# Multiuser Transmit Precoding Design for Dimming Compatible Visible Light Communications
## 3 Joint Multiuser TPC Design and Dimming Control
\begin{split}&\sum_{j\
eq k}\sum_{i\
eq k,j}b_{i}b_{j}\mathbf{h}_{k} ^{T}\mathbf{p}_{i}\mathbf{h}_{k}^{T}\mathbf{p}_{j}=\left(\sum_{j\
eq k}b_{j} \mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}-\sum_{j\
eq k}b_{j}^{2}\left( \mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}\\ &=\left(\sum_{j}b_{j}\mathbf{h}_{k}^{T}\mathbf{p}_{j}-b_{k} \mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}-\sum_{j\
eq k}b_{j}^{2}\left( \mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}\\ &=\left(\mathbf{h}_{k}^{T}\sum_{j}b_{j}\mathbf{p}_{j}-b_{k} \mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}-\sum_{j\
eq k}b_{j}^{2}\left( \mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}.\end{split}  \tag{24}\]  
Since we have:  
\[\mathbf{Pb}=\left[\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{N_{t}} \right]\mathbf{b}=\sum_{j}b_{j}\mathbf{p}_{j}, \tag{25}\]  
(24) can be further expressed as:  
\[\sum_{j\
eq k}\sum_{i\
eq k,j}b_{i}b_{j}\mathbf{h}_{k}^{T}\mathbf{p}_{i} \mathbf{h}_{k}^{T}\mathbf{p}_{j}=\left(\mathbf{h}_{k}^{T}\mathbf{Pb}-b_{k} \mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}-\sum_{j\
eq k}b_{j}^{2}\left( \mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}. \tag{26}\]  
Additionally, since we have:  
\[ \begin{split}&\sum_{j\
eq k}b_{j}^{2}\left(\mathbf{h}_{k}^{T} \mathbf{p}_{j}\right)^{2}=\sum_{j}b_{j}^{2}\left(\mathbf{h}_{k}^{T}\mathbf{p}_ {j}\right)^{2}-b_{k}^{2}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2},\\ &\sum_{j\
eq k}u_{j}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^ {2}=\sum_{j}u_{j}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}-u_{k}\left( \mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2},\end{split}  \tag{27}\]  
the left-hand side of (23) can be equivalently rewritten as:  
\[ \begin{split}&\sum_{j\
eq k}\left[u_{j}\left(\mathbf{h}_{k}^{T} \mathbf{p}_{j}\right)^{2}+\sum_{i\
eq k,j}b_{i}b_{j}\mathbf{h}_{k}^{T}\mathbf{ p}_{i}\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right]+\sigma_{k}^{2}\\ &=\sum_{j}u_{j}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}- u_{k}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}+\left(\mathbf{h}_{k}^{T} \mathbf{Pb}-b_{k}\mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}-\sum_{j}b_{j}^{2} \left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}+b_{k}^{2}\left(\mathbf{h}_{ k}^{T}\mathbf{p}_{k}\right)^{2}+\sigma_{k}^{2}\\ &=\sum_{j=1}^{N_{t}}\left(u_{j}-b_{j}^{2}\right)\left(\mathbf{h}_{ k}^{T}\mathbf{p}_{j}\right)^{2}+\left(\mathbf{h}_{k}^{T}\mathbf{Pb}-b_{k} \mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}+\sigma_{k}^{2}-\left(u_{k}-b_{k}^{2 }\right)\left(\mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}.\end{split}  \tag{28}\]  
Thus, the inequality in (23) can be reformulated as:  
\[ \begin{split}&\sum_{j=1}^{N_{t}}\left(u_{j}-b_{j}^{2}\right) \left(\mathbf{h}_{k}^{T}\mathbf{p}_{j}\right)^{2}+\left(\mathbf{h}_{k}^{T} \mathbf{Pb}-b_{k}\mathbf{h}_{k}^{T}\mathbf{p}_{k}\right)^{2}+\sigma_{k}^{2}\\ &\leq\frac{u_{k}}{\gamma}\left(\mathbf{h}_{k}^{T}\mathbf{p}_{k} \right)^{2}+\left(u_{k}-b_{k}^{2}\right)\left(\mathbf{h}_{k}^{T}\mathbf{p}_{k }\right)^{2}.\end{split}  \tag{29}\]  
Using the equalities:  
\[ \begin{split}&\mathbf{p}_{i}=\mathrm{vec}(\mathbf{Pe}_{i})=( \mathbf{e}_{i}^{T}\otimes\mathbf{I}_{N_{t}})\mathbf{p}\triangleq\mathbf{W}_{i }\mathbf{p},\\ &\mathbf{h}_{k}^{T}\mathbf{Pb}=\mathrm{vec}\left(\mathbf{h}_{k}^{T }\mathbf{Pb}\right)=(\mathbf{b}^{T}\otimes\mathbf{h}_{k}^{T})\mathbf{p}, \end{split}  \tag{30}\]  
where \(\mathbf{W}_{i}\triangleq\mathbf{e}_{i}^{T}\otimes\mathbf{I}_{N_{t}}\), (29) can be formulated in the form of \(l_{2}\) norm, which is given by:  
\[||\mathbf{B}_{k}\mathbf{p}+\mathbf{v}_{k}||_{2}\leq\varepsilon_{k}\mathbf{h}_ {k}^{T}\mathbf{W}_{k}\mathbf{p}, \tag{31}\]where \(\mathbf{B}_{k},\mathbf{v}_{k},\) and \(\varepsilon_{k}\) are defined as:  
\[\mathbf{B}_{k} =\left[ \begin{array}{cc}\mathbf{h}_{k}^{T}\mathbf{W}_{1}(u_{1}- b_{1}^{2})\\ \vdots\\ \mathbf{h}_{N_{t}}^{T}\mathbf{W}_{N_{r}}(u_{N_{r}}-b_{N_{r}}^{2})\\ \mathbf{b}^{T}\otimes\mathbf{h}_{k}^{T}-b_{k}\mathbf{h}_{k}^{T}\mathbf{W}_{k} \\ \mathbf{0}_{1\times N_{t}N_{r}}\end{array} \right], \tag{32}\] \[\mathbf{v}_{k} =\left[ \begin{array}{cc}0,&\cdots,&0,&\sigma_{k},\end{array}