# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
Noting that \(\text{rank}(\mathbf{H}_{1})=N_{\text{T,1}}\), by substituting the ordered SVD of \(\mathbf{H}_{1}=\mathbf{U}_{\text{h,1}}\mathbf{\Lambda}_{\text{h,1}}\mathbf{U} _{\text{h,1}}^{\text{H}}\) in this equation, after some algebraic manipulations, one has that the minimum-norm solution [24] of the matrix equation \(\mathbf{B}_{2}\mathbf{U}_{\text{h,1}}\mathbf{\Lambda}_{\text{h,1}}=\mathbf{Q} _{2}\mathbf{\Lambda}_{\text{a}}^{1/2}\mathbf{U}_{\text{a}}^{\text{H}}\mathbf{V }_{\text{h,1}}\) is

\[\mathbf{B}_{2}=\mathbf{Q}_{2}\mathbf{\Lambda}_{\text{a}}^{1/2}\widetilde{ \mathbf{U}}_{\text{a}}\mathbf{\Lambda}_{\text{h,1,right}}^{-1}\mathbf{U}_{ \text{h,1,right}},\] (A3)

where \(\mathbf{U}_{\text{h,1,right}}\) collects the \(N_{\text{T,1}}\) rightmost columns of \(\mathbf{U}_{\text{h,1}}\), whereas \(\widetilde{\mathbf{U}}_{\text{a}}\triangleq\mathbf{U}_{\text{a}}^{\text{H}} \mathbf{V}_{\text{h,1}}\in\mathbb{C}^{N_{\text{T,1}}\times N_{\text{T,1}}}\) and the diagonal a \(\mathbf{\Lambda}_{\text{h,1,right}}\in\mathbb{R}^{N_{\text{T,1}}\times N_{ \text{T,1}}}\) gathers the \(N_{\text{T,1}}\) nonzero singular values of \(\mathbf{H}_{1}\) in increasing order. The aim is now to further determine (A3) by properly choosing \(\widetilde{\mathbf{U}}_{\text{a}}\) such that \(\text{tr}(\mathbf{B}_{2}\mathbf{B}_{2}^{\text{H}})=\text{tr}[(\widetilde{ \mathbf{U}}_{\text{a}}\mathbf{\Lambda}_{\text{h,1,right}}^{-2}\widetilde{ \mathbf{U}}_{\text{a}})\mathbf{\Lambda}_{\text{a}}]\) has the smallest value. It is readily seen that \(\text{tr}(\mathbf{B}_{2}\mathbf{B}_{2}^{\text{H}})\) is invariant to the choice of \(\mathbf{Q}_{2}\). By observing that \(\widetilde{\mathbf{U}}_{\text{a}}^{\text{H}}\widetilde{\mathbf{U}}_{\text{a} }=\mathbf{I}_{N_{\text{T,1}}}\) and using a known trace inequality, one has

\[\text{tr}[(\widetilde{\mathbf{U}}_{\text{a}}\mathbf{\Lambda}_{\text{h,1,right} }^{-2}\widetilde{\mathbf{U}}_{\text{a}})\mathbf{\Lambda}_{\text{a}}]\geq\sum_ {\ell=1}^{N_{\text{T,1}}}\lambda_{\text{h,1,\ell}}^{-2}\lambda_{\text{a,\ell}},\] (A4)

where \(\lambda_{\text{h,1,\ell}}\) and \(\lambda_{\text{a,\ell}}\) denote the \(\ell\)th diagonal entry of \(\mathbf{\Lambda}_{\text{h,1,right}}\) and \(\mathbf{\Lambda}_{\text{a}}\), respectively. The equality in (A4) holds when

\[\widetilde{\mathbf{U}}_{\text{a}}=\mathbf{U}_{\text{a}}^{\text{H}}\mathbf{V}_{ \text{h,1}}=\mathbf{I}_{N_{\text{T,1}}}.\] (A5)

Substituting (A5) in (A3), after some algebraic manipulations, one obtains \(\mathbf{B}_{2}=\mathbf{Q}_{2}\,\mathbf{\Lambda}_{2}\mathbf{U}_{\text{h,1,right}}^ {\text{H}}\), with \(\mathbf{\Delta}_{2}\triangleq\mathbf{\Lambda}_{\text{a}}^{1/2}\mathbf{\Lambda }_{\text{h,1,right}}^{-1}\in\mathbb{R}^{N_{\text{T,1}}\times N_{\text{T,1}}}\). Solution (15) comes from substituting in (A2) the minimum-norm solution [24] of (A5), that is, \(\mathbf{U}_{\text{a}}=\mathbf{V}_{\text{h,1,right}}\). ## References

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