## 2 Principle
### RBFNN Nonlinear Equalization  
Neural networks are utilized to process complex input data and make corresponding predictions. They consist of numerous neurons and the connections between them. Each neuron receives input from the neurons in the previous layer, which is weighted and then combined with a bias term. The output is subsequently computed through a specified activation function, including the sigmoid function, the hyperbolic tangent (tanh) function, and the rectified linear unit (ReLU) function. The input data undergoes forward propagation, passing layer-by-layer through the neural network to generate the output. Subsequently, optimization techniques, including gradient descent, are employed to optimize the weights and biases of the network in order to minimize the error. RBFNN is a form of feedforward neural network with a three-layer topology. Figure 3 illustrates the construction of the RBFNN. The input layer consists of nodes that receive the input signal and serve solely as conduits for data transmission without making any transformations to the input information. The second layer, known as the hidden layer, completes a spatial mapping transformation for the input data, with the number of nodes determined based on specific requirements. The activation function in the hidden layer neurons is RBF, typically a Gaussian kernel function, which depends only on the Euclidean distance between the input data and the center point. The corresponding formula is as follows:  
\[\phi_{i}(x)=\exp\left(-\frac{\|x-c_{i}\|^{2}}{2\sigma_{i}^{2}}\right) \tag{1}\]  
Figure 2: **(a)** Mapping of the second quadrant of 64 QAM based on LUT-PS, where points on the outer circle are mapped to the inner circle according to their corresponding colors. **(b)** Probability distribution of the 64 QAM signal. where \(x\) is the input vector, \(c_{i}\) is the \(i\)th center point of the RBF, and \(\sigma_{i}\) is the width parameter of this RBF, which controls the sphere of action of the function. The third layer is the output layer, which employs a linear activation function. It computes the weighted sum from the outputs of the neurons in the hidden layer and generates the neural network's final output. The expression is as follows:  
\[y_{i}=\sum_{j=1}^{N}\omega_{jk}\phi_{i}(x) \tag{2}\]  
where \(\omega_{jk}\) represents the weight between the hidden layer and the output layer and \(N\) signifies the quantity of hidden layer nodes. A complete RBFNN operates in two stages: the training stage and the testing stage. During the training stage, the network learns the nonlinear functional relationship between the input and the target by training on a large dataset without requiring prior information and subsequently generates the network model. In the testing stage, the trained network model is used to make predictions on the entire dataset. To effectively train the RBFNN, the Levenberg-Marquardt algorithm is employed. This is an iterative optimization technique commonly used to minimize the error function. This algorithm combines the advantages of gradient descent and [PERSON]'s method, demonstrating excellent performance in solving small-scale nonlinear problems. The specific training process is as follows: In this work, 40% of the received data is randomly selected as the training set, and an additional 20% is used as the testing set. The corresponding original data serve as the target set. The complex-valued signal is separated into its real and imaginary parts, which are fed independently into the RBFNN-based nonlinear equalizer. Based on the constructed network model, the output is computed and compared with the target values to evaluate the prediction error. If the error is below the predefined threshold, the training is terminated; otherwise, the algorithm checks whether the maximum number of iterations has been reached. If so, training stops; if not, the parameters are further optimized until convergence is achieved.