## 2 System Model and Basic Assumptions
### A&F NR-RN Signal Model  
In this subsection, we consider the two-hop MIMO A&F NR-RN configuration from Figure 1. Then, let \(\mathbf{X}\) denotes the symbols matrix to be transmitted by gNodeB. In the first-phase, the received signal at the relay is represented as  
\[\mathbf{Y}_{\text{AF}}^{\text{I}}=\mathbf{H}^{\text{I}}\mathbf{X}+\mathbf{N}_{ \text{AF}}^{\text{I}}, \tag{2}\]  
where \(\mathbf{H}^{\text{I}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \text{N}_{\text{T}}^{\text{S}}}\) denotes a source-to-relay node FR1 (<6 GHz) channel matrix. \(\mathbf{N}_{\text{AF}}^{\text{I}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R} }\times\text{T}}\) indicates noise matrix with independent and identical distributed random variables \(N\sim(0,\sigma_{\text{N}_{\text{AF}}^{\text{I}}}^{2})\), where \(\sigma_{\text{N}_{\text{AF}}^{\text{I}}}^{2}\) denotes the noise variance. The signal \(\mathbf{Y}_{\text{AF}}^{\text{I}}\) is received through FR1-Rx chains \(\mathbf{M}_{\text{RR}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \text{N}_{\text{R}}^{\text{R}}}\) at the relay. Then, it is precoded by precoder matrices \(\mathbf{R}_{\text{BBR}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \text{N}_{\text{R}}^{\text{R}}}\) and \(\mathbf{R}_{\text{BBT}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{R}}\times \text{N}_{\text{R}}^{\text{R}}}\), and RF chains \(\mathbf{M}_{\text{RT}}\in\mathbb{C}^{\text{N}_{\text{T}}^{\text{R}}\times \text{N}_{\text{R}}^{\text{R}}}\). In fact, the product \(\mathbf{R}_{\text{BBT}}\mathbf{R}_{\text{BBR}}=\mathbf{G}_{\text{AF}}\), which depicts a diagonal matrix that on your principal diagonal contains a gain vector (\(\mathbf{g}_{\text{AF}}\)). Considering the processing above described, the transmit signal at the A&F RN is expressed as  
\[\mathbf{X}_{\text{AF}}=\mathbf{M}_{\text{RT}}\mathbf{G}_{\text{AF}}\mathbf{M} _{\text{RR}}^{\text{H}}\mathbf{Y}_{\text{AF}}^{\text{I}}, \tag{3}\]  
where \(\mathbf{M}_{\text{RR}}^{\text{H}}\) denotes the hermitian matrix of \(\mathbf{M}_{\text{RR}}\). In the second-phase, the received signal at the UE can be written as  
\[\mathbf{Y}_{\text{AF}}^{2}=\mathbf{H}^{2}\mathbf{X}_{\text{AF}}+\mathbf{N}_{ \text{AF}}^{2}, \tag{4}\]  
where \(\mathbf{H}^{2}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times\text{N}_{ \text{T}}^{\text{R}}}\) describes relay-to-destination mm-Wave channel matrix, and the matrix \(\mathbf{N}_{\text{AF}}^{2}\) is the components respective of Additive White Gaussian Noise (AWGN), which \(\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times\text{T}}\) with \(N\sim(0,\sigma_{\text{N}_{\text{AF}}^{2}}^{2})\). At the destination node, the received signal (4) is processed by the RF matrix \(\mathbf{M}_{\text{DR}}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times \text{N}_{\text{R}}^{\text{D}}}\) and baseband matrix \(\mathbf{G}_{\text{BB}}\in\mathbb{C}^{\text{N}_{\text{Ad}}\times\text{N}_{\text{ R}}^{\text{D}}}\), which are equivalent to inverse processing performed by the gNodeB. The received signal is converted to baseband frequency, and the baseband signal and physical channels are  
Figure 2: Baseband gNodeB structure. demodulated and decoded, respectively. Finally, the estimated signal of the destination can be expressed as  
\[\mathbf{\hat{S}} =\mathbf{G}_{\text{BB}}^{\text{H}}\mathbf{M}_{\text{DR}}^{\text{H}} (\mathbf{H}^{2}\mathbf{X}_{\text{AF}}+\mathbf{N}_{\text{AF}}^{2})\] \[=\mathbf{G}^{\text{H}}\mathbf{H}^{2}\mathbf{M}_{\text{RT}}\mathbf{ G}_{\text{AF}}\mathbf{M}_{\text{RR}}^{\text{H}}\mathbf{H}^{1}\mathbf{TS}+\mathbf{N}_{ \text{D}}\enspace, \tag{5}\] \[=\mathbf{S}+\mathbf{N}_{\text{D}}\]  
where \(\mathbf{G}=\mathbf{G}_{\text{BB}}\mathbf{M}_{\text{DR}}\), \(\mathbf{T}=\mathbf{M}_{\text{ST}}\mathbf{T}_{\text{BB}}\), and \(\mathbf{N}_{\text{D}}=\mathbf{G}^{\text{H}}(\mathbf{H}^{2}\mathbf{M}_{\text{RT }}\mathbf{G}_{\text{AF}}\mathbf{M}_{\text{RR}}^{\text{H}}\mathbf{N}_{\text{AF }}^{1}+\mathbf{N}_{\text{AF}}^{2})\).