# Performance Analysis of Two-Hop mmWave Relay Nodes over the 5G NR Uplink Signal
## 2 System Model and Basic Assumptions
### A&F Signal Model  
A single-user uplink mmWave MIMO new radio transmission model for an A&F relay node is introduced in this subsection. The main functionality of this protocol is to capture the mmWave signal from NR-UE and, after that, amplify it and forward it to the gNodeB without performing any additional processing. Additionally, the 3 GPP classification can be defined as a Layer 1 (L1) relay node. Therefore, consider the uplink of an SU-mmWave MIMO co-operative system in Figure 1 and the A&F architecture in Figure 3, consisting of an A&F protocol that has \(\text{N}_{\text{R}}^{\text{R}}\) receiver antennas which correspond with the number of RF mmWave chains (FR2-Rx) and \(\text{N}_{\text{T}}^{\text{R}}\) represents the FR1 transmitter antennas (FR1-Tx), where we consider that \(\text{N}_{\text{R}}^{\text{R}}=\text{N}_{\text{T}}^{\text{R}}\). Figure 2: Baseband mmWave NR-UE architecture. Let \(\mathbf{X}\) denote the symbol matrix to be transmitted by the NR-UE. In the first hop, the received \(\mathbf{N}_{\mathrm{R}}^{\mathrm{R}}\times\mathrm{T}-\)dimensional signal \(\mathbf{Y}_{\mathrm{AF}}^{\mathrm{I}}\) at the mmWave MIMO A&F strategy can be expressed as  
\[\mathbf{Y}_{\mathrm{AF}}^{\mathrm{I}}=\mathbf{H}^{\mathrm{I}}\mathbf{X}+ \mathbf{N}_{\mathrm{AF}}^{\mathrm{I}}, \tag{2}\]  
where \(\mathbf{H}^{\mathrm{I}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\) describes an NR-UE-to-A&F node FR2 (26 GHz) mmWave channel matrix. \(\mathbf{N}_{\mathrm{AF}}^{\mathrm{I}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}} ^{\mathrm{R}}\times\mathrm{T}}\) denotes Additive White Gaussian Noise (AWGN) matrix with independent and identical distributed random variables \(N\sim(0,\sigma_{\mathrm{N}_{\mathrm{AF}}^{\mathrm{I}}}^{2})\), where \(\sigma_{\mathrm{N}_{\mathrm{AF}}^{\mathrm{I}}}^{2}\) is the noise variance. The signal \(\mathbf{Y}_{\mathrm{AF}}^{\mathrm{I}}\) is received through FR2-Rx chains \(\mathbf{M}_{\mathrm{RR}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\) in the A&F relaying technology. Nevertheless, it is precoded by precoder matrices \(\mathbf{R}_{\mathrm{BBR}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\) and \(\mathbf{R}_{\mathrm{BBT}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\), and RF chains \(\mathbf{M}_{\mathrm{RT}}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{R}} \times\mathrm{N}_{\mathrm{R}}^{\mathrm{R}}}\). In addition, the product \(\mathbf{R}_{\mathrm{BBT}}\mathbf{R}_{\mathrm{BBR}}=\mathbf{G}_{\mathrm{AF}}\), which depicts a diagonal matrix that on the principal diagonal contains a gain vector (\(\mathbf{g}_{\mathrm{AF}}\)). Considering the processing described above, the transmit signal at the Uplink A&F RN is written as  
\[\mathbf{X}_{\mathrm{AF}}=\mathbf{M}_{\mathrm{RT}}\mathbf{G}_{\mathrm{AF}} \mathbf{M}_{\mathrm{RR}}^{\mathrm{H}}\mathbf{Y}_{\mathrm{AF}}^{\mathrm{I}}, \tag{3}\]  
where \(\mathbf{M}_{\mathrm{RR}}^{\mathrm{H}}\) denotes the hermitian matrix of \(\mathbf{M}_{\mathrm{RR}}\). In the secon -phase, the received signal at the gNodeB can be given by  
\[\mathbf{Y}_{\mathrm{AF}}^{2}=\mathbf{H}^{2}\mathbf{X}_{\mathrm{AF}}+\mathbf{ N}_{\mathrm{AF}}^{2}, \tag{4}\]  
where \(\mathbf{H}^{2}\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{D}}\times \mathrm{N}_{\mathrm{T}}^{\mathrm{R}}}\) describes the relay-to-destination channel matrix, and the matrix \(\mathbf{N}_{\mathrm{AF}}^{2}\) represents the components of AWGN, where \(\in\mathbb{C}^{\mathrm{N}_{\mathrm{R}}^{\mathrm{D}}\times\mathrm{T}}\) with \(N\sim(0,\sigma_{\mathrm{N}_{\mathrm{AF}}^{2}}^{2})\).