## 2 System Model and Basic Assumptions
### D&F NR-RN Signal Model  
In this subsection, the system model of the two-hop MIMO mm-Wave D&F NR-RN will be addressed. Taking into account Figure 1 and transmitted signal (1) from the source, in the backhaul link, the received data at the D&F protocol in a block can be expressed as  
\[\mathbf{Y}_{\text{DF}}^{\text{1}}=\mathbf{H}^{1}\mathbf{X}+\mathbf{N}_{\text{ DF}}^{\text{1}}, \tag{6}\]  
where matrix \(\mathbf{N}_{\text{DF}}^{\text{1}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R} }\times\mathbf{T}}\) with \(N\sim(0,\sigma_{\text{N}_{\text{DF}}}^{2})\) represents the AWGN in the first-hop, where \(\sigma_{\text{N}_{\text{DF}}}^{2}\) is the noise variance. \(\mathbf{H}^{1}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times\mathbf{N}_{ \text{f}}^{\text{R}}}\) denotes the sub-6 GHz MIMO channel matrix between the source and relay. In the D&F RN, the signal (6) is captured through the matrix \(\mathbf{M}_{\text{RR}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \mathbf{N}_{\text{R}}^{\text{R}}}\), which in the real environment can be emulated by an SDR as has been proposed in Reference [17]. Worthy of highlighting is that D&F strategy performs two stages: firstly, decode the received signal, and, before forward the signal to the destination, it encodes the estimated symbols, as shown in Figure 3, which supposes the major difference in comparison with A&F protocol.  
In fact, the decoding and encoding stages are performed by the matrices \(\mathbf{R}_{\text{BBR}}\in\mathbb{C}^{\text{N}_{\text{R}}\times\text{N}_{ \text{R}}^{\text{R}}}\) and \(\mathbf{R}_{\text{BBT}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \text{N}_{\text{R}}}\), respectively. Consequently, the transmitted signal by the RN can be given as  
\[\mathbf{\hat{X}}=\mathbf{M}_{\text{RT}}\mathbf{R}_{\text{BBT}}\mathbf{\tilde {S}}, \tag{7}\]  
where matrix \(\mathbf{M}_{\text{RT}}\in\mathbb{C}^{\text{N}_{\text{T}}^{\text{R}}\times \text{N}_{\text{T}}^{\text{R}}}\) denotes the transmitter RF mmWave chains of the RN. On the other hand, \(\mathbf{R}_{\text{BBT}}\in\mathbb{C}^{\text{N}_{\text{T}}^{\text{R}}\times \text{N}_{\text{R}}}\) is the baseband matrix and \(\mathbf{\tilde{S}}\) represents the estimated data  
Figure 3: mmWave MIMO NR Decode-and-Forward (D&F) Relay Node (RN) structure.  
symbols matrix at the D&F protocol. Taking into account (7), the received signal at the UE can be expressed by  
\[\mathbf{Y}_{\text{DF}}^{2}=\mathbf{H}^{2}\mathbf{\hat{X}}+\mathbf{N}_{\text{DF}}^ {2}, \tag{8}\]  
where \(\mathbf{H}^{2}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times\mathbf{N}_{ \text{T}}^{\text{R}}}\) is the channel matrix of the access link (relay-to-destination link) and \(\mathbf{N}_{\text{DF}}^{2}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times \mathbf{N}_{\text{T}}^{\text{R}}}\) denotes the AWGN with \(N\sim(0,\sigma_{\mathbf{N}_{\text{DF}}^{2}}^{2})\), where \(\sigma_{\mathbf{N}_{\text{DF}}^{2}}^{2}\) is the noise variance in the access link. Considering the data processing at the UE through \(\mathbf{M}_{\text{DR}}\) and \(\mathbf{G}_{\text{BB}}\), which the estimated received signal at the destination can be written as  
\[\mathbf{\hat{S}} =\mathbf{G}_{\text{BB}}^{\text{H}}\mathbf{M}_{\text{DR}}^{\text{ H}}(\mathbf{H}^{2}\mathbf{\hat{X}}+\mathbf{N}_{\text{DF}}^{2}) \tag{9}\] \[=\mathbf{G}^{\text{H}}\mathbf{H}^{2}\mathbf{R}_{\text{RT}}\mathbf{ \hat{S}}+\mathbf{N}_{\text{D}}\qquad,\]  
where \(\mathbf{N}_{\text{D}}=\mathbf{G}^{\text{H}}\mathbf{N}_{\text{DF}}^{2}\), \(\mathbf{G}=\mathbf{G}_{\text{BB}}\mathbf{M}_{\text{DR}}\), and \(\mathbf{R}_{\text{RT}}=\mathbf{M}_{\text{RT}}\mathbf{R}_{\text{BBT}}\). In (9), the first term is the desired signal and the second term is the noise, which represents the noise in access link at the Decode-and-Forward (D&F) cooperative system, respectively.