# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 2 System Description
### Sampling Distortion: Zero-Bias  
In addition to resolution error, the sampling distortion causes a zero-bias in the phase estimation of the digital discriminator. In the commensurate sampling case, the resolution error is much larger than the zero-bias. While in the noncommensurate sampling case the effect of the zero-bias phase is more arresting since resolution error is suppressed. The zero-bias of the discriminator arises due to the asymmetry of the correlation curve and is related to the randomness of the spreading sequence. In a  
Figure 1: Block diagram of binary offset subcarrier (BOC) signal receiving structure based on double estimation loop.  
real-world scenario, the zero-bias of the code phase discriminator is prominent, but the zero-bias of a sub-carrier phase discriminator is not that obvious.  
The zero-bias of the digital phase discriminator is defined as the phase delay of a stable convergence point of the discrimination curve, i.e.,  
\[S(\tau_{0})=-\epsilon\big{|}_{D(\tau_{0},\epsilon)=0}. \tag{8}\]  
In the case of noncommensurate sampling, regardless of the influence of phase resolution error, the zero-bias of the discriminator can be equivalent to the phase discrimination value at zero phase, i.e.,  
\[S(\tau_{0})\approx D(\tau_{0},0). \tag{9}\]  
Figure 2 shows the simulation results of the code phase discriminator and subcarrier phase discriminator of the BOCs (1, 1) signal. The simulation selects a GPS LIC signal whose pseudo-random number (PRN) is 1. The noncommensurate sampling frequency \(\tilde{F}_{s}\) is set to 4.999 MHz, the spreading code rate \(F_{c}\) is 1.023 MHz, the subcarrier symbol rate \(F_{sc}\) is 2.046 MHz and the coherent integration time \(T_{coh}\) is 1 ms. The correlation interval of the discriminator is an integer multiple of the phase resolution and is close to a typical value. The correlation interval, \(2d_{0}^{c}\), of the code phase discriminator is 0.5001 chips, and the correlation interval of the subcarrier phase discriminator is 0.5\(d_{0}^{c}\) chips. Thus, the zero-bias of the digital code phase discriminator is 2.5 ns (\(2.5\times 10^{-3}\) chips), which is evidently larger than the phase resolution of 0.2 ns (\(2.0\times 10^{-4}\) chips). Meanwhile, the zero-bias of the digital subcarrier phase discriminator is similar to the phase resolution.  
Figure 3 simulates the code phase discriminator and subcarrier phase discriminator of the BOCs (1, 1) signal, and analyzes the effect of the signal's initial phase. The initial phase of the signal is set to a range of \([-T_{s}/2,T_{s}/2]\), and the simulation interval is 1/10 of the resolution of the subcarrier phase discriminator \(P_{sc}\). The simulation results show that: (1) The zero-bias of the digital subcarrier phase discriminator is significantly smaller than the digital code phase discriminator, (2) the resolution of the zero-bias of the digital code phase is equal to the phase resolution of the discrimination curve, (3) the variation of the zero-bias can be approximately continuous, reaching a maximum at a specific phase.