## 4 Joint User Scheduling and Hybrid Beamforming Design for Maximizing EE
### SDP Algorithm  
It is worth noting that the objective function and constraints \(C_{2}\) and \(C_{5}\) in the problem \(Q_{1}\) involve the quadratic form of the variable \(\mathbf{f}_{l}\), therefore, \(Q_{1}\) is the QCQP form problem. To handle this problem, we invoke the SDP algorithm, a new variable \(\mathbf{W}_{l}\triangleq f_{l}\mathbf{f}_{l}^{H}\) is introduced, and the positive semidefinite matrix \(\mathbf{W}_{l}\) needs to meet the constraints of \(\mathbf{W}_{l}\underline{\succeq}0\) and \(rank(\mathbf{W}_{l})=1\). Then, the problem \(Q_{1}\) can be equivalent to:  
\[Q_{2}:\max_{\mathbf{\eta},\mathbf{W}_{l},\text{SINR}^{\text{min}}}\text{EE}=\frac{B \sum\limits_{l=1}^{L}\log_{2}\left(1+\text{SINR}^{\text{min}}_{l}\right)}{ \sum\limits_{l}^{L}Tr(\mathbf{W}_{l})+P_{0}}, \tag{26}\]  
\[s.t.\ C_{1},C_{2},\ C_{3},\ C_{4}\ in\ Q_{1}, \tag{27}\]  
\[C_{5}:\sum\limits_{l}^{L}Tr(\mathbf{W}_{l})\leq P_{T}, \tag{28}\]  
\[C_{6}:\mathbf{W}_{l}\underline{\succeq}0,\ \forall l, \tag{29}\]  
\[C_{7}:rank(\mathbf{W}_{l})=1,\ \forall l, \tag{30}\]  
Similarly, the SINR and the communication rate of the \(k\)th user in the \(l\)th multicast group can be equivalently converted to:  
\[\text{SINR}_{k,l}\approx\frac{\text{E}\left\{Tr(\mathbf{H}_{k,l}\mathbf{W}_{l})\right\} }{\text{E}\left\{\sum\limits_{j=1,j\
eq l}^{L}Tr(\mathbf{H}_{k,l}\mathbf{W}_{j}) \right\}+\sigma^{2}}, \tag{31}\]  
\[R_{k,l} =B\log_{2}\left(1+\frac{\text{E}\left\{Tr(\mathbf{H}_{k,l}\mathbf{W}_{l}) \right\}}{\text{E}\left\{\sum\limits_{j=1,j\
eq l}^{L}Tr(\mathbf{H}_{k,l}\mathbf{W}_{ j})\right\}+\sigma^{2}}\right)=B\log_{2}\left(1+\frac{\frac{Tr(\text{E}\left\{H _{k,l}\mathbf{W}_{l}\right\})}{\sum\limits_{j=1,j\
eq l}^{L}Tr(\text{E}\left\{H _{k,l}\mathbf{W}_{j}\right\})+\sigma^{2}}}\right) \tag{32}\] \[=B\log_{2}\left(1+\frac{\frac{Tr(\mathbf{H}_{k,l}\mathbf{W}_{l})}{\sum \limits_{j=1,j\
eq l}^{L}Tr(\mathbf{H}_{k,l}\mathbf{W}_{j})+\sigma^{2}}}{\sum\limits_ {j=1,j\
eq l}^{L}Tr(\text{E}\left\{H_{k,l}\mathbf{W}_{j}\right\})+\sigma^{2}}\right)\]  
where \(\mathbf{H}_{k,l}\in C^{M\times M}\) is the instantaneous channel autocorrelation matrix of the \(k\)th user in the \(l\)th multicast group and \(\mathbf{\overline{H}}_{k,l}\in C^{M\times M}\) is the long-term channel autocorrelation matrix. The relationship between the two can be expressed as:  
\[\mathbf{\overline{H}}_{k,l}=\text{E}\left\{\mathbf{H}_{k,l}\right\}\triangleq\text{ E}\left\{\mathbf{\hat{h}}_{k,l}\mathbf{h}_{k,l}^{H}\right\}=diag\left(\mathbf{\hat{h}}_{k,l} \right)P_{\frac{\sigma^{2}}{l,\lambda,l}}Q_{\tau_{k,l}}diag\left(\mathbf{\hat{h}}_{ k,l}^{H}\right), \tag{33}\]  
where \(\mathbf{P}_{\frac{\sigma^{2}}{l,\lambda,l}}\) and \(\mathbf{Q}_{\tau_{k,l}}\) can be expressed as follows:\[P_{f_{d,k,l}^{\text{rad}}} =E\Bigg{\{}v_{\phi_{f_{d,k,l}^{\text{rad}}}}v_{\phi_{f_{d,k,l}^{ \text{rad}}}}^{H}\Bigg{\}} \tag{34}\] \[=E\Bigg{\{}\Bigg{[} \begin{array}{cccc}\begin{array}{c}\hat{ \psi}_{f_{d,k,l}^{\text{rad}}},\end{array}e^{\hat{\psi}_{f_{d,k,l}^{\text{rad}} },\end{array} e^{-\hat{\psi}_{f_{d,k,l}^{\text{rad}}}}\Big{]}^{T}\bigg{[}e^{- \hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array}e^{-\hat{\psi}_{f_{d,k,l}^{ \text{rad}}}},\end{array}e^{-\hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array}e^{ -\hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array}e^{-\hat{\psi}_{f_{d,k,l}^{ \text{rad}}},\end{array}e^{-\hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array} \right\}}\Bigg{\}}\] \[=E\Bigg{\{} \begin{array}{cccc}\begin{array}{cccc}1&\cdots&e^{ \hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array}e^{-\hat{\psi}_{f_{d,k,l}^{ \text{rad}}},\end{array} e^{-\hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array}\] \[\vdots\ddots\vdots\vdots\Bigg{\}}\\ \begin{array}{cccc}\hat{\psi}_{f_{d,k,l}^{\text{rad}}},\end{array} e^{-\hat{ \psi}_{f_{d,k,l}^{\text{rad}}},\end{array}\cdots& \begin{array}{cccc}1\\ \end{array} \Bigg{\}}\] \[=\left\{ \begin{array}{cccc}1&\cdots&E\bigg{\{}e^{\hat{\psi}_{f _{d,k,l}^{\text{rad}}},\end{array} e^{-\hat{\psi}_{f_{d,k,l}^{\text{rad}}}, \end{array}\right\}}\\