# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 2 Tracking Error Estimation
### Traditional Discriminator
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal** & **Useful Signal** & **Noise** & **Noise Distribution** \\ \hline \(I_{\rm E}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i}-\frac{d}{2})\sin\c(\delta f_{i}T_{ \rm coh})\cos(\overline{\delta\varphi_{i}})\) & \(n_{\rm IE}\) & \\ \(Q_{\rm E}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i}-\frac{d}{2})\sin\c(\delta f_{i}T_{ \rm coh})\sin(\overline{\delta\varphi_{i}})\) & \(n_{\rm QE}\) & \\ \(I_{\rm P}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i})\sin\c(\delta f_{i}T_{\rm coh}) \cos(\overline{\delta\varphi_{i}})\) & \(n_{\rm IP}\) & \\ \(Q_{\rm P}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i})\sin\c(\delta f_{i}T_{\rm coh}) \sin(\overline{\delta\varphi_{i}})\) & \(n_{\rm QP}\) & Gaussian white noise \(N(0,\sigma_{\rm noise}{}^{2})\) \\ \(I_{\rm L}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i}+\frac{d}{2})\sin\c(\delta f_{i}T_{ \rm coh})\cos(\overline{\delta\varphi_{i}})\) & \(n_{\rm IL}\) & \\ \(Q_{\rm L}\) & \(A_{i}\cdot D_{i}\cdot R(\hat{\sigma}\tau_{i}+\frac{d}{2})\sin\c(\delta f_{i}T_{ \rm coh})\sin(\overline{\delta\varphi_{i}})\) & \(n_{\rm QL}\) & \\ \hline \(E^{2}\) & \(A^{2}|\sin\c(\delta fT_{\rm coh})|^{2}\sum\limits_{j=1}^{N_{\rm E}}R^{2}(\delta \tau_{j}-\frac{d}{2})\) & \(N_{\rm nc}n_{\rm E^{2}}\) & \\ \(P^{2}\) & \(A^{2}|\sin\c(\delta fT_{\rm coh})|^{2}\sum\limits_{j=1}^{N_{\rm P}}R^{2}(\delta \tau_{j})\) & \(N_{\rm nc}n_{\rm IP^{2}}\) & White noise with unknown distribution \\ \(L^{2}\) & \(A^{2}|\sin\c(\delta fT_{\rm coh})|^{2}\sum\limits_{j=1}^{N_{\rm P}}R^{2}(\delta \tau_{j}+\frac{d}{2})\) & \(N_{\rm nc}n_{\rm I^{2}}\) & \\ \hline \(E\) & \(A|\sin\c(\delta fT_{\rm coh})|\sum\limits_{j=1}^{N_{\rm NP}}R(\delta\tau_{j}- \frac{d}{2})\) & \(N_{\rm nc}n_{\rm E}\) & \\ \(P\) & \(A|\sin\c(\delta fT_{\rm coh})|\sum\limits_{j=1}^{N_{\rm NP}}R(\delta\tau_{j})\) & \(N_{\rm nc}n_{\rm IP}\) & White noise with unknown distribution \\ \(L\) & \(A|\sin\c(\delta fT_{\rm coh})|\sum\limits_{j=1}^{N_{\rm NP}}R(\delta\tau_{j}+ \frac{d}{2})\) & \(N_{\rm nc}n_{\rm L}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The mathematical model of coherent/non-coherent integration results. outputs are expanded in the form of the summation of real tracking error and estimation noise in Equations (15)-(17). Thus, it is clear that outputs of ATAN/ATAN2/NC-EMLE discriminator contain the real tracking error and estimation noise (i.e., \(n_{\rm ATAN}\), \(n_{\rm ATAN2}\), and \(n_{\rm NC-EMLE}\)). These discriminator algorithms make non-linearity processing for useful signal and noise of coherent/non-coherent integration (e.g., division and arctangent operation). Thus it is difficult to get the analytic solution for the probability distribution of estimation noise, which is the barrier to implementing the non-coherent pre-filter. The estimation noise variance of the ATAN discriminator has been derived in [10], with some necessary simplification, and is shown in Equation (18). The estimation noise variance of the ATAN2 discriminator is derived in Appendix B based on Equation (18), and the result is shown in Equation (19). However, the accuracy of variance estimation of Equations (18) and (19) would decrease with decreasing \(C/N_{0}\), as will be shown in Section 3.1. Besides, the estimation noise variance of the NC-EMLE discriminator is difficult to derive. \[\sigma_{\rm ATAN}{}^{2}=\mathrm{D}(n_{\rm ANAN})=(\frac{1}{2\pi})^{2}\frac{1}{ 2\cdot c/n_{0}\cdot T_{\rm coh}}(1+\frac{1}{c/n_{0}\cdot T_{\rm coh}})[\rm cycle ^{2}] \tag{18}\]  
\[\sigma_{\rm ATAN2}{}^{2}=\mathrm{D}(n_{\rm ANAN2})=(\frac{1}{2\pi})^{2}\frac{ 1}{c/n_{0}\cdot T_{\rm coh}{}^{3}}(1+\frac{1}{c/n_{0}\cdot T_{\rm coh}})[\rm Hz ^{2}] \tag{19}\]