# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
Similarly to (14), problem (17) can be decomposed into two separate problems involving disjoint subsets of variables. It can be shown, with straightforward manipulations, that the objective function in (17) is convex if and only if  
\[z_{i,\ell}\,w_{i,\ell}\geq\frac{\sigma_{n,i}^{2}}{3\lambda_{\ell}^{2}(\mathbf{ H}_{i})} \tag{18}\]  
\(\forall\ell\in\{1,2,\ldots,N_{\mathrm{S},i}\}\), with \(i\in\{1,2\}\). It is also seen that, based on (a2), one has \(\lambda_{\min}(\mathbf{H}_{i})\gg 1\) in the large \(N_{\mathrm{C}}N_{\mathrm{R}}\) limit, with \(i\in\{1,2\}\). Thus, condition (18) boils down to \(z_{i,\ell}\), \(w_{i,\ell}>0\), for all \(\ell\in\{1,2,\ldots,N_{\mathrm{S},i}\}\), with \(i\in\{1,2\}\), which is already included in the constraints of (17). Therefore, convex programming can be used to find a global minimum of (17). To calculate the relaying matrices, let us partition solution (16) as \(\mathbf{B}_{i}=\left[\mathbf{B}_{i\downarrow 1},\mathbf{B}_{i\downarrow 2}, \cdots,\mathbf{B}_{i\downarrow N_{\mathrm{C}}}\right]\), with \(\mathbf{B}_{i\downarrow k}\in\mathbb{C}^{N_{\mathrm{T}}\times N_{\mathrm{R}}}\), \(i\in\{1,2\}\). Defining \(\widetilde{\mathbf{G}}_{k}\triangleq[\mathbf{G}_{1,k}^{\mathrm{T}},\mathbf{G}_ {2,k}^{\mathrm{T}}]^{\mathrm{T}}\in\mathbb{C}^{2N_{\mathrm{T}}\times N_{ \mathrm{R}}}\) and \(\widetilde{\mathbf{B}}_{k}\triangleq[\mathbf{B}_{1,k}^{\mathrm{T}},\mathbf{B}_ {2,k}^{\mathrm{T}}]^{\mathrm{T}}\in\mathbb{C}^{2N_{\mathrm{T}}\times N_{ \mathrm{R}}}\), and assuming that \(\widetilde{\mathbf{G}}_{k}\) is full-row rank, that is, \(\text{rank}(\widetilde{\mathbf{G}}_{k})=2N_{\mathrm{T}}\leq N_{\mathrm{R}}\), with \(k\in\{1,2,\ldots,N_{\mathrm{C}}\}\), the \(k\)th relay can construct its own relaying matrix by solving the matrix equation \(\widetilde{\mathbf{G}}_{k}\mathbf{F}_{k}=\widetilde{\mathbf{B}}_{k}\), whose minimum-norm solution is given by  
\[\mathbf{F}_{k}=\widetilde{\mathbf{G}}_{k}^{\mathrm{T}}\widetilde{\mathbf{B}}_ {k}, \tag{19}\]  
where the superscript \(\dagger\) denotes the Moore-Penrose inverse. ```
Input quantities: \(\{\mathbf{H}_{i},\,\mathbf{G}_{i},\,\sigma_{n,i}^{2},\mathcal{P}_{\mathrm{T},i },\,\widetilde{\mathcal{P}}_{\mathrm{R},i}\}_{i=1}^{2}\) Output quantities: \(\{\mathbf{P}_{i},\mathbf{D}_{i,\text{mmse}}\}_{i=1}^{2}\) and \(\{\mathbf{F}_{k}\}_{k=1}^{N_{\mathrm{C}}}\)
1. Choose arbitrary \(\{\mathbf{Q}_{i}\}_{i=1}^{2}\) such that \(\mathbf{Q}_{i}^{\mathrm{



efficiently carried out using standard techniques, such as the interior-point method. We observe that the worst-case theoretical complexity of the interior-point method is proportional to \(\sqrt{N_{\mathrm{T}}}\). Hence, for a realistic setting of the system parameters, the computational complexity of the proposed algorithm, is dominated by the SVD computation (in step 2), which is of order \(\mathcal{O}(N_{\mathrm{C}}\,N_{\mathrm{R}}\,N_{\mathrm{T}}^{2})\) and, thus, it _linearly_ grows with the number \(N_{\mathrm{C}}\) of relays. It is noteworthy that, even though the alternating algorithm proposed in Reference [16] allows to solve a nonconvex problem by solving convex subproblems, it is more complex than calculating the solution of (17); moreover, it requires proper initialization to monotonically converge to (at least) a local optimum. ## 4 Simulation Results

In this section, to assess the performance of the considered design, we present the results of Monte Carlo computer simulations, aimed at evaluating the average (with respect to channel realizations) bit-error-rate (BER) of the proposed cooperative two-way MIMO system. We consider a network encompassing two terminals equipped with \(N_{\mathrm{T}}=2\) antennas, and transmitting QPSK symbols with \(N_{\mathrm{S},1}=N_{\mathrm{S},2}=2\). The \(N_{\mathrm{C}}\) relays are equipped with \(N_{\mathrm{R}}=4\) antennas.