## 4 Maximization of Weighted Sum Rate Based on ZF Scheme
### SCA ZF-Based Precoding Design Solution
At this point, C7 and C8 can be merged and organized into a single convex constraint, i.e.,  
\[\frac{\lambda_{k,m}}{2\theta_{k,m}}+\frac{\gamma_{k,m}^{2}\theta_{k,m}}{2}-n_ {\text{s}}\mathbf{g}_{K}^{T}\mathbf{w}_{k}\leq 0,\quad\forall m=1,2,\quad\forall k=1, \cdots,K, \tag{22}\]where \(w_{k},\gamma_{k,m},\lambda_{k,m},\theta_{k,m},\forall m,\forall k\) can be iteratively updated using convex approximation. Problem **P2** can be reformulated as Problem **P3**. \[\begin{split}\textbf{P3}:\max_{\textbf{W}}&\quad \sum_{k=1}^{K}\mu_{k}\Big{[}1-\frac{1}{2\ln 2}-\frac{\sum_{m=1}^{2}\mathcal{F}( \gamma_{k,m})}{4\ln 2}\Big{]}\\ \text{s.t.}&\quad\text{C1:}\quad n_{\text{s}}\sum_{ k=1}^{K}|\mathbf{w}_{k}|-\mathbf{d}\preccurlyeq\textbf{0},\\ &\quad\text{C2:}\quad n_{\text{s}}\sum_{k=1}^{K}|\mathbf{w}_{k}|+\mathbf{d} \preccurlyeq\frac{P_{\text{max}}\tau}{\hbar v}\textbf{1},\\ &\quad\text{C3:}\quad\|\textbf{W}\|_{\infty}\leq 1,\\ &\quad\text{C4:}\quad g_{k}^{T}\mathbf{w}_{j}=0,\forall j\
eq k,\quad \forall j,k=1,\cdots,K,\\ &\quad\text{C5:}\quad\lambda_{k,1}=-\zeta n_{\text{s}}g_{k}^{T} \mathbf{w}_{k}+\zeta g_{k}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\\ &\quad\text{C6:}\quad\lambda_{k,2}=\zeta n_{\text{s}}g_{k}^{T} \mathbf{w}_{k}+\zeta g_{k}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\\ &\quad\text{C9:}\quad\frac{\lambda_{k,m}}{2\theta_{k,m}^{(t-1)} }+\frac{\gamma_{k,m}^{2}\theta_{k,m}^{(t-1)}}{2}-n_{\text  
\[\mathbf{4}:\max_{\mathbf{W}} \sum_{k=1}^{K}\mu_{k}\Big{(}1-\frac{1}{2\ln 2}\Big{)}-\frac{1}{4 \ln 2}\sum_{\gamma_{k,m}\in\mathcal{V}}\mu_{k}\mathcal{F}(\gamma_{k,m})\] s.t. \[\text{C1:}\quad n_{s}\sum_{k=1}^{K}|\mathbf{w}_{k}|-\mathbf{d}\prec \mathbf{0},\] \[\text{C2:}\quad n_{s}\sum_{k=1}^{K}|\mathbf{w}_{k}|+\mathbf{d}\prec \frac{P_{\max}\tau}{\hbar\upsilon}\mathbf{1},\] \[\text{C3:}\quad\|\mathbf{W}\|_{\infty}\leq 1,\] \[\text{C4:}\quad g_{k}^{T}\mathbf{w}_{j}=0,\forall j\
eq k,\quad \forall j,k=1,\cdots,K,\] \[\text{C5:}\quad\lambda_{k,1}=-\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+ \zeta g_{k}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\] \[\text{C6:}\quad\lambda_{k,2}=\zeta n_{s}g_{k}^{T}\mathbf{w}_{k}+ \zeta g_{k}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\] \[\gamma_{k,m}\sqrt{\lambda_{k,m}}\geq\frac{\lambda_{k,m}}{2\theta_{k,m}}+\frac {\gamma_{k,m}^{2}\theta_{k,m}}{2},\quad\forall m=1,2,\quad\forall k=1,\cdots,K, \tag{21}\]0 \[\gamma_{k,m}\sqrt{\lambda_{k,m}}\geq\frac{\lambda_{k,m}}{2\theta_{k,m}}+\frac {\gamma_{k,m}^{2}\theta_{k,m}}{2},\quad\forall m=1,2,\quad\forall k=1,\cdots,K, \tag{21}\]1 \[\gamma_{k,m}\sqrt{\lambda_{k,m}}\geq\frac{\lambda_{k,m}}{2\theta_{k,m}}+\frac {\gamma_{k,m}^{2}\theta_{k,m}}{2},\quad\forall m=1,2,\quad\forall k=1,\cdots,K, \tag{21}\]2  
where C10-C12 are adjusted according to the area where \(\gamma_{k,m},\forall m,\forall k\) is located, and C11 is because of \(\gamma_{k,m}\leq\frac{n_{s}g_{k}^{T}\mathbf{w}_{k}}{\sqrt{\lambda_{k,m}}}\) and \(\gamma_{k,m}\leq 1.6362\). After transformation, Problem **P4** becomes a well-formed convex optimization problem. This means we can utilize the existing MATLAB CVX toolbox [33], particularly the interior-point method, for its solution. It is worth noting that for the \(2K\)\(\gamma_{k,m}\)s, their interval combinations could total up to \(2^{2K}\) different cases. Hence, it is necessary to discuss each scenario to attain the optimal solution. We use \(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\},\forall t=1,\cdots,2^{2K}\) to distinguish different potential scenarios. Here, \(\hat{\mathcal{U}}_{t}\) and \(\hat{\mathcal{V}}_{t}\) are sets, where \(t\) denotes the \(t\)-th case. \(\hat{\mathcal{U}}_{t}\) comprises all \(\gamma_{k,m}\) within the interval \(\mathcal{U}\), and \(\hat{\mathcal{V}}_{t}\) contains all \(\gamma_{k,m}\) within the interval \(\mathcal{V}\). No \(\gamma_{k,m}\) exists that does not belong to either \(\hat{\mathcal{U}}_{t}\) or \(\hat{\mathcal{V}}_{t}\).