## 2 System Model
### Learning the Power Amplifier with Neural Network  
In this article, firstly, we conducted research to reduce the PAPR in underwater acoustic OFDM communication. Secondly, unnecessary distortion, which is caused by high PAPR, was reduced at the receiver. It affects the BER performance of the overall system; for this, we need to learn the PA model at the receiver. To learn a model we need to train the Machine learning algorithm with a proper set of data. At the beginning of this communication, a set of QAM symbols is generated by the transmitting transducer with different amplitudes over the range of PA characteristics. The data are transmitted through a noisy underwater acoustic BELLHOP Gaussian beam tracking model with several delays and multipath effects. The PA machine learning model at the receiver side is explicated in Figure 2. Hence, at the receiver's data acquisition, this model is used. If there are better and more appropriate data sets, one can use various deep learning algorithms and train the model before using this distortion mitigation process. Here, we have implemented a neural network (nonparametric) model.  
The neural networks are being classified under the nonparametric models. The weight of the hidden neurons that is learned does not provide physical meaning for the undergoing consideration of the problem. The main aim of training a neural network is to estimate the whole function, not only approximating the weights (as in the parametric model). The proposed distortion algorithm demands the knowledge of the PA model at the transmitter side. In this paper, we implement a feed-forward machine learning (neural network) method that estimates the PA characteristics model at the receiver side. A neural network has the advantage of using the method as a universal appropriator. It can realize an arbitrary mapping of a single vector space onto different vector spaces. The training process is regarded as the leaning of weights; supervised and unsupervised are the two types of training processes. The supervised training process is used when the NN knows the output, and it adjusts the weights accordingly. An example is a feed-forward network. In the unsupervised training process, the NN does not know about the output; it recognizes a random pattern and develops a certain relationship. The multilayer feed-forward neural network (MLFNN) is used in this article as we know about the desired output values. Hence, it is the correct decision.  
The neurons are ordered as input, hidden, and output layers in a MLFNN, as shown in Figure 3. In each layer, the neurons relate to another neuron in the upcoming layer. There is a connection between \(i^{th}\) and \(j^{th}\) neuron, which is characterized by \(w_{ij}\) and threshold coefficients of \(\vartheta_{i}\) and \(\vartheta_{j}\), as described in  
Figure 2: The framework of the OFDM receiver.  
\[y_{j}=f(\xi_{j}) \tag{12}\] \[\xi_{j}=\delta_{j}+\sum_{i\in\Gamma_{j}^{-1}}\omega_{ji}y_{i} \tag{13}\]  
In Equation (12), \(\xi_{j}\) denotes the potential of the \(i^{th}\) neuron function \(f(\xi_{i})\) or the transfer function. The transfer function is applied to all neurons \(i^{th}\), the transferring signal to the \(j^{th}\) neuron. The threshold coefficient is the weight coefficient of the connection between i neuron, where \(y_{i}=1\); it is called bias. Then this transfer function can be a sigmoid, defining a nonlinear solution.  
\[f(\xi_{i})=\frac{1}{1+e^{-\xi_{i}}} \tag{14}\]  
The threshold \(\delta_{j}\) and weight \(w_{ji}\) coefficients are changed to reduce the sum of the square difference between the actual and desired outputs. After this, the minimum cost function can be written as  
\[Q=\sum_{j}\left(y_{j}-\hat{y}_{j}\right)^{2} \tag{15}\]  
The desired output and actual output runs overall j are denoted by vector \(y_{j}\) and \(\hat{y}_{j}\). There is a different training algorithm described for calculating the weight and threshold values in different researches. The most common algorithm is the backpropagation algorithm.