# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 3 The Proposed Closed-Form Design
: _Assume that: (a4) \(\mathbf{P}_{i}\in\mathbb{C}^{N_{T,i}\times N_{\mathbb{S},i}}\) is full-column rank, that is, \(\text{rank}(\mathbf{P}_{i})=N_{\mathbb{S},i}\leq N_{T,i}\), \(i\in\{1,2\}\); (a5) \(\mathbf{B}_{\underline{i}}\mathbf{H}_{i}\in\mathbb{C}^{N_{T,i}\times N_{T,i}}\) is full-column rank, that is, \(\text{rank}(\mathbf{B}_{\underline{i}}\mathbf{H}_{i})=N_{T,i}\leq N_{T, \underline{i}}\), for \(i\in\{1,2\}\). Moreover, let \(\mathbf{H}_{i}=\mathbf{U}_{h,i}\mathbf{\Lambda}_{h,i}\mathbf{V}_{h,i}^{H}\) denote the singular value decomposition (SVD) of \(\mathbf{H}_{i}\), where \(\mathbf{U}_{h,i}\in\mathbb{C}^{N_{\mathbb{C}}N_{\mathbb{R}}\times N_{\mathbb{ C}}N_{\mathbb{R}}}\) and \(\mathbf{V}_{h,i}\in\mathbb{C}^{N_{T,i}\times N_{T,i}}\) are the unitary matrices of left/right singular vectors, and \(\mathbf{\Lambda}_{h,i}\in\mathbb{C}^{N_{\mathbb{C}}N_{\mathbb{R}}\times N_{T, i}}\) is the rectangular diagonal matrix of the corresponding singular values arranged in increasing order. Then, the solution of (14) has the following general form:_  
\[\mathbf{P}_{i} =\mathbf{V}_{h,i,\text{right}}\;\mathbf{\Omega}_{i} \tag{15}\] \[\mathbf{B}_{\underline{i}} =\mathbf{Q}_{\underline{i}}\,\mathbf{\Lambda}_{\underline{i}} \mathbf{U}_{h,i,\text{right}}^{H}, \tag{16}\]  
_where \(\mathbf{V}_{h,i,\text{right}}\) contains the \(N_{\mathbb{S},i}\) rightmost columns of \(\mathbf{V}_{h,i}\), \(\mathbf{U}_{h,i,\text{right}}\) contains the \(N_{T,i}\) rightmost columns of \(\mathbf{U}_{h,i}\), the diagonal matrices \(\mathbf{\Omega}_{i}\in\mathbb{R}^{N_{\mathbb{S},i}\times N_{\mathbb{S},i}}\) and \(\mathbf{\Delta}_{\underline{i}}\in\mathbb{R}^{N_{T,i}\times N_{T,i}}\) will be specified soon after, and \(\mathbf{Q}_{\underline{i}}\in\mathbb{C}^{N_{T,\underline{i}}\times N_{T,i}}\) is an arbitrary semi-unitary matrix, that is, \(\mathbf{Q}_{\underline{i}}^{H}\mathbf{Q}_{\underline{i}}=\mathbf{I}_{N_{T,i}}\)._  
Proof. : See Appendix A. **Remark 1**. : _(a4) implies that \(N_{\mathbb{S},i}\leq N_{T,i}\), \(i\in\{1,2\}\)._  
**Remark 2**. : _(a5) implies that \(\mathbf{H}_{i}\) is full-column rank too, that is, \(\text{rank}(\mathbf{H}_{i})=N_{T,i}\) and \(N_{T,1}=N_{T,2}\). Hence, in the following we set \(N_{T}\triangleq N_{T,1}=N_{T,2}\)._  
Under (a4) and (a5), the dual-hop channel matrices \(\{\mathbf{C}_{i,\underline{i}}=\mathbf{B}_{i}\mathbf{H}_{\underline{i}} \mathbf{P}_{\underline{i}}\}_{i=1}^{2}\) are full-column rank, that is, \(\text{rank}(\mathbf{C}_{i,\underline{i}})=N_{\mathbb{S},\underline{i}}\leq N_ {\mathbb{T},\underline{i}}\), for \(i=1,2\): this ensures perfect recovery of the source symbol vectors \(\{\mathbf{s}_{i}\}_{i=1}^{2}\) at the terminals in the absence of noise by means of linear equalizers. Although Theorem 1 holds for any value of \(N_{\mathbb{S},1}\) and \(N_{\mathbb{S},2}\), we will assume herein that \(N_{\mathbb{S},1}=N_{\mathbb{S},2}=N_{\mathbb{T}}\), which allows the terminals to transmit as many symbols as possible with an acceptable performance in practice. Theorem 1 allows one to rewrite the optimization problem (14) in a simpler scalar form:  
\[\min_{ \begin{subarray}{c}\{z_{1,\ell},w_{2,\ell}\}_{\ell=1},\\ \{z_{2,\ell},w_{1,\ell}\}_{\ell=1}^{N_{\mathrm{T}}}\\ \end{subarray} }\sum_{i=1}^{2}\sum_{\ell=1}^{N_{\mathrm{T}}}\frac{1}{1+\sigma_{ n,i}^{-2}\,\lambda_{\ell}^{2}(\mathbf{H}_{i})z_{i,\ell}\,w_{i,\ell}} \tag{17}\] \[\text{s.to}\quad \begin{cases}\sum_{\ell=1}^{N_{\mathrm{T}}}z_{i, \ell}\leq\mathcal{P}_{\mathrm{T},i}\\ \sum_{\ell=1}^{N_{\mathrm{T}}}w_{i,\ell}\leq\widetilde{\mathcal{P}}_{\mathrm{ R},i}\\ w_{i,\ell},z_{i,\ell}>0\qquad\forall\ell\in\{1,2,\ldots,N_{\mathrm{S},i}\}\end{cases} \]  
with \(z_{i,\ell}\) and \(w_{i,\ell}\) representing the \(\ell\)th squared diagonal entry of \(\mathbf{\Omega}_{i}\) and \(\mathbf{\Delta}_{i,\ell}\) respectively, whereas \(\lambda_{\ell}(\mathbf{H}_{i})\) denotes the \(\ell\)th nonzero singular value of \(\mathbf{H}_{i}\), for \(\ell\in\{1,2,\ldots,N_{\mathrm{T}}\}\).