Optimal time and rate allocation for a network-coded bidirectional two-hop communication

We consider a bidirectional communication between two terminals with the help of a relay. The transmissions of the three nodes are divided in time and there is no direct link between the terminals. The allocation of the transmission time and of the rates in both directions can be asymmetric. We first derive a closed form expression of the optimal time allocation for a given ratio of the rates in both directions and for given signal-to-noise ratios of all channels. Then, we derive a closed form expression of the optimal rate ratio such that the sum-rate is maximized under the assumption that the time allocation is optimally chosen. This also leads to a condition based on the channel SNRs, when to communicate with or without network coding in order to maximize the sum-rate. Then, we consider the design of the network code and compare several code designs by simulation whereas we assume that a turbo code is used as channel code. It turns out that efficient asymmetric two-way communication requires to perform network coding at the physical layer and can be realized by network-encoding the channel-encoded bits.