# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design  
With reference to model (4), the problem at hand is to find optimal values of \(\{\mathbf{P}_{i}\}_{i=1}^{2}\), \(\mathbf{F}\), and \(\{\mathbf{D}_{i}\}_{i=1}^{2}\) for recovering \(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\) according to a certain cost function and subject to suitable power constraints at the terminals and relays. A common performance measure of the accuracy in recovering the symbol vector \(\mathbf{s}_{i}\) at terminal \(i\) is the mean-square value of the error \(\mathbf{e}_{i}\triangleq\hat{\mathbf{s}}_{i}-\mathbf{s}_{i}\): \(\mathrm{MSE}_{i}\triangleq\mathbb{E}[\|\mathbf{e}_{i}\|^{2}]=\mathrm{tr}( \mathbf{K}_{\mathbf{e}_{i}\mathbf{e}_{i}})\), where \(\mathbf{K}_{\mathbf{e}_{i}\mathbf{e}_{i}}\triangleq\mathbb{E}[\mathbf{e}_{i} \mathbf{e}_{i}^{\mathrm{H}}]\) is the error covariance matrix, which depends on \((\mathbf{P}_{i},\mathbf{F},\mathbf{D}_{i})\). As a global cost function for the overall two-way transmission, we consider as in References [15, 16, 17, 18] the _sum-MSE_, defined as \(\mathrm{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F},\{\mathbf{D}_{i}\}_{i=1 }^{2})=\mathrm{MSE}_{1}+\mathrm{MSE}_{2}\). It is well-known that, for fixed values of \(\{\mathbf{P}_{i}\}_{i=1}^{2}\) and \(\mathbf{F}\), the matrices \(\{\mathbf{D}_{i}\}_{i=1}^{2}\) minimizing the sum-MSE are the Wiener filters  
\[\mathbf{D}_{i,\mathrm{mmse}}=\mathbf{C}_{i,i}^{\mathrm{H}}(\mathbf{C}_{i,i}^{ \mathrm{H}}\mathbf{C}_{i,i}^{\mathrm{H}}+\mathbf{K}_{\mathbf{v}_{i}\mathbf{v}_{ i}})^{-1} \tag{6}\]  
for \(i\in\{1,2\}\), thus yielding  
\[\mathrm{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F}) \triangleq\mathrm{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F},\{ \mathbf{D}_{i,\mathrm{mmse}}\}_{i=1}^{2})\] \[=\sum_{i=1}^{2}\mathrm{tr}[(\mathbf{I}_{\mathrm{N}_{\mathrm{S}_{i} }}+\mathbf{C}_{i,i}^{\mathrm{H}}\mathbf{K}_{\mathbf{v}_{i}^{\mathrm{H}}}^{-1} \mathbf{C}_{i,i})^{-1}]\,. \tag{7}\]  
It is noteworthy that the variables \(\mathbf{P}_{1}\), \(\mathbf{P}_{2}\), and \(\mathbf{F}\) are coupled in (7) and, hence, the two terms in (7) cannot be minimized independently. Herein, we relax the original problem so as to _separate_ the minimization of the two terms in (7). As a first step, we observe that minimizing (7) is complicated by the presence of \(\mathbf{K}_{\
u_{\mathbf{Y}}i^{\prime}}^{-1}\) which depends non-trivially on \(\mathbf{F}\). For such a reason, we consider instead minimization of the following high signal-to-noise ratio (SNR) approximation:  
\[\text{MSE}(\{\mathbf{P}_{i}\}_{i=1}^{2},\mathbf{F})\approx\sum_{i=1}^{2}\text{ tr}[(\mathbf{I}_{N_{\text{S},i}}+\sigma_{n,i}^{-2}\,\mathbf{C}_{i;i}^{\text{H}} \mathbf{C}_{i;i})^{-1}], \tag{8}\]  
which turns out to be accurate when \(\sigma_{w}^{2}\ll\min(\sigma_{n,i}^{2},\mu_{\min})\), where \(\mu_{\min}\) is the smallest eigenvalue of \(\mathbf{G}_{i}\,\mathbf{F}\,\mathbf{F}^{\text{H}}\,\mathbf{G}_{i}^{\text{H}}\). Suitable constraints must be set to avoid trivial solutions in minimizing (8). It is customary to impose power constraints to limit the average transmit power at the terminals:  
\[\mathbb{E}[\|\mathbf{P}_{i}\mathbf{s}_{i}\|^{2}]=\text{tr}(\mathbf{P}_{i}\, \mathbf{P}_{i}^{\text{H}})\leq\mathcal{P}_{\text{T},i}>0 \tag{9}\]  
for \(i\in\{1,2\}\). In order to limit \(\mathbf{F}\), we impose a constraint on the average power received at the terminals in the second time slot, that is, with reference to (2), we attempt to limit, for \(i\in\{1,2\}\), the following quantities:  
\[\mathbf{E}[\|\mathbf{G}_{i}\mathbf{F}\,\mathbf{y}\|^{2}]=\text{tr}(\mathbf{G} _{i}\mathbf{F}\mathbf{K}_{\mathbf{y}\mathbf{y}}\mathbf{F}^{\text{H}}\mathbf{G }_{i}^{\text{H}}), \tag{10}\]  
where \(\mathbf{K}_{\mathbf{y}\mathbf{y}}\triangleq\mathbb{E}[\mathbf{y}\mathbf{y}^{ \text{H}}]=\sum_{i=1}^{2}\mathbf{H}_{i}\mathbf{P}_{i}\mathbf{P}_{i}^{\text{H} }\mathbf{H}_{i}^{\text{H}}+\sigma_{w}^{2}\mathbf{I}_{N_{\text{C}}N_{\text{R}}}\) is the covariance matrix of \(\mathbf{y}\).