## 2 Principle
### SVM-Based Soft Decision for 4-D Constellation  
Given the transmitting \(m\)-QAM sequence \(s(n)\) and the receiving sequence \(r(n)\), previous works about SVM-based soft decision only consider the separate symbol (\(r_{t}(n)\), \(r_{Q}(n)\)) as the 2-D feature vector (\(x_{1}\), \(x_{2}\)), whereas the correlation between consecutive symbols is ignored. By concatenating the I/Q components of two consecutive symbols, the original 2-D constellation is expanded to the 4-D constellation. Generally, considering more consecutive symbols may provide better performance, the additional computational complexity should also be considered. For the \(m\)-order QAM signal, the combinations of consecutive symbols will increase to \(m^{n}\) with \(n\)-consecutive-symbol-expanded high-dimensional constellation. The complexity grows rapidly as the number of consecutive symbols increases. However, strong interference usually comes from adjacent symbols, and most of the ISI can be eliminated by the equalization process. Therefore, we mainly consider 2-consecutive-symbol-expanded 4-D constellation for soft decoding. The 4-D feature vector (\(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\)) for SVM is (\(r_{t}(n)\), \(r_{Q}(n)\), \(r_{t}(n+1)\), \(r_{Q}(n+1)\)). The \(2m\)-bit binary labels of the 4-D feature vectors \begin{table}
\begin{tabular}{c c} \hline \hline
**Multi-Class Strategy** & **Number of Classifiers** \\ \hline OVO SVM & \(m(m-1)/2\) \\ OVR SVM & \(m\) \\ Bit-based binary SVM & \(\log_{2}m\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of required classifiers for different multi-class strategy. are generated by concatenating the original \(m\)-bit binary labels of the two consecutive labels, so that we can ensure that if the bit-to-symbol mapping of the original 2-D constellation follows gray coding, the bit-to-symbol mapping of the expanded 4-D constellation also follows gray coding. Thus, the proposed scheme can be employed for \(m\)-order QAM signal with an even \(\log_{2}m\). The combinations of consecutive symbols for 64-QAM are \(64^{2}\), the required training data size is too large for implementation, while the spectral efficiency QPSK seems not sufficient for a high-speed UVLC system. As a result, we choose 16-QAM for validation. As Figure 1a shows, we decompose the 4-D constellation into four 3D diagrams according to the \(s_{Q}(n+1)\) level. Different colors denote different labels. There are 256 (\(16^{2}\)) labels, corresponding to the 8-bit binary sequence. As for the multi-class SVM strategy, the common OVO and OVR methods are not suitable for the expanded high-dimensional constellation anymore, because the number of classes increases from 16 to 256. The numbers of required classifiers for OVO and OVR become 32,640 and 256, respectively, whereas the number of required classifiers for the bit-based binary SVM is \(\log_{2}m\). If bit-based binary SVM is employed as the multi-class strategy, only 8 classifiers are needed, corresponding to the binary label of the 4-D constellation. Therefore, the complexity can be significantly reduced. Furthermore, in order to go through enough points in each class, a large amount of training data is needed. However, as mentioned above, the SVM classification hyperplane only depends on a few support vectors. There are redundant training data which have no contribution to the SVM training phase but instead bring high computational complexity. It can be more efficient if the edge points are picked out in advance. As the support vectors are the closest to the hyperplane, we can infer that these support vectors are included in the edge points of each training data cluster. Therefore, we propose to use a data preprocessing method to reduce the redundant training data. As Figure 2a shows, the 4-D training feature vectors \((x_{1},x_{2},x_{3},x_{4})\) can be decomposed to six 2-D constellations. It is noted that the I/Q scale of the constellation is ranging from \([-4,\,4]\). Then, the constellations are converted to six binary images through pixel division.