## 3 Rate Analysis of MU-PhC-MIMO VLC Systems
### Achievable Weighted Sum Rate  
Starting from the fundamental definition of mutual information, we can obtain the achievable rate \(R_{k}\) for \(\text{User}_{k},\forall k\) as follows:  
\[ \begin{split} R_{k}=-\sum_{y_{k}=0}^{\infty}\Pr(Y_{k}=y_{k}) \times\log_{2}(\Pr(Y_{k}=y_{k}))+\\ \sum_{y_{k}=0}^{\infty}\sum_{s_{k}=-1}^{1}\Pr(Y_{k}=y_{k},S_{k}= s_{k})\times\log_{2}(\Pr(Y_{k}=y_{k}\mid S_{k}=s_{k}))\\ =-\sum_{y_{k}=0}^{\infty}\biggl{[}\sum_{s_{k}=-1}^{1}\Pr(y_{k} \mid s_{k})\times\Pr(s_{k})\biggr{]}\times\log_{2}\biggl{[}\sum_{s_{k}=-1}^{1 }\Pr(y_{k}\mid s_{k})\times\Pr(s_{k})\biggr{]}+\\ \sum_{y_{k}=0}^{\infty}\sum_{s_{k}=-1}^{1}\Bigl{[}\Pr(y_{k}\mid s _{k})\times\Pr(s_{k})\Bigr{]}\times\log_{2}(\Pr(y_{k}\mid s_{k}))\\ =\frac{1}{2}\sum_{y_{k}=0}^{\infty}\sum_{s_{k}=-1}^{1}\Pr(Y_{k}= y_{k}\mid S_{k}=s_{k})\times\log_{2}\frac{2\Pr(Y_{k}=y_{k}\mid S_{k}=s_{k})}{ \sum_{s_{k}=-1}^{1}\Pr(Y_{k}=y_{k}\mid S_{k}=s_{k})}.\end{split}  \tag{7}\]  
The first term in the second equality in (7) is derived from the law of total probability, i.e., \(\Pr(Y_{k}=y_{k})=\sum_{s_{k}=-1}^{1}\Pr(Y_{k}=y_{k}\mid S_{k}=s_{k})\times\Pr( S_{k}=s_{k})\), and the second term is obtained using the multiplication rule in probability theory. The third equality in (7) results from substituting \(\Pr(S_{k})=\frac{1}{2}\) and subsequent rearrangements. It can be found from (7) that the achievable rate \(R_{k}\) of User\({}_{k}\) is only related to the conditional probability \(\Pr(Y_{k}=y_{k}\mid S_{k}=s_{k})\), and according to (5), the conditional probabilities for \(S_{k}=-1\) and \(S_{k}=1\) can be obtained by  
\[ \begin{split}&\Pr(Y_{k}=y_{k}\mid S_{k}=-1)\\ &=\sum_{S_{1}}\cdots\sum_{S_{j,j\
eq k}}\cdots\sum_{S_{K}}\Pr(Y_{ k}=y_{k}\mid S=\mathbf{s})\times\Pr(S_{1})\times\cdots\times\Pr\Bigl{(}S_{j,j\
eq k }\Bigr{)}\times\cdots\times\Pr\Bigl{(}S_{K}\Bigr{)}\\ &=\frac{1}{2^{K-1}}\sum_{S_{1}}\cdots\sum_{S_{j,j\
eq k}}\cdots \sum_{S_{K}}\frac{\Bigl{(}-\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+\sum_{j=1,j\
eq k}^{ K}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}S_{j}+\zeta g_{k}^{T}\mathbf{d}+n_{\text{b}}\Bigr{)}^{y_{ k}}}{y_{k}!}\times\\ &\qquad\qquad\exp\Biggl{(}-\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+\sum_{j =1,j\
eq k}^{K}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}S_{j}+\zeta g_{k}^{T}\mathbf{d}+n_{ \text{b}}\Biggr{)}.\end{split}  \tag{8}\]  
Similarly, we can obtain:  
\[ \begin{split}&\Pr(Y_{k}=y_{k}\mid S_{k}=1)\\ &=\frac{1}{2^{K-1}}\sum_{S_{1}}\cdots\sum_{S_{j,j\
eq k}}\cdots \sum_{S_{K}}\frac{\Bigl{(}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+\sum_{j=1,j\
eq k}^{ K}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}S_{j}+\zeta g_{k}^{T}\mathbf{d}+n_{\text{b}} \Bigr{)}^{y_{k}}}{y_{k}!}\times\\ &\qquad\qquad\exp\Biggl{(}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+\sum_{j =1,j\
eq k}^{K}\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}S_{j}+\zeta n_{s}g_{k}^{T}\mathbf{d}+n_ {\text{b}}\Biggr{)}.\end{split}  \tag{9}\]  
The first equality in (8) is derived from the chain rule of conditional probability along with the definition of marginal probability, and the second equality in (8) results from substituting (5) and (6) into the first equality in (8). For convenience, let us denote \(\Pr(Y_{k}=y_{k}\mid S_{k}=-1)\) as \(\zeta_{k,1}\) and \(\Pr(Y_{k}=y_{k}\mid S_{k}=1)\) as \(\zeta_{k,2}\). Substituting (8) and (9) into (7), we obtain  
\[ \begin{split} R_{k}&=\frac{1}{2}\sum_{y_{k}=0}^{ \infty}\zeta_{k,1}\times\log_{2}\frac{2\zeta_{k,1}}{\zeta_{k,1}+\zeta_{k,2}}+ \frac{1}{2}\sum_{y_{k}=0}^{\infty}\zeta_{k,2}\times\log_{2}\frac{2\zeta_{k,2} }{\zeta_{k,1}+\zeta_{k,2}}\\ &=\frac{1}{2}\sum_{y_{k}=0}^{\infty}(\zeta_{k,1}+\zeta_{k,2})+ \frac{1}{2\ln 2}\sum_{y_{k}=0}^{\infty}(\zeta_{k,1}\times\ln\zeta_{k,1}+\zeta_{k,2} \times\ln\zeta_{k,2})\\ &\quad-\frac{1}{2\ln 2}\sum_{y_{k}=0}^{\infty}(\zeta_{k,1}+\zeta_{k,2} )\times\ln(\zeta_{k,1}+\zeta_{k,2})\\ &=1+\frac{1}{2\ln 2}\sum_{y_{k}=0}^{\infty}(\zeta_{k,1}\times\ln\zeta_{k,1 }+\zeta_{k,2}\times\ln\zeta_{k,2}-(\zeta_{k,1}+\zeta_{k,2})\times\ln(\zeta_{k,1 }+\zeta_{k,2})).\end{split}