## 2 Communication System Model and Theory
### Transmitter and Underwater Acoustic Channel Transmission Process  
Without loss of generality, this section uses BPSK as an example. The detailed steps of the transmitter are illustrated in Figure 1, where the binary bit stream data at the transmitter are divided into polarity data \(a[0]\) and index data \(a[n-1]\), and further segmented into i groups corresponding to the data streams \(a_{1}[0]\)\(a_{2}[0]\ldots a_{i}[0]\) and \(a_{1}[n-1]\)\(a_{2}[n-1]\ldots a_{i}[n-1]\), where n denotes the number of bits in each group, and its value is related to the size of the SS code index library, corresponding to \(M=2^{n-1}\), where M represents the number of entries in the SS code index library. First, the \(i\times(n-1)\) index data \(a_{i}[n-1]\) are converted into decimal data \(b_{i}(1\leq b_{i}\leq M)\) and the corresponding PN code \(c_{i,b_{i}}(t)\) is selected from the SS code index library as the SS code for spreading, based on the position of \(b_{i}\) in the SS code index table, the activated PN code \(c_{i,b_{i}}(t)\) is represented as follows:  
\[c_{i,b_{i}}(t)=\sum_{m=0}^{L-1}c_{i,b_{i},m}g_{c}(t-mT_{c}) \tag{1}\]  
\[g_{c}(t-mT_{c})=\left\{\begin{array}{ll}1,&\leq t\leq T_{c}\\ 0,&\text{otherwise}\end{array}\right. \tag{2}\]  
where \(L\) represents the length of the activated PN code, \(T_{c}\) denotes the chip duration, \(c_{i,b_{i}}=[c_{1,b_{1}}c_{2,b_{2}}...c_{i,b_{i}}]\) represents the sequence of the active code. The transmitted data \(x(t)\) after merging with the polarity data can be expressed as follows:  
\[x(t)=\sum_{i=1}^{\infty}a_{i}[0]c_{i,b_{i}}(t) \tag{3}\]  
The transmitted data \(x(t)\) pass through a pulse-shaping filter prior to modulation, and the baseband signal \(\bar{s}(t)\) can be expressed as follows:  
\[\bar{s}(t)=\sum_{i=0}^{\infty}x(t)\omega(t-iT_{s}) \tag{4}\]  
Here, \(T_{s}\) represents the symbol interval, \(\omega(t)\) is the pulse-shaping function, and \(\beta\) denotes the roll-off factor. The transmitted signal, suitable for underwater acoustic channels after carrier modulation, can be expressed as follows:  
\[s(t)=\sum_{i=0}^{\infty}\bar{s}(t)e^{\beta\pi f_{c}t} \tag{5}\]  
where \(f_{c}\) is a carrier center frequency.  
It is well-known that the underwater acoustic (UWA) channel is a typical double-diffusion channel [20; 21]. The impulse response \(h_{i}(t,\tau)\) of the time-varying channel for the \(i\)-th symbol in the transmitted data can be expressed as follows [22; 23]:  
\[h_{i}(t,\tau)=\sum_{q=0}^{Q-1}A_{q,i}(t)\delta(\tau-\tau_{q,i}(t)) \tag{6}\]  
Here, \(Q\) denotes the number of channel taps, while the amplitude and delay of the \(q\)-th path for the \(i\)-th symbol can be represented as \(A_{q,i}(t)\) and \(\tau_{q,i}(t)\).  
Ultimately, the received signal \(r(t)\) can be described as the convolution of the transmitted signal and the channel impulse response, and its final expression is as follows:  
\[r(t)=\Re\left\{\sum_{i=1}^{\infty}\sum_{q=0}^{Q-1}A_{q,i}(t)a_{i}[0]c_{i,b_{i }}\omega e^{2\pi f_{c}(t-\tau_{q,i}(t))}\right\}+\eta(t) \tag{7}\]  
where \(\eta(t)\) is the noise in the transmission process, \(c_{i,b_{i}}=c_{i,b_{i}}(t-\tau_{q,i}(t))-iT_{s})\), \(\omega=\omega(t-\tau_{q,i}(t)-iT_{s})\).