# Energy-Efficient Multicast Precoding for Massive MIMO Transmission with Statistical CSI
## 3 Energy-Efficient Multicast Precoding for Massive MIMO  
In this section, we investigate energy-efficient multicast precoding for massive MIMO where the BS only has knowledge of the statistical CSI of all multicast UTs and all the UTs know their own instantaneous CSI perfectly. In particular, with multicast transmit covariance matrix \(\mathbf{Q}\), the achievable ergodic multicast can be written as follows  
\[R_{\mathrm{mc}}=\min_{k}\ R_{k}, \tag{5}\]  
where \(R_{k}\) is UT \(K\)'s achievable ergodic rate which can be written as  
\[R_{k} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{k}}+\frac{1}{ \sigma^{2}}\mathbf{H}_{k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\] \[\overset{\mathrm{(a)}}{=}\mathsf{E}\left\{\log\det\left\{\mathbf{ I}_{N_{k}}+\frac{1}{\sigma^{2}}\mathbf{G}_{k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V} \mathbf{G}_{k}^{H}\right\}\right\}, \tag{6}\]  
where (a) follows from the massive MIMO beam domain channel model in (4), and the determinant identity \(\det\left\{\mathbf{I}+\mathbf{A}\mathbf{B}\right\}=\det\left\{\mathbf{I}+ \mathbf{B}\mathbf{A}\right\}\). Before proceeding, we present the power consumption model. In massive MIMO systems, the BS consumes most of the power and we focus on the power consumption at the BS. Specifically, we adopt the power consumption model as follows [11; 13]  
\[P=\zeta\mathsf{tr}\left\{\mathbf{Q}\right\}+MP_{\mathrm{c}}+P_{ \mathrm{s}}, \tag{7}\]  
where \(\mathsf{tr}\left\{\mathbf{Q}\right\}\) is the multicast transmit power, which is scaled by a coefficient \(\zeta\geq 1\) modeling the reciprocal of the transmit amplifier drain efficiency, \(P_{\mathrm{c}}\) denotes the circuit power consumption of each BS antenna, and \(P_{\mathrm{s}}\) denotes the static power consumption at the BS. Then, the system EE of multicast transmission with bandwidth \(W\) is given by  
\[\mathrm{EE}_{\mathrm{mc}}\triangleq\frac{WR_{\mathrm{mc}}}{P}. \tag{8}\]  
In the following, we investigate energy-efficient multicast precoding with statistical CSI. We aim to design the optimal transmit covariance \(\mathbf{Q}\) that can maximize the multicast EE in (8). The problem to be dealt with can be formulated as follows  
\[\operatorname*{arg\,max}_{\mathbf{Q}} \mathrm{EE}_{\mathrm{mc}},\] \[\mathrm{s.t.} \mathsf{tr}\left\{\mathbf{Q}\right\}\leq P_{\mathrm{max}},\ \mathbf{Q}\succeq\mathbf{0}, \tag{9}\]  
where \(P_{\mathrm{max}}\) denotes the multicast power budget at the BS. We denote by \(\mathbf{Q}=\mathbf{\Phi}\mathbf{\Lambda}\mathbf{\Phi}^{H}\) the eigenvalue decomposition of the transmit covariance matrix \(\mathbf{Q}\). Please note that the columns of \(\mathbf{\Phi}\) denote the eigenvectors of \(\mathbf{Q}\) and can represent the multicast signaling directions. Meanwhile, diagonal elements of \(\mathbf{\Lambda}\) denote the eigenvalues of \(\mathbf{Q}\) and the powers allocated to each beam can be reflected in \(\mathbf{\Lambda}\). We first investigate the eigenvectors of the optimal multicast transmit covariance matrix in the following theorem. **Theorem 1**. : _The eigenvectors of the optimal transmit covariance matrix \(\mathbf{Q}^{\mathrm{opt}}\) in terms of maximizing the systems EE are constituted by the columns of the matrix \(\mathbf{V}\) in (4), i.e.,_  
\[\mathbf{Q}^{\mathrm{opt}}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^{ H}. \tag{10}\]  
Proof. : Please refer to Appendix A. The result presented in Theorem 1 shows that the optimal energy-efficient multicast signaling directions should align with the eigenvectors of the transmit correlation matrices at the BS. Thus, the optimal multicast precoding in terms of maximizing the systems EE should be performed in the beam domain. With Theorem 1, the matrix-valued energy-efficient transmit covariance matrix design can be simplified into a vector-valued beam domain power allocation problem with significantly reduced optimization variables.