# Separable MSE-Based Design of Two-Way Multiple-Relay Cooperative MIMO 5G Networks
## 2 Network Model and Basic Assumptions
Moreover, by taking into account (1), the vector \(\mathbf{r}_{i}\) can also be directly written in terms of \(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\) as  
\[\mathbf{r}_{i}=\sum_{j=1}^{2}\mathbf{C}_{i,j}\,\mathbf{s}_{j}+\mathbf{v}_{i}, \tag{3}\]  
where \(\mathbf{C}_{i,j}\triangleq\mathbf{G}_{i}\,\mathbf{F}\,\mathbf{H}_{j}\,\mathbf{ P}_{j}\in\mathbb{C}^{N_{\text{T}j}\times N_{\text{S}j}}\) is the _dual-hop_ matrix from terminal \(j\) to \(i\), for \(i,j\in\{1,2\}\), and \(\text{ vector }\mathbf{v}_{i}\triangleq\mathbf{G}_{i}\,\mathbf{F}\,\mathbf{w}+\mathbf{n}_{i} \in\mathbb{C}^{N_{\text{T}j}}\) is the overall noise. We assume customarily [14; 18] that each terminal can estimate and subtract the self-interference deriving from its own symbols. To do this, terminal \(i\) has to first acquire the matrix \(\mathbf{C}_{i,i}\), which can be obtained by resorting to standard training-based identification methods. Specifically, each data transmission can be preceded by a training period, wherein the two terminals transmit orthogonal pilot sequences to the relays. In this case, by redefining \(\mathbf{r}_{i}\) with a slight abuse of notation as \(\mathbf{r}_{i}-\mathbf{C}_{i,i}\,\mathbf{s}_{i}\), for \(i\in\{1,2\}\), we write explicitly  
\[\mathbf{r}_{i}=\mathbf{C}_{i,i}\,\mathbf{s}_{i}+\mathbf{v}_{i}=\mathbf{G}_{i} \mathbf{F}\mathbf{H}_{i}\,\mathbf{P}_{i}\,\mathbf{s}_{i}+\mathbf{v}_{i}, \tag{4}\]  
where \(i=2\) when \(i=1\), whereas \(i=1\) when \(i=2\). At terminal \(i\), vector \(\mathbf{r}_{i}\) is subject to linear equalization through matrix \(\mathbf{D}_{i}\in\mathbb{C}^{N_{\mathrm{S}_{i}}\times N_{\mathrm{T}_{i}}}\), thus yielding a soft estimate \(\hat{\mathbf{s}}_{i}\triangleq\mathbf{D}_{i}\,\mathbf{r}_{i}\) of the symbols \(\mathbf{s}_{i}\) transmitted by terminal \(i\), whose entries are then subject to minimum-distance hard decision. In the sequel, we consider the common assumptions: (a1) \(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\) are mutually independent zero-mean circularly symmetric complex (ZMCSC) random vectors, with \(\mathbb{E}[\mathbf{s}_{i}\,\mathbf{s}_{i}^{\mathrm{H}}]=\mathbf{I}_{\mathrm{N }_{\mathrm{S}_{i}}}\), for \(i\in\{1,2\}\); (a2) the entries of \(\mathbf{H}_{i}\) and \(\mathbf{G}_{i}\) are independent identically distributed ZMCSC Gaussian unit-variance random variables, for \(i\in\{1,2\}\); (a3) the noise vectors \(\mathbf{w}\), \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) are mutually independent ZMCSC Gaussian random vectors, statistically independent of \(\{\mathbf{s}_{i},\mathbf{H}_{i},\mathbf{G}_{i}\}_{i=1}^{2}\), with \(\mathbb{E}[\mathbf{w}\mathbf{w}^{\mathrm{H}}]=\sigma_{w}^{2}\,\mathbf{I}_{ \mathrm{N}_{\mathrm{C}}N_{\mathrm{R}}}\) and \(\mathbb{E}[\mathbf{n}_{i}\mathbf{n}_{i}^{\mathrm{H}}]=\sigma_{n,i}^{2}\, \mathbf{I}_{\mathrm{N}_{\mathrm{T}_{i}}}\), for \(i\in\{1,2\}\). Full channel-state information (CSI) is assumed to be available at both the terminals and the relays. Particularly, we assume that: (i) \(\{\mathbf{H}_{i}\}_{i=1}^{2}\) are known at the terminals and at the relays; (ii) the \(k\)th second-hop channel matrices \(\mathbf{G}_{1,k}\) and \(\mathbf{G}_{2,k}\) are known only to the \(k\)th relay, for \(k\in\{1,2,\ldots,N_{\mathrm{C}}\}\); (iii) the dual-hop channel matrix \(\{\mathbf{C}_{i,i}\}\) and the covariance matrix  
\[\mathbf{K}_{\mathbf{v}_{i}\mathbf{v}_{i}}\triangleq\mathbb{E}[\mathbf{v}_{i} \,\mathbf{v}_{i}^{\mathrm{H}}]=\sigma_{w}^{2}\,\mathbf{G}_{i}\,\mathbf{F} \mathbf{F}^{\mathrm{H}}\,\mathbf{G}_{i}^{\mathrm{H}}+\sigma_{n,i}^{2}\, \mathbf{I}_{\mathrm{N}_{\mathrm{T}_{i}}} \tag{5}\]  
of \(\mathbf{v}_{i}\) are known at the \(i\)th terminal, for \(i\in\{1,2\}\). It should be noted that, hereinafter, all the ensemble averages are evaluated for fixed values of the first- and second-hop channel matrices.