## 1 Introduction  
With the development of ocean exploration, the demand for high-speed underwater wireless data transmission is increasing. Since radio frequency is not suitable for underwater communication because of the great power attenuation [1], acoustic communication is widely used, but due to the low transmission speed and large latency, the typical data rate of acoustic communication is extremely limited. Due to the low attenuation window [2, 3] at blue-green spectrum, underwater visible light communication (UVLC) is supposed to provide long-distance and high-speed transmission, and abundant investigations on light-emitting diodes (LEDs) have been reported [1, 4, 5, 6, 7]. However, inter-symbol interference (ISI) and nonlinear impairment are the two main problems that restrict the performance of UVLC system. On the one hand, the UVLC system is bandwidth-limited due to the low modulation bandwidth of LED [5], and the low-pass filtering effect will induce severe ISI. On the other hand, the UVLC channel is faced with absorption, scattering, and turbulence [8]. Therefore, large driving power is required to ensure the desired signal-to-noise ratio (SNR) at the receiver. The nonlinearityfrom optoelectronic devices such as LEDs, photodiodes, and electrical amplifiers (EAs) [6; 7] becomes severe at high amplitude. In order to eliminate the combination of ISI and nonlinear impairment, digital signal processing (DSP) is required. Generally, DSP includes equalization and decoding. Constant modulo algorithm (CMA), least mean square (LMS)-based linear equalizers, and Volterra series or polynomial-based nonlinear equalizers [9; 10; 11] have been widely investigated. As to decoding, hard decision based on Euclidean distance is most commonly used, whereas soft decision is supposed to provide better performance. In this context, the latest researches have employed machine learning algorithms such as support vector machine (SVM) [12; 13; 14; 15] and K-means [16; 17] for soft decision. SVM-based soft decision is especially effective for quadrature amplitude modulation (QAM) when there is strong nonlinearity [14; 15], because the decision boundary can be adjusted according to the distribution characteristics of constellation points. However, previous works mainly concentrate on 2-dimensional (2-D) separate symbol [12; 13; 14; 15], whereas the correlation between consecutive symbols is ignored. Inspired by [18; 19], the constellation can be expanded to a higher dimensional constellation by concatenating consecutive symbols. For QAM signal, the original 2-D constellation is expanded to a 4-D constellation. The nonlinear effect and ISI interact with each other. It is difficult to remove the correlation between adjacent symbols thoroughly with a feedforward equalizer, and the high-dimensional clusters will be distorted due to the combination of residual ISI and nonlinear impairment. In this paper, we propose to employ SVM to find the optimal decision hyperplane for the consecutive-symbol-expanded 4-D constellation in an experimental UVLC system. The proposed scheme is used for soft decision, independent of the post-equalization process. As the intensity modulation-direct detection (IM-DD) system requires a real and positive signal, a pair of carrierless amplitude and phase (CAP) shaping filters are used for CAP modulation [20; 21] so that the complex-valued QAM signal can be converted into a real-valued signal. Different from existing works that employ SVM for 2-D normal constellation [12; 13; 14; 15], the correlation between consecutive symbols is considered in this work. However, for the consecutive-symbol-expanded 4-D constellation, the computational complexity of SVM training phase becomes higher because of the demand for more classifiers and larger training data size. To deal with the high computational complexity resulting from the demand for more classifiers, the bit-based binary SVM is applied as the multi-class SVM strategy, so that the number of required classifiers can remain small [15].