## 2 Principle
### Principle of CAP Modulation and Conventional CAP Receiver
Principle of a Cascaded Receiver Consisting of a DNN-Based Waveform-to-Symbol Converter and Modified NN-Based DD-LMS  
Thanks to the artificial neural network's universal approximation theorem, it is feasible to convert the received signal waveform with severe ISI and distortions into complex symbols or constellation points directly by using a well-trained neural network [23]. It is called a waveform-to-symbol converter that replaces conventional waveform-level equalization, matched filtering, down-sampling, and symbol-level post-equalization. Compared with conventional multi-stage processing at the receiving end, a DNN-based waveform-to-symbol converter optimizes the receiver as a whole and achieves better performance due to the powerful ability of a neural network to find out implicit relationships between received waveforms and symbols [24]. However, the converter must be of large enough complexity to convert received waveforms into symbols perfectly. To achieve a better tradeoff between communication performance and computation complexity, a cascaded receiver consisting of a DNN-based waveform-to-symbol converter and modified NN-based DD-LMS is proposed in this paper. Although with fewer taps and nodes than traditional converter [24], we will optimize in this paper the front-stage DNN-based waveform-to-symbol converter could mitigate the majority of ISI and signal nonlinear distortions. Then modified NN-based DD-LMS is cascaded to eliminate remaining ISI and distortions and improve signal quality even further. Thus, the cascaded receiver could achieve excellent communication performance with relatively low complexity. The schematic of the proposed algorithm is shown in Figure 1. At the transmitter end, an original pseudorandom binary sequence is generated and then mapped to 64 QAM or 64 APSK complex symbols using GS technology. The constellation points of 64 QAM are square, while those of 64 APSK consists of four rings. These symbols are converted to transmitted waveforms after CAP modulation. After traveling from transmitter to receiver, the received signals have experienced severe inter-symbol interference (ISI) and distortions due to bandwidth limitation of LED, various noises, and linear and nonlinear damages. The transmitted and received signals are displayed in Figure 1, where the waveform distortions can be observed both in the time domain and frequency domain, respectively. The received signals are then sent into a DNN-based waveform-to-symbol converter, whose structure is shown in Figure 1a. It consists of one input layer with N nodes, three hidden layers with n\({}_{1}\), n\({}_{2}\), and n\({}_{3}\) nodes, respectively and one output layer with two nodes that correspond to the real and imaginary parts of the output. Considering inter-symbol interference, a sliding window is applied to received signals \(x(t)\). The \(j^{th}\) slice can be expressed as:  
\[X(j)=\left\{x\left(j-\frac{l-1}{2}\right),\ldots,x(j-1),x(j),x(j+1),\ldots,x \left(j+\frac{l-1}{2}\right)\right\} \tag{5}\]  
where \(l\) is an odd number that represents the length of the sliding window. The step size of the sliding window equals the number of up-sampling. For example, four times up-sampling is adopted in this paper, so the step size would be set as four. Then each slice will be calculated by three hidden layers in turns and connected to two output nodes to produce the real and imaginary parts, respectively. The \(j^{th}\) output can be expressed as:  
\[y_{out}(j)=\left[y_{r}(j),y_{i}(j)\right]^{T} \tag{6}\]  
where \(y_{r}(j)\) and \(y_{i}(j)\) denote the real and imaginary parts of \(j^{th}\) output. The relationship between input and output can be expressed as:  
\[y_{out}(j)=W^{4}f\left(W^{3}f\left(W^{2}f\left(W^{1}f(X(j))+b^{1}\right)+b^{2} \right)+b^{3}\right)+b^{4} \tag{7}\]  
Figure 1: The schematic diagram of the UVLC system using the cascaded receiver. (**a**) shows the details of the front-stage DNN-based waveform-to-symbol converter at the front stage.