## 2 Proposed Synchronization Algorithm Based on LPA
### Synchronization Algorithm Based on LPA  
In order to reduce the complexity of the synchronization besides the phase states trellis, we use the LPA to rederive the synchronization error detector. LPA is a method to approximate the phase response of the CPM as a linear phase response (or a truncated REC phase response), expressed as  
\[q_{0}(t)= \begin{cases}0,&t\leq 0\\ \dfrac{t}{2L^{\prime}T},&0<t\leq L^{\prime}T\\ \dfrac{1}{2},&t>L^{\prime}T\end{cases}  \tag{18}\]  
where \(L^{\prime}\) is the length of the linear phase response. Note that \(L^{\prime}\) also stands for the truncated length of PT. LPA is similar to the PT method, and we use PT as one of the reduced-complexity methods for MLSD. Therefore, the number of phase states in the trellis is reduced to \(pM^{L^{\prime}-1}\). For a 4-ary \(h=\{\frac{4}{16},\frac{5}{16}\}\) CPM, \(N_{state}\) becomes \(pM^{L^{\prime}-1}=64\), and the number of MFs becomes \(N_{h}M^{L^{\prime}}=32\). The signal \(s(t,\mathbf{\alpha})\) using TPT and LPA can be rewritten as  
\[s_{0}(t,\mathbf{\alpha})=\sqrt{\dfrac{E_{s}}{T}}e^{j\phi_{0}(t,\mathbf{\alpha})} \tag{19}\]  
with  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]  
where  
\[\theta^{\prime}_{n}=2\pi\sum_{i=0}^{n-L^{\prime}}h_{i}u_{i} \tag{21}\]  
Figure 1: Comparison of synchronization algorithm: (**a**) Schematic diagram of conventional EL-based synchronization algorithm. (**b**) Schematic diagram of LPA-based synchronization algorithm. and  
\[\eta_{0}(t;\mathbf{l^{\prime}_{n}},\alpha_{n})=2\pi\sum_{i=n-L^{\prime}+1}^{n}h_{i} \alpha_{i}q_{0}(t-iT),\]  
\[\mathbf{l^{\prime}_{n}}=[\alpha_{n-L^{\prime}+1},...,\alpha_{n-2},\alpha_{n-1}]. \tag{22}\]  
Substituting the phase response with Equation (18) into Equation (22), we have the correlative phase with LPA as  
\[\eta_{0}(t;\mathbf{l^{\prime}_{n}},\alpha_{n}) =\pi\sum_{i=n-L^{\prime}+1}^{n}h_{i}\alpha_{i}\frac{t-iT}{L^{ \prime}T}, \tag{23}\] \[=\pi\frac{b_{n}t-c_{n}T}{L^{\prime}T}\]  
where  
\[b_{n}=\sum_{i=n-L^{\prime}+1}^{n}h_{i}\alpha_{i} \tag{24}\]  
and  
\[c_{n}=\sum_{i=n-L^{\prime}+1}^{n}ih_{i}\alpha_{i}. \tag{25}\]  
Note that  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]0  
Then, the output of the matched filter is  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]1 \[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]2  
The joint log-likelihood function is  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]3  
The maximum likelihood estimate takes the partial derivative of the phase error and timing error  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]4 \[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]5  
and  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]6  
Then, the error signal expression is  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]7  
and  
\[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]8 \[\phi_{0}(t,\mathbf{\alpha})=\theta^{\prime}_{n}+\phi_{n}+\eta_{0}(t;\mathbf{l}^{ \prime}_{n},\alpha_{n}), \tag{20}\]9In Equation (32), \(\frac{\pi\hat{b}_{n-D}}{LT}\) is mainly related to the polarity of the estimated error. To further reduce the complexity of Equation (32), we use the sign of \(\frac{\pi\hat{b}_{n-D}}{LT}\) instead of itself, and Equation (32) becomes  
\[ \begin{split} e_{\tau}^{s}(n-D)&=\text{sign}\Bigg{(} \frac{\pi\hat{b}_{n-D}}{L^{\prime}T}\Bigg{)}e_{\varphi}(n-D)\\ &=\text{sign}\Big{(}\hat{b}_{n-D}\Big{)}e_{\varphi}(n-D).\end{split}