# Channel Estimation and Iterative Decoding for Underwater Acoustic OTFS Communication Systems
## 2 Related Work  
To date, research on OTFS modulation technology has primarily focused on the realm of radio communication, and it remains in a nascent stage at present [15; 16; 17; 24; 25; 26; 27; 28]. In recent years, numerous enhancements to the original OTFS modulation scheme have been proposed by researchers. For example, [PERSON] et al. [15; 16; 24; 25] established the mathematical framework of OTFS signals and derived the modulation and demodulation input-output equations in the DD domain. In [16], an OTFS modulation model for multi-channel dual dispersion channels was established, accounting for time delays within the maximum delay and limited Doppler shift range. However, the original OTFS input-output relationship assumes ideal pulse shaping waveforms to be biorthogonal in both time and frequency, which is unattainable due to the Heisenberg uncertainty principle [16]. To address this, the authors of [24] derived the input-output relationship for actual pulse-shaping waveforms based on the DD domain, extending it to any waveform. Moreover, in [15], it was demonstrated that using the message passing (MP) algorithm for symbol detection in OTFS results in a lower symbol error probability (SEP) compared to OFDM. Building on this, the authors of [24] proposed a scheme that adds a cyclic prefix (CP) only at the end of the OTFS frame, significantly reducing the overhead caused by the loss of biorthogonality of pulse shaping waveforms. Additionally, in [25], a channel estimation method utilizing embedded pilot symbols was introduced, showing that OTFS with non-ideal channel estimation outperforms OFDM with ideal channel estimation in terms of error performance. Further research has delved into the sparsity of the DD domain in OTFS systems [26; 27; 28]. For example, [PERSON] et al. proposed a low-complexity iterative decision feedback equalizer (DFE) based on maximum ratio combining (MRC), demonstrating its favorable performance and complexity compared to state-of-the-art message passing detectors [17]. These advancements represent significant progress in enhancing OTFS technology, paving the way for its broader application in communication systems. Compared with the field of terrestrial wireless communication, research on OTFS modulation technology in the field of UWA communication remains relatively limited [17; 18; 19; 20; 21; 22; 23]. Some scholars have explored its applicability in UWA communication through leveraging research findings from wireless communication and have achieved notable results, primarily focusing on receiver complexity reduction, channel estimation, and signal detection. Researchers at Xiamen University have proposed a single-input single-output (SISO) UWA communication scheme based on OTFS [19]. Simulation results comparing OTFS and OFDM in UWA channel environments in terms of bit error rate (BER), spectral efficiency, and peak-to-average power ratio (PAPR) indicated the superior performance of the OTFS-based UWA communication scheme in time-varying multipath UWA channels, compared to OFDM and discrete Fourier transform spread (DFTS) OFDM schemes. In 2019, [PERSON] et al. from the University of Bristol, U.K., introduced an OTFS system based on large-scale multiple-input multiple-output (MIMO) OFDM to investigate multi-user UWA communication schemes [20]. In the studies [19; 20], OTFS-UWA communications in SISO and MIMO scenarios were explored, respectively, with simulation results demonstrating that OTFS outperforms OFDM. However, these studies assume that accurate channel information is known to the receiver. Researchers from Dalian University of Technology have proposed a time-reversal OTFS receiver with adaptive channel estimation [21] and evaluated its system performance in both analog and measured channels. Their results indicated that the proposed two-dimensional passive time reversal (PTR) receiver achieves improved system performance with low complexity. [PERSON] et al.