## 6 Numerical Simulations
(**d**–**f**) are three of the CIRs caught in these spots, which show large delays. It should be noted that the optimal \(N_{\text{c}}\) of both modulations in Figure 8 is 1. This is because the training set contained multiple CIRs, including Rayleigh channels, with different delays. In this situation, the coherence bandwidth of the channel samples was limited to a small value. To further demonstrate the influence of the coherence bandwidth, the trained networks were further used to detect the symbol from the channels with specific maximum delays. The pool channels in Figure 7a were cut off with lengths of 48, 64, and 77 points, the corresponding \(N_{\text{w}_{\text{c}}}\)s of which were 8, 6, and 5. Figure 9a-f show the BERs with different \(N_{\text{c}}\) in BPSK-OFDM and QPSK-OFDM systems. The BER curves of \(SNR=[15,25]\) dB are enlarged in Figure 9b,d,f. The MSE of the channels estimated was set to 0.01. It can be seen from the figures that the optimal \(N_{\text{c}}\) varies with modulation and \(N_{\text{w}_{\text{c}}}\). According to Algorithm 1, the optimal \(N_{\text{c}}\)s are listed in Table 6. It can be found that in all situations, the networks with \(N_{\text{c}}>N_{\text{w}_{\text{c}}}\) showed poor performance. These results prove the analysis in Section 4, and further demonstrate the feasibility of Algorithm 1. In addition, the networks with optimal \(N_{\text{c}}\)s were compared with the ZF and MMSE equalizers. The pool channel with \(N_{\text{w}_{\text{c}}}=8\) was used. Figure 10 shows the BERs of different equalizers. The proposed network showed lower BERs than the ZF and MMSE equalizers both under perfect channel estimation (MSE = 0) and MSE = 0.01 of the estimated channel. In particular, when \(\text{SNR}=20\) dB and MSE = 0, the BER of QPSK detection network was 18.66% lower than that of the ZF and MMSE equalizers, while the BER of BPSK detection network was 14.23% lower than that of the ZF and MMSE equalizers. When \(\text{SNR}=20\) dB and MSE = 0.01, the BER of QPSK detection network was 29.26% lower than that of the ZF and MMSE equalizers and the BER of BPSK detection network was 22.16% lower than that of the ZF and MMSE equalizers. In addition, it can be seen from Figure 10 that the BER curve of MMSE equalizer for each estimation error is almost the same as the ZF equalizer, which confirms the discussion in Section 2. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **BPSK** & **QPSK** \\ \hline \(N_{\text{w}_{\text{c}}}=8\) & 2 & 2 \\ \(N_{\text{w}_{\text{c}}}=6\) & 1 & 3 \\ \(N_{\text{w}_{\text{c}}}=5\) & 4 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The optimal \(N_{\text{c}}\) for different systems. Figure 8: BER of different \(N_{\text{c}}\) varying with \(N_{\text{L}}\). Further, the above equalizers were tested with the LS estimation. The ELM-based detector in [35] was also compared. All receivers apply the minimum Euclidean distance method to detect symbols. Figure 11 shows the BER for these detectors. It can be seen that the ELM-based detector showed the worst performance in pool channels. The NN-based receiver still showed low BERs. When SNR = 20 dB, the BERs of NN were lower than ZF and MMSE equalizers for 25.92% and 30.99% under BPSK and QPSK modulations. Figure 9: BER of different \(N_{\rm c}\) in channels with specific maximum delays. (**a**,**c**,**e**) are complete BER curves of different \(N_{\rm w_{\rm c}}\)s. The BER in the range of [15, 20] dB are enlarged in (**b**,**d**,**f**), respectively. The above results have proved the efficiency of the proposed NN-based detector. Furthermore, the underwater trials introduced in Section 6 show the practicability of the proposed detector.