# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks  
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1 Department of Engineering, University of Naples Parthenope, I-80143 Naples, Italy;  
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2 National Inter-University Consortium for Telecommunications (CNIT), I-80125 Naples, Italy;  
2 [EMAIL_ADDRESS]
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[PERSON]  
2 National Inter-University Consortium for Telecommunications (CNIT), I-80125 Naples, Italy;  
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[PERSON]  
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transmission, due to the inability of the relays to receive and transmit at the same time. To overcome the inherent halving of spectral efficiency, a possible remedy for 5G applications is to adopt _two-way_ relaying [14] (see Figure 1), which works as follows: (i) in the first slot, the two terminal nodes simultaneously transmit their precoded signals to the relays; (ii) in the second slot, the relays precode and forward the received signals to the terminals. Since each terminal knows its own transmitted signal, the effects of self-interference can be subtracted from the received signal at the terminals, and the data of interest can be decoded. On the other side of the coin, with respect to the one-way relaying setting, the optimization of two-way cooperative networks is complicated by fact that terminal precoders/decoders and relay forwarding matrices are coupled among themselves. Design and performance analysis of two-way cooperative MIMO networks encompassing multiple _amplify-and-forward_ (AF) or _non-regenerative_ relays has been considered in References [15; 16; 17; 18; 19]. Compared with the single-relay case [20], the multiple-relay scenario generally leads to more challenging _nonconvex_ constrained optimization problems, which are usually solved by burdensome iterative procedures. In Reference [15], by adopting a weighted sum-mean-square-error (MSE) or a sum-rate cost function, iterative gradient descent optimization algorithms are proposed, with transmit-power constraints imposed at both the terminals and the relays. A similar scenario is considered in Reference [16] and Reference [17]. In Reference [16], the original constrained minimum sum-MSE nonconvex optimization problem is iteratively solved. Specifically, the algorithm of Reference [16] starts by randomly choosing the terminal precoders and the relay forwarding matrices satisfying the transmission power constraints at the source terminals and the relay nodes. In each iteration, the terminal precoders, the relay forwarding matrices, and the decoders are alternatingly updated in Reference [16] through solving convex subproblems: first, with given precoders and relay forwarding matrices, the optimal decoders are obtained in closed-form by solving an unconstrained convex problem; second, with fixed precoders and decoders, the relaying matrix of all the relays are updated in closed-form one-by-one by freezing the relaying matrices of the other relays; finally, given the decoders and relaying matrices, the precoders are updated by solving a convex quadratically constrained quadratic programming problem. A different iterative optimization procedure is proposed in Reference [17], based on the matrix conjugate gradient algorithm, which is shown to converge faster  
Figure 1: Model of the considered two-way relaying multiple-input multiple-output (MIMO) 5G network. than conventional gradient descent methods. Finally, some recent papers [18; 19] propose architectures for two-way relaying based on relay/antenna selection strategies. In this paper, we propose an optimization algorithm for two-way AF MIMO relaying 5G networks, where terminal precoders/decoders and relay forwarding matrices are jointly derived under power constraints on the transmitted/received power at the terminals. Rather than attempting to solve it iteratively, we derive a relaxed version of the original minimum sum-MSE nonconvex optimization, which allows one to decompose it in two separate problems that admit a closed-form, albeit suboptimal, solution.