# Posterior Probability-Based Symbol Detection Algorithm for CPM in Underwater Acoustic Channels
## 3 Proposed Algorithms
### PS-FDE for CPM Signals  
Considering the sparse property of the UWA channel, the orthogonal matching pursuit algorithm [28] is used for the channel estimation. The channel impulse response estimated by CP is expressed as \(\hat{h}=\left\{\hat{h}_{0},\hat{h}_{1},\ldots,\hat{h}_{N_{d}-1}\right\}\). According to classic FDE for CPM signal [15], the equalized signal in the frequency domain is given as follows:  
\[\hat{U}_{k,q}=Y_{k,q}W_{k,q}\hat{H}_{k,q}, \tag{15}\]  
where \(Y_{k,q}\) and \(\hat{H}_{k,q}\) are the frequency domain representations of \(y_{n,q}\) and \(\hat{h}_{n}\), respectively. MMSE equalization coefficient is expressed as  
\[\hat{F}_{k,q}=\frac{\hat{H}_{k}^{*}W_{k,q}}{\left|\hat{H}_{k}W_{k,q}\right|^{2 }+N_{0}}, \tag{16}\]  
where \(N_{0}\) is the noise power spectral density. The noise-whitening filter \(W_{k,q}\) is given as follows:  
\[W_{k,q}=\frac{1}{\sqrt{C(p,q;k)}}, \tag{17}\]  
where \(C(p,q;k)\) denotes the frequency domain representations of \(c(p,q;n)\). Subsequently, symbol detection is performed on the equalized signal \(\hat{a}_{q}\). To streamline the computation, the equalized signal is mapped according to the employed modulation scheme. The symbol information corresponding to different \(l\) values is then separated based on phase and amplitude characteristics. The mapped symbol sequences \(\mathbf{b}_{q}^{(l)}\) and \(\hat{\mathbf{v}}_{p,q}^{(l)}\) are defined, where \(b_{n,q}\) is assumed to follow a Gaussian distribution. Under this assumption, the a priori probability of the \(l\)th symbol at the \(n\)th symbol interval is expressed as follows:  
\[\Pr(b_{n,q}^{(l)}=\sum_{p=0}^{K-1}\phi_{n,p,q}^{(l)})=\frac{1}{\sqrt{2\pi} \sigma_{q}^{(l)}}\exp(-\frac{\left(\left|b_{n,q}^{(l)}-\sum_{p}\hat{\mathbf{v}}_{ n,p,q}^{(l)}\right|\right)^{2}}{2\left(\sigma_{q}^{(l)}\right)^{2}}), \tag{18}\]  
where the equivalent noise variance \(\sigma_{q}^{(l)}\) is calculated using CP because the theoretical value \(\sum_{p}\sigma_{n,p,q}^{(l)}\) of CP after FDE is known at the \(n\)th symbol interval:  
\[\left(\sigma_{q}^{(l)}\right)^{2}=\frac{1}{N_{c}}\sum_{n=N-N_{c}+1}^{N}\left( \left|b_{n,q}^{(l)}-\sum_{p}\hat{\mathbf{v}}_{n,p,q}^{(l)}\right|\right)^{2}. \tag{19}\]  
\(u_{n,q}\) contains information of the consecutive \(l=\{1-L,\ldots,-1,0,1,\ldots,L-1\}\) symbols; hence, the symbol information at times \(n+l\) and \(n+2l\) can be delayed and merged to estimate the \(n\)th symbol. Generally, the \(n\)th symbol information carries the dominant energy within the \(n\)th symbol interval. Therefore, the initial estimated likelihood value is expressed by \(b_{n,q}^{(0)}\):  
\[\mathrm{L}(r_{n,q})=\ln[\Pr(b_{n,q}^{(0)}=\sum_{p}\hat{\mathbf{v}}_{n,p,q}^{(0)})]. \tag{20}\]  
During the information merging, the estimated likelihood value \(\hat{\mathrm{L}}(r_{n,q})\) of the symbol is computed using a joint probability according to the symbol information of \(l\
eq 0\):  
\[\hat{\mathrm{L}}(r_{n,q})^{(k)}=\mathrm{L}(r_{n,q})+\sum_{l=\{1-L,\ldots,-1,1,\ldots L-1\}}\left\{\ln[\sum_{(\mathcal{I},q)\in\sum_{p}\hat{\mathbf{v}}_{n,p,q}^{ (0)}}\Pr(b_{n-l,q}^{(l)}=\xi)\Pr(r_{n-2l,q}=\eta)]\times\kappa_{q}^{(k,l)}\right\} \tag{21}\]where \(\kappa_{q}^{(k,l)}\) is related to the noise variance \(\sigma_{q}^{(k,l)}\) of \(k\)-times information merging and energy of \(\sum_{p}\vartheta_{n,p,q}^{(l)}\). According to \(c(p,q;l)\), \(r_{n,q}\triangleq\sum_{p}\vartheta_{n,p,1}^{(0)}\) when \(\left\{b_{n-l,q}^{(l)},r_{n-2l,q}\right\}\) satisfies the state \(\{\xi,\eta\}\). This relationship is defined as \((\xi,\eta)\propto\sum_{p}v_{n,p,q}^{(0)}\). The likelihood value \(\mathrm{L}(r_{n,q})\) can be replaced by the estimated likelihood value \(\hat{\mathrm{L}}(r_{n,q})^{(k)}\) after the information merging. The derived probabilities \(\Pr(b_{n,q}^{(l)})\) and \(\Pr(r_{n,q})\) are utilized to enhance symbol detection performance. The equivalent detection symbols \(r_{n,q}\) are obtained from \(\hat{\mathrm{L}}(r_{n,q})^{(k)}\) by hard decision.