## 3 System Model and Basic Assumptions
### D&F MIMO Signal Model  
In this subsection, the D&F Co-operative System will be addressed. In the first-hop, the signal is received by the D&F Relay Node which, before re-transmitting (second-hop), decodes the transmitted symbols from eNB, encodes these symbols into an OSFBC (having the same structure as used by eNB) and, then, transmits it over the access link (i.e., the D&F-UE link), as shown in Figure 4.  
Let \(\mathbf{X}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times\text{T}}\) denotes the symbols matrix by the eNB in the first-hop of data transmission, where T is the number of used subcarriers for the transmission and \(\text{N}^{\text{S}}_{\text{T}}\) is the number of transmit antennas. Therefore, the received data at the D&F relay node in a block can be expressed as  
\[\mathbf{Y}^{1}_{\text{DF}}=\mathbf{H}^{1}\mathbf{X}+\mathbf{N}^{1}_{\text{DF}}, \tag{17}\]  
where \(\mathbf{H}^{1}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{R}}\times\text{N}^{ \text{S}}_{\text{T}}}\) represents the channel frequency responses of the eNB-D&F link. On the other hand, \(\mathbf{N}^{1}_{\text{DF}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{R}} \times\text{T}}\) represents the Additive White Gaussian Noise (AWGN) matrix of the Backhaul Link (i.e., the eNB-D&F link). Furthermore, the elements of the noise matrix are complex Gaussian random variables with \(N\sim(0,\sigma^{2}_{\text{N}^{1}_{\text{DF}}})\) and \(\sigma^{2}_{\text{N}^{1}_{\text{DF}}}\) is the noise variance. In the D&F Relay Node, the decoding and encoding of the received signal (17) is carried out by two stages, as can be seen in Figure 4. Firstly, the decoding processing is performed through the receive baseband matrix \(\mathbf{R}_{\text{BBR}}\in\mathbb{C}^{\text{N}_{\text{R}}\times\text{N}^{ \text{R}}_{\text{R}}}\) and RF receiver chains matrix \(\mathbf{M}_{\text{RR}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{R}}\times \text{N}^{\text{R}}_{\text{R}}}\). Therefore, as result of the first stage, an estimated data symbols matrix \(\hat{\mathbf{S}}\in\mathbb{C}^{\text{N}_{\text{R}}\times\text{T}}\) is obtained. Thus, the encoding processing is carried out by the transmit baseband matrix \(\mathbf{R}_{\text{BBT}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times \text{N}_{\text{R}}}\) and RF transmit chains matrix \(\mathbf{M}_{\text{RT}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times\text{N}^{ \text{R}}_{\text{T}}}\), consequently, the transmitted signal by the Relay Node is given by  
\[\hat{\mathbf{X}}=\mathbf{M}_{\text{RT}}\mathbf{G}_{\text{DF}}\mathbf{R}_{\text{BBT }}\hat{\mathbf{S}}, \tag{18}\]  
where \(\mathbf{G}_{\text{DF}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times \text{N}^{\text{R}}_{\text{T}}}\) represents a diagonal matrix with principal diagonal containing a gain vector (\(\mathbf{g}_{\text{DF}}\)). On the other hand, the received signal at the UE after performing demodulation, modulation, and re-transmission at the relay node can be expressed by  
\[\mathbf{Y}^{2}_{\text{DF}}=\mathbf{H}^{2}\hat{\mathbf{X}}+\mathbf{N}^{2}_{\text{ DF}}, \tag{19}\]where \(\mathbf{\hat{X}}\) is the \(\mathrm{N}_{\mathrm{T}}^{\mathrm{R}}\times\mathrm{T}\) matrix forwarded by the relay node after the decoding of \(\mathbf{X}\), \(\mathbf{H}^{2}=[h_{i,j}^{1}]\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{D} }\times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\) is the channel matrix of the access link, \(\mathbf{N}_{\mathrm{DF}}^{2}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{D} }\times\mathrm{T}}\) contains the AWGN with \(N\sim(0,\sigma_{\mathrm{N}_{\mathrm{DF}}^{2}}^{2})\), and \(\sigma_{\mathrm{N}_{\mathrm{DF}}^{2}}^{2}\) indicates the noise variance.