## 4 Results and Discussions
### Performance Evaluation of Neural Network and Bit Error Rate  
In this implementation, the data set of 2000 \(\times\) 612, having subcarriers 512 with 100 cyclic prefixes, is shaped again into smaller sets of data. In this way, the six data scalar inputs are added to the neural network. Here, the subcarriers received are 612. The software uses only 516. Then, we form the 2000 \(\times\) 512 data set matrix into smaller sets. For example, 25,800 \(\times\) 600 data sets will have 300 OFDM symbols, which are evaluated as 25,800 \(\times\) 6/516 = 300, as shown in Table 3. For better estimation of the PA model, the data which have been fed to neurons are important with regard to quality and quantity. When implementing the neural network, only one hidden layer is used with 12 neurons, which has six input and six output neurons. The sigmoid transfer function is used to define neurons. If we add another hidden layer that has enough numbers of neurons, this will fit any finite input and output problem. Increasing the number of neurons will demand more training time and computation capability. As we know, the received OFDM symbols \(X\) used for training the proposed PA model are not dependent.  
Figure 8: Simulation result of the channel impulse response.  
Figure 7: Complementary cumulative distribution function (CCDF) of PAPR performance for proposed and conventional schemes.  
Therefore, we can reduce the number of inputs in the neural network. In contrast, the number of neurons at the input is less than the length of OFDM symbols. By doing this, the convergence becomes a lot faster and more accessible when learning the relationship between 6 rather than 612 input symbols (in this simulation, the OFDM symbol length was 612). In a feature vector, the actual number of neurons is the same as the columns in a neural network. Here, we have reshaped the feature vector from 612 to 6. Hence, the computational complexity is reduced. We selected the number 6 for simulation. It is possible to choose a different number of less than 10 feature vectors for evaluating and analyzing the performance of this model. The bad estimation was observed beyond the PA saturation region when the number of hidden neurons was less than 12. Therefore, we studied the algorithm results and simulated the whole NN with 12 hidden neurons. The learning algorithm is shown in Table 3.  
The performance of the FFB algorithm was analyzed when the distortion was only due to the PA, and distortion was also verified with the signal clipping. Let us show how the BER reduces when we apply FFB to the overall OFDM system, with and without clipping. For this case, we used the simulation parameters given in Table 1. In Figure 9, the blue line shows the BER curve without applying clipping and the FFB algorithm. At the same time, the BER curve (red line) is shown for the TFNN method. By adding the clipping, there is an improvement in the BER performance with the FFB algorithm; it can be seen in the green and yellow lines. The nonlinearity distortion is caused by a PA when it gets into a saturation region. From Figure 9, the proposed FFB algorithm outperforms the rest when it is trained with the neural network. The neural network gives us the best result at all SNRs above 10 dB.