## 4 Maximization of Weighted Sum Rate Based on ZF Scheme
### SCA ZF-Based Precoding Design Solution  
The objective of Problem **P1** is complex and non-concave, and the constraints also exhibit uncertainty, making the entire problem a complex and challenging non-convex optimization problem. Below we employ methods such as variable substitution and SCA to transform Problem **P1** equivalently, deriving an effective approach to solve it. Constraints C1 and C2 are uncertain due to the inclusion of a random signal \(S_{k},\forall k\), but considering \(S_{k}\in\{1,-1\},\forall k\), we can replace C1 as  
\[n_{\text{s}}\sum_{k=1}^{K}|\mathbf{w}_{k}|-\mathbf{d}\preccurlyeq\mathbf{0}. \tag{17}\]  
After this processing, C1 no longer depends on the values of \(S_{k},\forall k\), as they apply universally across all \(S_{k},\forall k\) scenarios. Moreover, this transformation ensures the convexity of the constraint, which is advantageous for the subsequent solving steps. Similarly, we can rewrite C2 as  
\[n_{\text{s}}\sum_{k=1}^{K}|\mathbf{w}_{k}|+\mathbf{d}\preccurlyeq\frac{P_{\max}\tau}{ \hbar v}\mathbf{1}. \tag{18}\]The objective of Problem **P1** involves nested functions, making it challenging to handle. Here, we use variable substitution and introduce two sets of auxiliary variables \(\gamma_{k,1}\), \(\forall k\) and \(\gamma_{k,2}\), \(\forall k\) for processing. At this point, Problem **P1** can be restated as Problem **P2**. \[ \begin{split}\textbf{P2}:&\max_{\textbf{W}}\quad \sum_{k=1}^{K}\mu_{k}\Big{[}1-\frac{1}{2\ln 2}-\frac{\mathcal{F}(\gamma_{k,1})+ \mathcal{F}(\gamma_{k,2})}{4\ln 2}\Big{]}\\ \text{s.t.}&\text{C1:}\quad n_{\text{s}}\sum_{k=1}^{ K}|\mathbf{w}_{k}|-\mathbf{d}\preccurlyeq\mathbf{0},\\ &\text{C2:}\quad n_{\text{s}}\sum_{k=1}^{K}|\mathbf{w}_{k}|+\mathbf{d} \preccurlyeq\frac{P_{\text{max}}\tau}{\hbar v}\textbf{1},\\ &\text{C3:}\quad\|\textbf{W}\|_{\infty}\leq 1,\\ &\text{C4:}\quad\mathbf{g}_{K}^{T}\mathbf{w}_{j}=0,\forall j\
eq k,\quad \forall j,k=1,\cdots,K,\\ &\text{C5:}\quad\lambda_{k,1}=-\zeta n_{\text{s}}\mathbf{g}_{K}^{T} \mathbf{w}_{k}+\zeta\mathbf{g}_{K}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\\ &\text{C6:}\quad\lambda_{k,2}=\zeta n_{\text{s}}\mathbf{g}_{K}^{T} \mathbf{w}_{k}+\xi\mathbf{g}_{K}^{T}\mathbf{d}+n_{\text{b}},\quad\forall k=1,\cdots,K,\\ &\text{C7:}\quad-|\frac{n_{\text{s}}\mathbf{g}_{K}^{T}\mathbf{w}_{k}}{ \sqrt{\lambda_{k,1}}}|\leq\gamma_{k,1}\leq|\frac{n_{\text{s}}\mathbf{g}_{K}^{T}\bm {w}_{k}}{\sqrt{\lambda_{k,1}}}|,\quad\forall k=1,\cdots,K,\\ &\text{C8:}\quad-|\frac{n_{\text{s}}\mathbf{g}_{K}^{T}\mathbf{w}_{k}}{ \sqrt{\lambda_{k,2}}}|\leq\gamma_{k,2}\leq|\frac{n_{\text{s}}\mathbf{g}_{K}^{T}\bm {w}_{k}}{\sqrt{\lambda_{k,2}}}|,\quad\forall k=1,\cdots,K,\end{split}  \tag{19}\]  
where C7 and C8 are derived from the equivalence after variable substitution, and the \(|\cdot|\) is considered due to the potential positivity or negativity of \(n_{\text{s}}\mathbf{g}_{K}^{T}\mathbf{w}_{k}\). **Proposition 2**. : _The optimal solution for \(\mathbf{g}_{K}^{T}\mathbf{w}_{k}\)' must indeed be non-negative._  
Proof of Proposition 2.: See Appendix B. Without losing optimality, the '\(|\cdot|\)' on the right-hand side (RHS) of C7 and C8 can be removed and transformed as follows:  
\[\gamma_{k,m}\sqrt{\lambda_{k,m}}\leq n_{\text{s}}\mathbf{g}_{K}^{T}\mathbf{w}_{k}, \quad\forall m=1,2,\quad\forall k=1,\cdots,K. \tag{20}\]  
For convenience, we use the variable '\(m\)' for a unified representation. When \(m=1\), it signifies the transformed version of C7, whereas \(m=2\) represents the transformed version of C8. However, the left-hand side (LHS) of (20) is also a non-convex term. Here, we employ the sequential parametric convex approximation (SPCA) method, utilizing its convex lower bounds for processing, resulting in  
\[\gamma_{k,m}\sqrt{\lambda_{k,m}}\geq\frac{\lambda_{k,m}}{2\theta_{k,m}}+\frac {\gamma_{k,m}^{2}\theta_{k,m}}{2},\quad\forall m=1,2,\quad\forall k=1,\cdots,K, \tag{21}\]  
where (21) reaches equality only when \(\theta_{k,m}=\frac{\sqrt{\lambda_{k,m}}}{\gamma_{k,m}},\forall m,\forall k\).