## 4 Results and Discussion  
This section introduces the experimental results and discussion. First, the performance of SVM-based soft decision for normal 2-D constellation and consecutive-symbol-expanded 4-D constellation at different training data size is given in Figure 4. It should be noted that the system is working at 1.1 V and 1.6 Gbps, and data preprocessing is not employed. As Figure 4a shows, the convergence speed for normal constellation with SVM is faster because the number of classes is smaller, whereas, if there are enough training data, 4-D constellation with SVM can provide better bit error rate (BER) performance. However, as Figure 4b shows, the percentage of support vector (SV) in the training data is decreasing when there is more training data. It indicates that there are considerable redundant training data. The complexity of the training phase scales between O(dNt2) and O(dNt3) for SVM with d-dimensional feature vectors and Nt training points. Therefore, data preprocessing is necessary for reducing computational complexity. To make a trade-off between BER performance and SVM training complexity, the training data size is set as 26,214. The trained SVM models can be applied for a period of time, because the system is approximately deterministic at a fixed operating condition. In a practical system, the training process can be operated offline for some typical operating conditions. Figure 3: Experimental setup of the underwater visible light communication system. To search for the optimal parameters for the data preprocessing method, we measure the BER performance and the compression ratio (\(N_{\text{edge}}/N\)) with different pixel size Rs and Threshold of the Sobel operator. First, the experimental results are measured at a data rate of 1.6 Gbps and V\({}_{\text{pp}}\) of 1.1 V, as shown in Figure 5a,b. Figure 5a shows the BER performance under different parameter conditions. The area with larger value indicates a better BER performance. However, not all the Threshold and Rs pairs with large-log (BER) are appropriate to simplify the training data set in the edge detection process. Figure 5b shows the compression ratio of the training data after pre-processing. The area with a lower value indicates that the Sobel edge detector worked and the training set becomes simplified, to a certain extent. To make a tradeoff between the BER performance and compression ratio, we prefer the parameters that correspond to lower BER and lower \(N_{\text{edge}}/N\). As a result, at the data rate of 1.6 Gbps, the optimal Threshold ranges from 0.1 to 0.23, while the optimal Rs ranges from 0.057 to 0.08. To ensure the generalization of the optimal parameters, the BER performance is also measured at the operating condition of 1.1 V, 1.4 Gbps. The results are shown in Figure 5c,d. Similarly, at the data rate of 1.4 Gbps, the optimal threshold ranges from 0.1 to 0.23, meanwhile the optimal Rs ranges from 0.057 to 0.08. The optimal ranges are coincident with that at the data rate of 1.6 Gbps. Figure 4: (**a**) BER versus training data size. (**b**) The percentage of SV size versus training data size for 4-D constellation with SVM. Figure 5: Parameter selection based on grid search. (**a**) BER performance, and (**b**) at 1.6 Gbps. (**c**) BER performance, and (**d**) at 1.4 Gbps. More specifically, as Figure 6a shows, we illustrate the BER performance versus Rs in the case of two fixed Threshold values under the condition of a data rate of 1.6 Gbps and V\({}_{\text{PP}}\) of 1.1 V. When Threshold equals 0.2, the BER keeps a low value. The constellation diagrams after edge detecting under certain values of Rs are inserted as subfigures. We can observe that the constellation F obtains a clear profile, which effectively reduces the training data size. By contrast, constellation B shows that the edge detection algorithm has not obtained a good result and that there are many redundant data even though BER performance is barely satisfactory.