# Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission
## 2 Channel Model  
We consider a massive MIMO secure multicast downlink transmission system, in which public messages sent by the BS are transmitted to the legitimate UTs. There is one BS with \(M\) antenna transmits and \(K\) UTs, each with \(N_{r}\) antennas. The public messages are secret to the eavesdropper with \(N_{e}\) antennas.  
\(\mathbf{x}\in\mathbb{C}^{M\times 1}\) denotes the signal that is sent from the BS in multicast transmission. \(\mathbb{E}\left\{\mathbf{x}\right\}=\mathbf{0}\) and \(\mathbb{E}\left\{\mathbf{x}\mathbf{x}^{H}\right\}=\mathbf{Q}\in\mathbb{C}^{M \times M}\), where \(\mathbf{Q}\) denotes the transmit covariance matrix. We model the signals received by legitimate UT \(k\) and the eavesdropper as  
\[\mathbf{y}_{k} =\mathbf{H}_{k}\mathbf{x}+\mathbf{n}_{k}\in\mathbb{C}^{N_{e} \times 1}, \tag{1}\] \[\mathbf{y}_{\mathrm{ev}} =\mathbf{H}_{\mathrm{ev}}\mathbf{x}+\mathbf{n}_{\mathrm{ev}}\in \mathbb{C}^{N_{e}\times 1}, \tag{2}\]  
respectively, where \(\mathbf{H}_{k}\) and \(\mathbf{H}_{\mathrm{ev}}\) are defined as the channel matrices of BS-UT \(k\) and BS-eavesdropper, respectively.  
We consider our MIMO channel model to follow jointly spatially correlated Rayleigh fading, which means there are joint correlation characteristics between the transmitter and the receivers [37]. Thus, the channel matrices \(\mathbf{H}_{k}\) and \(\mathbf{H}_{\mathrm{ev}}\) in (1) and (2) can be expressed as  
\[\mathbf{H}_{k} =\mathbf{U}_{k}\mathbf{G}_{k}\mathbf{V}_{k}^{H}\in\mathbb{C}^{N_{ k}\times M}, \tag{3}\] \[\mathbf{H}_{\mathrm{ev}} =\mathbf{U}_{\mathrm{ev}}\mathbf{G}_{\mathrm{ev}}\mathbf{V}_{ \mathrm{ev}}^{H}\in\mathbb{C}^{N_{e}\times M}, \tag{4}\]  
respectively, where \(\mathbf{G}_{k}\in\mathbb{C}^{N_{k}\times M}\) and \(\mathbf{G}_{\mathrm{ev}}\in\mathbb{C}^{N_{e}\times M}\) are random matrices with zero mean and independent elements, which are defined as the channel matrices of BS-UT \(k\) and BS-eavesdropper in the beam domain, respectively [34; 38]. \(\mathbf{U}_{k}\in\mathbb{C}^{N_{k}\times N_{t}}\), \(\mathbf{V}_{k}\in\mathbb{C}^{M\times M}\), \(\mathbf{U}_{\text{ev}}\in\mathbb{C}^{N_{k}\times N_{\text{e}}}\), and \(\mathbf{V}_{\text{ev}}\in\mathbb{C}^{M\times M}\) are unitary matrices of transmitter and receivers, respectively. The statistics of the beam domain channels can be described as  
\[\mathbf{\Omega}_{k} =\mathsf{E}\left\{\mathbf{G}_{k}\odot\mathbf{G}_{k}^{*}\right\} \in\mathbb{R}^{N_{k}\times M}, \tag{5}\] \[\mathbf{\Omega}_{\text{ev}} =\mathsf{E}\left\{\mathbf{G}_{\text{ev}}\odot\mathbf{G}_{\text{ ev}}^{*}\right\}\in\mathbb{R}^{N_{\text{e}}\times M}, \tag{6}\]  
respectively. It can be proved that as the number of antennas in the BS, \(M\), increases to infinity, the eigenvectors of the correlation matrices in (5) and (6) tend to be equal, which can be denoted by a deterministic unitary matrix \(\mathbf{V}\)[38]. \(\mathbf{V}\) is only effected by the topology of transmit array at the BS, which means the channel matrices can be rewritten as  
\[\mathbf{H}_{k} \stackrel{{ M\rightarrow\infty}}{{=}}\mathbf{U}_{k} \mathbf{G}_{k}\mathbf{V}^{H}, \tag{7}\] \[\mathbf{H}_{\text{ev}} \stackrel{{ M\rightarrow\infty}}{{=}}\mathbf{U}_{ \text{ev}}\mathbf{G}_{\text{ev}}\mathbf{V}^{H}, \tag{8}\]  
respectively. It has been widely used in previous works and shows considerable accuracy for a finite number of BS antennas [35; 36]. Thus, we adopt the channel models in (7) and (8) in this paper.