# Design of PAM-8 VLC Transceiver System Employing Neural Network-Based FFE and Post-Equalization
## 3 Analysis of Equalization Methods
### DNN and RBF-NN  
Both a DNN and an RBF-NN were adopted in the VLC transceiver system as the post-equalization for comparing their performances. The target of adopting the NN-based post-equalization was to equalize the received signal to make the output signal as approximate to the original signal as possible. Since NNs can approximate any nonlinear function, this superior characteristic made it possible to fit the inverse matrix of the VLC link limitations. Therefore, by applying the trained NNs in the baseband of the receiver, the effects of the VLC link limitations could be eliminated by the inverse matrix. The structure of the three-layer DNN is presented in Figure 8, and it contained one input layer, three hidden layers, and one output layer. The input signal of this DNN was \(\mathrm{X}=[\mathrm{x}_{1},\ \dots,\ \mathrm{x}_{n}]\), which was generated from the received signal with delay. In the hidden layer, the input signals \(\mathrm{X}\) were multiplied with weight \(W_{m}\) and then summed together with bias weight \(b_{m}\) for the following process through activation function f. The output of an m-hidden neuron was \(f(X\cdot W_{m}+b_{m})\), and the final output was:  
\[\mathrm{y}(\mathrm{n})\mathrm{=f}\Big{(}\mathrm{f}\Big{(}\mathrm{f}\Big{(} \mathrm{X}\cdot W_{\mathrm{m}}^{1}+\mathrm{b}_{\mathrm{m}}^{1}\Big{)}\cdot W _{\mathrm{m}}^{2}+\mathrm{b}_{\mathrm{m}}^{2}\Big{)}\cdot W_{\mathrm{m}}^{3}+ \mathrm{b}_{\mathrm{m}}^{3}\Big{)}\cdot\mathrm{W_{m}}, \tag{10}\]  
where \(W_{\mathrm{m}}^{1}\), \(W_{\mathrm{m}}^{2}\), and \(W_{\mathrm{m}}^{3}\) are the neuron weights of first, second, and third hidden layers, respectively; and \(W_{\mathrm{m}}\) is the weight of the output layer. The typical activation function in a DNN is the sigmoid function, and its training target was the originally transmitted signal. After the training process, the weights in this three-layer DNN were determined, and this trained DNN could be utilized as a post-equalizer to compensate for the newly received signals. Figure 8: Structure of DNN. Figure 7: Structure of NN-based FFE. The radial basis function utilized the distance between the input and fixed center point to determine the value of the output, and the common expression of this function is:  
\[\phi(\mathrm{x})=\phi(\|\mathrm{x}-\mathrm{c}\|), \tag{11}\]  
where the function \(\phi(\cdot)\) is the radial function, the distance is called Euclidean distance, and \(\mathrm{c}\) is the radial kernel center. With this definition, the RBF-NN is a neural network that utilizes RBF as the activation function [32]. A conventional RBF-NN only consists of three neuron layers: the input layer, the hidden layer, and the output layer, as shown in Figure 9. The input layer sends input data to the hidden layer, where the activation function is the radial basis function. In this hidden layer, the Euclidean distance between the input data and the kernel center is calculated and utilized to determine the output of the hidden layer. There are various choices for the activation function, such as the Gaussian kernel function, multiquadric function, and inverse quadric function. The output layer linearly combines the output from the hidden layer and provides the final output of the RBF-NN. The weights of this linear layer are also changed during the training process. The RBF-NN adopted in the proposed VLC system is presented in Figure 9. The input signals were received from the TIA after down-sampling and synchronization, and then the sampled data were delayed, forming a matrix with a scale equal to the window length of (L + 1) in which the window length represented the delay time of the original signal and was related to the size of the training dataset. Therefore, it could impact the training performance, and a moderate value was required for the tradeoff of training performance and training time.