0090-6778 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCOMM.2017.2664866, IEEE Transactions on Communications 1 Degrees of Freedom of the Full-Duplex Asymmetric MIMO 3-Way Channel with Unicast and Broadcast Messages Adel M. Elmahdy 1,6 , Amr El-Keyi 1,2 , Yahya Mohasseb 3 , Tamer ElBatt 1,4 , Mohammed Nafie 1,4 , Karim G. Seddik 5 , and Tamer Khattab 6 1 Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egypt. 2 Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada. 3 Dept. of Communications, The Military Technical College, Cairo, Egypt. 4 Dept. of EECE, Faculty of Engineering, Cairo University, Giza, Egypt. 5 Electronics and Communications Engineering Dept., American University in Cairo, New Cairo, Egypt. 6 Electrical Engineering Dept., Qatar University, Doha, Qatar. adel@umn.edu, amr.elkeyi@sce.carleton.ca, {mohasseb, telbatt, mnafie}@ieee.org, kseddik@aucegypt.edu, tkhattab@ieee.org Abstract—In this paper, we characterize the total degrees of freedom (DoF) of the full-duplex asymmetric multiple-input multiple- output (MIMO) 3-way channel. Each node has a separate-antenna full-duplex MIMO transceiver with a different number of antennas, where each antenna can be configured for either signal transmission or reception. We study this system under two message configurations; the first configuration is when each node has two unicast messages to be delivered to the two other nodes, while the second configuration is when each node has two unicast messages as well as one broadcast message to be delivered to the two other nodes. For each configuration, we first derive upper bounds on the total DoF of the system. Cut-set bounds in conjunction with genie-aided bounds are derived to characterize the achievable total DoF. Afterwards, we analytically derive the optimal number of transmit and receive antennas at each node to maximize the total DoF of the system, subject to the total number of antennas at each node. Finally, the achievable schemes for each configuration are constructed. The proposed schemes are mainly based on zero-forcing and null-space transmit beamforming. We show that the derived outer and inner bounds on the total DoF are tight for each message configuration. I. I NTRODUCTION Interference-limited wireless communication networks have been extensively investigated over recent years. Despite the fact that uncoordinated interference decreases the achievable data rates in wireless networks, novel paradigms have emerged to sagaciously harness interference and, hence, efficiently uti- lize the scarce spectrum and enhance the network performance. Full-duplex systems have attracted a great deal of attention recently due to their potential benefits to significantly enhance The research work of A. M. Elmahdy and T. ElBatt was made possible by grants number NPRP 4-1034-2-385 and NPRP 5-782-2-322 from the Qatar National Research Fund, QNRF (a member of Qatar Foundation, QF). The research work of A. El-Keyi, M. Nafie and T. Khattab was made possible by grant number NPRP 7-923-2-344 from the QNRF. The statements made herein are solely the responsibility of the authors. Part of this work has been published in preliminary form in the IEEE Information Theory Workshop (ITW), Cambridge, UK, Sept. 2016 [1]. the throughput and spectral efficiency of conventional half- duplex systems [2]. Existing wireless communication systems operate in either a time-division duplex or a frequency-division duplex mode to separate the downlink and uplink traffic. However, recent results from academia [3]–[10] and industry [11] have proposed various practical designs to implement in-band full-duplex radios by cancelling or suppressing the self-interference signal, generated during simultaneous trans- mission and reception, at the RF and baseband level. There are two possible methods of antenna interfacing for full- duplex MIMO transceivers; separate-antenna architecture [3]– [7], and shared-antenna architecture [8]–[10]. In separate- antenna architecture, each antenna is dedicated to either signal transmission or reception. In shared-antenna architecture, each antenna simultaneously transmits and receives signals on the same channel with the aid of a circulator that routes the transmitted signal from the TX signal chain to the antenna and the received signal on the antenna to the RX signal chain. Full-duplex systems are envisioned to have an enormous impact on the evolution of future 5G generations of wireless communication systems. A. Related Work Several works have been dedicated to study the DoF of wireless networks with full-duplex operation [12]–[14]. The capacity region of a full-duplex wireless network with relay nodes, feedback and cooperation is characterized in [12]. The authors in [13] investigate the achievable total DoF of cellular networks in which a base station has full-duplex operation to simultaneously communicate with half-duplex mobile stations. In [14], the authors study the DoF of full-duplex bidirectional interference networks with and without a MIMO relay. The two-way communication channel was introduced in the seminal paper by Shannon [15]. The capacity regions of several multi-user two-way networks have been studied in [16]. The extension of the two-way channel to the case