Two-Way Relay Channels: Error Exponents and Resource Allocation

In a two-way relay network, two terminals exchange information over a shared wireless half-duplex channel with the help of a relay. Due to its fundamental and practical importance, there has been an increasing interest in this channel. However, surprisingly, there has been few work that characterizes the fundamental tradeoff between the communication reliability and transmission rate across all signal-tonoise ratio (SNR) regimes. In this paper, we consider amplify-and-forward (AF) two-way relaying due to its simplicity. We first derive the random coding error exponent for the link in each direction. From the exponent expression, the capacity and cutoff rate for each link are also deduced. We then define the minimal error exponent for the two-way AF relay fading channel, which is the worst exponent decay between the two links and gives us insight into the fundamental tradeoff between the rate pair and information-exchange reliability of the two terminals. As applications of the error exponent analysis for the design of two-way relay networks, we present two optimal resource allocations to maximize the minimal error exponent: i) the optimal rate allocation under a sum-rate constraint and its closed-form quasi-optimal solution that requires only the knowledge of the capacity and cutoff rate of each link; and ii) the optimal power allocation under a total power constraint of two terminals in the presence of global channel state information, which can be efficiently determined via a sequence of convex feasibility problems in the form of second-order cone programs. Numerical results verify our analysis and the effectiveness of the optimal rate and power allocations in maximizing the minimal error exponent.