# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
We also assume that \(\mathcal{P}_{\mathrm{T},1}=\mathcal{P}_{\mathrm{T},2}=\mathcal{P}_{k}= \mathcal{P}\), for all \(k\in\{1,2,\ldots,N_{\mathrm{C}}\}\), where \(\mathcal{P}_{k}\) represents the average transmitted power at the \(k\)th relay, and set \(\sigma_{w}^{2}=\sigma_{n,1}^{2}=\sigma_{n,2}^{2}=1\). Consequently, the energy per bit to noise power spectral density ratio \(E_{b}/N_{0}\) is a measure of the per-antenna link quality of both the first- and second-hop transmissions. The BER is evaluated by carrying out \(10^{3}\) independent Monte Carlo trials, with each run using independent sets of channel realizations and noise, and an independent record of \(10^{6}\) source symbols. We compare the performances of our design (labeled as \"Proposed\") to those of the iterative technique proposed in Reference [16], which has been shown [16] in its turn to outperform other iterative techniques, such as the gradient-descent technique of Reference [15]. It is worthwhile to note that both the strategies under comparison require the same amount of CSI. Furthermore, since the method of Reference [16] imposes different power constraints on the design of the relaying matrices, our solutions for \(\{\mathbf{F}_{k}\}_{k=1}^{N_{\mathrm{C}}}\) are properly scaled so as to ensure that the average power transmitted by each relay is the same for both methods. In Figures 2-4, we report the BER for different values of the number \(N_{\mathrm{C}}\in\{2,3,4\}\) of relays. Results in Figure 2 for \(N_{\mathrm{C}}=2\) show that the proposed closed-form design, based on the solution of the relaxed problem (14), exhibits performances comparable with the iterative solution of Reference [16] in the considered range of \(E_{b}/N_{0}\) values only when the latter employs more than 5 iterations. Specifically, when the method of Reference [16] employs 10 iterations, a crossover can be observed in Figure 2 between the BER curve of the proposed algorithm and that of Reference [16]. This behavior is due to the fact that the rate of convergence of Reference [16] strongly depends on the SNR. Figures 3 and 4 show that, as the number of relays increases, the proposed method clearly outperforms the method of Reference [16] even when the latter employs 10 iterations. Performance improvement of Reference [16] is negligible after 10 iterations. In a nutshell, although the alternating iterative procedure [16] attempts to solve the nonconvex original two-way constrained minimum sum-MSE problem, its convergence behaviors are affected in practice by both the operative SNR and number of relays: in the low-SNR region and/or when the number of relays is sufficiently large, convergence to a local minimizer is not guaranteed in a reasonable number of iterations for all possible initializations. This is the price to pay for swapping a difficult joint optimization with a sequence of easier problems involving subsets of the variables. On the other hand, the proposed optimization strategy gives up the idea of solving the original nonconvex problem, by resorting to suitable relaxations of both the cost function and the relaying power constraint. This allows us to jointly optimize all the variables, without using burdensome iterative algorithms. Figure 3: BER versus \(E_{b}/N_{0}\) of the proposed design versus the iterative method of Reference [16] (\(N_{\rm C}=3\)). Figure 2: Bit-error-rate (BER) versus \(E_{b}/N_{0}\) of the proposed design versus the iterative method of Reference [16] (\(N_{\rm C}=2\)). ## 5 Discussion and Directions for Future Work

We tackled the joint sum-MSE design of terminal precoders/decoders and relay forwarding matrices for two-way AF MIMO 5G systems. We showed that a relaxed version of such a problem can be separated into two simpler ones, which can be solved in parallel by admitting closed-form solutions. The proposed technique exhibits a performance gain over the iterative method of Reference [16], exhibiting a better scaling with the number of relays and a reduced computational complexity.