## 5 Conclusions  
We provide two optimal beamforming schemes for the AP and the cooperative UTs in multi antenna cooperative NOMA systems resulting from the derived optimal BF criterion. One scheme with power limitation of cooperative UTs considers the practical implementation restriction, while the other one has no such limitation. These schemes have rank one structure and rely on the semi-definite programming (SDP) optimization to achieve the optimal points. Simulation results show that the proposed schemes efficiently utilize the spatial dimensions from the multiple antennas and improve the performances of cooperative NOMA systems both in the diversity and in the average transmit power of the system. The power limited cooperative UT scheme exhibits only a slight performance loss compared to the one without such power limitation. Here, we consider decode-and-forward relaying for the cooperative UTs, while the study on the amplify-and-forward UT can be considered in the future. In addition, a study on practical schemes with less computational complexity is an important future research challenge. Finally, the impact of antenna design [29; 30; 31] on the performance also attracts the research focus.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(M\) & AP antenna number. & \(N\) & cooperative UT antenna number. \\ \hline \(L\) & number of cooperative UTs. & \(J\) & Total number of UTs \\ \hline \(\mathbf{h}_{j}\) & AP to the \(j\)-th non cooperative & \(\mathbf{H}_{j}\) & AP to the \(j\)-th cooperative \\  & UT channel & & UT channel. \\ \hline \(\mathbf{g}_{m,n}\) & \(n\)-th cooperative UT to the \(m\)-th & \(\mathbf{G}_{m,n}\) & \(n\)-th cooperative UT to the \(m\)-th cooperative UT channel. \\ \hline \(50\),\(000\) & channel generation for outage & \(2000\) & channel generation for average \\  & probability. & & power. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definitions and simulation conditions.  
Figure 4: The average power of the proposed schemes against \(M\). Here, \(J=4\) and \(\gamma^{*}=10\) for all schemes; \(L=1\) and \(P_{d,1}=\frac{\gamma^{*}}{2}\) for the cooperative schemes. “coop.PC” represents the cooperative NOMA with UT power constraint.  
**Author Contributions:** [PERSON] proposed the beamforming principles and associated two algorithms for the NOMA MISO wireless networks; [PERSON] and S.S.N. supplemented in improving the two algorithms; derived and analyzed all simulation results. In addition, [PERSON] provided the experimental materials for better computational simulations and revised critical errors of the manuscript. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received external funding from the National Research Foundation of Korea.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The study did not report any data.  
**Acknowledgments:** This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science and ICT under Grant NRF-2017R1A2B4007398.  
**Conflicts of Interest:** The authors declare no conflict of interest.