# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 3 Statistical Characteristic of Zero-Bias
### Equivalent Reference Waveform Analysis  
The equivalent reference waveform is in a form of the local signal of the early-late phase discriminator [23; 24; 25]. The early-late phase discriminator estimates phase using the information difference between an early correlator and a late correlator. A phase discriminator based on an equivalent reference waveform utilizes the correlation of the local reference signal with the received signal to estimate the phase. As shown in Figure 5, the rectangular reference waveform \(w(t)\) is defined as the difference between the early local signal and the late local signal [26] as follows  
\[w(t)=0.5(s(t+d_{0})-s(t-d_{0})), \tag{11}\]  
where \(2d_{0}\) is the correlation interval of the discriminator, which is numerically equal to the gate width of the reference wavef orm. The rectangular reference waveform is the first reference waveform [27], and the valid symbol appears as the polarity inversion of the chip. \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{**Notation**} & \multicolumn{1}{c}{**Definition**} \\ \hline \(s(t)\) & Baseband signal \\ \(c(t)\) & Code signal \\ \(sc(t)\) & Subcarrier signal \\ \(w(t)\) & Rectangular reference signal \\ \(p(t)\) & Composite signal \\ \(\tau_{0^{\prime}}^{c}\), \(\tau_{1}^{c}\), \(\tau_{2}^{c}\) & Initial code phase \\ \(\tau_{0^{\prime}}^{sc}\), \(\tau_{1}^{sc}\), \(\tau_{2}^{sc}\) & Initial subcarrier phase \\ \(\tau_{0^{\prime}}^{c}\), \(\tau_{0}^{sc}\) & Phase estimation \\ \(T_{s_{\sigma}}\), \(T_{c}\), \(T_{sc}\), \(T_{cdh}\) & Sampling period, chip period and subcarrier period, coherent integration time \\ \(F_{s_{\sigma}}\), \(F_{c_{\sigma}}\), \(F_{sc}\), \(F_{0}\), \(\tilde{F}_{s}\) & Sampling frequency, code rate, subcarrier rate, reference frequency, noncommensurate \\  & sampling frequency \\ \(k\), \(k_{s_{\sigma}}\), \(k_{c_{\sigma}}\), \(k_{c_{\sigma}}\), \(\tilde{k}_{s}\) & Frequency coefficient \\ \(2d_{0}^{c}\), \(2d_{0}^{sc}\) & Early-late correlation interval of the phase discriminator \\ \(R^{c}\), \(R_{c}^{c}\), \(R_{c}^{c}\) & Correlation values corresponding to different code phases \\ \(R^{sc}\), \(R^{sc}\), \(R_{L}^{sc}\) & Correlation values corresponding to different subcarrier phases \\ \(D^{c}\), \(D^{c}\) & Phase discrimination results \\ \(P_{c}\), \(P_{sc}\) & Phase resolution \\ \(S\), \(S^{c}\), \(S^{c}\) & Zeros-bias of disrimination \\ \(\tilde{\tau}_{s}\) & Average number of samples of the composite waveform \\ \(N_{s}\), \(N_{p}\), \(\tilde{N}_{p}\) & Number of coherent integration points of the correlator, number of composite waveforms, number of non-zero composite waveforms \\ \(lcm(\cdot)\) & Least common multiple operation \\ \(round(\cdot)\) & Rounding operation \\ MAX(\(S^{c}\)) & The maximum deviation of zero-bias \\ STD(\(S^{c}\)) & The standard deviation of zero-bias \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequently used notations. An equivalent phase discriminator can be obtained by correlating the reference waveform with the received signal. Assuming that the local signal is synchronized with the received signal, the composite signal associated with the reference waveform and the received signal \(p(t)\) is  
\[p(t)=s(t)w(t). \tag{12}\]  
According to Equation (9), the zero-bias of the digital discriminator can be approximated as the output value of the discriminator when the phase error is zero, namely:  
\[S(\tau_{0})=\frac{2}{(1+2d_{0})\times N_{s}}\sum_{n=0}^{N_{p}-1}\big{(}\sum p (n)\big{)}, \tag{13}\]  
where \(\sum p(n)\) represents the \(n\)-th accumulated value of the composite waveform and \(N_{p}\) represents the number of waveforms of the composite signal during the coherent integration time \(T_{cah}\). When the reference signal is synchronized with the local signal, the composite waveform symbol \(p(n)\) is oddly symmetric about the center point. Considering the sampling effect, the accumulated value of the composite waveform's actual value may be \(+1\), \(-1\) or \(0\).