## 6 Numerical Simulations  
The configurations of the simulated system are listed in Table 3. An OFDM system with the bandwidth 5000 Hz was built. BPSK and QPSK were chosen as the digital modulations. The number of subcarriers, which was 384, should be divisible by \(N_{\text{c}}\), whereas several possible \(N_{\text{c}}\)s were chosen to build the networks. Before comparing the performance, the network was firstly trained with mixed samples. As mentioned in Section 4, besides Rayleigh channels generated with MATLAB, the real channels collected in the pool, artificial lake, and Wuyuanan Bay, Xiamen, were taken as samples. All channel samples were mixed randomly in the proportion 1:1:1:1. For Rayleigh channel samples, the maximum CFO was set to 100 Hz. Moreover, Figure 7a-c shows the real scenarios to collect channels. The average depths of the pool, artificial lake, and testing sea area were 1 m, 5 m, and 8 m, while the depths of the transmitter and the receiver in the three areas were 0.5 m, 0.8 m, and 1.5 m. The only factor which affected the pool channel was the hard wall and bottom made of tiles. In addition, the average wind power in the artificial lake and testing sea area was \(<\)level 3. The outdoor tests were all performed in sunny days. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Detector** & **Number of MACC** & **Time Complexity** \\ \hline ELM & \(N(\frac{N_{\text{L}}^{2}}{3}+2N_{\text{L}}^{2}+\frac{N_{\text{L}}^{2}}{2}+8N_{ \text{L}}+\frac{5N_{\text{L}}}{6}+6+3M)\) & \(O(N^{4})\) \\ ZF & \(7N+3 MN\) & \(O(N)\) \\ MMSE & \(3(N^{3}+3N^{2}+2N+MN)\) & \(O(N^{3})\) \\ Proposed NN & \((2+M)NN_{\text{L}}+3N\) & \(O(N^{3})O(NN_{\text{L}})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time complexity of different detectors. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & \\ \hline Bandwidth & \(B\) & 5000 Hz \\ Modulation & \(-\) & QPSK/BPSK \\ Number of subcarriers & \(N\) & 384 \\ Block size & \(N_{\text{c}}\) & 1/2/3/4/6/8/12/16 \\ Proportion of training/testing set & \(-\) & 3/1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the simulated OFDM system. Different distances were covered to obtain different maximum delays shown in Figure 7d-f. Furthermore, the power threshold \(C_{\text{Th}}\) was set to 0.01 to filter the paths with low power. Taking mixture of the AWGN and impulsive noise generated with the GAN in [41] as noise samples, the dataset was finally constructed. Table 4 shows the training parameters for the network. The networks were trained with dynamic SNR and mean square errors (MSE) of the estimated channels. To accelerate convergence, ReLU was taken as the activation function, along with the Adam optimizer. The whole training process was conducted in Python with TensorFlow. Figure 8 shows the BERs of the given \(N_{\text{c}}\) changing with \(N_{\text{L}}\). The SNR was 25 dB and the assumed MSE of the channel estimated was 0.004. The step size for \(N_{\text{L}}\) was 4. It can be seen that large input dimensions, such as \(N_{\text{c}}=12\) and \(N_{\text{c}}=16\), required more neurons to reach the best performance. In addition, \(N_{\text{c}}\) for QPSK was larger than that of BPSK. To be clear, the estimated \(N_{\text{Lopt}}\)s from Figure 8 are listed in Table 5. For each \(N_{\text{c}}\), QPSK needed at least 20 more neurons than BPSK. \begin{table}
\begin{tabular}{c c} \hline Item & Value \\ Input size & 2\(N_{\text{c}}\) \\ Number of neurons & \(N_{\text{L}}\) \\ output size & \(MN_{\text{c}}\) \\ Activation function & ReLU \\ Optimizer & Adam \\ SNR for training (dB) & [20; 30] \\ Predetermined channel estimation MSE & [0, 0.03] \\ Epoch & 300 \\ learning rate & 0.001 \\ Platform & Python with TensorFlow \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters for network training. Figure 7: Scenes and their channel impulsive response: (**a**,**b**) are the scenes of pool and artificial lake in Xiamen University, while (**c**) is the testing area in WuyuanWay.