# Posterior Probability-Based Symbol Detection Algorithm for CPM in Underwater Acoustic Channels
## 4 Numerical Simulation
### Gaussian Minimum Shift Keying  
A GMSK signal featuring a memory length of L = 3 and a bandwidth-time product of 0.3 was employed. The correlation function is represented as follows with Laurent decomposition: \(c(0,0;l)=\left\{1,0.447,0.028,8.4\times 10^{-5},\ldots,8.4\times 10^{-5},0.028,0. 447\right\}\) Then,  
\[c_{0}(t)= \begin{cases}\sin(\pi q(t)),&0\leq t\leq LT\\ \cos(\pi q(t-LT)),&LT\leq t\leq 2 LT\\ 0,&\text{else}\end{cases} , \tag{35}\]  
here  
\[q(t)= \begin{cases}1/2,&t>LT\\ \int_{0}^{t}g(\tau)d\tau,&0\leq t\leq LT\\ 0,&t<0\end{cases} , \tag{36}\]  
\[g(\tau)=\frac{1}{2T}\Bigg{\{} \begin{array}{ll}\frac{Q[2\pi B(\tau-\frac{T}{ 2})]-Q[2\pi B(\tau+\frac{T}{2})]}{\sqrt{ln2}},&0\leq\tau\leq LT\\ 0,&\text{else}\end{array} . \tag{37}\]  
where \(Q(x)\) is a complementary error function. The detection gain is low because the symbol energy of \(l=\pm 4\) is low. Thus, \(L_{m}=3\) was used to calculate the LLR(\(r_{n}\)) from (29):  
\[ \begin{array}{rl}\text{LLR}(r_{n})^{(k)}&=\ln\{\frac{\exp(\widehat{\Pr}(\text {Re}(b_{n})=1))}{\exp(\widehat{\Pr}(\text{Re}(b_{n})=-1))}\}\times\kappa^{(k,0)}-\frac{2(A_{n}^{(2)}+A_{n}^{(-2)})}{\left(\sigma^{(0)}\right)^{2}}\times \kappa^{(k,0)}\\ &+\ln\{\frac{\sum\limits_{(\xi,\eta)\propto\{1\}}\exp(\widehat{\Pr}(\text{Re }(b_{n-2})=\xi)+\widehat{\Pr}(\text{Im}(b_{n-1})=\eta))}{\sum\limits_{(\xi, \eta)\propto\{-1\}}\exp(\widehat{\Pr}(\text{Re}(b_{n-2})=\xi)+\widehat{\Pr}( \text{Im}(b_{n-1})=\eta))}\}\times\kappa^{(k,1)}\\ &+\ln\{\frac{\sum\limits_{(\xi,\eta)\propto\{1\}}\exp(\widehat{\Pr}(\text{Re }(b_{n+2})=\xi)+\widehat{\Pr}(\text{Im}(b_{n+1})=\eta))}{\sum\limits_{(\xi, \eta)\propto\{-1\}}\exp(\widehat{\Pr}(\text{Re}(b_{n+2})=\xi)+\widehat{\Pr}( \text{Im}(b_{n+1})=\eta))}\}\times\kappa^{(k,-1)}\end{array} , \tag{38}\]  
where \(A_{n}^{(l)}=c(0,0;l)\times[\Pr(\hat{u}_{n-l}=1)-1/2]\). Table 4 provides the state transitions to a different \(l\) with normalization. Figure 7 depicts the simulation results of the GMSK signal over ChA, ChB, and ChC. The dotted line represents the theoretical value of the GMSK signal over the AWGN channel, while the black line displays the BER of traditional FDE with simple phase detection [17]. The red, green, and blue curves correspond to the BER of PS-FDE after \(k=1\), \(k=2\), and \(k=3\) information merging, respectively. Figure 7a shows the simulation results of the GMSK signal within ChA. At a BER of \(10^{-4}\), the performance of FDE was 13 dB, while PS-FDE, after three iterations of information merging, exhibited improvements of 1.4 dB, 2 dB, and 2.3 dB, respectively. Figure 7b presents the BER curve within ChB. At a BER of \(10^{-4}\), FDE performance yielded 10.5 dB, whereas that of PS-FDE following three iterations of information merging improved by 0.2, 0.3, and 0.3 dB, respectively. Note the similarity of the performance in the third information merging to that in the second. Figure 7c displays the BER curve within ChC. At a BER of \(10^{-4}\), FDE performance was 11.2 dB, whereas PS-FDE resulted in enhancements of 0.3, 0.45, and 0.45 dB, respectively. Overall, the first two iterations of information merging resulted in significant improvements in symbol detection performance, whereas the third iteration provided only a marginal enhancement in BER performance. The simulations conducted on the aforementioned channels with MSK and GMSK signals effectively validated the performance of PS-FDE. PS-FDE achieved substantial improvements during the first two iterations of the information merging, while the incremental gains became significantly limited in the third iteration. Given that the information merging of PS-FDE was predicated on the memory inherent within CPM signals, the information provided by CPM in the information merging gradually converged as the iteration count increased; the error correction capability also gradually weakened. The inherent coding gain of CPM signals was comparably lower than the traditional channel coding, limiting the extent of the performance augmentation.