# Performance Analysis of Two-Hop mmWave Relay Nodes over the 5G NR Uplink Signal
## 2 System Model and Basic Assumptions
### Uplink mmWave MIMO D&F Implementation
It should be noted that several modulation options can be applied to the PUSCH, including Q-PSK, 16-QAM, 64-QAM, and 256-QAM, where the 64-QAM and 256-QAM modulation schemes have been used in the proposed link-level co-operative simulator demonstrating the flexibility of ours simulator. ```
Input :\(\mathbf{Y}=\mathbf{Y}_{\text{DF}}^{1},\hat{\mathbf{H}}=\hat{\mathbf{H}}^{1}, \text{P}^{conf},\text{d}\text{D}^{conf},\text{e}\text{D}^{conf}\) Output :\(\hat{\mathbf{X}}\) \([\mathbf{N}_{\text{sub}},\mathbf{N}_{\text{sym}},\mathbf{N}_{\text{T}}^{ \text{S}},\mathbf{N}_{\text{R}}^{1}]=\mathbf{size}(\mathbf{Y}_{\text{DF}}^{1})\); if\(\mathbf{N}_{\text{T}}^{\text{S}}==1\)&&\(\mathbf{N}_{\text{R}}^{R}==1\)then \(\hat{\mathbf{Y}}=(\hat{\mathbf{H}}^{1}\mathbf{H}+\hat{\mathbf{N}}_{0})^{-1}\hat{ \mathbf{H}}^{1}\mathbf{Y}\); else for\(\epsilon\gets 1:N_{\text{sub}}\)do \(\mathbf{f}\gets 1:N_{\text{sub}}\)do \(\mathbf{\hat{Y}}^{(:)}_{(\epsilon,\epsilon)}=\frac{(\hat{\mathbf{H}}^{( \mathbf{H})}_{(\epsilon,\epsilon)}\hat{\mathbf{H}}^{(:)}_{(\epsilon,\epsilon)}+ \hat{\mathbf{N}}_{0})^{-1}}{1}\hat{\mathbf{H}}^{(:)}_{(\epsilon,\epsilon)} \mathbf{Y}^{(:)}_{(\epsilon,\epsilon)}\); for\(i\gets 0:N_{\text{sl}}\)do \([\text{ulsch}^{b}\), \(\text{pusch}^{s}]=\mathbf{NR-PUSCH}^{\text{d}}(\hat{\mathbf{Y}}^{i},\hat{ \mathbf{H}}^{i},\text{P}^{conf})\); \([\text{data}^{b}]=\mathbf{NR-ULSCH}^{\text{d}}(\text{ulsch}^{b}\), \(\text{pusch}^{s}\), \(\text{P}^{conf}\), \(\text{d}\text{D}^{conf})\); \([\text{ulsch}^{b}_{t_{x}}]=\mathbf{NR-UUSCH}^{\text{c}}(\text{data}^{b}, \text{P}^{conf},\text{e}\text{D}^{conf})\); \([\text{pusch}^{s}_{t_{x}}]=\mathbf{NR-PUSCH}^{\text{e}}(\text{ulsch}^{b}_{t_{x }},\text{P}^{conf})\); \([\hat{\mathbf{X}}]=\mathbf{NR-P}(\text{pusch}^{s}_{t_{x}})\); \([\hat{\mathbf{X}}]=\mathbf{Cat}(\hat{\mathbf{X}},\hat{\mathbf{X}})\);
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**Algorithm 1**Uplink MIMO NR decode-and-encode with MMSE algorithm  
On the other hand, in order to maximize the SNR, the Mean Maximum Square Error (MMSE) equalizer has been considered, which employs the \(\mathbf{Y}_{\text{DF}}^{1}\), \(\hat{\mathbf{H}}^{1}\), and estimated noise for its implementation. Furthermore, in the proposed algorithm, the subindices \((\cdot)^{\text{d}}\) and \((\cdot)^{\text{c}}\) in the functions describe the processes of decoding and encoding, as well as the subindices \((\cdot)^{\text{b}}\) and \((\cdot)^{\text{s}}\) in the outputs of the functions represent data bits and data symbols, respectively. The \(\mathbf{NR-P}^{\text{c}}\) function, mapping each symbol, in the resource grid, and the \(\hat{\mathbf{x}}\) subframe to the output of the function, is obtained. Finally, concatenating of each subframe is performed by means of the \(\mathbf{Cat}(\cdot)\) function, from which all frames to be re-transmitted are stored. After that, \(\hat{\mathbf{X}}\) is processed through the DM-RS symbols mapping block, where reference signals are added to the resource grid. The resultant signal to the output of blocks is modulated considering the uplink NR CP-OFDM modulate block. Then, the resultant waveform is passed by the FR1-Tx transmitter \(\mathbf{M}_{\text{RT}}\) block, which performs the translation in frequency of baseband to the FR1 band and transmits the signal to gNodeB, as has been seen in Figure 6.