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.

中文翻译：

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MSE约束和速率约束功率优化方法在分布式双向波束成形中的等效性证明

本文认为是 异步单载波双向中继网络，其中两个收发器旨在通过一组放大转发（AF）中继进行通信。从两个收发器中的任何一个发送的信号通过不同的中继路径（可能具有不同的传播延迟）通过不同的中继路径到达另一个收发器的意义上，该网络是异步的，从而实现了多路径信道。在足够高的数据速率下，此多径端到端信道会在接收信号中引起符号间干扰（ISI）。考虑到这样的网络，本文提出了两个贡献。第一个贡献是在总功率预算下对两个收发器处的符号估计的均方误差（MSE）区域进行了严格的表征，当在两个收发器的接收器前端使用线性块后信道均衡时。MSE区域表征的重要性在于以下事实：知道该区域可以表征两个收发器的未编码错误概率区域。同样，该MSE区域特征为提出本文的第二个贡献铺平了道路。的确，在第二个贡献中，本文依靠该MSE区域特征来严格证明，如果MSE等效，则设计收发器功率分配和分布式波束成形的MSE约束的总功率最小化方法和速率约束的总功率最小化方法是等效的。在前一种方法的阈值和所述阈值速率在后一方法中适当地选择。