## 2 System Structure
### Channel Modeling  
According to the description in [38], the influence of the UWA channel on communication is divided into two time-varying parts. One is expressed by slow varied channel impulse response, and the other is the effect of time-variant Doppler. We express the UWA channel by the following two equations  
\[\overline{y}(t)=s(t)*h(t) \tag{7}\]  
\[r(t)=\overline{y}(t+\frac{v(t)t}{c})+n(t) \tag{8}\]  
where \(r(t)\) is the observation signal at the receiving end, \(h(t)\) is the impulse response of a slow time-varying channel, \(v(t)\) is the radial relative motion velocity of the transmitter and receiver, \(c\) is the sound propagation velocity, and \(n(t)\) is the additive white Gaussian noise.  
The UWA signals are affected by the multi-path effect. In general, different paths could have different Doppler factors. In [20], they use the assumption that all paths have the same Doppler scaling factor. They find that as long as the dominant Doppler shift is caused by the direct transmitter/receiver motion, the assumption seems to be justified. The same assumption is used in [15].  
Due to the excellent autocorrelation performance of the pseudo-random sequences, it is easy to distinguish each multi-path signal. In addition, we assume that the time delay is less than the duration of one spreading symbol and ignore the impact of intersymbol interference. Hence, without losing generality, we focus on the Doppler's distortion on the propagated signals [15; 26]. Define the Doppler factor as  
\[\Delta=\frac{v(t)}{c} \tag{9}\]  
The channel model is simplified as  
\[r(t)=s((1+\Delta)t)+n(t) \tag{10}\]  
The core of this paper is the received signal processing method under the change of motion over the duration of the transmitted signals. The duration of a one-symbol signal is relatively short, which is usually tens of milliseconds. Therefore, we assume that the relative motion velocity of transmitter/receiver remains changeless within one symbol, i.e., the Doppler factor of one symbol is a constant. During the whole transmission duration, the velocity of transmitter/receiver varies continuously, and the Doppler factors of different symbols are distinct. On this premise, the received signal model is modified, derived from Equations (4), (6), and (10)  
\[r(t)=\Re\left\{\sum_{n=0}^{L-1}\sum_{k=0}^{N-1}d_{n}[k]g((1+\Delta_{n})t-nT_{s }-kT_{c})e^{j2\pi f_{c}(1+\Delta_{n})t}\right\}+n(t) \tag{11}\]  
where \(\Delta_{n}\) represents the Doppler factor on the n_th_ symbol. It can be seen from Equation (11) that both the time-domain scale and carrier frequency of the signal are affected by Doppler effects. The time-domain scale variation on each symbol is \(\frac{-\Delta_{n}}{1+\Delta_{n}}T_{s}\), and the frequency-domain scale variation is \(\Delta_{n}f_{c}\). In terrestrial wireless communications, the propagation speed of electromagnetic waves is quite large. According to Equation (9), the Doppler is small in general, so the time-domain amplitude variation of signal is completely ignored.  
However, because the sound velocity in the water is only 1500 m/s, the Doppler factor underwater is five orders of magnitude higher than that on the land. Small relative motion causes a change of signal length that cannot be ignored. When each symbol is affected by different Doppler factors, the change of the symbol length is more complex. Coupled with the distortion of signal carrier frequency, it is not easy to recover data without corresponding processing at the receiving end.