## 3 Coordinated Multicast Transmission Design  
Assume that each BS only has access to statistical CSI of all UTs in the coordinated network, and each UT has knowledge of its instantaneous CSI in the associated cell. For each UT \((k,u)\), the aggregate interference-plus-noise \(\mathbf{z}_{k,u}^{\prime}=\sum_{\ell\
eq u}\mathbf{H}_{k,u,\ell}\mathbf{x}_{ \ell}+\mathbf{z}_{k,u}\) is treated as the worst-case Gaussian noise [33] in detection with covariance given by  
\[\mathbf{K}_{k,u}=\mathbf{I}_{N_{k,u}}+\sum_{\ell\
eq u}\mathbb{E}\left\{ \mathbf{H}_{k,u,\ell}\mathbf{Q}_{\ell}\mathbf{H}_{k,u,\ell}^{H}\right\}\in \mathbb{C}^{N_{k,u}\times N_{k,u}}. \tag{5}\]  
For multicast transmission, the transmitted multicast signals have to be decodable by all UTs in the corresponding multicast UT group. Then, the achievable ergodic multicast rate in cell \(u\) can be defined as a minimum of the UT rates across the cell [34], which can be explicitly written as  
\[\mathcal{R}_{u}= \underset{k\in\mathcal{K}_{u}}{\text{min}}\ \left\{\mathbb{E}\left\{\text{logdet}\left(\mathbf{K}_{k,u}+\mathbf{H}_{k,u, u}\mathbf{Q}_{u}\mathbf{H}_{k,u,u}^{H}\right)\right\}-\text{logdet}\left(\mathbf{K}_ {k,u}\right)\right\}\] \[\overset{\text{(a)}}{=} \underset{k\in\mathcal{K}_{u}}{\text{min}}\ \left\{\mathbb{E}\left\{\text{ logdet}\left(\overline{\mathbf{K}}_{k,u}+\mathbf{G}_{k,u,u}\mathbf{V}^{H} \mathbf{Q}_{u}\mathbf{V}\mathbf{G}_{k,u,u}^{H}\right)\right\}-\text{logdet} \left(\overline{\mathbf{K}}_{k,u}\right)\right\}, \tag{6}\]  
where (a) follows from the massive MIMO channel model in (4), and the following definition  
\[\overline{\mathbf{K}}_{k,u} \triangleq\mathbf{U}_{k,u,\ell}^{H}\mathbf{K}_{k,u}\mathbf{U}_{k, u,\ell}\] \[=\mathbf{I}_{N_{k,u}}+\sum_{\ell\
eq u}\mathbb{E}\left\{\mathbf{ G}_{k,u,\ell}\mathbf{V}^{H}\mathbf{Q}_{\ell}\mathbf{V}\mathbf{G}_{k,u,\ell}^{H} \right\}\in\mathbb{C}^{N_{k,u}\times N_{k,u}}. \tag{7}\]  
For notational convenience, we define a matrix-valued function as follows  
\[\mathbf{A}_{k,u,\ell}\left(\mathbf{X}\right)\triangleq\mathbb{E}\left\{ \mathbf{G}_{k,u,\ell}\mathbf{X}\mathbf{G}_{k,u,\ell}^{H}\right\}. \tag{8}\]  
Using the beam domain massive MIMO channel properties, it is not difficult to verify that \(\mathbf{A}_{k,u,\ell}\left(\mathbf{X}\right)\) defined in (8) is a diagonal matrix-valued function with the elements given by  
\[\left[\mathbf{A}_{k,u,\ell}\left(\mathbf{X}\right)\right]_{i,j}=\left\{ \begin{array}{ll}\text{tr}\left\{\text{diag}\left\{\left(\left[\mathbf{ \Omega}_{k,u,\ell}\right]_{i;}\right)^{T}\right\}\mathbf{X}\right\},&i=j,\\ 0,&i\
eq j.\end{array} \right. \tag{9}\]  
Then, \(\overline{\mathbf{K}}_{k,u}\) defined in (7) can be rewritten as  
\[\overline{\mathbf{K}}_{k,u}=\mathbf{I}_{N_{k,u}}+\sum_{\ell\
eq u}\mathbf{A} _{k,u,\ell}\left(\mathbf{V}^{H}\mathbf{Q}_{\ell}\mathbf{V}\right). \tag{10}\]  
We investigate coordinated multicast precoding design for multi-cell massive MIMO transmission in the following. Our design objective is to identify the optimal transmit covariance matrices \(\{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}\}\) of all BSs in the coordinated network that can maximize the sum of the achievable ergodic multicast rate under individual power constraints, which can be formulated as the following problem:\[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\] \[\mathrm{subject\ to} \mathrm{tr}\left(\mathbf{Q}_{u}\right)\leq P_{u},\] \[\mathbf{Q}_{u}\succeq\mathbf{0},\ u=1,\cdots,U, \tag{11}\]  
where \(\mathcal{R}_{u}\) is the ergodic multicast rate of cell \(u\) defined in (6), and \(P_{u}\) is the transmit power constraint of the BS in cell \(u\).