## 2 Signal Model and Conventional Algorithm Analysis  
### Signal Model  
Unlike GPS, the space constellation of the BeiDoi System consists of Geostationary Earth Orbit (GEO) satellites, Medium Earth Orbit (MEO) satellites and Inclined Geosynchronous Satellite Orbit (IGSO) satellites [21]. The open service signal consists of BII signal and B2I signal [22]. The signal is composed of the carrier frequency, ranging code and navigation message. The nominal frequency of the B1I signal is 1561.098 MHz, and the nominal frequency of the B2I signal is 1207.140 MHz. The chip rate of the B1I and B2I ranging code is 2.046 Mcps, and the length is 2046 chips. Navigation messages are formatted in D1 and D2 based on their rate and structure. MEO/IGSO and GEO satellites broadcast D1 and D2 navigation messages respectively. For D1 navigation message in format D1 of rate 50 bps, a secondary code of Neumann-Hoffman (NH) code is modulated on ranging code. The period of the NH code is selected to match the duration of a navigation message bit. The bit duration of NH code is the same as one period of the ranging code. The duration of one navigation message bit is 20 milliseconds and the ranging code period is 1 millisecond. Thus the NH code (\(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(1\), \(1\), \(-1\), \(1\), \(-1\), \(1\), \(-1\), \(-1\), \(1\), \(1\), \(1\), \(-1\)) with length of 20 bits, rate 1 kbps and bit duration of 1 millisecond is adopted. It is modulated on the ranging code synchronously with navigation message bit, as shown in Figure 1. In addition, the rate of D2 navigation message is 500 bps [22].  
This paper mainly discusses about the B1I and B2I signal broadcast by MEO/IGSO satellites which is modulated by NH code. The primary symbols and variables used are listed in Table 1.  
The signal at the input of a receiver is generally an intermediate frequency (IF) signal, obtained by down-converting, sampling and quantization. The IF signal can be written as follows [22]:  
\[s_{IF}(n)=AD(nT_{s})H(nT_{s})c(nT_{s}-\tau_{0})\exp[j\cdot 2\pi(f_{IF}+f_{d})nT_ {s}+\phi_{0}] \tag{1}\]  
where \(A\) stands for the carrier amplitude, \(D\) and \(H\) represent the navigation data and the NH code, and \(T_{s}\) is sampling time, \(c(\cdot)\) represents the ranging code sequence, \(\tau_{0}\), \(\phi_{0}\), \(f_{IF}\) and \(f_{d}\) are code propagation delay, initial carrier phase, the carrier IF and Doppler shift.  
After signal acquisition, the receiver obtains the approximate incoming frequency \(\hat{f}_{d,L}\) and ranging code phase \(\hat{\tau}_{L}\). The locally generated signal sequence is:  
\[s_{L}(n)=c(nT_{s}-\hat{\tau}_{L})\exp[j\cdot 2\pi(f_{IF}+\hat{f}_{d,L})nT_{s}] \tag{2}\]  
Assuming the coherent integration time is \(T_{c}\), the output of the kth coherent integration is [14]:  
\[Y(k)=\frac{1}{N_{c}}\sum_{n=kN_{c}}^{(k+1)N_{c}-1}s_{IF}(n)\cdot s_{L}(n)=0.5 AD_{k}H_{k}R(\delta\tau)\text{sinc}[\delta f_{d}(k)T_{c}]\exp[j\cdot\phi_{k}+j \pi\delta f_{d}(k)T_{c}] \tag{3}\]  
where k denotes the index of the coherent integration interval, \(N_{c}=T_{c}/T_{s}\) is the number of samples in the coherent integration time \(T_{c}\), \(D_{k}=\pm 1\) is navigation data, \(H_{k}=\pm 1\) is NH code, \(R(\cdot)\) represents the autocorrelation function of the ranging code, \(\delta\tau\) and \(\delta f_{d}\) are the errors of code delay estimation and the Doppler shift, \(\phi_{k}\) is the initial carrier phase error in the kth coherent integration interval.  
Figure 1: Signal structure of B1I and B2I signal broadcasted by MEO/IGSO and GEO satellites.