## 4 Results and Discussion
Similarly, as Figure 6b shows, we also illustrate the BER performance versus Threshold in the case of two fixed Rs values. We can observe that the constellation A is well edge detected, while the constellation B has a lot of redundant data. It can be seen from the aforementioned experimental results that the edge detection algorithm can achieve comparatively optimal performance when the threshold is 0.072 and Rs is 0.2, the parameters adopted in rear experiments. Next, we investigate the BER versus box constraint, which is a key parameter of SVM, corresponding to C in Equation (2). The BER performance is measured by changing box constraint from 0.1 to 100 at 1.4 Gbps and 1.6 Gbps. As shown in Figure 7, when the box constraint increases, the BER firstly decreases. When the box constraint reaches around 1, the BER drops to a minimum value, and the BER gradually increases as the box constraint increases. The constellation diagrams at 1.4 Gbps and 1.6 Gbps are inserted as inset (i) and inset (ii), respectively. The trend of the BER curve at 1.4 Gbps is slowly varying because the ISI is not severe at a low data rate, and the boundary of constellation clusters is clear. However, when the data rate exceeds 1.6 Gbps, the constellation is distorted more severely. BER increases significantly as box constraint becomes larger, indicating that it is more at risk of overfitting. When the box constraint is 1, the SVM-based soft decision performs well at the two data rates. We can infer that constellation with less distortion has higher tolerance for the variation of box constraint. The parameter can be obtained at the condition with a high data rate and applied to other operation condition with lower data rate. Figure 6: (a) BER versus the pixel-division size Rs and (b) BER versus the Sobel parameter Threshold at 1.6 Gbps. Figure 8 shows the projection of 4-D constellation points on a 2-D plane. The decision boundary obtained by SVM is also illustrated. Since the four axes are combined in pairs, six different projection planes can be illustrated (Figure 8a-f). As can be seen from Figure 8a,f, the constellation points spread out into a nearly circular shape because the axes of the projection plane are from the same time while the constellation clusters in Figure 8b-e spread out into an oval shape because the axes are from different time. The correlation between the two consecutive symbols will crosstalk each other and lead to the special ellipse constellation. Additionally, the outer constellation points spread more than the inner circle, which indicates the nonlinear effect in the UVLC system. The SVM-based decision boundary for elliptical constellation clusters is adjusted adaptively according to the distribution characteristics of constellation points. Therefore, the proposed method can deal with the distortion resulting from residual nonlinearity and ISI after post-equalization. The adaptive decision boundary is supposed to outperform the simple hard decision. Figure 8: (**a–f**) are the projection of 4-D constellation points on a 2-D plane and the SVM-based soft decision boundary. Figure 7: BER versus box constraint at 1.4 Gbps and 1.6 Gbps. (i) and (ii) are constellations at 1.4 Gbps and 1.6 Gbps respectively. Figure 9 shows the BER performance on different baud rare under conditions without SVM, normal 2-D constellation with SVM (abbreviated as 2-D SVM in Figure 9), and 4-D constellation with SVM (abbreviated as 4-D SVM in Figure 9) after post-equalization. It should be noted that the SVM-based soft decision is applied in the decoding process, independent of the post-equalization. The solid lines are results employing LMS-based linear equalization. Obviously, 4-D constellation with SVM performs best, followed by normal 2-D constellation with SVM and without SVM. When the baud rate is low, the nonlinear effect is the dominant impairment, normal 2-D constellation with SVM can provide a significant performance gain.