# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 3 Statistical Characteristic of Zero-Bias
### Standard Deviation of Zero-Bias  
According to the analysis in Section 3.1, a composite waveform with a non-zero accumulated value may affect the zero-bias of the digital phase discriminator. Whether the phase synchronization result is equal to zero depends on the parity of sampling point number of the composite waveform. Suppose the composite waveform appear continuously, the average of composite waveform samples defined during the coherent integration time is  
\[\overline{c}_{s}=2d_{0}\bar{F}_{s}/F_{p}, \tag{14}\]  
where \(F_{p}\) represents the symbol rate of the composite signal. Under noncommensurate sampling conditions, the sampling frequency is not proportional to the symbol rate, therefore the number of sampling points varies according to different symbol waveforms. Figure 5: Schematic diagram of time domain sampling of local reference signals and equivalent reference waveforms. The average number of samples reflect the statistical characteristics of the coherent integration time. When the average of sampling points is close to an odd number, the number of sampling points is large. However, when the number of composite waveforms is an odd, the number of sampling points is rather small. According to this relationship, the number of composite waveforms \(\tilde{N}_{p}\) where the cumulative value is non-zero can be determined by  
\[\tilde{N}_{p}=F_{p}T_{coh}\cdot\big{|}\overline{\xi_{s}}-2\cdot round( \overline{\varepsilon}_{s}/2)\big{|}. \tag{15}\]  
For subcarrier signals, each adjacent subcarrier symbol is flipped. Thus, the number of composite waveforms whose non-zero accumulated value is equal to \(\tilde{N}_{p}\). Since the composite waveform is continuous, a composite waveform with accumulated values of \(+1\) and \(-1\) will be canceled when coherently accumulated. This explains why the zero-bias of the digital subcarrier phase discriminator is very small. However, for a spreading code sequence, the polarity flip of the adjacent chips is random. In a statistical sense, the polarity inversion of a pseudo-random spreading code sequence can approximately satisfy a Bernoulli distribution with a probability of 1/2. Therefore, for the spreading code sequence, the number of symbol waveforms, whose accumulated values are non-zero, is smaller than \(\tilde{N}_{p}\). In the spread spectrum sequence, the position of the symbol waveform whose accumulated value is non-zero is related not only to the sampling frequency, the code rate, and the initial phase of the signal but also to the polarity inversion of the code sequence. For the most extreme case, assuming that there is a set of spreading code sequences \(\{c_{1},c_{2},\cdots,c_{n}\}\), a polarity inversion occurs for all composite waveforms with an accumulated value of \(+1\), and no polarity inversion occurs for any positions where the accumulated value is \(-1\). Then, the digital code phase discriminator corresponds to the maximum zero-bias \(\mathrm{MAX}(S^{c})\), which is characterized by  
\[\mathrm{MAX}(S^{c})=\frac{F_{c}}{\left(1+2d_{0}^{c}\right)\overline{F}_{s}} \big{|}\overline{\varepsilon}_{s}-2\cdot round(\overline{\varepsilon}_{s}/2) \big{|}. \tag{16}\]  
The maximum value of the zero-bias is just a special case. In order to apply the general case, a standard zero-bias deviation of the non-periodic spreading code sequence is defined below. With an assumption that the probability of polarity inversion of the non-periodic spreading code sequence is 1/2 and is independent of each other. According to Appendix A, the variance of the composite waveform accumulated value based on the aperiodic spreading code sequence is:  
\[\mathrm{VAR}(\mathbf{Z})=\frac{\tilde{N}_{p}}{4}. \tag{17}\]  
wherein, the statistic \(\mathbf{Z}\) is equivalent to the accumulated value \(\sum\limits_{n=0}^{N-1}(\sum p(n))\) of the composite waveform in the coherent integration time \(T_{coh}\).