Jointly Optimal Pre- and Post-Channel Equalization and Distributed Beamforming in Asynchronous Bidirectional Relay Networks

We consider an asynchronous bi-directional relay network (consisting of two single transceivers and multiple single-antenna relays) where the transceiver-relay paths are subject to different relaying and/or propagation delays. In such a network, the end-to-end link can be viewed as a multipath channel which can cause intersymbol-interference (ISI) in the signals received by the two transceivers. Assuming a block transmission/reception scheme, we consider both pre- and post-channel equalization at both transceivers to combat the interblock-interference (IBI) induced due to ISI. Considering amplify-and-forward relays, we study the problem of optimal design of pre- and post-channel linear equalizers and optimal power loading at the two transceivers as well as optimal network beamforming. To do so, assuming a limited total transmit power budget, we minimize the total mean square error of the linearly estimated signals at both transceivers by optimally obtaining the transceivers' powers and relay beamforming weights as well as block pre- and post-channel linear equalizers at the two transceivers. We prove rigorously that this minimization leads to a certain relay selection scheme, where only a subset of the relays will be turned on. We also provide semiclosed form solutions to the design parameters.