## 4 Results and Discussion
### Performance Analysis
(**i,j,k**) Classified constellations at elevated launched optical power (10 mW) for maximum likelihood detection, k-means, and HBC, respectively. A better understanding of the performance enhancement can be achieved by analysing the classified constellations (after undergoing maximum likelihood, k-means, or HBC) for different power levels. In particular, Figure 3c-e represents the classified constellations when using maximum likelihood, k-means, or HBC, respectively, for a launch optical power of 3 mW (green rectangle). Figure 3f-h shows the constellation for the optimum power levels, that is 5 mW for maximum likelihood, 6 mW for k-means, and 7 mW for HBC (orange polygon). The constellation degradation and classification for a relatively large power level, 10 mW, can be observed in Figure 3i-k (purple rectangle). Looking at the point dispersion for different power levels, e.g., 3, 5, and 10 mW, shown in the upper row, it can be seen that whereas for a low power level, the constellation shape remained, as the launch power increased, SPM distorted the constellation leading to non-rectangular constellations. The inner symbols seemed to be rotated clockwise, in contrast to the symbols in the periphery that were rotated counter-clockwise. In fact, all the symbols were rotated counter-clockwise by the Kerr effect, but the phase noise compensation stage in the DSP inverted the average rotation, resulting in some points (those with smaller rotation corresponding to lower power) being rotated in the opposite direction. Another feature to be noted is that the noise variance at low power levels looked higher than for moderate power levels, but since the main noise mechanisms were the thermal and shot noises of the photodetectors, the noise level was virtually the same for the three power levels. This is typical in power limited coherent systems because the optical power arriving at the photodetectors is mainly that of the local oscillator laser. The apparently lower noise was then a consequence of the higher signal power and of the power normalization performed before clustering was carried out. Comparing the performance of maximum likelihood, k-means, and HBC, the reader can observe that for low power levels, the classifications obtained by the three methods were essentially identical, which agreed with the fact that same BER and Q-factor values were yielded. This made sense because, in the absence of non linear distortion, maximum likelihood was the optimum detection scheme [40]. As the power level increased, so did the SNR, but SPM led to the aforementioned symbol rotation. Is in this case, the rectangular decision regions of maximum likelihood were not optimum any more, and clustering with non-rectangular boundaries fit the distorted data better. At even higher power levels, SPM led to more complicated cluster sizes where decision regions with linear boundaries, as those obtained using k-means, may result in sub-optimal classification. The differences between the resultant decision regions using maximum likelihood, k-means, and HBC can be better observed in Figure 4, where to make the contrast more clear, data for a launch optical power of 10 mW were employed. First of all, we show the histogram of the received constellation in Figure 4a to demonstrate how the distortion especially affected the symbols with higher amplitude. A detailed view of two of the constellation points that were critically affected by SPM (identified by a white rectangle) can be seen under the constellation plot. As can be observed, the two clusters presented a complex shape, and therefore, an intricate border was necessary to classify them. Figure 4b shows the constellation of the received data superimposed on the decision regions calculated using maximum likelihood. As expected, the rectangular grid led to multiple constellation point to invade adjacent regions even for low power symbols. The zoom-in figure clearly reveals that the decision boundary was not optimal for the symbol distribution.