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Citation information: DOI 10.1109/TWC.2014.2367503, IEEE Transactions on Wireless Communications IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Ergodic Rate Analysis for Multi-pair Massive MIMO Two-way Relay Networks Shi Jin, Member, IEEE, Xuesong Liang, Kai-Kit Wong, Senior Member, IEEE, Xiqi Gao, Senior Member, IEEE, and Qi Zhu Abstract—This paper considers a multi-pair massive multiple- input multiple-output (MIMO) two-way relay network, in which multiple pairs of users are served by a relay station with a large number of antennas which uses maximum ratio com- bining/maximum ratio transmission (MRC/MRT) and a fixed amplification factor for reception/transmission. First, the users’ ergodic rates are derived for the case with a finite number of antennas, and then the rate gain is analyzed when the transmit power of the senders and the relay is sufficiently large. We show that the ergodic rates increase with the number of antennas at the relay, N , but decrease with the number of user pairs, K, both logarithmically. The energy efficiency for the network is also investigated when the number of antennas grows to infinity. It is further revealed that the ergodic sum-rate can be maintained while the users’ transmit power is scaled down by a factor of 1/N or the relay power by a factor of 2K/N . This indicates that users obtain an energy efficiency gain of N but the relay has an energy efficiency gain of N divided by the number of users, i.e., 2K. Index Terms—Ergodic achievable rate, Energy Efficiency, Massive MIMO, Two-way relaying, Maximum ratio combining, maximum ratio transmission. I. I NTRODUCTION I N recent years, massive multiple-input multiple-output (MIMO) antenna system has become a very active area of research in wireless communications, e.g., [1–4]. For cellular networks, by employing hundreds or more antennas at a base station, it is predicted that the spectral and energy efficiencies can be improved by several orders of magnitude compared to the conventional MIMO systems [2, 3]. Due to these im- pressive gains and the simplicity of transceiving techniques compared to traditional multiuser MIMO systems, massive Manuscript received January 14, 2014; revised June 12, 2014, August 24, 2014, and October 16, 2014. This work was supported by National Natural Science Foundation of China under Grants (61222102, 61320106003, and 61171094), the Natural Science Foundation of Jiangsu Province under Grant BK2012021, the National Science and Technology Major Project of China under Grant 2013ZX03001032-004, the Program for Jiangsu Innovation Team, and the Talents Start Foundation of Nanjing University of Posts and Telecommunications (NY213062). It was presented in part at the IEEE International Conference on Communications (ICC), Sydney, Australia, June 2014. Shi Jin and Xiqi Gao are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China. (e-mail: jinshi,xqgao@seu.edu.cn). Xuesong Liang and Qi Zhu are with the College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommuni- cations, Nanjing 210003, China. (e-mail: liangxs,zhuqi@njupt.edu.cn). Kai-Kit Wong is with the Department of Electronic and Electrical Engineering, University College London, United Kingdom. (e-mail: kai- kit.wong@ucl.ac.uk). MIMO has become one of the most sought-after technologies for the next-generation wireless communications networks [4]. Massive MIMO was first proposed in multi-cellular nonco- operative networks [1] in which a base station can serve many users via linear processing and approach the performance of using the optimal receiver, when a large antenna array is used at the base station with independent and identically distributed (i.i.d.) channel vectors. However, the system performance of massive MIMO is limited by inter-cellular interference due to the reuse of pilot sequences in other cells (known as pilot contamination) when the number of antennas grows to infinity. Subsequently, the energy efficiency for massive MIMO was examined when the number of antennas, N , is much greater than the number of users, K [5]. It is shown that the system will be more robust when N becomes large, and the total transmit power can be scaled down by 1/N with the achievable rate unchanged. The energy efficiency for massive multiuser MIMO was also studied in a single-cell environment consid- ering perfect channel state information (CSI) and imperfect CSI acquisition [6]. In [6], it was revealed that to maintain the uplink rates at a constant level, users can reduce their transmit power by a factor of 1/N in the perfect CSI scenario and a factor of 1/ √ N in the imperfect CSI scenario, when N grows to infinity. Furthermore, massive MIMO has also been extensively studied in the areas of performance analysis [7– 9], CSI acquisition and training [10–12], pilot contamination mitigation [13–15], and downlink precoding [16–18]. On the other hand, recently, MIMO relay systems have been intensively studied. Precoding matrices and power allocation designs have been actively investigated, including one-way relay systems [19–22], two-way relay systems [23, 24], in- terference relay systems [25–27] and MIMO multiple access channel [28]. It was illustrated in the aforementioned works that precoding optimization is extremely complex and the algorithms for optimal power allocation require considerable complexity. Encouraged by the results in massive MIMO systems that when large numbers of antennas were used, low-complexity linear precoding would suffice to achieve near-optimal rate performance, and massive antennas have also been introduced into relaying systems. It was proved in [29] that the system power efficiency is proportional to the number of antennas without degrading the quality-of-service in one-way relay systems where the relay has a very large antenna array, whilst the energy efficiency was also derived in two-way relay networks with a large number of relays [30]. Moreover, the achievable rates for K-hop MIMO relay channels have been discussed in the large-system regime in