# Model-Driven Deep-Learning-Based Underwater Acoustic OTFS Channel Estimation
## 1 Introduction  
Underwater acoustic (UWA) communications have been widely applied in various domains, including marine environmental monitoring [1], resource exploration [2], underwater target localization [3], etc. However, its practical implementation is constrained by characteristics of UWA communication channels [4]. The UWA communication channels exhibit multipath delay and Doppler shift characteristics that are significantly greater in magnitude than radio communication channels, leading to severe time-selective fading and frequency-selective fading [5]. In UWA communication, orthogonal frequency division multiplexing (OFDM) has the advantages of high spectrum utilization and robustness against frequency selective fading [6]. However, the severe Doppler shift in the UWA channel will harm the orthogonality of OFDM subcarriers, thereby degrading the communication performance. Different from the time-frequency (TF)-domain-modulated OFDM, orthogonal time-frequency space (OTFS) communication modulates the signals in the delay-Doppler (DD) domain. The DD domain channel that corresponds to the TF dual-selective fading channel does not exhibit significant selective fading characteristicsin the DD domain. Therefore, the symbols within a DD domain frame experience nearly time-invariant fading [7]. In OTFS channel estimation, an exploratory work [8] proposes a threshold-based channel estimation method. The pilot symbols and guard symbols are embedded in the DD domain OTFS frame. The receiver employs a threshold criterion to estimate a finite number of channel impulsive responses through the least squares (LS) algorithm. The threshold-based channel estimation method often exhibits limited performance in low SNR channels. To address this limitation, ref. [9] proposes an enhanced OTFS channel estimation method that employs a highly correlated pseudorandom noise sequence as the pilot. And it estimates three parameters, delay offset, Doppler offset, and channel fading coefficient for OTFS channel estimation. An OTFS channel estimation approach is proposed in [10], utilizing the orthogonal matching pursuit (OMP) algorithm with reduced requirements on the two-dimensional correlation of pilot sequences. Moreover, the channel sparsity in the DD domain is exploited within OMP to enhance estimation performance. The OMP channel estimation is further developed into three dimensions [11]. In addition to the DD domain, the angle domain has also been introduced. Leveraging the channel sparsity in this three-dimensional domain, a compressed sensing algorithm has been proposed to effectively address the challenge of recovering sparse signals in OTFS channel estimation based on prior knowledge of their sparsity conditions. The effective design of pilot and guard space for OTFS channel estimation is also studied. A superimposed pilot pattern is proposed [12], which superimposes one pilot symbol with one data symbol. This approach can address the low spectrum efficiency issue in embedded pilot patterns. A multiple superimposed pilots scheme is further proposed [13], which assumes constant delay and Doppler effects across consecutive OTFS frames. Especially in the UWA channel, the channel information usually cannot strictly meet the sparsity assumptions. The UWA channel exhibits significant multipath propagation and Doppler effects. Upon transforming the channel from the time domain to the DD domain, the delay and Doppler grids of the channel undergo dispersion along their corresponding axes in the DD domain. Consequently, the channel distribution in the DD domain becomes less sparse. In this situation, the performance of typical OMP-based channel estimation methods will degrade. In communication systems, deep learning (DL) has the potential to extract hidden correlation features of signals and fit the complex nonlinear system for performance improvement. The DL-based communication can be mainly categorized into data-driven and model-driven methods. Data-driven DL-based OFDM channel estimation [14; 15] does not require prior information.