## 4 Simulations Results  
In this section, we present the numerical results obtained through MATLAB simulations of SN-SC, FCM, and PSO-FCM. Table 1 provides an overview of the simulation parameters employed in our study. Figure 2: 16 QAM modulation constellation with inner radius (\(r_{min}\)) and outer radius (\(r_{max}\)). In the low-SNR scenario (\(\text{SNR}=4\) dB), the FCM algorithm is susceptible to becoming trapped in local minima, as depicted in Figure 3. When the FCM algorithm, which relies on a fuzzy clustering objective function, is used, this tendency can cause some cluster centers to shift within the constellation. This sensitivity to initialization and the resulting misplacement can lead to substantial errors during the decision-making and demodulation processes. While the PSO-FCM algorithm demonstrates superior clustering performance compared to FCM, it still encounters challenges in fully mitigating the issue of local optimization, as shown by the presence of local optima in Figure 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Parameters** & **Values** \\ \hline SN-SC & \(K_{a}\) & 1 \\  & \(K_{b}\) & 1.5 \\  & \(\lambda\) & 0.4 \\  & \(N\) & 3000 \\ PSO & \(c_{1}\),\(c_{2}\) & 1.5 \\  & \(Q\) & 30 \\  & \(\omega_{0},\omega_{1}\) & 1.1, 0.5 \\  & \(\varepsilon\) & \(10^{-5}\) \\  & \(t_{max}\) & 300 \\ FCM & \(m\) & 2 \\  & \(W\) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values used in the simulations. Figure 4: _Cont._  
Figure 3: 16 QAM received signal modulation at SNR = 4 dB with clustering centers for SN-SC and FCM. In Table 2, we present the average number of cluster centers obtained by SN-SC (#1), SC with \(r_{a}=0.22\) (#2), and SC with \(r_{a}=0.33\) (#3) for 4 QAM, 16 QAM, 32 QAM, and 64 QAM digitally modulated signals under varying SNR scenarios. Algorithms #2 and #3 based on the SC method perform well in clustering 16 QAM and 4 QAM signals; however, their reliable classification performance cannot be guaranteed for other modulation schemes. In contrast, algorithm #1 demonstrates its suitability for diverse modulation signals and outperforms algorithms #2 and #3 in overall classification performance. Table 3 presents the clustering performance of the FCM and PSO-FCM algorithms. The results demonstrate that the PSO-FCM exhibits a reduced number of occurrences of local optima compared to the FCM algorithm across all modulation schemes. This suggests that the PSO-FCM algorithm is more effective in identifying the global optimum compared to the FCM algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline
**Type** & \multicolumn{3}{c}{**0 dB**} & \multicolumn{3}{c}{**2 dB**} & \multicolumn{3}{c}{**4 dB**} & \multicolumn{3}{c}{**6 dB**} & \multicolumn{3}{c}{**8 dB**} & \multicolumn{3}{c}{**10 dB**} \\  & **\#1** & **\#2** & **\#3** & **\#1** & **\#2** & **\#3** & **\#1** & **\#2** & **\#3** & **\#1** & **\#2** & **\#3** & **\#1** & **\#2** & **\#3** & **\#1** & **\#2** & **\#3** \\ \hline
4 QAM & **2.9** & 13.1 & 4.9 & **3.8** & 12.6 & 4 & **4** & 11.9 & 4 & **4** & 5.9 & 4 & **4** & 4 & 4 & 4 & 4 & 4 \\
8 QAM & **6.8** & 14.5 & 6.2 & **7.6** & 15.4 & 6.2 & **7.9** & 14.2 & 6.5 & **8** & 10.2 & 6.8 & **8** & 8 & 7.2 & **8** & 8 & 8 \\
16 QAM & **13.6** & 18.9 & 7.6 & **15.2** & 16 & 9.1 & **15.8** & 16 & 10.2 & **16** & 11.8 & **16** & 11.8 & **16** & 12.8 & **16** & 16 & 14.5 \\
32 QAM & **19.4** & 19.9 & 8.3 & **23.8** & 22.1 & 8.8 & **27.4** & 24.8 & 10.2 & **30.5** & 27.8 & 11.6 & **31.6** & 29.9 & 12.2 & **32** & 31.7 & 12.5 \\
64 QAM & **23.5** & 23.5 & 10.2 & **26.6** & 26.4 & 12.1 & **29.9** & 27.7 & 12.8 & **33.1** & 29.2 & 12.8 & **45.8** & 32.1 & 13.1 & **63.6** & 33.9 & 13.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of clustering results for different-order QAM signals under various SNR situations using the SC algorithm and the SN-SC algorithm.