## 2 Proposed Synchronization Algorithm Based on LPA
### Conventional Synchronization Algorithm Based on EL-Matched Filtering  
The matched filter (MF) output is  
\[Z(\tilde{\tau},\alpha_{n})=\int_{(n-1)T}^{nT}r(t)e^{-j\{\eta(t-\tau I_{n}, \alpha_{n})+\phi_{n}\}}dt. \tag{10}\]  
Assuming that \(\varphi\) and \(\tau\) are known, the expression of the joint log-likelihood function is [15]  
\[\Lambda[r(t);\tilde{\varphi},\tilde{\tau}]=\text{Re}\Bigg{\{}\sum_{n=0}^{N-1} e^{-j(\tilde{\varphi}+\theta_{n})}Z(\tilde{\tau},\alpha_{n})\Bigg{\}}. \tag{11}\]  
The estimated value of the synchronization parameter can be obtained by setting the partial derivative of the likelihood function equal to zero with respect to \(\varphi\) and \(\tau\). Thus, we obtain  
\[\frac{\partial\Lambda[r(t);\tilde{\varphi},\tilde{\tau}]}{\partial\varphi}= \text{Im}\Bigg{\{}\sum_{n=0}^{N-1}e^{-j(\tilde{\varphi}+\theta_{n})}Z(\tilde{ \tau},\hat{\alpha}_{n})\Bigg{\}}=0 \tag{12}\]  
and  
\[\frac{\partial\Lambda[r(t);\tilde{\varphi},\tilde{\tau}]}{\partial\tau}= \text{Re}\Bigg{\{}\sum_{n=0}^{N-1}e^{-j(\tilde{\varphi}+\theta_{n})}Y(\tilde {\tau},\hat{\alpha}_{n})\Bigg{\}}=0, \tag{13}\]  
where \(Y(\tilde{\tau},\hat{\alpha}_{n})\) is the derivative of \(Z(\tilde{\tau},\hat{\alpha}_{n})\) with respect to \(\tilde{\tau}\). The value of \(\hat{\alpha}_{n}\) is taken from the best survivor in the Viterbi algorithm or related decoding algorithm.  
The carrier phase and timing error signal can be expressed as  
\[e_{\varphi}(n-D)=\text{Im}\{Z(\hat{\tau}_{n-D},\hat{\alpha}_{n-D})e^{-j(\hat {\varphi}_{n-D}+\hat{\theta}_{n-D})}\} \tag{14}\]  
and  
\[e_{\tau}(n-D)=\text{Re}\{Y(\hat{\tau}_{n-D},\hat{\alpha}_{n-D})e^{-j(\hat{ \varphi}_{n-D}+\hat{\theta}_{n-D})}\}. \tag{15}\]  
The iterative signal expressions of time error and carrier error are as follows  
\[\hat{\varphi}_{n+1}=\hat{\varphi}_{n}+\gamma_{\varphi}e_{\varphi}(n-D) \tag{16}\]  
and  
\[\hat{\tau}_{n+1}=\hat{\tau}_{n}+\gamma_{\tau}e_{\tau}(n-D), \tag{17}\]  
where \(\gamma\) is the step size, \(\gamma=4 BT/k_{p}\), \(BT\) is the normalized equivalent noise bandwidth, and \(k_{p}\) is derived from the S-curve. \(D\) is an introduced delay and \(D=1\) produces satisfactory results in many cases (see [21]).  
The decision-directed (DD) joint phase and timing synchronization, following the error signal detectors of Equations (14) and (15), is shown in Figure 1a. The received signal is synchronized by the estimated \(\hat{\phi}_{n}\) and \(\hat{\tau}_{n}\). The synchronized signal is fed to the on-time-, early-, and late-MF banks. The results of the on-time-MF bank are processed by the Viterbi algorithm (VA). The detected \(\hat{\alpha}_{n}\) assists in estimating the phase error and timing error signals through a phase error detector (PED) and a timing error detector (TED), respectively. Note that the derivative matched filtering in Equation (15) is implemented by the difference between early- and late-MF banks. Therefore, three MF banks are required: the on-time-, early-, and late-MF banks. Finally, the first-order loop filters update the \(\hat{\phi}_{n}\) and \(\hat{\tau}_{n}\), respectively from Equations (16) and (17). As mentioned in Section 1, such an EL-based timing synchronizer has been widely used in digital receivers for CPM.