## 8 Results and Discussion
Meanwhile, we can see that the SE of the joint user scheduling and hybrid beamforming design algorithm is higher than that of the decoupling design algorithm. In addition, the more preselected users, the higher the system SE. Compared with the traditional algorithm, the proposed robust joint design algorithm can obtain higher system SE at the same transmission power, and thus can improve the system EE. Under the conditions of \(\sigma_{f_{d}}^{2}=25\), \(\sigma_{\phi_{\tau}}^{2}=10\) and \(P_{T}=30\) W, when \(U_{l,p}/U_{s}=2,U_{l,p}=4\) and \(U_{l,p}/U_{s}=3,U_{l,p}=6\), with the improvement of system SE performance, the EE performance gain of the proposed joint design algorithm is 26.16% and 37.85% higher than that of the traditional decoupling design algorithm. Figure 10 shows the SE comparison of different multicast groups. It can be seen that the SE of each multicast group of the proposed robust algorithm is higher than that of the nonrobust algorithm. Meanwhile, with the increase in the number of preselected users, the diversity of users increases, and the performance of the proposed joint user scheduling and hybrid beamforming design algorithm also improves. This is because with the increase in the number of preselected users, the range of users that can be scheduled and selected increases. By scheduling different users in each multicast group, the SE can be further improved. Meanwhile, with the improvement of the system SE, the system EE performance gain also increases, which verifies the effectiveness of the joint user scheduling and hybrid beamforming design algorithm. Figure 8: Comparison of system EE of different algorithms with different transmission power thresholds \(P_{T}\). Figure 11 shows the change trajectory of the system EE and SE versus the different \(U_{s}\). It can be seen that with the increase in \(U_{s}\), the system EE and SE show a downward trend. This is because the communication rate of each multicast group is constrained by the user with the worst SINR in the multicast group. With the increase in \(U_{s}\), if the users' channel vectors in the multicast group remain collinear, the EE and SE would remain unchanged. However, according to the rules of the user preselection and scheduling, with the increase in \(U_{s}\), the collinearity among users in the multicast group would decrease, which can result in the increase in interference and the decrease in the worst SINR in each multicast group. In other words, with the decrease in \(U_{s}\), the users' SINR will be improved. Therefore, with the improvement of SINR, the system EE performance gain also increases, as shown in Figure 10(a). Under the condition of \(U_{l,p}/U_{s}=2,U_{l,p}=4\), when \(U_{s}=2\), the EE performance gain is 21.05% higher than when \(U_{s}=4\). Figure 10: Comparison of SE of different multicast groups. Figure 9: Change trajectory of system SE versus different transmission power thresholds \(P_{T}\). Figure 12 shows the performance comparison of different algorithms for solving \(\mathbf{F}_{BB}\) and \(\mathbf{F}_{RF}\). In this simulation, we set three comparison algorithms, i.e., the optimal design algorithm, the alternating minimization algorithm based on the phase extraction (PE-Altmin) algorithm and the orthogonal matching pursuit (OMP) algorithm. The optimal design algorithm refers to the numerical simulation result of the hybrid beamforming matrix \(\mathbf{F}\). It can be seen that the system performance of the MM-AltOpt algorithm is slightly lower than that of the optimal design algorithm. Meanwhile, in Figure 12a, when \(P_{T}=10\) W, we can see that the system EE performance gain of the proposed MM-AltOpt algorithm is improved by about 2% and 5%, respectively, compared with the PE-Altmin algorithm and the OMP algorithm.