## 4 Simulation Results and Analyses
### Comparison between Different Equalizations  
An EXIT chart [32] is adopted to analyze the convergence of the turbo equalization under the four-channel underwater acoustic channel. It is a powerful semi-analytical tool to analyse and design iteratively decoded systems with soft information exchange. The exchange of extrinsic information between the constituent decoders is verified using EXIT charts, which characterise the flow of extrinsic information exchange between the constituent decoders of a concatenated structure, such as the turbo equalization. It determines the order of soft-information exchange among the three components of the two constituent decoders and equalizer. Furthermore, it also exhibits beneficial decoding convergence after a fixed number of iterations, which resulted in a complexity reduction. The SNR is set to 1 and 3 dB, and the setting of other parameters is set the same as the above subsection of the comparison of channel estimation. The channel estimation adopts the SZA-IPNLMS in the initial stage and the SM SZA-DIPNLMS in the main stages of iterative channel estimation. The length of the forward filter for the SDFE and LE is set to 148, and the length of the feedback filter is set to 74. After numerical simulations, the results of the EXIT charts of different equalization schemes in SPACE'08 and MACE'10 are shown in Figures 8 and 9, respectively. In the EXIT chart, \(\mathbf{I}_{i}^{E}\) and \(\mathbf{I}_{o}^{E}\) denote the input and output mutual information of the equalization, and \(\mathbf{I}_{i}^{D}\) and \(\mathbf{I}_{o}^{D}\) denote those of the decoder. Information theory states that a larger amount of mutual information leads to a more reliable system [33]. Figures 8 and 9 are in accordance with the analyses of EXIT charts in SPACE'08 and MACE'10. At 1 or 3 dB, the IFDTE is better than other equalization schemes in terms of early iteration due to the self-iteration of the EP. It has better performance under low SNR and it also has a wide \"tunnel\" area, and the width of \"tunnel\" is proportional to \(\mathbf{I}_{o}^{D}\). Thus, the wider the width is, the better the performance. Thus, the performance is ranked as IFDTE \(>\) SDFE \(>\) Exact-LE \(>\) FDDF-FDTE under 1 dB and IFDTE \(>\) SDFE \(>\) FDDF-FDTE \(>\) Exact-LE under 3 dB. Therefore, the IFDTE has a faster convergence and better performance than other equalization schemes. The BER curves show the recovery ability of the received signals. The range of SNR is from \(-1\) dB to 3 dB with 0.5 dB intervals. The BER curves are simulated and shown in Figures 10 and 11. With the increase in SNR, the BER performance of all equalization schemes is improved significantly. In Figure 10, the IFDTE has 1, 0.5, and 0.4 dB gains compared with FDDF-FDTE, Exact-LE, and SDFE, respectively, under the time-varying UWAC channel of SPACE'08 with a BER of \(10^{-3}\). In this simulation, the proposed Bi-EPIC-FDTE obtains the best BER performance of \(3.35\times 10^{-2}\), \(4.74\times 10^{-4}\), and \(1.98\times 10^{-5}\) at SNR of 0 dB, 0.5 dB, and 1 dB among all contrasted schemes, respectively. In Figure 11, LE-FDTE in the time-varying UWAC channel of MACE'10 cannot converge well in the set SNR under the same BER of \(10^{-3}\). The IFDTE has 0.5, 0.4, and 0.2 dB gains compared with the Exact-LE, FDDF-FDTE, and SDFE, respectively. In this simulation, the proposed Bi-EPIC-FDTE obtains the best BER performance of \(4.12\times 10^{-2}\), \(1.87\times 10^{-4}\), and \(7.98\times 10^{-6}\) at SNR of \(-0.5\) dB, 0 dB, and 0.5 dB among all contrasted schemes, respectively. The result can be explained as follows. The IFDTE estimates the a posteriori probability iteratively with the Gaussian distribution in (13), (16), and (17) to estimate the actual a posteriori distribution iteratively with moment matching. Thus, the IFDTE outperforms its counterparts in terms of other equalization schemes. In Figures 10 and 11, the bidirectional structure of equalization provides 0.25 and 0.2 dB for the IFDTE in SPACE'08 and MACE'04 with bidirectional gain.