## 3 Proposed Differential Coherent Bit Synchronization Algorithm
### Differential Coherent Bit Synchronization Algorithm  
In order to eliminate the influence of frequency deviation, the differential coherent algorithm is combined with the maximum likelihood bit synchronization method. For BeiDou weak signals, bit  
Figure 3: The amplitude attenuation of ML bit synchronization method due to frequency deviation. The coherent integration time \(T_{coh}\) is 20 ms.  
transitions are more frequent with NH codes. The differential coherent process of two consecutive correlation outputs is not suitable for BeiDou signals. According to the BeiDou signal structure and NH code modulation, the period of NH code is as long as the duration of a navigation message bit, so \(H_{k}=H_{k-20m}\), where m is an integer. In Equation (10), the signs of \(Q(k)\) depend on \(D_{k}D_{k-1}H_{k}H_{k-1}\). If the delay time of differential coherent is selected to be an integral multiple of 20 ms, \(H_{k}H_{k-20m}\) will be equal to 1 and \(D_{k}D_{k-20m}H_{k}H_{k-20m}\) will be equal to\(D_{k}D_{k-20m}\) which means that the differential coherent result will only depend on transitions of the navigation data and have nothing to do with NH code transitions. Therefore, in the proposed scheme, the differential coherent delay time is modified to an integral multiple of 20 ms to eliminate the influence of NH code. The concrete steps of the proposed scheme are as follows:  
* Step 1. Differential coherent algorithm with long delay time The delay time differential coherent is modified to an integral multiple of 20 ms and \(H_{k}~{}=~{}H_{k-20m}\). The mathematical representation is: \[ \begin{array}{ll}Z(k)&=Y(k)\cdot Y(k-20m)^{*}=\left[0.5 AR(\delta\tau)\right] ^{2}\exp(j\cdot\pi\delta f_{d}mT_{c})\\ &\times D_{k}D_{k-20m}H_{k}H_{k-20m}\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc }[\delta f_{d}(k-20m)T_{c}]\\ &=\left[0.5 AR(\delta\tau)\right]^{2}\exp(j\cdot\pi\delta f_{d}mT_{c})\\ &\times D_{k}D_{k-20m}\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc}[\delta f_{ d}(k-20m)T_{c}]\end{array} \] (12) where \(m\) is an integral number, \(\pi\delta f_{d}mT\) is the carrier phase deviation between \(Y(k)\) and \(Y(k-20m)\) and it does not change with time. Because:  
\[|\exp(j\cdot\pi\delta f_{d}mT_{c})|=|\cos(\pi\delta f_{d}mT_{c})+j\cdot\sin( \pi\delta f_{d}mT_{c})|=1 \tag{13}\]  
\(\exp(j\cdot\pi\delta f_{d}mT_{c})\) has no effect on the amplitude of \(Z(k)\). \(\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc}[\delta f_{d}(k-20m)T_{c}]\) represents the amplitude attenuation of the differential coherent result due to the frequency deviation. In stable tracking stage, \(\delta f_{d}\) changes slightly, so \(\delta f_{d}(k)\) can be approximately equal to \(\delta f_{d}(k-20m)\) and \(\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc}[\delta f_{d}(k-20m)T_{c}]\) can be approximately equal to \(\sin c[\delta f_{d}(k)T_{c}]^{2}\). Here, \(T_{c}\) is 1 ms, and the frequency deviation is limited in \([-250\) Hz, 250 Hz]. Under the worst frequency deviation, the maximum loss \(10\lg[\text{sinc}[\delta f_{d}(k)T_{c}]^{2}]\) is about \(-0.9121\) dB. Figure 4 is the plot of \(\text{sinc}[\delta f_{d}(k)T_{c}]^{2}\) which stands for the amplitude attenuation due to the frequency deviation. The frequency deviation effect is greatly eliminated compared with Figure 3.  
Figure 5 shows the signs of data in the differential coherent algorithm with long delay time. It can be seen that after differential coherent process, the signs of \(Z(k)\) only depend on transitions of the navigation data and have nothing to do with NH code. * Step 2. Maximum likelihood bit synchronization method Based on differential coherent algorithm with long delay time, the ML bit synchronization method uses the sum of the absolute values of cross-correlation function between \(Z(k)\) and the window function.