1 Outage-Optimal Power and Time Allocation for Rate-Aware Two-Way Relaying with a Decode-and-Forward Protocol Thinh Phu Do and Yun Hee Kim, Senior Member, IEEE Abstract—We consider a two-way relaying (TWR) network, where two communicating nodes exchange their data with the help of a decode-and-forward relay. For the network, opti- mization of power allocation (PA) and time allocation (TA) is explored to minimize the system outage probability in supporting asymmetric data rates. We tackle the joint optimization of PA and TA with full channel state information by dividing the problem into two convex optimization problems, which find the optimal PA in a closed form for a given TA and then find the optimal TA with the binary search method. To lower the complexity of the joint optimization, we devise sub-optimal schemes, one selecting the TA value adaptively from two pre-determined TA values and the other employing a single fixed TA value. We analyze the outage probabilities and asymptotic behaviors of the proposed PA-TA schemes in generalized Nakagami-m fading channels. The results show that PA optimization improves the performance for both symmetric and asymmetric rate transmission but TA optimization improves the performance only for asymmetric rate transmission. In addition, the sub-optimal schemes get close to the optimal scheme in the outage performance when two relay channels are asymmetric with unequal fading parameters and asymmetric relay position. Index Terms—Two-way relay, decode-and-forward, time allo- cation, power allocation, outage probability I. I NTRODUCTION Cooperative and relaying communication has drawn a lot of attention from wireless researchers for its capabilities of extending the transmission coverage and improving the system robustness in severe fading environments [1]–[4]. In the case of one-way relaying (OWR) by half-duplex nodes, such gains are usually obtained at the cost of the spectral inefficiency since additional wireless resources are required for relay transmission [5], [6]. To alleviate the loss in spectral efficiency, two-way relaying (TWR) protocols based on network coding have been proposed for bidirectional communications [7]– [12]. The TWR protocols based on network coding spend only two time phases for the information exchange although the conventional OWR protocols spend four time phases to accomplish the same task. The most commonly used TWR protocols accomplish the data exchange in two time phases, based on amplify-and- forward (AF) [7], [9] and decode-and-forward (DF) [8], [10]– [12], which are called AF-TWR and DF-TWR, respectively: the protocols are called in different names in the literature Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. T. P. Do and Y. H. Kim (corresponding author) are with the Department of Electronics and Radio Engineering, Kyung Hee University, Yongin, Korea (e-mail: dopthinh@gmail.com,yheekim@khu.ac.kr). This work was supported by the National Research Foundation of Ko- rea (NRF) grant funded by the Korea government (MSIP) (No. NRF- 2015R1A2A2A01005390). such as multiple access broadcasting (MABC) with AF and DF, analog network coding and digital network coding, and so on. The protocols allow two communicating nodes to transmit their information simultaneously in the multiple access (MAC) phase so that a superimposition of the two node signals is received at a relay. In the broadcast (BC) phase, the AF-TWR relays simply forward the amplified version of the received signal while the DF-TWR relays decode the received signal and forward a regenerated signal based on the network coding. Although the DF-TWR renders relays more complex than the AF-TWR in general, the DF-TWR makes the communicating nodes exempt from the self-interference cancelation, which is essential in the AF-TWR. Therefore, the DF-TWR is preferred for the network allowing more expensive relays while the AF- TWR is preferred for the network constructed with cheaper relays. The advent of the TWR protocols has also given rise to various resource allocation problems to leverage various performance objectives of the TWR networks under different system conditions [13]–[26]. The main issues are how to share the power among the communicating and relay nodes and how to divide the time for MAC and BC phases. Only power allocation (PA) problems have been studied for the AF- TWR networks since the MAC and BC phases should have an equal time duration to relay a non-regenerative signal [13]– [18]. On the other hand, both PA and time allocation (TA) are possible for the DF-TWR networks using a regenerative relay signal [19]–[26], which makes the DF-TWR networks more advantageous than the AF-TWR networks from the performance perspective. In detail, the PA problems studied for the AF-TWR net- works aim at maximizing the sum rate [13], [14], minimizing the outage probability [15]–[17], and minimizing the symbol error probability [18]. The solutions for the PA problems are derived either with average channel state information (CSI) [13], [16], [17] or with instantaneous CSI [14], [15], [18]. The rates of the bidirectional data flows are assumed to be equal earlier [15], [16] and then asymmetric [17] in the optimization. In the case of the DF-TWR networks, the PA optimization is considered in a way of minimizing the symbol error probability with instantaneous CSI [19], maximizing the sum rate with instantaneous CSI [20], and minimizing the coded bit error rate with average CSI [21]. The TA optimization without PA is investigated for the DF-TWR networks to maximize the achievable rate region of a network with multi-antenna nodes [22] and to maximize the sum rate of a network with or without a constraint on the quality-of-service [23], [24]. A joint PA and TA optimization for the DF-TWR networks is tackled in a limited case of maximizing the weighted sum rate [25], [26], which is solved by an interior-point method without an explicit