Jointly optimal pre- and post-channel equalization and distributed beamforming in asynchronous bi-directional relay networks

We consider an asynchronous bi-directional relay network (consisting of two single-antenna 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 multi-path channel which can cause inter-symbol-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 inter-block-interference (IBI) induced due to ISI. Considering amplify-and-forward (AF) relays, we study the problem of optimal design of pre- and post-channel linear equalizers and power loading at the two transceivers as well as the relay network beamforming. To do so, assuming a limited total transmit power budget, we minimize the total mean square error (MSE) of the linearly estimated signals at both transceivers by optimally obtaining the transceivers' powers and relay beamforming weights as well as pre- and post-channel linear equalizers at the two transceivers. We rigorously prove that this minimization leads to a certain relay selection scheme, where only a subset of the relays will be turned on and the rest switched off. We also provide semi-closed form solutions to the design parameters.