# Energy-Efficient Multicast Precoding for Massive MIMO Transmission with Statistical CSI
## 2 Massive MIMO Beam Domain Channel Model  
We consider a single-cell massive MIMO system consisting of one BS which is equipped with \(M\) antennas, and \(K\) UTs, where UT \(k\) is equipped with \(N_{k}\) antennas. In this paper, we focus on the single group multicast transmission case where the BS sends one common message to all the UTs in the multicast group. We denote by \(\mathbf{x}\in\mathbb{C}^{M\times 1}\) the multicast signal intended for the UTs in the multicast group, then the signal received at UT \(k\) at a time instant can be written as  
\[\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{x}+\mathbf{n}_{k}\in\mathbb{C}^{N_{k} \times 1}, \tag{1}\]  
where \(\mathbf{H}_{k}\) represents the downlink channel matrix from the BS to UT \(k\) at the given time instant, and \(\mathbf{n}_{k}\sim\mathcal{CN}\left(\mathbf{0},\sigma^{2}\mathbf{I}_{N_{k}}\right)\) denotes the additive Gaussian noise with \(\sigma^{2}\) being the noise power. We assume that the multicast signal \(\mathbf{x}\) satisfies \(\mathsf{E}\left\{\mathbf{x}\right\}=\mathbf{0}\) and \(\mathsf{E}\left\{\mathbf{x}\mathbf{x}^{H}\right\}=\mathbf{Q}\in\mathbb{C}^{M \times M}\) where \(\mathbf{Q}\) is the multicast signal transmit covariance matrix. Before investigating the energy-efficient massive MIMO multicast transmission design, we first introduce the channel model adopted in our work which has a great impact on the considered multicast design. Specifically, [PERSON]'s channel model [24; 25] is adopted in our work. Please note that in the adopted [PERSON]'s channel model, the channel correlation properties between the transmitter (BS) and the receiver (UT) are jointly taken into account, which is different from the [PERSON] channel model where only the correlations at both ends are considered. With [PERSON]'s jointly correlated channel model, the channel matrix \(\mathbf{H}_{k}\) in (1) can be written as follows  
\[\mathbf{H}_{k}=\mathbf{U}_{k}\mathbf{G}_{k}\mathbf{V}_{k}^{H}\in\mathbb{C}^{N_ {k}\times M}, \tag{2}\]  
where \(\mathbf{U}_{k}\in\mathbb{C}^{N_{k}\times N_{k}}\) and \(\mathbf{V}_{k}\in\mathbb{C}^{M\times M}\) are both deterministic unitary matrices, and \(\mathbf{G}_{k}\in\mathbb{C}^{N_{k}\times M}\) is a random matrix where the elements are all zero-mean independently distributed. Please note that in massive MIMO related literature, \(\mathbf{G}_{k}\) is usually referred to as the beam domain channel matrix [19; 21; 23]. The second-order channel statistics of \(\mathbf{G}_{k}\) can be fully characterized by the following matrix  
\[\mathbf{\Omega}_{k}=\mathsf{E}\left\{\mathbf{G}_{k}\odot\mathbf{G}_{k}^{*} \right\}\in\mathbb{R}^{N_{k}\times M}. \tag{3}\]  
As the \((i,j)\)th element of \(\mathbf{\Omega}_{k}\) corresponds to the average power of the \((i,j)\)th element of the beam domain channel matrix, \(\mathbf{\Omega}_{k}\) is usually referred to the beam domain channel power matrix. Please note that \(\mathbf{\Omega}_{k}\) varies in a larger time scale than instantaneous CSI. In addition, the channel statistics are approximately constant for a wide range of frequencies [23; 26]. Therefore, statistical CSI can be obtained via averaging the samples over time and frequency in a realistic wideband wireless transmission system with guaranteed accuracy. In this paper, we assume that the statistical CSI of the multicast UTs can be perfectly known at the BS. In massive MIMO systems with a sufficiently large number of BS antennas, the channels usually exhibit new properties. One particular property of the massive MIMO channels is that the eigenvector matrices of the correlation matrices at the BS of different transmission links (UTs) tend to be the same and can be uniquely determined by the array topology equipped at the BS asymptotically [19; 21; 23].