Proof of the Equivalency of MSE-Constrained and Rate-Constrained Power Optimization Approaches to Distributed Bi-Directional Beamforming

Considered in this article is an
asynchronous
single-carrier two-way relay network, where two transceivers aim to communicate through a set of amplify-and-forward (AF) relays. This network is asynchronous in the sense that the signal transmitted by any of the two transceivers arrives the other transceiver through different relaying paths, with possibly different propagation delays, thereby materializing a multipath channel. At sufficiently high data rates, this multi-path end-to-end channel causes inter-symbol-interference (ISI) in the received signals. Considering such a network, this paper presents two contributions. The first contribution is the rigorous characterization of the region of the mean-squared errors (MSEs) of the symbol estimates at the two transceivers under a total power budget, and when linear block post-channel equalization is used at the receiver front-end of the two transceivers. The importance of this MSE region characterization resides in the fact that knowing this region allows for characterization the region of un-coded probabilities of error at the two transceivers. Also, this MSE region characterization paves the way towards presenting the second contribution in this paper. Indeed, in the second contribution, this article relies on this MSE region characterization to rigorously prove that an MSE-constrained total power minimization approach and a rate-constrained total power minimization approach to design transceiver power allocation and distributed beamforming are equivalent, if the MSE thresholds in the former approach and the rate thresholds in the latter approach are properly chosen. The equivalence of these two approaches implies that the un-coded MSE performance of the network can be inferred from the rate-constrained problem, and conversely, the coded rate performance of the network can be inferred from the MSE-constrained total power minimization problem.