## 4 Numerical Results and Discussion
### BER Performance over a UWA Channel  
In this section, the data of the MSSS-I/Q-IM system are modulated on the carrier for simulation experiment, and the BER performance is analyzed by changing various \begin{table}
\begin{tabular}{c c c c} \hline \hline
**MSSS** & \multicolumn{3}{c}{**MSSS-I/Q-IM**} \\ \hline \multirow{2}{*}{18} & \(J=8\) & \(J=6\) & \(J=3\) \\ \cline{2-4}  & 22 & 24 & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of transmitted bits by the two systems within a spreading sequence duration. Figure 3: BER performance of the two systems with \(M=1023\), \(L=100\), and \(J=8,6,3\) over an AWGN channel. parameters of the system according to different UWA channels. The experiment is carried out on the Watermark 1.0 simulation platform [18]. Two kinds of UWA channels are adopted in the simulation experiment. The first channel was measured in a shallow sea of Oslofjorden, with a fixed single source and a fixed single hydrophone. The second channel was measured on a Shelf in Norway, and the transmitter and receiver were also fixed during the test. The channel impulse response (CSI) of the two channels are shown in Figure 4. The left figure characterizes the Fjord channel, whose delay profile decays quickly, and a cluster of discrete path arrivals is observed between 50 ms and 80 ms, however, these paths only carry a small fraction of the total energy. The stable channel paths carry about 80% of received signal energy. It is a benign channel on the whole. In contrast, the Shelf channel has a larger Doppler shift. In the Monte Carlo simulation for the two UWA channels, the sampling rate is 100 kHz, the carrier frequency is 14 kHz, the bandwidth is 8 kHz, the cyclic prefix length is set as 101 chips, and quadrature phase shift keying (QPSK) modulation is adopted. Table 2 compares the effective bit rate of the MSSS-I/Q-IM and MSSS systems when taking different superposition sequence numbers \(J\), and the signal duration in the simulation is 7.868 s. It can be seen from the table that when \(J=3\), the effective bit rate of the two systems are consistent. With the increasing of \(J\), the communication rate of the MSSS-I/Q-IM system increases continuously. Figure 5 analyzes the influence of parameter \(J\) on the BER performance of the systems under the two UWA channels. The length of spreading sequence \(M=1023\), and the cyclic shift step size \(L+1=101\) chips. From the overall trend of the curves, the BER over the Fjord channel is better than that over the Shelf channel. When \(J=4\), compared with the MSSS system, the MSSS-I/Q-IM system has been able to achieve better BER performance in both UWA channels. In addition, the BER performance gain is larger over the Fjord channel, which means that with the increasing of channel complexity, the performance advantage \begin{table}
\begin{tabular}{c c c} \hline \hline
**MSSS** & \multicolumn{2}{c}{**MSSS-I/Q-IM**} \\ \hline  & \(J=3\) & 228.77 bit/s \\ \cline{2-3}
228.77 bit/s & \(J=4\) & 254.19 bit/s \\ \cline{2-3}  & \(J=5\) & 279.61 bit/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of effective bit rates of the two systems. Figure 4: The CIRs of the Fjord and Shelf channels. that the MSSS-I/Q-IM system achieved will decrease. However, according to the influence of \(J\) on the BER performance, it indicates that the MSSS-I/Q-IM system has more flexibility. Figure 6 shows the effect of different cyclic shift step sizes on the BER performance of the proposed system. Other parameters are set to \(M=1023\) and \(J=3\). In the simulation, the cyclic prefix length is set to 100, and under different \(L\) values, the effective bit rates of the MSSS-I/Q-IM system are the same, at 228.77 bit/s. As shown in the figure, there is a gap in the BER performance under different shift step sizes. Comparing the BER curves when \(L=100\) and \(L=80\) over the two UWA channels, the difference between them is small. However, for \(L=60\), the BER performance drops significantly.