Single-Carrier Equalization for Asynchronous Two-Way Relay Networks

We consider an asynchronous bi-directional amplify-and-forward relay network, where two single-antenna transceivers communicate with the help of several single-antenna relay nodes using a single-carrier communication scheme. The propagation delay of each relaying path, (which originates from one transceiver, goes through a certain relay, and ends at the other transceiver) is assumed to be different from those of the other relaying paths. This assumption turns the end-to-end link into a frequency selective channel which can have multiple taps. As such, intersymbol interference (ISI) is inevitable at the two transceivers. Assuming a block transmission/reception scheme, ISI results in interblock interference (IBI) between successive transmitted blocks. To combat IBI, cyclic prefix insertion and deletion as well as block postchannel equalization are used at the two transceivers. Assuming a limited total transmit power budget, we minimize the total mean squared error (MSE) of the estimated received signals at both transceivers by optimally obtaining the transceivers´ transmit powers and the relay beamforming weight vector as well as the block post-channel equalizers at the two transceivers. We prove that this optimization problem leads to a relay selection scheme, where only the relays contributing to one tap of the end-to-end channel impulse response are turned on and the remaining relays are switched off. Moreover, we present a semi-closed-form solution for the optimal relay weight vector. Our numerical results show that the proposed algorithm significantly outperforms an equal power allocation scheme, where all nodes receive the same level of transmit power.