# Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission
## 3 Energy Efficiency Optimization of Secure Multicast Transmission  
We study the energy efficiency multicast optimization of secure massive MIMO in this section. We assume that the BS can only capture the statistical CSI of all legitimate UTs and the eavesdropper [34]. Besides, legitimate UTs and the eavesdropper can obtain the instantaneous CSI. Before constructing the energy efficiency model, we first analyze the secure transmit rate. The received signal transmission rate needs to be higher for any user terminal than that received by the eavesdropper. Focusing on delay-tolerant scenarios [39], the ergodic secure multicast rate can be expressed as  
\[R_{\text{se}}\triangleq\left[R_{\text{mc}}-R_{\text{ev}}\right]^{+}, \tag{9}\]  
where the ergodic multicast rate \(R_{\text{mc}}\) is expressed as  
\[R_{\text{mc}} =\min_{k} \mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{t}}+\mathbf{H}_{ k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\] \[=\min_{k} \mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{t}}+\mathbf{G}_{ k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V}\mathbf{G}_{k}^{H}\right\}\right\}, \tag{10}\]  
and the ergodic rate for eavesdropper \(R_{\text{ev}}\) is expressed as  
\[R_{\text{ev}} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{\text{e}}}+ \mathbf{H}_{\text{ev}}\mathbf{Q}\mathbf{H}_{\text{ev}}^{H}\right\}\right\}\] \[=\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{\text{e}}}+ \mathbf{G}_{\text{ev}}\mathbf{V}^{H}\mathbf{Q}\mathbf{V}\mathbf{G}_{\text{ev} }^{H}\right\}\right\}. \tag{11}\]  
In (10) and (11), we exploit the identity \(\det\left\{\mathbf{I}+\mathbf{A}\mathbf{B}\right\}=\det\left\{\mathbf{I}+ \mathbf{B}\mathbf{A}\right\}\). Meanwhile, we adopt a model of power consumption as follows. For large-scale MIMO systems, most of the power is consumed by the BS, we then focus mainly on the power consumption of the BS. Besides, the static and dynamic power consumption of the radio frequency chains is considered. Specifically, our power consumption model is adopted as  
\[P(\mathbf{Q})\triangleq\zeta\text{tr}\left\{\mathbf{Q}\right\}+M\text{P}_{ \text{c}}+P_{\text{s}}, \tag{12}\]  
where \(\text{tr}\left\{\mathbf{Q}\right\}\) denotes the transmit power of multicast transmission. Note that the coefficient \(\zeta\geq 1\) is the reciprocal of the transmit amplifier drain efficiency, \(P_{\text{s}}\) and \(P_{\text{c}}\) are the static power consumption and the hardware dissipated power of the BS, respectively. We denote \(W\) by the bandwidth of secure transmission, the energy efficiency of massive MIMO secure multicast transmission is formulated by the ratio of secure transmit rate to power consumption as follows,  
\[\text{EE}_{\text{se}}=\frac{WR_{\text{se}}}{P}. \tag{13}\]  
Our aim is optimizing the transmit covariance matrix \(\mathbf{Q}\), leading to maximizing the \(\text{EE}_{\text{se}}\) of downlink secure multicast in (13). The problem is addressed as  
\[\underset{\mathbf{Q}}{\arg\max} \frac{WR_{\text{se}}}{P},\] s.t. \[\text{tr}\left\{\mathbf{Q}\right\}\leq P_{\max},\mathbf{Q}\succeq \mathbf{0}, \tag{14}\]  
where \(P_{\max}\) denotes the maximum multicast power budget at the transmitter. \(R_{\text{ev}}\) in (11) can get its upper bound via using [PERSON]'s inequality, which is expressed as follows,  
\[R_{\text{ev}}\leq R_{\text{ev,ub}}\triangleq\log\det\left\{\mathbf{I}_{N_{ \text{e}}}+\underbrace{\mathsf{E}\left\{\mathbf{G}_{\text{ev}}\mathbf{V}^{H} \mathbf{Q}\mathbf{V}\mathbf{G}_{\text{ev}}^{H}\right\}}_{\triangleq\mathbf{A}_ {\text{ev}}\left(\mathbf{V}^{H}\mathbf{Q}\mathbf{v}\right)}\right\}, \tag{15}\]  
where \(\mathbf{A}_{\text{ev}}\left(\mathbf{X}\right)\triangleq\mathsf{E}\left\{ \mathbf{G}_{\text{ev}}\mathbf{X}\mathbf{G}_{\text{ev}}^{H}\right\}\in\mathbb{ C}^{N_{\text{e}}\times N_{\text{e}}}\) denotes a function of matrix-valued form, which can output a diagonal matrix.