# Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission
## 3 Energy Efficiency Optimization of Secure Multicast Transmission
### Deterministic Equivalent  
Due to a large number of samples need to be averaged during the expectation operation in (30), the optimization complexity is still high, although each subproblem in (28) is a concave program. Thus, without using Monte Carlo averaging channel realizations, we calculate the deterministic equivalent of the secure transmit rate by adopting the large-dimensional random matrix theory, which can reduce the implementation complexity [45; 46]. Note that the method of deterministic equivalent can keep the convexity of the transmission rate [47]. Therefore, the concave subproblem in (28) is given as  
\[\mathbf{\Gamma}^{(t+1)}=\underset{\mathbf{\Gamma}}{\arg\max} \quad\underset{\mathbf{\Gamma}}{\min}\quad W\overline{R}_{k}\left(\bm {\Gamma}\right)-W\text{tr}\left\{\left(\frac{\partial}{\partial\mathbf{\Lambda}}R _{\text{ev,ub}}\left(\mathbf{\Lambda}^{(\ell)}\right)\right)^{T}\mathbf{\Gamma}\right\}- \eta^{(t)}P(\mathbf{\Gamma}),\] s.t. \[\quad\text{tr}\left\{\mathbf{\Gamma}\right\}\leq P_{\text{max}},\ \mathbf{\Gamma}\succeq\mathbf{0},\ \mathbf{\Gamma}\text{ diagonal}. \tag{32}\]  
In (32), \(\overline{R}_{k}\left(\mathbf{\Gamma}\right)\) denotes the deterministic equivalent expression of \(R_{k}\left(\mathbf{\Gamma}\right)\), which is expressed as  
\[\overline{R}_{k}\left(\mathbf{\Gamma}\right)=\log\det\left\{\mathbf{I}_{M}+\mathbf{ \Xi}_{k}\mathbf{\Gamma}\right\}+\log\det\left\{\widetilde{\mathbf{\Phi}}_{k}\right\}- \text{tr}\left\{\widetilde{\mathbf{\Xi}}_{k}\widetilde{\mathbf{\Phi}}_{k}^{-1}\right\}, \tag{33}\]where \(\Xi_{k}\in\mathbb{C}^{M\times M}\), \(\widetilde{\Xi}_{k}\in\mathbb{C}^{N_{t}\times N_{t}}\), and \(\widetilde{\Phi}_{k}\in\mathbb{C}^{N_{t}\times N_{t}}\) can be obtained by solving the following three fixed-point equations,  
\[\Xi_{k} =\mathbf{B}_{k}\left(\widetilde{\Phi}_{k}^{-1}\right), \tag{34a}\] \[\widetilde{\Xi}_{k} =\mathbf{C}_{k}\left(\mathbf{\Gamma}\left(\mathbf{I}_{M}+ \mathbf{\Gamma}\Xi_{k}\right)^{-1}\right),\] (34b) \[\widetilde{\Phi}_{k} =\mathbf{I}_{N_{t}}+\widetilde{\Xi}_{k}, \tag{34c}\]  
where \(\mathbf{B}_{k}\left(\mathbf{X}\right)\triangleq\mathsf{E}\left\{\mathbf{G}_{k }^{H}\mathbf{X}\mathbf{G}_{k}\right\}\in\mathbb{C}^{M\times M}\) and \(\mathbf{C}_{k}\left(\mathbf{X}\right)\triangleq\mathsf{E}\left\{\mathbf{G}_{ k}\mathbf{X}\mathbf{G}_{k}^{H}\right\}\in\mathbb{C}^{N_{t}\times N_{t}}\) are matrix-valued functions, which can output matrices in diagonal forms, the corresponding \(i\)th diagonal elements are respectively obtained by  
\[\left[\mathbf{B}_{k}\left(\mathbf{X}\right)\right]_{i,i} =\text{tr}\left\{\text{diag}\left\{\left[\mathbf{\Omega}_{k} \right]_{;i}\right\}\mathbf{X}\right\}, \tag{35}\] \[\left[\mathbf{C}_{k}\left(\mathbf{X}\right)\right]_{i,i} =\text{tr}\left\{\text{diag}\left\{\left(\left[\mathbf{\Omega}_ {k}\right]_{;i}\right)^{T}\right\}\mathbf{X}\right\}. \tag{36}\]  
Via the deterministic equivalent approach instead of exhaustive averaging, the expression \(\overline{R}_{k}\left(\mathbf{\Gamma}\right)\) is efficiently calculated by adopting the channel statistics \(\mathbf{\Omega}_{k}\) in a small number of fixed-point iterations. As a result, the computational complexity to optimize the convex program in (28) can be reduced significantly.