TWO-WAY RELAY BEAMFORMING DESIGN: PROPORTIONAL FAIR AND MAX-MIN RATE FAIR APPROACHES USING POTDC Arash Khabbazibasmenj † and Sergiy A. Vorobyov †,* † Dept. of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada * School of Electrical Engineering, Aalto University, Espoo, Finland Emails: {khabbazi,svorobyo}@ualberta.ca ABSTRACT The challenge in designing relay beamforming in two-way relaying systems is the non-convex nature of the correspond- ing optimization problem. In this work, we concentrate on the mathematical issues of such design for the cases when the max-min rate and proportional fairness are used as the design criteria. We show that the corresponding optimiza- tion problems belong to the class of difference-of-convex functions (DC) programming problems. Due to the spe- cific structure of the corresponding DC problems, they can be efficiently addressed by using the polynomial-time DC (POTDC) algorithm which guarantees to find the Karush- Kuhn-Tucker (KKT) optimal point in polynomial-time. We have also shown earlier that the question of global optimality of the POTDC algorithm boils down to a simple numerical convexity check for a certain one-dimensional optimal value function. Index Terms— Difference-of-convex functions optimiza- tion, Max-min rate fairness, Proportional fairness, Two-way relaying. 1. INTRODUCTION Two-way relaying (TWR) is a certain realization of the net- work coding [1] in which both terminals transmit their signals to the relay simultaneously through a multiple access channel (MAC) [2]. After receiving the transmitted signals corrupted by the additive noise, relay processes the mixture and then broadcasts it to the terminals. The most common relaying protocols are amplify-and-forward (AF) [3] and decode-and- forward (DF) [4]. In this work, it is assumed that the relay uses the AF relaying protocol which is more practical com- pared to other protocols in terms of the processing delay and processing energy consumption. One fundamental problem associated with TWR systems is the relay beamforming design based on the available chan- nel state information (CSI) [5]–[12]. It is usually designed so that a specific performance criterion is optimized under con- straints on the available resources and/or quality of service This work was supported in parts by the Natural Science and Engineer- ing Research Council (NSERC) of Canada. (QoS) requirements. The optimization criterion for most of the relay beamforming methods is the sum-rate [5]-[10]. The importance of the user fairness in asymmetric TWR systems has been recently demonstrated in [2], [11] and [12]. The authors of [2] study the optimal power allocation problem for single antenna users and single antenna relay where the sum-rate is maximized under the fairness constraint. Relay beamforming and optimal power allocation for a pair of single antenna users and several single antenna relays based on max- min signal-to-noise ratio (data rate) has been also considered in [11] and [12]. The main difficultly of the relay beamforming design in TWR system is the non-convex nature of the correspond- ing optimization problem. In this work, we study the relay beamforming for two single antenna users and one AF multi- antenna (multiple-input multiple-output (MIMO)) relay when the max-min rate and proportional fairness are used as the de- sign criteria. It is shown that the corresponding optimization problems can be recast as difference-of-convex functions (DC) programming problems. Although DC problems do not generally have polynomial-time solution, we handle the corresponding optimization problems using the so-called polynomial-time DC (POTDC) algorithm, which guarantees to find the Karush-Kuhn-Tucker (KKT) optimal point in poly- nomial time. Moreover, the global optimality can be checked by a simple numerical test. 2. SYSTEM MODEL Consider two single antenna terminals that communicate via a MIMO AF relay equipped with M R antennas through frequency-flat quasi-static block fading channels. Every data transmission between the terminals takes place in two phases. In the first phase, both terminals transmit their signals to the relay simultaneously. Then the received signal at the relay, which is a combination of both transmitted signals, can be expressed as r = h 1 x 1 + h 2 x 2 + n R (1) where h i =[h i,1 ,...,h i,MR ] T ∈ C MR denotes the chan- nel vector between terminal i and the relay, x i is the trans- mitted symbol from terminal i, n R ∈ C MR is the additive 4997 978-1-4799-0356-6/13/$31.00 ©2013 IEEE ICASSP 2013