## 3 Scattering Matrix with Grouping Scheme and Power Allocation
### Power Allocation of the RS  
Finally, the RS power ratio should be optimized, but the objective function is a nonconvex function. To optimize the RS power ratio simply, the sub-problem based on fractional programming (FP) by introducing four auxiliary variables is considered as follows:  
\[\begin{array}{ll}\underset{\alpha,\
u_{c},\mu_{c},\
u_{k},\mu_{k},\
u_{k},\
u_ {k}}{\max}&\log_{2}(1+\
u_{c})-\
u_{c}+2\mu_{c}\sqrt{(1+\
u_{c})P_{c}\big{|} \mathbf{h}_{\widehat{k}}\mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{c}\big{|}^{2}}-\mu_ {c}^{2}P_{c}\big{|}\mathbf{h}_{\widehat{k}}\mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{c }\big{|}^{2}\\ &-\sum_{k=1}^{N_{K}}\mu_{c}^{2}P_{p}\big{|}\mathbf{h}_{\widehat{k}}\mathbf{\[PERSON] }\mathbf{G}\mathbf{p}_{k}\big{|}^{2}+\mu_{c}^{2}\sigma_{\widehat{k}}^{2}+\sum_{ k=1}^{N_{K}}\{\log_{2}(1+\
u_{k})-\
u_{k} \tag{16}\] \[&+2\mu_{k}\sqrt{(1+\
u_{k})P_{p}\big{|}\mathbf{h}_{k}\mathbf{\[PERSON] }\mathbf{G}\mathbf{p}_{k}\big{|}^{2}}-\sum_{j\
eq k}\mu_{k}^{2}P_{p}\big{|} \mathbf{h}_{k}\mathbf{\Phi}\mathbf{G}\mathbf{p}_{j}\big{|}^{2}+\mu_{k}^{2} \sigma_{\widehat{k}}^{2}\}\\ &\text{s.t.\ (\ref{eq:1})-(\ref{eq:1}),}\end{array} \tag{17}\]  
where \(\widehat{k}=\ arg\ \underset{k}{\min}\gamma_{c,k}\) is the UE that has the minimum SINR of the common stream. Then, the problem can be optimized by updating one variable while fixing the others. The optimal variables can be determined by setting \(\vartheta(\ref{eq:1})/\vartheta x\) to 0, where \(x=\{\
u_{c},\mu_{c},\
u_{k},\mu_{k},\alpha\},\forall k\), as follows:  
\[\
u_{c}=\frac{P_{c}\big{|}\mathbf{h}_{\widehat{k}}\mathbf{\Phi}\mathbf{G} \mathbf{p}_{c}\big{|}^{2}}{\sum_{k=1}^{N_{K}}P_{p}\big{|}\mathbf{h}_{\widehat{ k}}\mathbf{\Phi}\mathbf{G}\mathbf{p}_{k}\big{|}^{2}+\sigma_{\widehat{k}}^{2}},\ \mu_{c}=\frac{\sqrt{(1+\
u_{c})P_{c}\big{|}\mathbf{h}_{\widehat{k}}\mathbf{ \[PERSON]}\mathbf{G}\mathbf{p}_{c}\big{|}^{2}}}{P_{c}\big{|}\mathbf{h}_{\widehat{k}} \mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{c}\big{|}^{2}+\sum_{k=1}^{N_{K}}P_{p}\big{|} \mathbf{h}_{\widehat{k}}\mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{k}\big{|}^{2}+ \sigma_{\widehat{k}}^{2}}, \tag{18}\]  
\[\alpha=\widehat{\alpha}, \tag{19}\]  
\[\begin{array}{ll}-\frac{\sqrt{(1+\
u_{c})\mu_{c}^{2}P_{c}\big{|}\mathbf{h}_{ \widehat{k}}\mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{c}\big{|}^{2}}}{(15-\alpha)/ \sqrt{2}}+\frac{P_{c}\sum_{k=1}^{N_{K}}\mu_{k}\sqrt{(1+\
u_{k})\mathbf{h}_{ \widehat{k}}\mathbf{\[PERSON]}\mathbf{G}\mathbf{p}_{k}\big{|}^{2}}}{(\alpha-0.5)/ \sqrt{2}}+\mu_{c}^{2}P_{c}\big{|}\mathbf{h}_{\widehat{k}}\mathbf{\[PERSON]}\mathbf{G }\mathbf{p}_{c}\big{|}^{2}\\ -\frac{P_{c}}{N_{K}}\sum_{k=1}^{N_{K}}\mu_{c}^{2}\big{|}\mathbf{h}_{\widehat{k}} \mathbf{\Phi}\mathbf{G}\mathbf{p}_{k}\big{|}^{2}-P_{T}\sum_{k=1}^{N_{K}}\sum_{ j\
eq k}\mu_{k}^{2}\big{|}\mathbf{h}_{k}\mathbf{\Phi}\mathbf{G}\mathbf{p}_{j}\big{|}^{2}+ \mu_{k}^{2}\sigma_{\widehat{k}}^{2}=0,\end{array} \tag{20}\]  
where \(\widehat{a}\) is the solution of Equation (20), which is provided in Appendix A in detail. Equation (20) can be solved by a simple quadratic equation. The overall optimization is described in the following Algorithm 2.  
```
0: RS power-splitting ratio: \(\alpha\)
1: Initialize \(\alpha\)
2:while The value of the sum-rate converges. do
3: Update \(\
u_{c}\) by (17).
4: Update \(\mu_{c}\) by (17).
5: Update \(\
u_{k}\), \(\forall k\) by (18).
6: Update \(\mu_{k}\), \(\forall k\) by (18).
7: Update \(\alpha\) by (19).
8:endwhile
```  
**Algorithm 2** Optimization RS power-splitting ratio.  
In Algorithm 2, instead of calculating the initial value of \(\alpha\) according to the channel information, the calculation of the algorithm is reduced by selecting the median value in the range of possible numbers. Through the simulation results in the next section, it was confirmed that the sum-rate performance can be optimized without additional calculation.