# Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission
## 1 Introduction
Besides, as a vast antenna array with high spatial resolution, massive MIMO channel typically exhibits new statistical characteristics [35; 36]. We study the energy efficiency of massive MIMO secure multicast transmission with the assumption that only the statistical CSI of legal UTs and the eavesdropper can be obtained at the BS. We summarize the contributions of the current paper as follows. * We first considered a secure multicast transmission scenario, in which the BS can only obtain the statistical CSI of the legitimate UTs and eavesdropper, and formulate our energy efficiency maximization problem. A tight lower bound rate of secure multicast is presented to simplify the optimization objective. Therefore, our problem is transformed into finding the optimal solution of this equivalent problem. * Via making the transmit signaling directions into a closed-form, we simplify the energy efficiency optimization of matrix-valued strategy design to a beam domain power allocation problem, reducing the optimization variables significantly. * By invoking the Minorize-Maximize (MM) method and Dinkelbach's transform, we propose a two-level iterative algorithm to solve the equivalent problem. In the outer iteration, we decompose the original problem into several fractional programming subproblems, where the numerator is concave and the denominator is linear. In the inner iteration, we introduce related variables to remove the denominator and transform each subproblem into a common convex optimization problem to obtain the optimal solution of the equivalent problem. Our simulation results reveal that this solution is almost equal to the original problem. * According to the large-dimensional random matrix theory, we use the deterministic equivalent of the secure transmit rate instead of massive sample averaging to reduce the computational complexity, which is proved to be almost the same as the original result. The rest of this paper is arranged as follows. Section 2 presents the massive MIMO channel model. In Section 3, we divide the algorithm into three parts. The energy efficiency of secure multicast transmission is investigated. The optimal energy efficiency multicast transmit covariance matrix is decomposed into eigenvectors and eigenvalues as pretreatment. We propose an iterative algorithm for some fractional programming subproblems. Each subproblem is also solved by the inner iterative algorithm of sequential beam domain power allocation. The last part is to simplify the complexity of the problem. In Section 4, numerical simulation results are given. Finally, we summarize the full work in Section 5. In our work, the use of symbols is also an important part. The main adopted symbols are as follows. * \(\mathbb{C}^{M\times N}\) denotes \(M\times N\) dimensional complex-valued vector space and \(\mathbb{R}^{M\times N}\) denotes \(M\times N\) dimensional real-valued vector space. * \([\mathbf{A}]_{m,n}\) denotes the \((m,n)\) th element of matrix \(\mathbf{A}\). * \(\det\left\{\cdot\right\}\) denotes the matrix determinant operation, tr \(\left\{\cdot\right\}\) denotes the matrix trace operation, and diag \(\left\{\cdot\right\}\) denotes the diagonalization of the matrix, respectively. * \(\mathbb{E}\left\{\cdot\right\}\) denotes the expectation operation and \(\left[x\right]^{+}\) denotes \(\max\{x,0\}\). * \((\cdot)^{T},(\cdot)^{*}\), \((\cdot)^{H}\) and \((\cdot)^{-1}\) denote the matrix transpose, conjugate, conjugate-transpose and inverse, respectively. * \(\odot\) denotes the Hadamard product. * \(\mathbf{X}\succeq\mathbf{0}\) denotes that \(\mathbf{X}\) is a positive semidefinite matrix.