## 2 Theory of TRM-SBDC UWA Communication
### Principles of SBDC  
In the presence of relative motion between the transmitter and receiver in fluctuating underwater environments, the signals will be compressed or dilated due to Doppler effects. As a result, the time delay shift of the encoded pulse will be reduced or increased, that is, a time delay error will occur in each symbol. Moreover, the time delay error will be superimposed in PDS communication scheme, eventually leading to decoding errors. We propose a SBDC method to improve the performance of PDS communication in this paper. Since the received signal has been channel equalized by TRM, we use the copy-correlation technique for decoding [15; 24]. First, the position of the sync signal can be found by correlating the received signal with the entire know sync signal, thereby the start position of the information code is determined. Subsequently, the received signal is subjected to copy-correlation processing using the pattern pulse signal and a series of correlation peaks are obtained. According to the positions of the correlation peaks, the time delay shift of each symbol can be calculated, thereby the information is demodulated. Figure 4 illustrates the schematic diagram of decoding. As shown in Figure 4, \(T_{s}\) is the pulse width of the sync signal, \(T_{ISI}\) is the ISI guard time between the sync signal and the first symbol, \(T_{i}\) denotes the start position of \(i\)th symbol, and \(T_{pi}\) denotes the position of the pattern pulse. When there is no Doppler, \(T_{i}\) can be determined by the position of the sync signal, and \(T_{pi}\) can be obtained by the copy-correlation processing. So the time delay shift of the symbol is:  
\[\tau_{di}=T_{pi}-T_{i}=k_{i}\cdot\Delta\tau \tag{8}\]  
where \(k_{i}\) represents the digital information carried by the \(i\)th symbol, and \(k_{i}=0,1,\ldots,2^{n}-1\). However, \(T_{i}\) and \(T_{pi}\) will change if there is a Doppler effect, resulting in time delay errors. Assuming that the time delay error induced by the Doppler effect on the \(i\)th symbol is \(\Delta_{Di}\), it can be defined as:  
\[\Delta_{Di}=\tau_{di}-\tau^{\prime}_{di} \tag{9}\]  
In addition, it is considered that within the duration of each symbol the Doppler time delay error remains almost invariable, so the changed position of the pattern pulse can be expressed as:  
\[T^{\prime}_{pi}=T_{pi}-\sum_{j=1}^{i}\Delta_{Dj}-\Delta_{D0} \tag{10}\]  
where \(\Delta_{D0}\) is the time delay error caused by Doppler during the time period of the sync code. If the time delay shift is still calculated according to the start position of each symbol determined by the sync signal, the time delay shift of the \(i\)th symbol should be rewritten as:  
Figure 4: The diagram of decoding. \[ \begin{array}{ll}\tau_{di}^{\prime}&=T_{pi}^{\prime}-T_{i}\\ &=T_{pi}-\sum\limits_{j=1}^{i}\Delta_{Di}-\Delta_{D0}-T_{i}\\ &=\tau_{di}-\sum\limits_{j=1}^{i}\Delta_{Di}-\Delta_{D0}\end{array}  \tag{11}\]  
As can be seen from Equation (11), the time delay error will gradually increase. Once the time delay error accumulation exceeds \(\Delta\tau/2\), the decoding will generate errors. According to the decoding characteristics of the PDS communication scheme, the paper puts forward SBDC method. The proposed method uses the Doppler time delay error of the current symbol to compensate the time delay shift of the latter symbol, and has real-time performance. The decoding process using SBDC is explained below.