# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 3 Enhanced Non-Coherent Pre-Filter Design
### Observation Noise Characteristics Analysis of Non-Coherent Pre-Filter  
Considering the complexity of the probability distribution of discriminator's estimation noise, Monte Carlo simulation is introduced as the analysis tool. Roughly 40,000 groups of correlation values are simulated as Equation (4). Each group of correlation values contains six values, which are enough to execute the ATAN/ ATAN2/NC-EMLE discriminator algorithms. The real tracking error, \(T_{\text{coh}}\), \(C/N_{0}\) can be preset before the discriminator works. The values of estimation noise are calculated by subtracting the discriminator's outputs from real tracking error. Thus, it is convenient for us to study the estimation noise characteristics of the ATAN/ATAN2/NC-EMLE discriminator in different \(T_{\text{coh}}\) and \(C/N_{0}\) without executing the intact tracking loop algorithms. Here we consider the discriminator and pre-filter as estimators, just a part of the tracking loops. It also should be noted that the distribution of estimation noise of the ATAN/ATAN2/NC-EMLE discriminator changes with tracking error changing according to Equations (15)-(17). However, in this paper, we only analyze the condition where tracking error is zero, because this condition will simplify the analysis and will always be met when the tracking loop is in locking state. Figures 2-4 respectively depict the estimation noise distribution of the ATAN/ATAN2/NC-EMLE discriminator in different \(C/N_{0}\) and \(T_{\text{coh}}\). The blue histograms in these figures indicate the statistical results of estimation noise. The statistical results have been divided by the number of simulated I/Q correlation groups so that units of y-axis can be transformed to probability density. The red lines in these figures represent the shape of Gaussian probability density distribution function whose mean and variance are same as those of corresponding statistical results. These figures imply that there is an obvious difference between the actual estimation noise distribution of the ATAN/ATAN2/NC-EMLE discriminator and a Gaussian distribution. That is, the estimation noise of the ATAN/ATAN2/NC-EMLE discriminator does not satisfy Gaussian distribution, which is consistent with the conclusion that can be given by Equations (15)-(17). Figures 5-7 show the mean values and standard deviation (STD) values of estimation noise for the ATAN/ATAN2/NC-EMLE discriminator in different \(C/N_{0}\) and \(T_{\text{coh}}\). On the one hand, the mean values of estimation noise, given by statistical analysis, are very close to zero in all conditions. On the other hand, these figures suggest that the variances of estimation noise of ATAN/ATAN2/NC-EMLE discriminator decrease with the \(C/N_{0}\), or \(T_{\text{coh}}\), or both increasing. Above all, these results are obtained by statistics of roughly 40,000 outputs of ATAN/ATAN2/NC-EMLE discriminator so that they are more close to the actual statistical characteristics of estimation noise. Figure 2: The estimation noise distribution of ATAN discriminator in different \(T_{\text{coh}}\) and \(C/N_{0}\). Figure 4: The estimation noise distribution of NC-EMLE discriminator in different \(T_{\rm coh}\) and \(C/N_{0}\). Figure 5: Mean values and standard deviations of estimation noise for ATAN discriminator in different \(C/N_{0}\) and \(T_{\rm coh}\). Figure 3: The estimation noise distribution of ATAN2 discriminator in different \(T_{\rm coh}\) and \(C/N_{0}\). Figure 8a,b shows the difference between the estimation noise variance obtained by Monte-Carlo simulation and estimation noise variance calculated by Equations (18) and (19). This figure indicates that these two equations can only provide relatively accurate estimation noise variances for the ATAN/ATAN2 discriminator when \(C/N_{0}\) is high or \(T_{\text{coh}}\) is long. However, these conditions cannot be satisfied at all times. Thus, Equations (18) and (19) are not accurate enough to calculate estimation noise variances of ATAN and ATAN2 discriminators.