## 3 Proposed Differential Coherent Bit Synchronization Algorithm
### Differential Coherent Bit Synchronization Algorithm
The 20 ms width window function is: \[W(k)=1,~{}(k=1,2,\ldots 20).\] (14) The cross-correlation between \(Z(k)\) and \(W(k)\) is given by: \[C(n,l_{b})=\sum_{k=1}^{20}Z(20n+l_{b}+k)W(k),~{}(n=0,2,\ldots N-1).\] (15)The sum of the absolute values of cross-correlation is:  
\[S(l_{b})=\sum\limits_{n=0}^{N-1}|C(n,l_{b})|. \tag{16}\]  
Then the ML estimate of bit boundaries is obtained as:  
\[\hat{l}_{b}=\underset{l_{b}\in[1:20]}{\mathrm{argmax}}S(l_{b}). \tag{17}\]  
The maximum sum of the absolute values of cross-correlation can be expressed as:  
\[ \begin{array}{ll}S(\hat{l}_{b})&=\sum\limits_{n=0}^{N-1}|C(n,l_{b})|=\sum \limits_{n=0}^{N-1}|\sum\limits_{k=1}^{20}Z(20n+\hat{l}_{b}+k)|\\ &=\sum\limits_{n=0}^{N-1}|[0.5 AR(\delta\tau)]^{2}\exp(j\cdot\pi\delta f_{d}mT_{ c})\sum\limits_{k=1}^{20}D_{k}D_{k-20m}\mathrm{sinc}[\delta f_{d}(k)T_{c}] \mathrm{sinc}[\delta f_{d}(k-20m)T_{c}]|\end{array} , \tag{18}\]  
here, \(\sum\limits_{k=1}^{20}\sin c[\delta f_{d}(k)T_{c}]\sin c[\delta f_{d}(k-20m)T _{c}]\) represents the frequency deviation effect of the proposed algorithm and \(\sin c[\delta f_{d}(k)T_{coh}]\) in Equation (9) represents the frequency deviation effect of the conventional ML algorithm. In order to compare the frequency deviation effects of the two algorithms, we should make approximations to \(\sum\limits_{k=1}^{20}\sin c[\delta f_{d}(k)T_{c}]\sin c[\delta f_{d}(k-20m)T _{c}]\). Because \(\delta f_{d}\) changes slightly in stable tracking stage, \(\delta f_{d}(k)\) can be approximately equal to \(\delta f_{d}(k-20m)\) and \(\sin c[\delta f_{d}(k)T_{c}]\sin c[\delta f_{d}(k-20m)T_{c}]\) can be approximately equal to \(\sin c[\delta f_{d}(k)T_{c}]^{2}\) as:  
\[\sum\limits_{k=1}^{20}\mathrm{sinc}[\delta f_{d}(k)T_{c}]\mathrm{sinc}[ \delta f_{d}(k-20m)T_{c}]\approx\sum\limits_{k=1}^{20}\{\mathrm{sinc}[\delta f _{d}(k)T_{c}]\}^{2}\approx 20\{\mathrm{sinc}[\delta f_{d}(k)T_{c}]\}^{2}. \tag{19}\]  
In Figure 6, the red line stands for 10 \(\times\) lg(20\(\{\)sinc\([\delta f_{d}(k)T_{c}]\}^{2}\)) and the blue line stands for 10 \(\times\) lg(sinc\([\delta f_{d}(k)T_{coh}]\)). It can be seen that the absolute value of the former is much smaller than the latter especially with a frequency deviation of an integral multiple of 50 Hz, because \(T_{c}\) is 1 ms and \(T_{coh}\) is 20 ms.  
Figure 4: The amplitude attenuation of differential coherent algorithm due to frequency deviation. The integration time \(T_{c}\) is 1 ms. Therefore, \(\sum\limits_{k=1}^{20}\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc}[\delta f_{d}(k-2 0m)T_{c}]\) in the Equation (19) is much smaller than \(\text{sinc}[\delta f_{d}(k)T_{coll}]\) in the Equation (9). Therefore, the proposed bit synchronization algorithm should outperform the conventional ML algorithm with a frequency deviation of an integral multiple of 50 Hz. As above, the proposed bit synchronization algorithm eliminates the influence of frequency deviation by adopting differential coherent algorithm and eliminates the influence of NH code by modifying the differential delay time to an integral multiple of 20 ms.