# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
In this paper, we assumed the availability of full-CSI at both terminals and the relays. In this respect, an interesting research subject consists of considering the use of partial CSI to extend network lifetime and reduce the complexity burden. Moreover, since channel estimation errors occur in practical situations, an additional research issue is to develop robust optimization designs. Conceptualization, D.D., G.G., I.I. and F.V. ; methodology, D.D., G.G., I.I. and F.V. ; writing--original draft preparation, G.G. and F.V. ; writing--review and editing, I.I. and F.V. ; supervision, F.V. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. ## Appendix A Proof of Theorem 1

We focus on the optimization (14) with indexes \(i=1\) and \(i=2\), that is, we consider

\[\min_{\mathbf{P}_{1},\mathbf{B}_{2}} \text{tr}[\left(\mathbf{I}_{\text{N}_{\text{S},1}}+\sigma_{n,2}^{ -2}\,\mathbf{P}_{1}^{\text{H}}\mathbf{H}_{1}^{\text{H}}\mathbf{B}_{2}^{\text{ H}}\mathbf{B}_{2}\mathbf{H}_{1}\mathbf{P}_{1}\right)^{-1}]\] \[\text{s.to}\quad \begin{cases}\text{tr}(\mathbf{P}_{1}\,\mathbf{P }_{1}^{\text{H}})\leq\mathcal{P}_{\text{T},1}\\ \text{tr}(\mathbf{B}_{2}\mathbf{B}_{2}^{\text{H}})\leq\widetilde{\mathcal{P}} _{\text{R},2}\end{cases} . \tag{15}\]

Figure 4: BER versus \(E_{b}/N_{0}\) of the proposed design versus the iterative method of Reference [16] (\(N_{\text{C}}=4\)). We note that under (a4) and (a5), one has \(\text{rank}(\mathbf{B}_{2}\mathbf{H}_{1}\mathbf{P}_{1})=N_{\text{S,1}}\leq N_{ \text{T,1}}\). Let \(\mathbf{U}_{\text{a}}\mathbf{\Lambda}_{\text{a}}\mathbf{U}_{\text{a}}^{\text{H}}\) be the eigenvalue decomposition (EVD) of \(\mathbf{A}\triangleq\mathbf{H}_{1}^{\text{H}}\mathbf{B}_{2}^{\text{H}}\mathbf{ B}_{2}\mathbf{H}_{1}\in\mathbb{C}^{N_{\text{T,1}}\times N_{\text{T,1}}}\), where the diagonal matrix \(\mathbf{\Lambda}_{\text{a}}\in\mathbb{R}^{N_{\text{T,1}}\times N_{\text{T,1}}}\) and the unitary matrix \(\mathbf{U}_{\text{a}}\in\mathbb{C}^{N_{\text{T,1}}\times N_{\text{T,1}}}\) collect the eigenvalues, arranged in increasing order, and the eigenvectors of \(\mathbf{A}\), respectively. The objective function in (A1) is a Schur-concave function of the diagonal elements of \((\mathbf{I}_{N_{\text{S,1}}}+\sigma_{n,2}^{-2}\mathbf{P}_{1}^{\text{H}}\mathbf{ A}\mathbf{P}_{1})^{-1}\). In this case, it can be shown [23] that there is an optimal \(\mathbf{P}_{1}\) such that \(\mathbf{P}_{1}^{\text{H}}\mathbf{A}\mathbf{P}_{1}\) is diagonal, whose diagonal elements are assumed to be arranged in increasing order, and such an optimal matrix, which also minimizes \(\text{tr}(\mathbf{P}_{1}\mathbf{P}_{1}^{\text{H}})\), is given by

\[\mathbf{P}_{1}=\mathbf{U}_{\text{a,right}}\mathbf{\Omega}_{1},\] (A2)

where \(\mathbf{U}_{\text{a,right}}\in\mathbb{C}^{N_{\text{T,1}}\times N_{\text{S,1}}}\) contains the \(N_{\text{S,1}}\leq N_{\text{T,1}}\) rightmost columns from \(\mathbf{U}_{\text{a}}\), and \(\mathbf{\Omega}_{1}\in\mathbb{C}^{N_{\text{S,1}}\times N_{\text{S,1}}}\) is a diagonal matrix. Let \(\mathbf{Q}_{2}\in\mathbb{C}^{N_{\text{T,2}}\times N_{\text{T,1}}}\) be an arbitrary semi-unitary matrix, that is, \(\mathbf{Q}_{2}^{\text{H}}\mathbf{Q}_{2}=\mathbf{I}_{N_{\text{T,1}}}\), it follows from the EVD of the matrix \(\mathbf{A}\) that \(\mathbf{B}_{2}\mathbf{H}_{1}=\mathbf{Q}_{2}\mathbf{\Lambda}_{\text{a}}^{1/2} \mathbf{U}_{\text{a}}^{\text{H}}\).