## 4 Analysis of the MCSK-VTRM System  
### Simulations  
This part studies the improvement of reliability after VTRM is applied to the MCSK system. The CIR of the UWA channel used in the simulation is shown in Figure 9. It can be seen that VTRM is suitable for this multipath channel to achieve channel equalization. The peak-sidelobe ratio of the matched channel is better than that of the original channel. That is, the energy of the direct path is enhanced while the energy of other paths is relatively suppressed, which shows a great channel focusing characteristic.  
The simulation conditions are given in Table 3. Assuming that the CIR is known, the reliability of the traditional DSSS system, MCSK system, and MCSK-VTRM system is compared in Figure 10. In all these systems, the spread spectrum sequences are composed of kasami sequences and their cyclic shift sequences. Each system transmits 1000 bits of data each time and repeats 200 experiments. In the MCSK system and the MCSK-VTRM system, one spread spectrum sequence carries 5 bits of information.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value and Unit** \\ \hline sampling rate & 48 kHz \\ carrier frequency & 10 kHz \\ communication bandwidth & 4 kHz \\ the length of spread spectrum & 15 \\ sequences & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation parameters.  
In Figure 10, it can be seen that the reliability of the MCSK system is lower than that of the DSSS system. When the BER is \(10^{-2}\), the SNR of MCSK is 7 dB higher than that of DSSS. It is known that the increase in the amount of information carried by each spread spectrum sequence improves the effectiveness of the MCSK communication system, but the cross-correlation between different spread spectrum sequences interfere with the decision of the receiver. However, when the VTRM technique is applied to the MCSK system, the reliability of communication is further improved, for VTRM can effectively suppress the multipath interference caused by UWA channels. Therefore, the MCSK-VTRM communication system proposed in this paper can not only improve the transmission rate of traditional spread spectrum communication but also improve the anti-noise ability and anti-multipath interference ability of UWA communication systems. When the length of spread spectrum sequences is kept the same, compared with the hybrid spread spectrum method in the Reference [18], this MCSK-VTRM system only uses a single carrier and can ensure higher efficiency and reliability while the system complexity is lower.  
In practice, the CIR of the UWA channel is unknown. To utilize VTRM to compensate the influence of the UWA channel, the received detection signal is first used to estimate the CIR of the UWA channel at the receiver. As shown in Figure 11, a transmitted signal is composed of a synchronous signal, a detection signal, some protection intervals and a spread spectrum signal which carries information. The synchronous signal is usually the linear frequency modulation signal, and the detection signal can choose the linear frequency modulation signal or other training sequences.  
Figure 10: BER Curves of three systems.  
Figure 9: CIR of the UWA channel in simulation. (**a**) Original channel; (**b**) matched channel.  
The BER curves of the MCSK-VTRM system with the MP algorithm and SpaRSA algorithm based on BPDN estimating CIR are shown in Figure 12. It can be seen that when the SNRs are the same, the BER of the system using BPDN is slightly lower. Compared with BER of MP, that of BPDN is lower about 0.5 dB when BER is \(10^{-2}\). The performance of BPDN channel estimation is better than that of MP channel estimation, but the BPDN algorithm has a large amount of computation and a long operation time.