## 3 System Model and Basic Assumptions
### A&F MIMO Signal Model  
In this subsection, we consider the co-operative system with an A&F relaying protocol, as shown in Figure 2.  
Figure 1: Baseband eNodeB architecture. Figure 2: Two-hop A&F MIMO Relay Node Co-operative System. Let \(\mathbf{X}\in\mathbb{C}^{\mathbb{N}_{\mathrm{T}}^{\mathrm{S}}\times\mathrm{T}}\) denotes the symbols matrix to be transmitted by eNodeB. Therefore, in the first-phase, the received signal by the A&F MIMO RN can be expressed as  
\[\mathbf{Y}_{\mathrm{AF}}^{1}=\mathbf{H}^{1}\mathbf{X}+\mathbf{N}_{\mathrm{AF}}^ {1}, \tag{2}\]  
where \(\mathbf{H}^{1}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{R}}\times \mathbb{N}_{\mathrm{T}}^{\mathrm{S}}}\) is the channel frequency responses between the eNodeB and A&F RN, and \(\mathbf{N}_{\mathrm{AF}}^{1}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{R }}\times\mathrm{T}}\) denotes noise matrix with independent and identical distributed random variables \(N\sim(0,\sigma_{\mathrm{N}_{\mathrm{AF}}^{1}}^{2})\), where \(\sigma_{\mathrm{N}_{\mathrm{AF}}^{1}}^{2}\) indicates the noise variance. The signal \(\mathbf{Y}_{\mathrm{AF}}^{1}\) is received through RF receiver chains \(\mathbf{M}_{\mathrm{RR}}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathbb{N}_{\mathrm{R}}^{\mathrm{R}}}\). After that, it is precoded by matrices \(\mathbf{R}_{\mathrm{BB}}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathbb{N}_{\mathrm{R}}^{\mathrm{R}}}\) and RF transmitter chains \(\mathbf{M}_{\mathrm{RT}}\in\mathbb{C}^{\mathbb{N}_{\mathrm{T}}^{\mathrm{R}} \times\mathbb{N}_{\mathrm{R}}^{\mathrm{B}}}\). It should be noticed that the employed strategy is the A&F, therefore, \(\mathbf{R}_{\mathrm{BB}}=\mathbf{G}_{\mathrm{AF}}\), which depicts a diagonal matrix that on its principal diagonal contains a gain vector \(\mathbf{g}_{\mathrm{AF}}\). Taking into account the processing above described, the transmitted signal from A&F MIMO RN is expressed by  
\[\mathbf{X}_{\mathrm{AF}}=\mathbf{M}_{\mathrm{RT}}\mathbf{G}_{\mathrm{AF}} \mathbf{M}_{\mathrm{RR}}^{\mathrm{H}}\mathbf{Y}_{\mathrm{AF}}^{1}, \tag{3}\]  
where \(\mathbf{M}_{\mathrm{RR}}^{\mathrm{H}}\) denotes the hermitian matrix of \(\mathbf{M}_{\mathrm{RR}}\). In second-phase, the relay forwards the signal which is received at the destination (UE), and can be given by  
\[\mathbf{Y}_{\mathrm{AF}}^{2}=\mathbf{H}^{2}\mathbf{X}_{\mathrm{AF}}+\mathbf{N }_{\mathrm{AF}}^{2}, \tag{4}\]  
where \(\mathbf{H}^{2}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{D}}\times \mathbb{N}_{\mathrm{T}}^{\mathrm{R}}}\) denotes the channel matrix in the A&F-UE link and \(\mathbf{N}_{\mathrm{AF}}^{2}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{D}} \times\mathrm{T}}\) is the components respective of Additive White Gaussian Noise (AWGN), with \(N\sim(0,\sigma_{\mathrm{N}_{\mathrm{AF}}^{2}}^{2})\) and \(\sigma_{\mathrm{N}_{\mathrm{AF}}^{2}}^{2}\) is the noise variance. At the UE, the received signal (4) is processed by the RF matrix \(\mathbf{M}_{\mathrm{DR}}\in\mathbb{C}^{\mathbb{N}_{\mathrm{R}}^{\mathrm{D}} \times\mathbb{N}_{\mathrm{R}}^{\mathrm{R}}}\) and baseband matrix \(\mathbf{G}_{\mathrm{BB}}\in\mathbb{C}^{\mathbb{N}_{\mathrm{d}}\times\mathbb{N }_{\mathrm{R}}^{\mathrm{B}}}\), which are equivalent to inverse processing performed by the eNodeB. The received signal is converted to baseband frequency, and the baseband signal and physical channels are demodulated and decoded, respectively. Finally, the estimated signal of the destination can be expressed as  
\[\mathbf{\hat{S}} =\mathbf{G}_{\mathrm{BB}}^{\mathrm{H}}\mathbf{M}_{\mathrm{DR}}^{ \mathrm{H}}(\mathbf{H}^{2}\mathbf{X}_{\mathrm{AF}}+\mathbf{N}_{\mathrm{AF}}^ {2}), \tag{5}\] \[=\mathbf{S}+\mathbf{N}_{\mathrm{D}}\]  
where \(\mathbf{G}=\mathbf{G}_{\mathrm{BB}}\mathbf{M}_{\mathrm{DR}}\) and \(\mathbf{N}_{\mathrm{D}}=\mathbf{G}^{\mathrm{H}}(\mathbf{H}^{2}\mathbf{M}_{ \mathrm{RT}}\mathbf{G}_{\mathrm{AF}}\mathbf{M}_{\mathrm{RR}}^{\mathrm{H}} \mathbf{N}_{\mathrm{AF}}^{1}+\mathbf{N}_{\mathrm{AF}}^{2})\).