## 1 Introduction  
Underwater acoustic communication technology is widely used in civilian activities and military activities. Therefore, reliable UAC technology is of great importance to national maritime rights and interests [1]. However, the UAC channel is complex due to multipath, time variation, a limited bandwidth, and high environmental noise. At the same time, because of the slow transmission speed of underwater sound waves, the transceiver's relative motion and water flow during signal transmission will lead to a serious Doppler Effect [2; 3; 4; 5]. In order to ensure the reliability of the communication system, the receiver must accurately estimate and compensate for the Doppler shift [6]. Effective methods to solve the Doppler Effect can be roughly divided into several categories: time-domain resampling [7; 8; 9; 10], sparse channel estimation [11; 12; 13; 14; 15; 16], and channel equalization [17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Time-domain resampling algorithms such as match filter and time-domain cross-correlation use auxiliary or correlated signals to estimate the Doppler factor. Most of these methods are based on resampling to deal with timescale changes in UAC, combined with ambiguity functions [7], synchronous signals [8], and two-step methods [9; 10]. One of the most classic algorithms takes advantage of the fact that the linear frequency modulation (LFM) signal is not susceptible to Doppler shift and has good autocorrelation. It estimates the Doppler factor (scale) through the correlation results of two LFM signals, which is less complex and easy to implement [8]. However, due to insufficient bandwidth and the limitation of the sampling rate in the communication process, the temporal resolution of the signal is poor, resulting in the ambiguity of the relevant peaks in the calculated output and the inability to identify them accurately. Moreover, due to the influence of a complex time-varying channel such as the multipath effect and attenuation, there is a possibility of multiple correlation peaks. These conditions lead to the failure of traditional timescale-based estimation methods. Figure 1 shows the cross-correlation between the received signal and the LFM signal, which can be used for synchronization and time estimation by judging where the maximum peak occurs before and after the signal. As shown in the highlighted parts of the figure, the pre-correlated signal has an obvious peak. However, due to the influence of the time-varying channel, the peak of the post-correlated signal is not obvious, and even multiple correlated peaks appear, resulting in the inability to accurately estimate the time, which seriously affects the performance of UAC. In addition, sparse channel estimation algorithms model the Doppler estimation as a representation and reconstruction of sparse signals [11]. Using the sparse channel estimation technique based on delay-Doppler-spread function representation, the Doppler factor can be estimated using the correlation principle and sparse reconstruction [12; 13]. The two-stage sparse channel estimation technique estimates the delay and Doppler scale sequentially, greatly reducing the computational complexity [14]. A novel orthogonal matching pursuit approach is proposed to quickly explore time-varying sparsity in a two-step sequential manner [15]. However, accurate channel parameter estimation plays an important role in these methods. At the same time, there are relatively high requirements for the sparsity of underwater acoustic channels, which have certain limitations. In addition, a new Doppler compensation method is proposed in [16] based on time reversal focusing. Still, it is unsuitable for the actual system because of the high requirements for the communication equipment and the cumbersome process. Combining the above methods with channel equalization technology can effectively alleviate the impact of the Doppler Effect and obtain better communication results.