## 1 Introduction  
Implementing efficient and reliable communication in the underwater acoustic (UWA) channel is a challenging task due to the complex and time-varying characteristics of the UWA channel in terms of the limited bandwidth, strong multipath, serious Doppler effect, and long propagation delay. The Doppler effect in the UWA channel is severe since the sound wave propagating in the water is slow (about 1500 m/s). And Doppler estimation and compensation become a critical issue in the UWA communication system. Moreover, the large multipath delay spread also brings great challenges to UWA communication [1]. Hence, channel estimation and equalization should be used to mitigate the performance degradation resulting from the multipath. When the UWA communication system with a high data rate is equipped on a mobile underwater node, Doppler estimation and compensation are essential before demodulation at the receiver [2, 3]. For the traditional Doppler method, a set of correlators are adopted to compute the correlation between various Doppler replicas of the transmitted signal and the received signal. And then the Doppler shift is determined according to the maximum correlation [4]. Although the resolution of the method can be improved with an increased number of correlators, the computational complexity also significantly increases. To reduce the computational complexity, an efficient block Doppler estimation method was proposed in [5]. In this method, two linear frequency modulation (LFM) signals are respectively inserted before and after the data frame, and the time duration of the data frame can be estimated according to two obtained correlation peaks at the receiver. The time duration change can further be used to estimate the Doppler factor. This method has been widely used in UWA communication because of its efficiency and simplicity. Moreover, several improved methods have been proposed [6; 7; 8; 9; 10; 11; 12]. For example, an adaptive dual-stage Doppler estimation method was proposed in [12]. In the first stage, the block Doppler estimation method was used for a coarse estimation. In the second stage, a bank of fraction-delay Farrow filters was used to correct the Doppler factor. However, these methods need to store the whole data frame, which introduces a large storage overhead and disables the real-time estimation. A Doppler estimation method based on hyperbolic frequency modulation (HFM) signals, which can obtain real-time estimation, was proposed in [13; 14]. In this method, two HFM signals with opposite frequency sweeping directions were inserted before the data frame. At the receiver, two corresponding matched filters were utilized to estimate the Doppler factor. Although the accuracy of this method is independent of the frame length, the estimation accuracy should deteriorate sharply in the presence of a time-varying multipath. To improve the estimation performance in the time-varying multipath environment, a correlation peak matching method was proposed in [15]. However, two HFM signals need to have the same autocorrelation characteristics in this method. And the computational complexity was increased. A new preamble waveform based on an HFM signal was proposed in [16], which leaves a mute span between the up-sweeping HFM and the down-sweeping HFM signals. Apart from using the time-domain property of HFM signals, a speed spectrum estimation method utilizing the spectrum property of the signal was proposed to estimate the Doppler factor in [17]. Recently, numerous achievements in channel estimation for UWA communication have been obtained. According to prior knowledge for the algorithm, the channel estimation methods can be divided into the blind method and the training-sequence-based method. In the blind channel estimation method, the statistical property of received signals is utilized to estimate the channel state information without any prior information of the transmitted signal [18].