# Energy-Efficient Multicast Precoding for Massive MIMO Transmission with Statistical CSI
## 1 Introduction  
The wireless traffic demand of group-oriented services and applications such as mobile TV, satellite communications, etc., is predicted to increase significantly in future wireless networks, and physical-layer wireless multicast transmission is a promising approach for such rapidly increasing demand. Recently, physical-layer multicasting has been incorporated in the 3 GPP release known as eMBMS [1] and has received extensive research interest [2; 3; 4]. In massive multiple-input multiple-output (MIMO) systems [5; 6; 7], large numbers of antennas are equipped at the base station (BS) so that a smaller number (compared with the number of BS antennas) of user terminals (UTs) can be simultaneously served in the same time and frequency resources. Compared with conventional small-scale MIMO systems [8], massive MIMO systems can significantly improve the spectral efficiency and transmission reliability. Due to the ability of massive MIMO in shaping the multicast transmission signals, physical-layer multicasting combined with massive MIMO promises to improve the multicast transmission quality of service in the evolution of future wireless networks [9; 10]. Energy efficiency (EE) is an important performance metric in wireless transmission design and has received growing attention from both academia and industry [11; 12]. Energy-efficient transmission design for MIMO unicasting was investigated in some existing works, e.g., [13; 14; 15]. Meanwhile, energy-efficient multicasting transmission was also studied in several previous works. For example, energy-efficient coordinated multicast transmission design for multi-cell scenarios was studied in e.g., [16; 17]. In addition, energy-efficient joint unicast and multicast transmission design for multi-cell MIMO systems was investigated in [18]. Please note that in most of the existing works on energy-efficient physical-layer multicast transmission, e.g., [16; 17; 18], instantaneous channel state information (CSI) is assumed to be known at the BS and exploited for transmission design. For physical-layer massive MIMO multicast transmission, the transmission performance will be highly dependant on the quality of the available CSI at the BS. CSI acquisition at the BS is a challenging task in massive MIMO for both frequency-division duplex (FDD) and time-division duplex (TDD) protocols. For FDD massive MIMO systems, acquisition of instantaneous CSI at the BS will lead to a huge pilot and feedback overhead. Meanwhile, the hardware limitations might destroy the reciprocity between the uplink and the downlink channels for TDD massive MIMO systems, and thus instantaneous CSI acquisition is still not an easy task [19; 20; 21; 22]. Compared with the instantaneous CSI, the statistical CSI varies over much larger time scales and thus can be efficiently and accurately obtained. In addition, massive MIMO channels exhibit new statistical properties with a large number of antennas [23], which, then, can be exploited in energy-efficient massive MIMO multicast transmission design. In this paper, we investigate physical-layer multicast precoding for energy-efficient massive MIMO transmission. Our work differs from previous works in that only statistical downlink CSI of the UTs is available at the BS. Our major contributions are summarized as follows:  
* We identify the eigenvectors of the optimal multicast transmit covariance matrix in terms of maximizing the system EE in closed form, which reveals that optimal energy-efficient multicast transmission should be performed in the beam domain in massive MIMO and simplifies the large-dimensional matrix-valued energy-efficient massive MIMO multicast transmission design into a power allocation problem in the beam domain with significantly reduced optimization variables. * We propose a sequential beam domain power allocation approach with guaranteed convergence to the global optimum via exploiting Dinkelbach's transform.