# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 2 System Description
### Signal Reception Model  
The influence of subcarriers introduces the possibility of a lock ambiguity in the full match tracking of BOC signals [13; 14]. Double estimation loops, delay locked loops (DLLs) and subcarrier lock loops (SLLs) work independently, which eliminates lock ambiguity. Figure 1 shows the block diagram of the BOC signal's receiving structure based on a double estimation. One baseband BOC signal enters the early-late code phase correlator after stripping the subcarrier signal. Then, the other baseband signal enters the early-late subcarrier phase correlator after stripping the spread code signal. The code tracking loop and the subcarrier tracking loop estimate and filter the code phase error, the subcarrier phase error, and their feedback control the generation of the local code signal and the local subcarrier signal. As shown in Figure 1, \(s(t)\) is the baseband BOC input signal, \(c(t)\), \(c\big{(}t+d_{0}^{c}\big{)}\), and \(c\big{(}t-d_{0}^{c}\big{)}\) are punctual, early, and late local code signals respectively, \(d_{0}^{c}\) is the  
early or late code phase, \(sc(t)\), \(sc\Big{(}t+d_{0}^{sc}\Big{)}\), and \(sc\Big{(}t-d_{0}^{sc}\Big{)}\) are punctual, early, and late subcarrier signal respectively, \(d_{0}^{sc}\) is the early or late subcarrier phase, \(R_{E}^{c}\) and \(R_{L}^{c}\) are the correlation values of the early and late code phases respectively, \(R_{E}^{sc}\) and \(R_{L}^{sc}\) are the correlation values of the early and late subcarrier phases respectively, \(\tau_{0}^{c}\) and \(\tau_{0}^{sc}\) are code phase and subcarrier phase estimated value respectively. For code phase correlators, assuming that the subcarrier phases in the local and received signals are fully synchronized, the normalized code phase correlation function \(R^{c}\Big{(}\tau_{1}^{c},\tau_{2}^{c}\Big{)}\) can be expressed as:  
\[R^{c}\Big{(}\tau_{1}^{c},\tau_{2}^{c}\Big{)}=\frac{1}{N_{s}}\sum_{m=0}^{N_{s}- 1}c_{g_{m}^{c}(\tau_{1}^{c})}c_{g_{m}^{c}(\tau_{2}^{c})} \tag{1}\]  
where \(c_{k}\) is the \(k\)-th chip in the spreading code sequence \(\{c_{1},c_{2},\cdots,c_{k},\cdots\}\), \(g_{m}^{c}(\tau^{c})(=\text{mod}((mT_{s}-\tau^{c})/T_{c}))\) is the code index of the \(m\)-th sample in the signal with the initial phase \(\tau^{c}\), \(\tau_{1}^{c}\) and \(\tau_{2}^{c}\) denote the code phase deviation of the local signal and the received signal, respectively. \(N_{s}\) is the number of coherent integration points of the correlator, \(T_{s}(=1/F_{s})\) is the sampling interval of the baseband signal, \(F_{s}\) is the sampling frequency of the baseband signal, \(T_{c}(=1/F_{c})\) is the width of the spread spectrum chip and \(F_{c}\) is the spreading code rate. The code phase discriminator estimates the code phase deviation of the local signal from the received signal using an early-late code phase correlator. The normalized code phase discrimination function \(D^{c}\Big{(}\tau_{0}^{c},\varepsilon\Big{)}\) is:  
\[D^{c}\Big{(}\tau_{0}^{c},\varepsilon\Big{)}=\frac{1}{1+2d_{0}^{c}}\Big{[}R^{c }\Big{(}\tau_{0}^{c},\tau_{0}^{c}+\varepsilon+d_{0}^{c}\Big{)}-R^{c}\Big{(} \tau_{0}^{c},\tau_{0}^{c}+\varepsilon-d_{0}^{c}\Big{)}\Big{]} \tag{2}\]  
where \(\tau_{0}^{c}\) is the initial code phase of the received signal, \(\varepsilon\) is the code phase deviation of the local signal from the received signal and \(2d_{0}^{c}\) is the early-late correlation interval of the code phase discriminator.