## 4 Results  
### Simulations  
In the simulations, we concentrate on evaluating the BER performance of the proposed VSTD-TEQ compared with aspects of the fractionally spaced soft DFE-based DA-TEQ (VSD-TEQ) and soft DFE-based DA-TEQ (SDA-TEQ). Because SDA-TEQ and time-reversed structures have been proven to demonstrate significant superiority over the Hard-DA-TEQ and single direction SISO equalizer in several recent works [29,30], we do not investigate the Hard-DA-TEQ simulation here. In the following simulations, a single-carrier SISO system with quadrature phase shift keying (QPSK) modulation is studied. The source data are thrown into a 1/2-rate RSC encoder with a generating polynomial \(g=[5,7]\) and then passed through an S-random interleaver. The coded bits are modulated in QPSK with a 10 kHz emission carrier frequency, and the symbol rate is set to 2.5 kSymbols/s. The data frame format is illustrated in Figure 6, where the transmitting signal begins with a balanced gold sequence of length \(N_{\text{BG}}=1023\) and ends with a frame gap of suitable length. Here we have no gap between the header and training sequence, i.e., \(T_{g,1}=0\). The balanced gold signal is applied to coarse frame synchronization and the Doppler frequency shift estimation. The lengths of the bidirectional training sequences are both 400 symbols, i.e., \(N_{\text{trf}}=N_{\text{trb}}=400\). The length of useful data is 1024 symbols and the gap between adjacent frames is \(T_{g,2}=100\) ms. The simulations are performed on the computer with the following specifications: AMD(R) Ryzen 7 5800H CPU 3.2 GHz (8 cores), NVIDIA(R) GeForce RTX 3070 GPU, 16 GB RAM, 2 TB memory with Windows 11 and MATLAB R2023a installed. The UWA CIR for simulation is generated by Bellhop tools with a field collected sound speed profile of theThousand Island Lake shown in Figure 7a. As shown in Figure 7b, the depth of the transmitting transducer is 5 m with a \(-6\) dB directivity from \(-5^{\circ}\) to \(90^{\circ}\), the depth of the receiving transducer is 15 m with a 1 km horizontal range away from the emitter. The generated CIR with the Doppler effect is demonstrated in Figure 8, where the carrier Doppler frequency shift is 5 Hz. The fractionally spaced order is 1/2 and the lengths of FFTs and SIC are 30 and 20, respectively. The \(\ell_{2,0}\)-PRLS algorithm is exploited to update the equalizers. In the first turbo iteration, the FS-SE degrades into the linear equalizer. Figure 6: The frame format of transmit signal. The Doppler tracking curve of the first iteration is shown in Figure 9. The Doppler frequency shift is first searched cursorily using the delay-frequency search method, and the coarse capture result is 6 Hz, i.e., \(\hat{a}=6\times 10^{-4}\), with a 2 Hz equidistant search. The average estimation of the fine Doppler frequency shift after DPLL convergence is 4.994 Hz, and the error of Doppler factor estimation attains \(6\times 10^{-3}\) Hz. The performance results are based on 1000-repetition Monte Carlo simulations. Figure 10 demonstrates the BER performance and extrinsic information transfer (EXIT) charts under the CIR simulation shown in Figure 8. Figure 10a is the BER curve comparison varying with SNR, and both the fractionally spaced structure equalizer and the bidirectional combination using virtual time-reversed technology obtain BER performance improvement. The proposed VSTD-TEQ is capable of acquiring the two-way diversity gain, which results in better BER performance, especially compared to the traditional SDA-TEQ  
Figure 8: The CIR for simulation with Doppler effect. Figure 7: The environment configuration for simulation. Figure 9: Doppler frequency shift tracking of DPLL (SNR = 4 dB). with a higher SNR. Figure 10 illustrates that the SISO equalizer of the proposed scheme produces superior mutual information (MI), which accelerates iteration convergence speed. Additionally, the iterative turbo equalization scheme indeed results in obvious performance amelioration for UWA communications.