Optimal Relay Selection and Channel Allocation for Multi-User Analog Two-Way Relay Systems Bo (Bob) Bai ∗ , Member, IEEE, Wei Chen † , Member, IEEE, Khaled Ben Letaief ∗ , Fellow, IEEE, and Zhigang Cao † , Senior Member, IEEE ∗ Center for Wireless Information Technology, Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong † Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing 100084, China E-mail: eebob@ust.hk wchen@tsinghua.edu.cn eekhaled@ee.ust.hk czg-dee@tsinghua.edu.cn Abstract—Analog network coding is a promising technique which can greatly improve the transmission efficiency of wireless communications. In two-way relay systems with multiple sub- channels, multiple user pairs and multiple relays, however, the optimal joint relay selection and subchannel allocation problem has not been studied in a systematic way. In this paper, a random combinatorial optimization approach, referred to as the weighted random bipartite graph (WRBG) based minimum weighted matching (MWM) method, will be proposed to solve this problem. By analyzing the properties of the MWM on WRBG, we shall derive the outage probability and diversity- multiplexing tradeoff of each user after relay selection and channel allocation. Theoretical results will demonstrate that the outage probability, cooperative diversity, and frequency diversity of the proposed WRBG based MWM method for multi-user two- way relay systems is the same as that of two-way relay systems with only one user pair. The proposed algorithm for MWM also enjoys a low computation complexity of O(log 2 N ) for parallel implementations, where N is the number of subchannels. Simulation results will illustrate the potential of the proposed method and also verify the theoretical derivations. I. I NTRODUCTION Network coding is a promising technique for future com- munication networks, and various coding schemes have been proposed since the foremost works [1]. Recently, the applica- tion of analog (or physical-layer) network coding to wireless communications, i.e., two-way relay systems, has attracted much attention from both academia and industry, because it saves one time-slot compared to digital network coding. There are many studies on two-way relay systems. Some of the works analyzed and compared different two-way relay schemes [2], [3]. A lot of the works studied the resource allocation problem in two-way relay systems. Chen, Letaief, and Cao considered the resource allocation problem in two- way relay systems from the point of view of cross-layer design [4], [5]. Jitvanichphaibool, Zhang, and Liang studied the optimal subcarrier allocation, power allocation, and bit- loading for two-way relay assisted OFDMA systems by using convex optimization [6]. Joung and Sayed proposed a user This work was supported by RGC/NSFC under Grant No. N_HKUST622/06 and NSFC founded key project under Grant No. 60832008. selection method for two-way relay systems with multiple antennas at the relay [7]. Other works also studied the optimal power allocation for two-way relay systems with multiple channels [8], [9]. In [10], Yi, Ju, and Kim studied the optimal power allocation scheme and computed the outage probability for two-way relay systems with one relay and one user pair. In contrast to conventional works, this paper focuses on broadband two-way relay systems with multiple subchannels, multiple user pairs, and multiple relays, i.e., multi-user two- way relay systems. In these systems, each user pair occupies one subchannel through one relay to achieve a successful two-way communication by applying physical-layer network coding. In particular, the transmission of the considered system occurs in two phases. In Phase I (or multiple access phase), two users in the same pair will access the same relay through the same subchannel. In Phase II (or broadcasting phase), the relay broadcasts the decoded signal which has been received in Phase I to two users in the same pair through the same subchannel. How to guarantee that most of the users are not in outage and how to select the relay and allocate subchannels to each user pair in both phases is a key problem for multi- user two-way relay systems. Since the joint relay selection and subchannel allocation is a kind of random combinatorial optimization problem, a weighted random bipartite graph (WRBG) model is proposed to formulate the multi-user two- way relay system. Based on the WRBG formulation, the minimum weighted matching (MWM) method is proposed to select the best relay and allocate the best subchannel to each user pair. By analyzing the properties of the MWM on WRBG, some closed-form formulas for the outage probability of each user are derived in both high-SNR and low-SNR regimes. The diversity-multiplexing tradeoff is then obtained accordingly. These results demonstrate that the outage performance of the WRBG based MWM method for multi-user two-way relay systems is the same as that in the two-way relay system with only one user pair. Therefore, the optimal cooperative and frequency diversity-multiplexing tradeoff is achieved by the proposed method. By applying the parallel algorithm, the complexity of the proposed method is only O(log 2 N ), where N is the number of subchannels. 978-1-4244-9268-8/11/$26.00 ©2011 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.