## 4 Results  
The proposed spectral clustering algorithm applies the data streaming processing explained above with a window of 2048 points, which will allow us to design a hardware  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Modulation**} & \multirow{2}{*}{**SNR**} & \multirow{2}{*}{
\begin{tabular}{c} **Oversampling** \\ **Rate** \\ \end{tabular} } & \multirow{2}{*}{**Level**} & \multirow{2}{*}{**Frequency Offset**} & \multirow{2}{*}{**Sample Rate**} \\ \cline{5-5}  
architecture for processing large volumes of data in future work. The total number of symbols sent for testing was 100,000 randomly generated symbols, keeping the exact data for both k-means and spectral clustering, with an oversampling rate of 2 for 16-QAM, 4 for 32-QAM, and 8 for 64-QAM.  
Figures 6-8 show the results obtained for 16-QAM, 32-QAM, and 64-QAM modulation, respectively, when applying different levels of nonlinear phase noise of \(-90\), \(-100\), and \(-110\) dBc/Hz at 1 MHz offset. The plots are constructed on a semi-logarithmic scale, with the x-axis representing the SNR in dB and the y-axis representing the BER on a logarithmic scale. An inset shows the labels for the result curves for k-means as solid lines and spectral clustering represented by dotted lines.  
Figure 6 shows a better performance of the spectral clustering algorithm for dealing with nonlinear phase noise than the results obtained by k-means. Especially at noise levels below \(-100\) dBc/Hz, we observe that the result for values greater than 17 dB SNR is zero for BER, as represented in the graph with the curves going downwards. The behavior of k-means for all cases is close to that of spectral clustering for \(-90\) dBc/Hz, which is the worst-case scenario.  
In Figure 7, the nonlinear noise punishes the k-means behavior more strongly, and it does not go below a BER of \(10^{-1}\) for any of the SNR values. On the contrary, the spectral  
Figure 6: SNR vs. BER performance of spectral clustering and k-means algorithm for 16-QAM.  
Figure 7: SNR vs. BER performance of spectral clustering and k-means algorithms for 32-QAM.  
clustering algorithm performs well at phase noise levels of \(-100\) and \(-110\) dBc/Hz, even achieving zero BER at values greater than 17 dB SNR for \(-110\) phase noise. However, its performance is affected by \(-90\) dBc/Hz noise, and it does not manage to decrease the BER lower than \(10^{-3}\), which is a challenging scenario for 32-QAM modulation due to the fact that the distance between symbols is smaller than in 16-QAM.  
Figure 8 is the scenario for 64-QAM, in which k-means cannot decrease the noise to values less than \(10^{-2}\). Spectral clustering achieves a BER of \(10^{-5}\) at 20 dB SNR for \(-100\) dBc/Hz and zero BER for values greater than 17 dB SNR for a phase noise of \(-110\) dBc/Hz. As expected, the BER for \(-90\) dBc/Hz worsens compared to that of 32-QAM, achieving BER values around \(10^{-2}\) for values above 10 dB SNR.  
Moreover, Table 2 shows a comparison of the algorithmic complexity, execution time, and BER for the clustering techniques used on constellations with different modulation formats. In the table, \(n\) stands for the number of data, \(k\) is the number of groups, and \(t\) is the number of iterations executed. Although the OSC has higher complexity, and this leads to a higher execution time, it presents the lowest BER for all modulation formats.