## 2 System Structure  
### Transmitter Structure  
MSS is actually a kind of \((N,K)\) encoding, which uses a group of pseudo-random codes with a quantity of \(M\) (\(M=2^{k}\)) and a length of \(N\) to represent the \(k\)-bit binary data information. MSS obtains processing gain \(K\) times higher than that of conventional DSSS with the same bandwidth. At the same time, MSS uses \(2^{K}\) pseudo-random codes, and their corresponding relationship with \(K\) bits information is variable. Hence, the spectrum of the MSS signal is closer to white Gaussian noise and has stronger interception resistance and information confidentiality than the DSSS. Among various pseudo-random sequences used in the DSSS system, the chaotic sequences have an excellent correlation performance. Compared with classical pseudo-noise sequences, the length of the chaotic sequences can be adjusted unrestricted. For a fixed lengthsequence, there are many available chaotic sequences, which is especially suitable for the MSS system where there are more requirements for the number of codes [36]. After spread spectrum modulation, the signal is carrier modulated and transmitted into the UWA channel through a power amplifier and transducer. Assuming that the spreading sequences used by the MSS are \(c_{1},c_{2},\ldots,c_{M}\), and every \(c_{i}(1\leq i\leq M)\) is an \(N\times 1\) vector, the chaotic sequence set is defined as  
\[\textbf{P}=[c_{1},c_{2},\ldots,c_{M}] \tag{1}\]  
where **P** is an \(N\times M\) matrix. This denotes the sequences in **P** as \(1,2,\ldots,M\). The binary data, \(a[i]\), \(1\leq i\leq K\) are divided into a group every \(\lfloor\log_{2}(M)\rfloor\) bits. The bits of each group are converted into a decimal number, \(D[i](1\leq D[i]\leq M)\), by using a user-defined mapping relationship. We use the mapping method in [37]. For example, if \(M=8\), there are 3 bits in one group. If the data bits are [1 0 0], the corresponding decimal number is 4. Since the decimal number range is [0, 7], we need to change the range to [1, 8]. We add one to the decimal number and get \(D[i]\). Then, we choose the fifth spread sequence in **P**, i.e., \(c_{5}\). At the receiving end, we subtract one from the decoded number and convert it into the binary numbers to recover the original data. Using the obtained \(D[i]\), we choose the chaotic sequence from **P** to construct the MSS baseband signal, which is expressed as  
\[\textbf{d}=\textbf{PQ}^{D[i]}\alpha \tag{2}\]  
where **d** is an \(N\times 1\) vector, \(\alpha=[1,0,\ldots,0]^{T}_{M\times 1}\), \((\cdot)^{T}\) denotes transposition, and **Q** is the MSS mapping matrix defined as  
\[\textbf{Q}=\left[ \begin{array}{cc}0_{1\times(M-1)}&1\\ \textbf{I}_{M-1}&0_{(M-1)\times 1}\end{array} \right] \tag{3}\]  
where \(\textbf{I}_{M-1}\) is the \(M-1\) order unit matrix and **Q** is an \(M\times M\) matrix. The process of modulating data with \(M\) spreading sequences is achieved by left-multiplying **P** and **Q**. For example, if the decimal data converted by the mapping relationship is seven, the effect is that all the elements in **P** are shifted circularly from left to right seven times then multiplied by \(\alpha\), i.e., the sequence with the number of seven in vector \(\alpha\) is chosen. Subsequently, the transmission signal of the n_th_ symbol is expressed as  
\[s_{n}(t)=\Re\left\{\sum_{k=0}^{N-1}d_{n}[k]g(t-nT_{s}-kT_{c})e^{j2\pi f_{c}t}\right\} \tag{4}\]  
where \((n-1)T_{s}<t<nT\), \(T_{s}\) is the symbol duration, \(g(t)\) is the chip impulse shaping function, and \(f_{c}\) is the carrier center frequency. \[\textbf{d}=[d[1],d[2],\ldots,d[k],\ldots,d[N]]^{T} \tag{5}\]  
Because the pseudo-random sequence used in this paper is binary, the value of \(d[k]\) is \(-1\) or \(1\). It depends on the current spreading sequence. \(d_{n}\) is obtained by Equation (2) with a length of \(N\) and a chip duration of \(T_{c}\). \(\Re\{\cdot\}\) denotes the real part of the transmitted signal.