# Performance Analysis of Two-Hop mmWave Relay Nodes over the 5G NR Uplink Signal
## 2 System Model and Basic Assumptions
### D&F Signal Model  
In a similar way as in the previous subsection, the transmission model for the proposed uplink mmWave MIMO D&F strategy can be described. A decode-and-forward relay node  
Figure 3: Uplink mmWave MIMO new radio amplify-and-forward relay node architecture. is equivalent to a Layer 2 relay taking into account the standardization of the 3 GPP, in which the RF signal from NR-UE is first decoded and, before being forwarded to the destination source (gNodeB), it is encoded again. The receiver system model of the protocol is presented, followed by the transmitter model based on the structure described in Figure 4. It is worth highlighting that D&F strategy is performed in two stages: firstly, decoding of the received signal, and, before forwarding the signal to the destination, encoding of the estimated symbols, as shown in Figure 4, which supposes the major difference in comparison with other protocols, such as the amplify-and-forward strategy. It should be noted that, in this paper, two types of uplink D&F relaying protocols have been developed: (a) one with the channel knowledge that performs a Perfect Channel Estimation (PCE) and (b) one that performs a channel estimation through a Least Square (LS) estimator. Additionally, both strategies consider the equalization technique by means of the MMSE algorithm and the protocols are equipped with \(\rm{N}_{R}^{R}\) and \(\rm{N}_{T}^{R}\), respectively--receiver and transmitter antennas. Considering Figure 1 and the transmitted mmWave signal (1) from the NR-UE, in the access link, the received data at the out-band D&F strategy in a block can be expressed as  
\[\mathbf{Y}_{\rm{DF}}^{1}=\mathbf{H}^{1}\mathbf{X}+\mathbf{N}_{\rm{DF}}^{1}, \tag{6}\]  
where matrix \(\mathbf{N}_{\rm{DF}}^{1}\in\mathbb{C}^{\rm{N}_{R}^{R}\times T}\) with \(N\sim(0,\sigma_{\rm{N}_{\rm{DF}}}^{2})\) describes the AWGN in the first hop, \(\sigma_{\rm{N}_{\rm{DF}}}^{2}\) is the noise variance. \(\mathbf{H}^{1}\in\mathbb{C}^{\rm{N}_{R}^{R}\times\rm{N}_{T}^{R}}\) denotes the mmWave MIMO channel matrix between the NR-UE and mmWave MIMO D&F RN. In the D&F protocol, the signal (6) is captured through the matrix \(\mathbf{M}_{\rm{RR}}\in\mathbb{C}^{\rm{N}_{R}^{R}\times\rm{N}_{R}^{R}}\), as can be seen in Figure 4 and which in the real environment can be emulated by an mmWave RF receiver chains that will be described in Section 3. In fact, the decoding and encoding stages are performed by the matrices \(\mathbf{R}_{\rm{BBF}}\in\mathbb{C}^{\rm{N}_{R}^{s}\times\rm{N}_{R}^{R}}\) and \(\mathbf{R}_{\rm{BBF}}\in\mathbb{C}^{\rm{N}_{R}^{s}\times\rm{N}_{s}}\), respectively. Consequently, the signal transmitted by the RN can be given as  
\[\hat{\mathbf{X}}=\mathbf{M}_{\rm{RT}}\mathbf{R}_{\rm{BBF}}\hat{\mathbf{S}}, \tag{7}\]  
Figure 4: Uplink mmWave MIMO new radio decode-and-forward relay node architecture. 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 \(<6\) GHz 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 symbols matrix at the D&F protocol. Considering (7), the received signal at the gNodeB can be written by  
\[\mathbf{Y}_{\text{DF}}^{2}=\mathbf{H}^{2}\mathbf{\hat{X}}+\mathbf{N}_{\text{DF }}^{2}, \tag{8}\]  
where \(\mathbf{H}^{2}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times\text{N}_{ \text{T}}^{\text{R}}}\) is the channel matrix of the backhaul link (D&F-to-gNodeB link) and \(\mathbf{N}_{\text{DF}}^{2}\in\mathbb{C}^{\text{N}_{\text{R}}^{\text{D}}\times \text{N}_{\text{T}}^{\text{R}}}\) denotes the AWGN with \(N\sim(0,\sigma_{\text{N}_{\text{DF}}^{2}}^{2})\), where \(\sigma_{\text{N}_{\text{DF}}^{2}}^{2}\) is the noise variance in the backhaul link.