## 2 System Model and Problem Statement
### System Model  
A UWA communication system with binary phase shift keying (BPSK) modulation is considered in this paper. And the block diagram of the UWA communication system is illustrated in Figure 1.  
At the transmitter, the input binary data bit sequence is used to control the switching of the carrier phase in the BPSK modulator. And a pulse shaping filter is also included to reduce the inter-symbol interference and control the spectral characteristics of the modulated signal. And then a preamble signal, which is dedicatedly designed for both Doppler and channel estimation, is added to the modulated information. Finally, the data frame is transmitted.  
After encountering the UWA channel, the transmitted data frame reaches the receiver.  
Then, the receiver operates as follows. In the first step, the receiver performs coarse frame synchronization by utilizing the received preamble signal. Afterward, the received preamble signal is used for Doppler estimation, as detailed in Section 3.2. Towards Doppler compensation, a resampling process is performed where the resampling factor is determined by the estimated Doppler factor. At the same time, fine frame synchronization is realized. Then, channel estimation using the preamble signal, as described in Section 3.3, is performed. Following that, frequency-domain channel equalization is carried out, where the fast Fourier transform (FFT) of time-domain channel estimation is used. Finally, signal demodulation using a BPSK demodulator is performed to recover the data bits.  
The time-varying CIR of a multipath UWA channel can be described by  
\[h(\tau,t)=\sum_{p=0}^{P-1}A_{p}(t)\delta(\tau-\tau_{p}(t)), \tag{1}\]  
where \(A_{p}(t)\) and \(\tau_{p}(t)\) are the time-varying amplitude and delay of the \(p\)th path, respectively. \(P\) is the number of paths.  
For a wideband signal, each frequency component is translated by a different amount due to the Doppler factor. The Doppler effect can be modeled as a time scaling (expansion or compression) of the signal waveform [5]. Let \(\alpha\) denote the Doppler factor, which is defined as the ratio of the radial relative velocity (\(v\)) between the transmitter and the receiver to the sound speed (\(c\)) in the water. The radial relative velocity is considered to be positive if the transmitter and the receiver are moving closer; otherwise, it is negative.  
The following assumptions are considered here.  
Figure 1: The block diagram of the UWA communication system.  
**Assumption 1**.: _All the paths have a similar Doppler factor. That is,_  
\[\tau_{p}(t)=\tau_{p}-\alpha t. \tag{2}\]  
Generally, different paths could have different Doppler factors as demonstrated in the field test results [26; 27]. However, the issues of Doppler and channel estimation that need to be resolved are based on the assumption that all the paths have approximately the same Doppler factor. In this paper, we consider that the Doppler shift is caused by the relative transmitter/receiver motion. This assumption seems to be justified as long as the dominant Doppler shift is caused by direct transmitter/receiver motion [28].  
**Assumption 2**.: _The delay and amplitude of each path, \(\tau_{p}\) and \(A_{p}(t)\) (\(0\leq p\leq P-1\)), and the Doppler factor \(\alpha\) are constant over the duration of a data frame._  
The time duration of a data frame adopted in this paper is less than 500 ms. It is reasonable that Assumption 2 is justified since the channel coherence time is typically on the order of seconds.  
Hence, the received signal can be expressed as  
\[y(t)=\sum_{p=0}^{P-1}A_{p}s((1+\alpha)t-\tau_{p})+w(t), \tag{3}\]  
where \(s(t)\) is the transmitted signal and \(w(t)\) is additive white Gaussian noise (AWGN).