# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 2 Network Model and Basic Assumptions  
We consider the two-way MIMO 5G network configuration of Figure 1, where bidirectional communication between two terminals, equipped with \(N_{\text{T,1}}\) and \(N_{\text{T,2}}\) antennas, respectively, is assisted by \(N_{\text{C}}\) half-duplex relays, each equipped with \(N_{\text{R}}\) antennas. We assume that there is no direct link between the two terminals, due to high path loss values or obstructions. Even though our approach can be generalized, for simplicity, the considered physical layer is that of a single-carrier cooperative system where all the channel links are quasi static and experience flat fading. Let \(\mathbf{s}_{1}\in\mathbb{C}^{N_{\text{S,1}}}\) and \(\mathbf{s}_{2}\in\mathbb{C}^{N_{\text{S,2}}}\) denote the symbol vectors to be transmitted by terminal 1 and 2, respectively. In the first time slot, each terminal precodes its symbols with matrix \(\mathbf{P}_{i}\in\mathbb{C}^{N_{\text{T}j}\times N_{\text{S}j}}\), for \(i\in\{1,2\}\), before transmitting it to the relays, which thus receive \(\mathbf{y}_{k}=\sum_{i=1}^{2}\mathbf{H}_{i,k}\mathbf{P}_{i}\,\mathbf{s}_{i}+ \mathbf{w}_{k}\), for \(k\in\{1,2,\ldots,N_{\text{C}}\}\), where \(\mathbf{H}_{i,k}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{T}j}}\) is the _first-hop_ channel matrix (from terminal \(i\) to relay \(k\)), and \(\mathbf{w}_{k}\in\mathbb{C}^{N_{\text{R}}}\) models additive noise at \(k\)th relay. By defining \(\mathbf{y}\triangleq[\mathbf{y}_{1}^{\mathsf{T}},\mathbf{y}_{2}^{\mathsf{T}},\ldots,\mathbf{y}_{N_{\text{C}}}^{\mathsf{T}}]^{\mathsf{T}}\in\mathbb{C}^{N _{\text{C}}N_{\text{R}}}\), the overall signal received by the relays can be compactly written as  
\[\mathbf{y}=\sum_{i=1}^{2}\mathbf{H}_{i}\mathbf{P}_{i}\,\mathbf{s}_{i}+ \mathbf{w}, \tag{1}\]  
where \(\mathbf{H}_{i}\triangleq[\mathbf{H}_{i,1}^{\mathsf{T}},\mathbf{H}_{i,2}^{ \mathsf{T}}\ldots,\mathbf{H}_{i,N_{\text{C}}}^{\mathsf{T}}]^{\mathsf{T}}\in \mathbb{C}^{N_{\text{C}}N_{\text{R}}\times N_{\text{T}j}}\) gathers all first-hop channels and the vector \(\mathbf{w}\triangleq[\mathbf{w}_{1}^{\mathsf{T}},\mathbf{w}_{2}^{\mathsf{T}},\ldots,\mathbf{w}_{N_{\text{C}}}^{\mathsf{T}}]^{\mathsf{T}}\in\mathbb{C}^{N _{\text{C}}N_{\text{R}}}\) gathers all the noise samples. In the second time slot, the \(k\)th relay forwards its received signal \(\mathbf{y}_{k}\in\mathbb{C}^{N_{\text{R}}}\), by using the relaying matrix \(\mathbf{F}_{k}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{R}}}\), thus transmitting \(\mathbf{z}_{k}=\mathbf{F}_{k}\,\mathbf{y}_{k}\). The received signal at each terminal can be written, for \(i\in\{1,2\}\), as  
\[\mathbf{r}_{i}=\sum_{k=1}^{N_{\text{C}}}\mathbf{G}_{i,k}\mathbf{F}_{k}\, \mathbf{y}_{k}+\mathbf{n}_{i}=\mathbf{G}_{i}\,\mathbf{F}\,\mathbf{y}+\mathbf{ n}_{i}, \tag{2}\]  
where \(\mathbf{G}_{i,k}\in\mathbb{C}^{N_{\text{T}j}\times N_{\text{R}}}\) is the _second-hop_ channel matrix (from relay \(k\) to terminal \(i\)), and the vector \(\mathbf{n}_{i}\in\mathbb{C}^{N_{\text{T}j}}\) is additive noise at terminal \(i\). Additionally, we have defined in (2) the extended matrices \(\mathbf{G}_{i}\triangleq[\mathbf{G}_{i,1},\mathbf{G}_{i,2}\ldots,\mathbf{G}_{ i,N_{\text{C}}}]\in\mathbb{C}^{N_{\text{T}j}\times N_{\text{C}}N_{\text{R}}}\) and \(\mathbf{F}\triangleq\operatorname{diag}(\mathbf{F}_{1},\mathbf{F}_{2},\ldots, \mathbf{F}_{N_{\text{C}}})\in\mathbb{C}^{N_{\text{C}}N_{\text{R}}\times N_{ \text{C}}N_{\text{R}}}\).