## 2 Theory of UWA Communication Using CCVTRM
### Principles of CCVTRM  
In a long-range shallow water environment, it is always impossible to estimate the channel pulse response of each sensor to implement traditional PTRM because of the low SNR. Although the array signal processing (ASP) can achieve spatial gain, the ISI resulting from multipath propagation in shallow water cannot be suppressed. In view of this, CCVTRM is put forward to improve the VSA processing. Figure 2 illustrates the procedure of CCVTRM. Figure 1: Pattern time delay shift coding scheme. The point source (PS) transmits a probe pulse signal \(p(t)\) first, and then sends the data stream \(s(t)\). After the signals pass through the shallow water acoustic channel, they are time delay processed, so that the direct sounds of each sensor are thought to be received at the same time. Signal propagation through a shallow water acoustic channel is characterized by the channel impulse response function. And assuming that the ocean channel keeps invariable between the transmission of the probe signal and the reception of the data stream, so the received probe signal and data stream on the number \(i\) sensor can be expressed as:  
\[p_{ri}(t) =p(t)*h_{i}(t)+n_{pi}(t) \tag{1}\] \[s_{ri}(t) =s(t)*h_{i}(t)+n_{si}(t) \tag{2}\]  
where '\(*\)' denotes convolution, \(h_{i}(t)\) is the channel impulse response, and \(n_{pi}(t)\) and \(n_{si}(t)\) are ocean noises. Due to the large multipath spreads, the channel impulse response \(h_{i}(t)\) can be expressed as:  
\[h_{i}(t)=\sum\limits_{j=1}^{N_{i}}a_{\bar{\jmath}}\delta(t-\tau_{\bar{\imath}}) \tag{3}\]  
where \(N_{i}\) is the number of multipath between the source and the number \(i\) receiver, and \(a_{\bar{\imath}j}\) and \(\tau_{\bar{\imath}j}\) are the amplitude and time delay of the number \(j\) path. During the long-range UWA communication in shallow water, the SNR of received signals of each receiver is so low that \(h_{i}(t)\) cannot be estimated, so that the traditional PTRM cannot be realized. Therefore, ASP is necessary. Here, the conventional beam forming (CBF) is used, including time delay processing and uniform weighted summation. After the time delay processing, it is thought that the transmission time of the signal in the corresponding sub-channel of each receiver is the same, and suppose \(\tau_{i1}\) of each sub-channel equal to \(\tau_{1}\). So the outputs of the VSA can be expressed as:  
\[ \begin{array}{rl}p_{r}(t)&=\sum\limits_{i=1}^{M}p_{ri}(t)\\ &=p(t)*h(t)+\sum\limits_{i=1}^{M}n_{pi}(t)\end{array}  \tag{4}\]  
Figure 2: The procedure of CCVTRM. \[s_{r}(t) =\sum\limits_{i=1}^{M}s_{i}(t) \tag{5}\] \[=s(t)*h(t)+\sum\limits_{i=1}^{M}n_{si}(t)\]  
where:  
\[h(t)=\sum\limits_{i=1}^{M}h_{i}(t) \tag{6}\]  
Combining Equations (3) and (6), the composite channel can be expressed as:  
\[h(t) =\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N_{i}}a_{ij}\delta(t- \tau_{ij}) \tag{7}\] \[=\sum\limits_{i=1}^{M}a_{i1}\delta(t-\tau_{1})+\sum\limits_{i=1}^ {M}\sum\limits_{j=2}^{N_{i}}a_{ij}\delta(t-\tau_{ij})\] \[=A_{1}\delta(t-\tau_{1})+\sum\limits_{k=2}^{N}A_{k}\delta(t-\tau_ {k})\]  
where \(N=\sum\limits_{i=1}^{M}N_{i}-M+1\), \(A_{1}=\sum\limits_{i=1}^{M}a_{i1}\), and when \(k\geq 2\), \(A_{k}\) and \(\tau_{k}\) correspond to \(a_{ij}\) and \(\tau_{ij}\), respectively. The function \(h(t)\), called the composite channel of the VSA, is considered to be a virtual ocean acoustic channel which the signals propagate through. This composite channel is also a very complex channel as well as the actual UWA channel. ASP makes the direct sounds of each sensor coherently superimposed and provides a higher SNR to the output signals \(p_{r}(t)\) and \(s_{r}(t)\), benefitting from the spatial gain of the VSA. However, because of the existence of the composite channel, the output data stream is still affected by the multipath extension. Hence, through the copy-correlation processing of \(p_{r}(t)\), the composite channel estimation denoted as \(h^{\prime}(t)\) is output.