## 4 Results and Discussion  
### Performance Analysis  
In order to analyse the performance of the proposed HBC algorithm, in Figure 3a,b, we show the BER and the corresponding Q-factor at launch optical powers ranging from 2 to 12 mW for the three different approaches: maximum likelihood detection, clustering using k-means, and the novel HBC algorithm. At low power levels, the performance of all three techniques improved (BER reduced and Q-factor increased) as the launch optical power was increased, which made sense since the additive noise of the photodetectors was dominant. For high power levels, on the other hand, Kerr induced SPM was the main physical impairment, and consequently, increasing power led to higher BER and a lower Q-factor. Comparing the performance of maximum likelihood with k-means and HBC, it was clear that all of them converged for a low power level, while the latter two presented improved performances for high power levels. This was an indicator that k-means and HBC were indeed compensating nonlinear distortion and not any linear impairment such as residual phase noise or chromatic dispersion. These two clustering algorithms, however, showed different performances, for both medium and high power levels. It can be clearly observed that HBC outperformed k-means for launch optical powers above 5 mW, revealing that HBC could mitigate nonlinear distortion more efficiently than the traditional k-means clustering. As a result, the best achievable BER was reduced from \(1.1\times 10^{-3}\) when using maximum likelihood and \(0.8\times 10^{-3}\) when employing k-means to \(0.6\times 10^{-3}\) in the case of HBC (orange polygon of Figure 3a). Regarding the Q-factor, numerical results showed an improvement of 0.53 dB with respect to the optimum performance of maximum likelihood and 0.23 dB when contrasted with the optimum of k-means (orange polygon of Figure 3b). The Q-factor enhancement was higher if, instead of comparing optimum performances, we looked at a fixed power level in the nonlinear regime. Thus, for 10 mW, HBC outperformed k-means by 0.51 dB and maximum likelihood by 1.22 dB (purple polygon of Figure 3b). Additionally, the optimum launch optical power where the trade-off between noise and SPM was held shifted towards a higher power level, from 5 mW in maximum likelihood to 6 and 7 mW for k-means and HBC, respectively. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & \multicolumn{3}{c}{**System Parameters**} \\ \hline Laser linewidth & 0.5 MHz & Fibre lengths (L\({}_{1}\),L\({}_{2}\)) & 80 km, 0–20 km \\ Laser power & 1 mW & Fibre attenuation & 0.2 dB \\ MZM insertion loss & 6 dB & Fibre chromatic dispersion & 16 ps/nm/km \\ Amplifier gain & 20 dB & Fibre PMD & 3.16 fs/\(\sqrt{\text{km}}\) \\ Amplifier noise figure & 4 dB & Nonlinear coefficient (\(\gamma\)) & 1.3\(\cdot\)W\({}^{-1}\).km\({}^{-1}\) \\ Attenuator & 20 dB & Fibre effective area & 80 \(\upmu\)m\({}^{2}\) \\ PD thermal noise density & 10 pA/\(\sqrt{\text{Hz}}\) & Electrical filter bandwidth & 10.5 GHz \\ PD responsivity & 1 W/A & Electrical RX filter order & 4 \\  & \multicolumn{3}{c}{Signal parameters} \\ Modulation format & 16-QAM & No. of synchronization symbols & 64 \\ Electrical TX filter & 4\({}^{\text{th}}\)-order Bessel & Bit rate & 56 Gbps \\  & \multicolumn{3}{c}{Simulation parameters} \\ Number of simulated symbols & 16,384 & Sampling rate & 8.96 \(\times\) 10\({}^{11}\) s\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the simulation. Figure 3: Performance of the proposed. HBC compared to that of maximum likelihood and k-means in terms of (**a**) BER and (**b**) effective Q-factor derived from BER. (**c,d,e**) Classified constellations using maximum likelihood, k-means, and HBC, respectively, for a launch optical power of 3 mW. Classified constellations at optimum launch optical powers for the different detection schemes: (**f**) 5 mW for maximum likelihood, (**g**) 6 mW for k-means, and (**h**) 7 mW for HBC.