of real numbers; \(\mathcal{R}_{+}\) denotes the set of positive real numbers; lowercase bold type indicates column vectors; uppercase bold type indicates matrices; \((\cdot)^{T}\) signifies transpose; \(|\cdot|\) signifies absolute value; \(\|\cdot\|_{p}\) signifies the p-norm, where \(p=1,2,\cdots,\infty\); \(\text{log}_{2}(\cdot)\) represents a logarithm with base 2; \(\text{ln}(\cdot)\) represents a natural logarithm with base Euler's number; \(\Pr(\cdot)\) stands for probability; \(\Pr(\cdot|\cdot)\) stands for conditional probability; and \(\Pr(\cdot,\cdot)\) stands for joint probability.