# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 4 Simulation Results  
In this section, we present simulation results to evaluate the covert communication performance of the proposed scheme. We assumed that the ARIS-aided uplink NOMA system operated in the mmWave band with a carrier frequency of 28 GHz and a bandwidth of 251.1886 MHz [42]. We set the simulation parameters as \(M=4\), \(N=16\) with the size of the ARIS panel being 2 cm \(\times\) 2 cm [43], \(\sigma_{b}^{2}=\sigma_{w}^{2}=\sigma_{r}^{2}=-90\) dBm, \(\rho=0\), \(R_{c}^{\text{min}}=\hat{R}_{c}^{\text{min}}=1\) bps/Hz, \(\eta_{a}^{2}=\eta^{2}\), \(\forall n\), \(\xi_{1}=10^{-1}\), \(\xi_{2}=\xi_{3}=\xi_{4}=10^{-3}\), \(\rho_{1}=\rho_{2}=\rho_{3}=10^{-6}\), and \(c_{1}=c_{2}=c_{3}=3.6\)[29; 31]. The small-scale fading of all channels followed \(\mathcal{CN}(0,1)\). The distance-related path loss of each channel was modeled as \(\mathcal{L}=\mathcal{L}_{0}(\frac{d}{d_{0}})^{-\chi}\), where \(\mathcal{L}_{0}=-30\) dB, \(d_{0}=1\) m, and \(d\) is the distance between two nodes. We considered the path-loss exponential factor \(\chi=3.5\) for the direct links and \(\chi=2.2\) for the channels associated with ARIS. The Rician factor for the links associated with the ARIS was set to 3 dB. The locations of [PERSON], ARIS, [PERSON], [PERSON], and [PERSON] were set as \((0,15)\) m, \((50,45)\) m, \((80,30)\) m, \((70,15)\) m, and \((10,30)\) m, respectively. In the following discussions, we compare the impacts of element grouping on the covert communication performance of an ARIS-aided NOMA system against those without element grouping. Specifically, the impacts of the transmit power, the horizontal position of the ARIS, the covertness level, and the number of ARIS elements on the covert communication rate are investigated, respectively. To gain insights on the implementation of the proposed schemes, we recorded the running time of the AO algorithms for the w/o-EG and w-EG schemes, as shown in Table 1. The simulation platform was a Lenovo Lecoo PC consisting of an Intel i5-12450H CPU and 16 G of memory. The operating system was Windows 11, and the software was Matlab R2023a. The results in Table 1 show that the AO algorithms of both the w/o-EG and w-EG schemes could be performed within 0.35 s, which was acceptable regarding the applied general PC platform. As expected, if a specific hardware/software platform was applied, e.g., FPGA and DSP, the real-time running speed of the proposed AO algorithms would be faster than that of a general PC. The results in Figure 2 reveal the impacts of the transmit power budget on the covert communication rates for different schemes. From Figure 2, we can see that the covert communication rates achieved by all the schemes increased with the increasing \(p_{a}^{\text{max}}\). The curves in Figure 2 confirmed that the proposed AO algorithm without employing element grouping achieved the highest covert communication rate in the whole transmit power regions. On the contrary, compared to the AO algorithm without employing element grouping, a decreased covert communication rate was achieved by the element grouping scheme. Furthermore, a higher \(\eta^{2}\) rendered a higher covert communication rate for the proposed schemes, which indicated that the constraints (21) and (37) were inactive, respectively. The achieved covert communication rates of the randomized \(\mathbf{\Theta}\) and \(\{\mathbf{\Theta},\mathbf{W}\}\) tended to overlap with the increasing \(p_{a}^{\text{max}}\), while the two considered schemes achieved a higher covert communication rate than the AO algorithm with element grouping in the low-transmit-power regime. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Running Time (s)** & **System Parameters** & \(M=4\), \(N=16\) & \(M=8\), \(N=16\) & \(M=8\), \(N=32\) \\ \hline w-EG & 0.2821 & 0.2963 & 0.3212 \\ \hline w/o-EG & 0.2806 & 0.2954 & 0.3176 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Running time of the w/o-EG and w-EG schemes.