# Multiuser Transmit Precoding Design for Dimming Compatible Visible Light Communications
## 3 Joint Multiuser TPC Design and Dimming Control  
In this section, we propose a joint TPC design and dimming control scheme under the constraints of LED non-linearity and indoor illumination for the multiuser VLC systems. The objective of our design is to maximize the minimum SINR among users, as well as to achieve a desired dimming level. Accordingly, we can formulate our design as the following optimization problem:  
\[ \begin{split}\underset{\mathbf{P},\mathbf{d}}{\text{max}}& \underset{k=1,\cdots,N_{t}}{\text{min}}\text{SINR}_{k},\\ \text{s.t.}&\text{abs}\{\mathbf{P}\}\mathbf{\Delta }-\mathbf{P}\mathbf{c}\leq\mathbf{d}-\mathbf{d}_{L},\\ &\text{abs}\{\mathbf{P}\}\mathbf{\Delta}+\mathbf{P}\mathbf{c} \leq\mathbf{d}_{H}-\mathbf{d},\ \mathbf{P}\mathbf{b}+\mathbf{d}=\mathbf{d}_{t},\end{split}  \tag{17}\]  
where \(\text{SINR}_{k}\) is calculated according to (5). Explicitly, in the above problem, both \(\mathbf{P}\) and \(\mathbf{d}\) are jointly optimized. By reformulating the dimming constraint in (15), we have:  
\[\mathbf{d}=\mathbf{d}_{t}-\mathbf{P}\mathbf{b}. \tag{18}\]  
Thus, the optimization problem in (17) can be equivalently simplified to:  
\[ \begin{split}\underset{\mathbf{P}}{\text{max}}& \underset{k=1,\cdots,N_{t}}{\text{min}}\text{SINR}_{k},\\ \text{s.t.}&\text{abs}\{\mathbf{P}\}\mathbf{\Delta }-\mathbf{P}\widehat{\mathbf{c}}\leq\widehat{\mathbf{d}}_{L},\ \text{abs}\{\mathbf{P}\}\mathbf{\Delta}+\mathbf{P}\widehat{\mathbf{c}}\leq \widehat{\mathbf{d}}_{H},\end{split}  \tag{19}\]  
where \(\widehat{\mathbf{c}}=\mathbf{c}-\mathbf{b}\), \(\widehat{\mathbf{d}}_{L}=\mathbf{d}_{t}-\mathbf{d}_{L}\) and \(\widehat{\mathbf{d}}_{H}=\mathbf{d}_{H}-\mathbf{d}_{t}\). It is observed that the above optimization problem is non-convex with respect to (w.r.t.) \(\mathbf{P}\), which is difficult to solve. Next, we will equivalently transform the problem into a more tractable form and obtain the optimal solution by invoking the bisection methods. Let us introduce \(\mathbf{p}=\text{vec}(\mathbf{P})\). Then, we have:  
\[ \begin{split}\text{abs}\{\mathbf{P}\}\mathbf{\Delta}=\text{vec} (\text{abs}\{\mathbf{P}\}\mathbf{\Delta})=(\mathbf{\Delta}^{T}\otimes\mathbf{ I}_{N_{t}})\text{abs}\{\mathbf{p}\},\\ \mathbf{P}\widehat{\mathbf{c}}=\text{vec}(\mathbf{P}\widehat{ \mathbf{c}})=(\widehat{\mathbf{c}}^{T}\otimes\mathbf{I}_{N_{t}})\mathbf{p}. \end{split}  \tag{20}\]  
Thus, the constraints of (19) can be reformulated as:  
\[\widetilde{\mathbf{\Delta}}\text{abs}\{\mathbf{p}\}-\mathbf{C}\mathbf{p}\leq \widehat{\mathbf{d}}_{L},\ \widetilde{\mathbf{\Delta}}\text{abs}\{\mathbf{p}\}+\mathbf{C}\mathbf{p}\leq \widehat{\mathbf{d}}_{H}, \tag{21}\]  
where \(\widetilde{\mathbf{\Delta}}=\mathbf{\Delta}^{T}\otimes\mathbf{I}_{N_{t}}\) and \(\mathbf{C}=\widehat{\mathbf{c}}^{T}\otimes\mathbf{I}_{N_{t}}\). Then, by introducing the slack variable \(\mathbf{f}\)[19], we can transform the constraints to the following equivalent linear formulations:  
\[\widetilde{\mathbf{\Delta}}\mathbf{f}-\mathbf{C}\mathbf{p}\leq\widehat{ \mathbf{d}}_{L},\ \widetilde{\mathbf{\Delta}}\mathbf{f}+\mathbf{C}\mathbf{p}\leq\widehat{ \mathbf{d}}_{H},\ -\mathbf{f}\leq\mathbf{p}\leq\mathbf{f}. \tag{22}\]  
Furthermore, given a non-negative value \(\gamma\), \(\text{SINR}_{k}\geq\gamma\) leads to:  
\[\sum_{j\
eq k}\left[u_{j}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}+ \sum_{i\
eq k,j}b_{i}b_{j}\mathbf{h}_{k}^{T}\mathbf{p}_{i}\mathbf{h}_{k}^{T} \mathbf{p}_{j}\right]+\sigma_{k}^{2}\leq\frac{u_{k}}{\gamma}\left(\mathbf{h} _{k}^{T}\mathbf{p}_{k}\right)^{2}. \tag{23}\]Rewriting the second item on the left-hand side of (23) yields the equivalent expression of:  
\[