# Posterior Probability-Based Symbol Detection Algorithm for CPM in Underwater Acoustic Channels
## 4 Numerical Simulation
### Minimum Shift Keying  
MSK is a distinctive form of CPM signal characterized by a 1/2 modulation index and a memory length of \(L=2\). Through Laurent decomposition, the correlation function is expressed as \(c(0,0;l)\triangleq(1,1/\pi,0,\ldots,0,1/\pi)\). Then,  
\[c_{0}(t)=\begin{cases}\sin\frac{\pi t}{2T},&0\leq t\leq 2T\\ 0,&\text{else}\end{cases}. \tag{33}\]  
If \(L_{m}=1\), then LLR\((r_{n})\) is given as  
Figure 4: Measured shallow-sea channel ChB (4–8 kHz bandpass filtration): (**a**) time domain; (**b**) frequency domain.  
Figure 5: Measured deep-sea channel ChC (4–8 kHz bandpass filtration): (**a**) time domain; (**b**) frequency domain.  
\[\begin{array}{ll}\text{LLR}(r_{n})^{(k)}&=\ln\{\frac{\exp(\widetilde{\Pr}(\text{Re} (b_{n})=1))}{\exp(\widetilde{\Pr}(\text{Re}(b_{n})=-1))}\}\times\kappa^{(k,0)}\\ &+\ln\{\frac{(\xi,\prime)\kappa(1)}{\sum\limits_{(\xi,\prime)\kappa(-1)}\exp( \widetilde{\Pr}(\text{Re}(b_{n-2})=\xi)+\widetilde{\Pr}(\text{Im}(b_{n-1})= \eta))}\\ &\qquad\qquad\sum\limits_{(\xi,\prime)\kappa(-1)}\exp(\widetilde{\Pr}(\text{ Re}(b_{n-2})=\xi)+\widetilde{\Pr}(\text{Im}(b_{n-1})=\eta))\}\times\kappa^{(k,1)}\\ &+\ln\{\frac{(\xi,\prime)\kappa(1)}{\sum\limits_{(\xi,\prime)\kappa(-1)}\exp( \widetilde{\Pr}(\text{Re}(b_{n+2})=\xi)+\widetilde{\Pr}(\text{Im}(b_{n+1})= \eta))}\}\times\kappa^{(k,-1)}\end{array}, \tag{34}\]  
where the weight coefficients are \(\kappa^{(k,0)}\triangleq 2E(\text{Re}(b_{n}))/\sigma^{(k,0)}\kappa^{(k,1)}= \kappa^{(k,-1)}\triangleq E(\text{Im}(b_{n}))/\sigma^{(k,1)}\). Table 4 presents the state transitions to different \(l\) with normalization.  
Figure 6 presents the simulation results of the MSK signal over channels ChA, ChB, and ChC. The dotted line represents the theoretical bit error rate (BER) of the MSK signal in an AWGN channel, while the black line illustrates the BER performance of traditional FDE with simple phase detection [17]. The red, green, and blue curves correspond to the BER of PS-FDE for \(k=1\), \(k=2\), and \(k=3\), respectively.  
Figure 6a shows the performance of PS-FDE over ChA. During symbol detection, the first two information merges (i.e., \(k=1\) and \(k=2\)) can significantly enhance performance. At BER = \(10^{-4}\), \(k=1\) and \(k=2\) improved the performance by 0.6 and 0.4 dB, respectively, compared with the traditional FDE [15]. Compared to the first two information merges, the third merging (\(k=3\)) exhibited a modest performance enhancement of approximately 0.2 dB.  
As illustrated in Figure 6b, for ChB, the BER of the MSK signal exhibited a decreasing trend as the number of information merges increased. At BER = \(10^{-4}\), the performance of FDE was 11 dB, while PS-FDE outperformed FDE by 0.1 dB, 0.15 dB, and 0.2 dB after the first, second, and third information merging, respectively. Figure 6c presents the results for ChC, where the performance of PS-FDE after three iterations of information merging showed improvements of 0.1 dB, 0.25 dB, and 0.45 dB, respectively, compared with the FDE performance.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**State**} & \multicolumn{2}{c}{\(\boldsymbol{a_{n}=1}\)} & \multicolumn{2}{c}{\(\boldsymbol{a_{n}=-1}\)} \\ \hline \multirow{2}{*}{\(l=1\)} & \(\Xi\) & \(1\) & \(-1\) & \(1\) & \(-1\) \\  & \(H\) & \(0\) & \(-1\) & \(1\) & \(0\) \\ \hline \multirow{2}{*}{\(l=-1\)} & \(\Xi\) & \(1\) & \(-1\) & \(1\) & \(-1\) \\  & \(H\) & \(0\) & \(1\) & \(-1\) & \(0\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: State transition relations.  
Figure 6: BER for MSK with PS-FDE: (a) ChA; (b) ChB; (c) ChC.  
The simulations conducted across the three channels consistently demonstrated that the first two iterations of information merging resulted in a significant performance improvement, whereas the third iteration exhibited a comparatively smaller performance gain.