## 3 Coordinated Multicast Transmission Design
Denote by \(\mathbf{Q}_{u}=\mathbf{\Phi}_{u}\mathbf{\Lambda}_{u}\mathbf{\Phi}_{u}^{H}\) the eigenvalue decomposition of the multicast transmit covariance matrix in cell \(u\), where the columns of \(\mathbf{\Phi}_{u}\) are the eigenvectors of \(\mathbf{Q}_{u}\) and \(\mathbf{\Lambda}_{u}\) is a diagonal matrix composed of the eigenvalues of \(\mathbf{Q}_{u}\). In practice, \(\mathbf{\Phi}_{u}\) represents the directions in which signals are transmitted and \(\mathbf{\Lambda}_{u}\) represents the transmit power allocated onto each direction, respectively. We first identify the optimal multicast signalling directions in the following theorem. The eigenvectors of the optimal transmit covariance matrix \(\mathbf{Q}_{u}^{\mathrm{op}}\) for each cell \(u\) to problem (11) are given by the columns of the eigenvector matrices of the BS correlation matrix \(\mathbf{V}\), i.e.,  
\[\mathbf{Q}_{u}^{\mathrm{op}}=\mathbf{V}\mathbf{\Lambda}_{u}\mathbf{V}^{H}, \quad\forall u. \tag{12}\]  
Please refer to the Appendix A.  
Theorem 4.1 shows that the optimal multicast transmit directions should align with the eigenvectors of the transmit correlation matrix at the BS for each cell. This implies that optimal coordinated multicast transmission should be performed in the beam domain to maximize the achievable ergodic sum multicast rate. Then, we focus on the beam domain coordinated multicast transmission. In particular, based on Theorem 4.1, the matrix-valued coordinated multicast precoding optimization problem in (11) can be simplified to the following beam domain power allocation problem  
\[\underset{\mathbf{\Lambda}=\left\{\mathbf{\Lambda}_{1},\cdots, \mathbf{\Lambda}_{U}\right\}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\left(\mathbf{\Lambda}\right)\] \[\mathrm{subject\ to} \mathrm{tr}\left(\mathbf{\Lambda}_{u}\right)\leq P_{u},\] \[\mathbf{\Lambda}_{u}\succeq\mathbf{0},\ \mathbf{\Lambda}_{u}\ \mathrm{diagonal},\ u=1,\cdots,U, \tag{13}\]  
where  
\[\mathcal{R}_{u}\left(\mathbf{\Lambda}\right) \triangleq\underset{k\in\mathcal{K}_{u}}{\min}\ \left\{\mathcal{R}_{k,u,1}\left(\mathbf{\Lambda}\right)-\mathcal{R}_{k,u,2} \left(\mathbf{\Lambda}\right)\right\}, \tag{14}\] \[\mathcal{R}_{k,u,1}\left(\mathbf{\Lambda}\right) \triangleq\mathbb{E}\left\{\mathrm{logdet}\left(\overline{\mathbf{ K}}_{k,u}\left(\mathbf{\Lambda}\right)+\mathbf{G}_{k,u,u}\mathbf{\Lambda}_{u} \mathbf{G}_{k,u,u}^{H}\right)\right\},\] (15) \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]0 (16) \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]1  
Although the number of variables needs to be optimized in the beam domain power allocation problem in (13) is significantly reduced compared with the original precoding optimization problem in (11), it is still challenging to solve (13) due to the complexity of objective function. We adopt the MM algorithmic framework [35] to address this problem. The MM algorithmic framework is a sequential optimization approach, and the idea is to handle a difficult maximization problem via solving a sequence of maximization problems that are easy to handle. The key step of MM framework is to construct a surrogate lower-bound function of the objective so that the maximization problems are easy to handle. For the considered coordinated power allocation problem in (13), we construct the surrogate lower-bound function in each iteration by replacing \(\mathcal{R}_{k,u,2}\left(\mathbf{\Lambda}\right)\) for \(\forall k,u\) with their first-order Taylor expansions and then solve it, which further yields the next iteration.