# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
It is noteworthy that (10) is typically limited in those scenarios where a target performance has to be achieved and per-node fairness is not of concern [10; 12]. Moreover, the average power received at the terminals is an important metric measuring the human exposure to radio frequency (RF) fields generated by transmitters operating at mmWave frequencies [21] and, with respect to traditional per-relay transmit power constraints, it is more easily related to regulatory specifications [22]. To simplify (10), we exploit the following chain of inequalities:  
\[\text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{K}_{\mathbf{y}\mathbf{ y}}\mathbf{F}^{\text{H}}\mathbf{G}_{i}^{\text{H}})\leq\text{tr}(\mathbf{G}_{i} \mathbf{F}\mathbf{F}^{\text{H}}\mathbf{G}_{i}^{\text{H}})\text{tr}(\mathbf{K} _{\mathbf{y}\mathbf{y}})\] \[\leq\text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{F}^{\text{H}} \mathbf{G}_{i}^{\text{H}})\left[\sum_{i=1}^{2}\text{tr}(\mathbf{H}_{i}\mathbf{ H}_{i}^{\text{H}})\mathcal{P}_{\text{T},i}+\sigma_{w}^{2}N_{\text{C}}N_{ \text{R}}\right]\] \[\lesssim\text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{F}^{\text{H}} \mathbf{G}_{i}^{\text{H}})N_{\text{C}}N_{\text{R}}\left(\sum_{i=1}^{2}N_{ \text{T},i}\,\mathcal{P}_{\text{T},i}+\sigma_{w}^{2}\right), \tag{11}\]  
where the last approximate inequality holds noting that, for fixed values of \(N_{\text{T},i}\), by the law of large numbers one has \(\mathbf{H}_{i}^{\text{H}}\,\mathbf{H}_{i}/(N_{\text{C}}N_{\text{R}})\to \mathbf{I}_{N_{\text{T},i}}\) almost surely as \(N_{\text{C}}N_{\text{R}}\) gets large. Therefore, if we impose \(\text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{F}^{\text{H}}\mathbf{G}_{i}^{\text{ H}})\leq\widehat{\mathcal{P}}_{\text{R},i}>0\), we get the upper bound:  
\[\text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{K}_{\mathbf{y}\mathbf{ y}}\mathbf{F}^{\text{H}}\mathbf{G}_{i}^{\text{H}})\lesssim\underbrace{\widehat{ \mathcal{P}}_{\text{R},i}N_{\text{C}}N_{\text{R}}\left(\sum_{i=1}^{2}N_{ \text{T},i}\mathcal{P}_{\text{T},i}+\sigma_{w}^{2}\right)}_{\triangleq\mathcal{P }_{\text{R},i}}. \tag{12}\]  
Such a choice allows one to considerably simplify the system design. In summary, the optimization problem to be solved can be expressed as  
\[\mathrm{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F}) \triangleq\mathrm{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F},\{ \mathbf{D}_{i,\mathrm{mmse}}\}_{i=1}^{2})\]0 \[\text{s.to} \begin{cases}\text{tr}(\mathbf{P}_{i}\,\mathbf{P}_{i}^{\text{H}}) \leq\mathcal{P}_{\text{T},i}\\ \text{tr}(\mathbf{G}_{i}\mathbf{F}\mathbf{F}^{\text{H}}\mathbf{G}_{i}^{\text{H}} )\leq\widehat{\mathcal{P}}_{\text{R},i}\end{cases} i\in\{1,2\}\;. \tag{13}\]In order to find a closed-form solution of (13), we introduce the matrix \(\mathbf{B}_{i}\triangleq\mathbf{G}_{i}\mathbf{F}\in\mathbb{C}^{N_{T,i}\times N_{ \mathbb{C}}N_{\mathbb{R}}}\) with \(i\in\{1,2\}\), and rewrite (13) as follows  
\[\min_{\{\mathbf{P}_{i}\}_{i=1}^{2}\{\mathbf{B}_{i}\}_{i=1}^{2}} \sum_{i=1}^{2} \operatorname{tr}[(\mathbf{I}_{N_{\mathbb{S},i}}+\sigma_{n,\underline {i}}^{-2}\,\mathbf{P}_{i}^{H}\mathbf{H}_{i}^{H}\mathbf{B}_{\underline{i}}^{H} \mathbf{B}_{\underline{i}}\mathbf{H}_{i}\mathbf{P}_{i})^{-1}]\] \[\text{s.to} \begin{cases}\operatorname{tr}(\mathbf{P}_{i}\,\mathbf{P}_{i}^{H} )\leq\mathcal{P}_{\mathbb{T},i}\\ \operatorname{tr}(\mathbf{B}_{\underline{i}}\mathbf{B}_{\underline{i}}^{H}) \leq\widehat{\mathcal{P}}_{\mathbb{R},\underline{i}}\end{cases}  i\in\{1,2\}\;. \tag{14}\]  
Remarkably, the cost function is the sum of two terms: the former one depends only on the variables \(\{\mathbf{P}_{1},\mathbf{B}_{2}\}\), whereas the latter one involves only the variables \(\{\mathbf{P}_{2},\mathbf{B}_{1}\}\). Therefore, (14) can be decomposed in two problems involving \(\{\mathbf{P}_{1},\mathbf{B}_{2}\}\) and \(\{\mathbf{P}_{2},\mathbf{B}_{1}\}\) separately, which can be solved in parallel in a closed-form manner. Indeed, capitalizing on such a decomposition, the solution of (14) can be characterized by the following theorem. **Theorem 1**.