# Energy-Efficient Multicast Precoding for Massive MIMO Transmission with Statistical CSI
## 3 Energy-Efficient Multicast Precoding for Massive MIMO
By replacing the multicast rate with its DE, the sequence of the convex optimization problems in (15) can be rewritten as follows  
\[R_{k} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{k}}+\frac{1}{ \sigma^{2}}\mathbf{H}_{k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\]6 \[R_{k} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{k}}+\frac{1}{ \sigma^{2}}\mathbf{H}_{k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\]7In (17), \(\overline{R}_{k}\left(\mathbf{\Lambda}\right)\) is the DE of \(R_{k}\left(\mathbf{\Lambda}\right)\) given by  
\[R_{k} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{k}}+\frac{1}{ \sigma^{2}}\mathbf{H}_{k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\]8  
where \(\mathbf{\Gamma}_{k}\in\mathbb{C}^{M\times M}\), \(\widetilde{\mathbf{\Gamma}}_{k}\in\mathbb{C}^{N_{k}\times N_{k}}\), and \(\widetilde{\mathbf{\Phi}}_{k}\in\mathbb{C}^{N_{k}\times N_{k}}\) can be efficiently calculated via solving the following fixed-point equations  
\[R_{k} =\mathsf{E}\left\{\log\det\left\{\mathbf{I}_{N_{k}}+\frac{1}{ \sigma^{2}}\mathbf{H}_{k}\mathbf{Q}\mathbf{H}_{k}^{H}\right\}\right\}\]9 \[\overset{\mathrm{(a)}}{=}\mathsf{E}\left\{\log\det\left\{\mathbf{ I}_{N_{k}}+\frac{1}{\sigma^{2}}\mathbf{G}_{k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V} \mathbf{G}_{k}^{H}\right\}\right\}, \tag{6}\]0 (19b) \[\overset{\mathrm{(a)}}{=}\mathsf{E}\left\{\log\det\left\{\mathbf{ I}_{N_{k}}+\frac{1}{\sigma^{2}}\mathbf{G}_{k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V} \mathbf{G}_{k}^{H}\right\}\right\}, \tag{6}\]1  
It is worth noting that \(\mathbf{\mathsf{B}}_{k}\left(\mathbf{\mathsf{X}}\right)\triangleq\frac{1}{\sigma^{2}} \mathsf{E}\left\{\mathbf{\mathsf{G}}_{k}^{H}\mathbf{\mathsf{X}}\mathbf{\mathsf{G}}_{k} \right\}\in\mathbb{C}^{M\times M}\) and \(\mathbf{\mathsf{C}}_{k}\left(\mathbf{\mathsf{X}}\right)\triangleq\frac{1}{\sigma^{2} }\mathsf{E}\left\{\mathbf{\mathsf{G}}_{k}\mathbf{\mathsf{X}}\mathbf{\mathsf{G}}_{k}^{H} \right\}\in\mathbb{C}^{N_{k}\times N_{k}}\) appeared in (19) are both matrix-valued functions which both output diagonal matrices with the corresponding \(i\)th output diagonal elements given by  
\[\overset{\mathrm{(a)}}{=}\mathsf{E}\left\{\log\det\left\{\mathbf{ I}_{N_{k}}+\frac{1}{\sigma^{2}}\mathbf{G}_{k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V} \mathbf{G}_{k}^{H}\right\}\right\}, \tag{6}\]2 \[\overset{\mathrm{(a)}}{=}\mathsf{E}\left\{\log\det\left\{\mathbf{ I}_{N_{k}}+\frac{1}{\sigma^{2}}\mathbf{G}_{k}\mathbf{V}^{H}\mathbf{Q}\mathbf{V} \mathbf{G}_{k}^{H}\right\}\right\}, \tag{6}\]3  
respectively. The DE \(\overline{R}_{k}\left(\mathbf{\Lambda}\right)\) can be efficiently calculated using the channel statistics \(\mathbf{\Omega}_{k}\) in a few iterations without exhaustive averaging involved in the Monte-Carlo approach, and then the computational complexity of the optimization problem in (15) can be further reduced. Please note that the DE \(\overline{R}_{k}\left(\mathbf{\Lambda}\right)\) is a quite tight approximation of \(R_{k}\left(\mathbf{\Lambda}\right)\) for massive MIMO channels in typical settings [27; 28]. In addition, as \(\overline{R}_{k}\left(\mathbf{\Lambda}\right)\) in (18) is a concave function with respect to the power allocation matrix \(\mathbf{\Lambda}\), each sub-problem in (17) is a convex program and can be efficiently solved using standard techniques. Formally, we present the description of our proposed energy-efficient multicast beam domain power allocation algorithm for massive MIMO transmission using [PERSON]'s transform and large-dimensional random matrix theory in Algorithm 1. The computational complexity of Algorithm 1 is presented as follows. From [11], we can obtain that the convergence rate of Dinkelbach's transform-based iteration will exhibit a super-linear trend in the sub-problem sequence (15). In addition, each sub-problem in (15) is a convex program and thus the computational complexity of each sub-problem is polynomial in the numbers of variables and constraints [29].