## 4 Results  
A 40 Gb/s 16 QAM waveform is transmitted with +16 dBm LOP over 50 km. Two parameters are needed to optimise the DBSCAN algorithm to produce the lowest BER, namely \(\varepsilon\) and the minimum number of points. It is worth noting that in a real-time communication link, the optimisation process of DBSCAN should be run offline as a training stage. In this stage, the optimal Voronoi regions would be generated, where in the real-time processing the incoming data would be assigned directly to ideal symbols. It is envisaged that such an approach should be very effective in optical communication links, where linear and nonlinear effects remain relatively stable over time, and therefore the training process should be run only once. The calculated BER while scanning for \(\varepsilon\) and the minimum number of points is shown in Figure 3. From Figure 3, it is evident that the majority of the best BER values can be found for \(0.1<\varepsilon<0.45\). For these values of \(\varepsilon\), the minimum points do not affect the BER much except when \(0.1<\varepsilon<0.14\), where the minimum points should be less than 110. In Figure 4, the performance of clustering algorithms is shown for different LOPs and two values of received OSNR: 30 and 15 dB. In Figure 4, the performance benefit of machine learning clustering over linear equalisation is significant for both OSNR values, especially when using DBSCAN method-2, resulting in up to 8.8 dB Q-factor improvement. This is attributed to the compensation of SPM since single-channel transmission is carried out. Results indicate that DBSCAN-based BNLE is a robust soft-clustering method when very strong nonlinear phase noise is present and where linear equalisation fails completely. Moreover, DBSCAN method-2 has the highest Q-factor along the whole range of LOPs. Compared to DBSCAN method-1 and K-means, method-2 increases the Q-factor by up to about 0.7 and 0.83 dB, respectively,  
Figure 2: Experimental setup for a 40 Gb/s 16 quadrature amplitude modulation (QAM) coherent optical signal transmitted at 50 km, incorporating machine learning clustering. PC: polarisation controller, OBPF: optical band-pass filter, LO: local oscillator, CMA/MMA: constant/multi-modulus algorithm, CPR (DDPLL): carrier phase recovery (decision-directed phase-locked loop).  
by better handling highly rotated clusters that become almost elliptically shaped. This is because the overlapping (soft) clustering ability of method-2 is more powerful than the common hard (exclusive) clustering of K-means and method-2 (which also includes K-means for the noisy constellation points). This is confirmed by the received constellation diagrams of Figure 5b, related to +16 dBm of LOP (OSNR = 30 dB). On the other hand, the quite similar performances between DBSCAN algorithms and K-means at lower LOPs and OSNR is due to the existence of nearly circular Gaussian clusters, which are not rotated. This is corroborated in the received 16 QAM constellation diagrams of Figure 5a at +10 dBm of LOP (OSNR = 15 dB). In the left constellation diagram of Figure 5a, the DBSCAN \"noisy\" points are also presented in the 1 st loop of the algorithm.  
Figure 4: DBSCAN vs. K-means for 16 QAM transmission at 50 km for different launched optical powers (LOPs) when optical signal-to-noise ratio (OSNR) is 30, 15 dB.  
Figure 3: DBSCAN optimisation for 16 QAM transmission over 50 km at +10 dBm of launch power: bit-error-rate (BER) vs. \(\varepsilon\) (“Epsilon”), Min. Points = 4.