## 4 Validation Through Sea Trials
### Processing and Analysis of the First Group of Sea Trial Data  
Figure 8a displays the speed recorded during the first sea trial process by the BDS, while Figure 8b presents the results of the channel impulse response. It is evident that the channel exhibits significant multipath effects during the communication process, with complex background noise and frequent instantaneous noise occurrences. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Test Groups**} & \multirow{2}{*}{**Average SNR (dB)**} & \multicolumn{4}{c}{**Different Receptions BER**} \\ \cline{3-6}  & & **CCF-MSS** & **WDE-MSS** & **FDE-MSS** & **CDE-MSS** \\ \hline Group 1 (Section 4.1) & \(-\)3.4 & 0 & 0.7717 & 0.0055 & 0.0207 \\ Group 2 (Section 4.2) & \(-\)4.7 & 0 & 0.7763 & 0.0027 & 0.0167 \\ Group 3 (Section 4.3) & \(-\)6.5 & 0.0013 & 0.7745 & 0.0103 & 0.0324 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The average SNR of different groups of sea test data and the BER processed by proposed receptions. Figure 8: (**a**) is the speed of the first group processes recorded by BDS. (**b**) is the first group processes channel impulse response. The frame accuracy in the first group of sea trials is depicted in Figure 9a,b, where CCF-MSS exhibits a BER of 0, indicating superior decoding performance. WDE-MSS achieved zero bit errors only in frames 26 to 29, while achieving an accuracy of 0.9 in frame 25, resulting in an overall error rate of 0.7717. The reason for this can be attributed to the decrease in vessel speed during the later stages of the trial, which significantly reduced the Doppler effect, thus restoring decoding capability. When using CDE-MSS, several frames in the middle experienced bit errors, which was due to inaccurate coarse Doppler estimation during the variable speed process, resulting in incorrect compensation of the CFO value, with an overall error rate of 0.0275. The decoding performance of FDE-MSS is significantly better than that of the other two traditional methods, indicating that fine Doppler estimation is an effective means to overcome the Doppler shifts caused by complex time-varying channels, with a total BER of 0.0055. Except for the significant difference in the processing results of the first frame, the decoding performances of FDE-MSS and CCF-MSS were comparable during this group of sea trials. The following section will analyze the decoding results of the first frame for both FDE-MSS and CCF-MSS. The time duration of the first frame signal in the figure is approximately 0 to 10 s. It can be observed that the vessel speed is relatively high, reaching up to 3 m/s, with complex background noise and pronounced multipath effects. Figure 10a,c shows the final output of the CFO compensation values for CCF-MSS and FDE-MSS in the first frame, while Figure 10b presents the CFO values calculated from the vessel speed output by the BDS, which can be used as a reference for the actual CFO for comparison. It can be observed that the CFO estimated by CCF-MSS is largely consistent with the CFO calculated from the BeiDou output speed, indicating that the decoding performance is quite good. However, the CFO compensated by FDE-MSS differs significantly from the measured CFO during symbols 30 to 40. Upon inspection, decoding errors were found in FDE-MSS for symbols 31 to 37. The following section will analyze the despreading results for the specific symbols. We compared the despreading correlation peaks of the 32 nd and 33 rd symbols under both CCF-MSS and FDE-MSS reception methods. Figure 11a presents the despreading result for symbol 32 under CCF-MSS, where the blue circles indicate the maximum peaks under different CFO compensations, and the red pentagons represent the peaks under CFO compensation selected based on the correlation cost factor. We can observe that the position of the output maximum peak corresponds to a CFO compensation of \(-14.7\) Hz, while the correct CFO compensation position for the next symbol is \(-13.8\) Hz (see Figure 11c).