## 2 Signal Model and Conventional Algorithm Analysis
### Maximum Likelihood Bit Synchronization Method  
The maximum likelihood bit synchronization method uses a likelihood function to detect bit boundary locations. The likelihood function used is the sum of the absolute values of cross-correlation function between the prompt correlator output sequence and a window function. In this method, every bit transition can contribute to the detection of the bit boundary, and more bit transitions at the same bit boundary can help to improve the successful synchronization rate (SSR). The ML bit synchronization algorithm is summarized below.  
Firstly, there are some parameters that need to be explained. The coherent integration time \(T_{c}\) in the tracking stage is 1 ms. The ratio between data bit period and ranging code period is 20, so assume there are 20 possible bit locations. Besides, the length of data sequence to be processed in ML method is N bits. For GPS signals, the 20 ms width window function is defined as:  
\[W(k)=1,\ (k=1,2,\ldots 20) \tag{4}\]  
For BeiDou signals, the 20 ms width window function is the same with the NH code and is given by:  
\[W(k)=[-1,-1,-1,-1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,1,1,-1,1,1,-1],\ (k=1,2,\ldots 20). \tag{5}\]  
The cross-correlation between \(Y(k)\) and \(W(k)\) is given by:  
\[C(n,l_{b})=\sum_{k=1}^{20}Y(20n+l_{b}+k)W(k),\ (n=0,2,\ldots N-1), \tag{6}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Meaning** \\ \hline \(s_{IF}(n)\) & received IF signal \\ \(A\) & carrier amplitude \\ \(D(n)\) & navigation data \\ \(H(n)\) & NH code \\ \(c(\cdot)\) & ranging code \\ \(T_{s}\) & sampling time \\ \(f_{IF}\) & carrier IF frequency \\ \(f_{d}\) & Doppler frequency shift \\ \(\tau_{0}\) & code propagation delay \\ \(\phi_{0}\) & initial carrier phase \\ \(s_{L}(n)\) & locally generated signal \\ \(\tau_{L}\) & ranging code phase of local signal \\ \(\hat{f}_{d,L}\) & carrier frequency shift of local signal \\ \(Y(k)\) & output of the kth coherent integration \\ \(T_{c}\) & integration time \\ \(N_{c}\) & the number of samples in the integration time \\ \(R(\cdot)\) & autocorrelation function of the ranging code \\ \(\delta\tau\) & code phase estimation error \\ \(\delta f_{d}\) & Doppler frequency estimation error \\ \(\phi_{k}\) & carrier phase estimation error \\ \(W(k)\) & window function \\ \(C(n,l_{b})\) & cross-correlation between \(Y(k)\) and \(W(k)\) \\ \(S(\hat{l}_{b})\) & sum of the absolute values of cross-correlation \\ \(l_{b}\) & estimate of the bit boundary \\ \(T_{coh}\) & coherent integration time \\ \(Q(k)\) & differential coherent values \\ \(Z(k)\) & Differential coherent values with long delay time \\ \hline \hline \end{tabular}
\end{table}
Table 1: Primary symbols and variables.  
where \(l_{b}\) is the initial edge shift of the window function in one bit period. The sum of the absolute values of cross-correlation is:  
\[S(l_{b})=\sum\limits_{n=0}^{N-1}\lvert C(n,l_{b})\rvert \tag{7}\]  
Then the ML estimate of the bit boundary is obtained as:  
\[\hat{l}_{b}=\underset{l_{b}\in[1:20]}{\mathrm{argmax}}S(l_{b}) \tag{8}\]  
Figure 2 is the diagram of the ML bit synchronization method. The maximum sum of the absolute values of cross-correlation corresponding with the real bit boundary can be simply considered as coherent integration of 20 ms and non-coherent integration of N bits (20N ms). Therefore, with longer data sequences or integration times, the performance for weak signals starts to improve.