On Total Transmission Power Minimization Approach to Decentralized Beamforming in Single-Carrier Asynchronous Bidirectional Relay-Assisted Communication Networks

Considered in this paper is a single-carrier asynchronous bidirectional cooperative network with amplify-and-forward relays helping two transceivers to exchange information. The network is asynchronous in the sense that the delays of propagation along different relaying paths are significantly different from each other. As a result, the channel between the two transceivers is best modeled with a multi-tap impulse response, which can cause interference between adjacent symbols at the end nodes. In a block-wise communication scheme, a cyclic prefix can be appended to each transmitted block of information symbols to avoid interference between adjacent blocks. To suppress interference within each block, however, the network parameters, namely, the relay complex weights and the transceivers' transmit powers should be judiciously chosen. To do so, we herein minimize, over these parameters, the total transmission power consumption throughout the network, under two constraints which ensure that the transceivers' data rates are above two given thresholds. It is herein proved that solving this total transmission power minimization problem leads to the impulse response of end-to-end channel having only a single non-zero tap. Indeed, our analysis shows that only the relays associated with this non-zero tap have to be selected to participate in the information exchange between the two transceivers. We present a simple 1-D integer search algorithm for optimally determining the index of the non-zero tap of the channel impulse response of the end-to-end channel. We also present computationally simple semi-closed-form expressions for the optimal values of the design parameters. Our numerical results show that under identical rate thresholds, in our power allocation scheme, for any channel realization, half of the power budget is allocated to the two transceivers and the remaining half is shared among all the relay nodes.