# Message Passing Detectors for UAV-Based Uplink Grant-Free NOMA Systems
## 4 Simulation Results  
This section compares the performance of the proposed GAMP-BG-MC algorithm. As a benchmark, we consider the state-of-the-art algorithms GAMP-SBL [13,14] and GAMP-PCSBL [14], referred to as SBL and PCSBL in the following. Referring to [14], the BPSK modulation scheme is employed in all simulations. To ensure the reproducibility of the results, the simulation setup is described as follows. The simulations were conducted using MATLAB as the programming language, utilizing its built-in functions for matrix operations and signal processing. The parameters used in the experiments are summarizedin Table 3. The experiments are conducted in two scenarios compared with the benchmark algorithms. One is the UE activity with temporal correlation, i.e., the UE will transmit the data partially during the whole frame, here denoted as \"Partial\". The other is an extreme phenomenon where the UE keeps transmitting data fully to the frame, referred to as \"Full\".  
For the initialization, according to [15], the prior parameters are \(\{e_{k,j}=\eta_{k,j}=1,\forall k,j\}\), \(\{e_{k}=f_{k}=c_{k}=d_{k}=1,\forall k\}\), \(\hat{\tau}_{k,j}^{\parallel}=1/2,\forall k,j\), and the belief parameters are initialized with the same values of the priors. All the experiments are obtained by averaging \(\text{sim}=1000\) Monte Carlo realizations. The experiments use the symbol error rate (SER), defined as \(\text{SER}=\log_{10}(\frac{K_{err}}{\text{sm}\times k})\), as a performance metric; \(K_{err}\) is the number of UE whose symbol is erroneously decoded. The per-iteration complexity of the BG-MC is \(\mathcal{O}((N+1)KJ)\), and it is in the same order as the SBL and PCSBL. However, the iteration number of the BG-MC will always be small, as shown in Figure 3.  
Figure 3 gives the convergence speed of the three algorithms under \(\text{SNR}=9\) dB, with two scenarios. The proposed BG-MC algorithm has the fastest convergence speed compared with the SBL and PCSBL algorithms in both scenarios, which demonstrates that the proposed algorithm can capture the AS correlation more accurately than the other two algorithms. Especially in the \"Partial\" scenario, the proposed algorithm converges faster and performs better than the comparison algorithm.  
In Figure 4, the simulations compare the average SER performance as a function of the SNRs. The simulation results show that the BG-MC algorithm performs better in the two scenarios. Further, the results show that the PCSBL is better than the SBL in the second scheme and the PCSBL is efficient in characterizing the block sparse structure, while the BG-MC is more accurate, not only in the block sparse structure but also in the cluster sparse structure. The reason for this is that the BG-MC can update the statistic parameters automatically, which is suitable for the practice system.  
Figure 5 shows the SER performances as a function of the UE active rate \(p_{a}\) under \(\text{SNR}=9\) dB. The BG-MC algorithm outperforms the benchmark algorithms in almost all ranges of \(p_{a}\), especially in the first scenario, which validates the effectiveness of the proposed algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Name** & **Value** & **Parameter Name** & **Value** \\ \hline UE number \(K\) & 20 & Spreading factor \(N\) & 30 \\ Time slots & 6 & Algorithm iteration \(T\) & 20 \\ Modulation scheme & BPSK & Scenarios & Partial and Full \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter settings for the experiment.  
Figure 3: SER vs. Iteration with \(p_{a}=0.3\).