# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 2 Tracking Error Estimation
### Relevant Baseband Signal Model
\begin{array}{l}E^{2}(i)=I_{\rm E}{}^{2}(i)+Q_{\rm E}{}^{2}(i)=A_{i}{}^{2}R^{ 2}(\delta\tau_{i}-\frac{d}{2})\sin{\rm c}^{2}(\delta f_{i}T_{\rm coh})&+n_{ \rm E}{}^{2}\\ P^{2}(i)=I_{\rm P}{}^{2}(i)+Q_{\rm P}{}^{2}(i)=A_{i}{}^{2}R^{2}(\delta\tau_{i}) \sin{\rm c}^{2}(\delta f_{i}T_{\rm coh})&+n_{\rm P}{}^{2}\\ L^{2}(i)\ =L_{\rm L}{}^{2}(i)+Q_{\rm L}{}^{2}(i)=A_{i}{}^{2}R^{2}(\delta\tau_{i}+ \frac{d}{2})\sin{\rm c}^{2}(\delta f_{i}T_{\rm coh})&+n_{\rm L}{}^{2}\\ \end{array}  \tag{6}\]  
\[ \begin{array}{l}E(i)=\sqrt{I_{\rm E}(i)^{2}+Q_{\rm E}(i)^{2}}=A_{i}\cdot R( \delta\tau_{i}-\frac{d}{2})|\sin{\rm c}(\delta f_{i}T_{\rm coh})|&+n_{\rm E}\\ P(i)=\sqrt{I_{\rm P}{}^{2}(i)+Q_{\rm P}{}^{2}(i)}=A_{i}\cdot R(\delta\tau_{i})| \sin{\rm c}(\delta f_{i}T_{\rm coh})|&+n_{\rm P}\\ L(i)=\sqrt{L_{\rm L}{}^{2}(i)+Q_{\rm L}{}^{2}(i)}=A_{i}\cdot R(\delta\tau_{i}+ \frac{d}{2})|\sin{\rm c}(\delta f_{i}T_{\rm coh})|&+n_{\rm L}\\ \end{array}  \tag{7}\]where  
\[ \begin{array}{rl}n_{\text{E}^{2}}&=n_{\text{IE}}{}^{2}+n_{\text{QE}}{}^{2}+2n_{ \text{IE}}A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}-\tfrac{d}{2})\sin\text{c}(\delta f _{i}T_{\text{coh}})\cos(\overline{\delta\varphi})&+\\ &2n_{\text{QE}}A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}-\tfrac{d}{2})\sin\text{c}( \delta f_{i}T_{\text{coh}})\sin(\overline{\delta\varphi})&\\ &\approx n_{\text{IE}}{}^{2}+n_{\text{QE}}{}^{2}+n_{\text{IE}}\\ n_{\text{P}^{2}}&=n_{\text{IP}}{}^{2}+n_{\text{QP}}{}^{2}+2n_{IP}A_{i}\cdot D _{i}\cdot R(\delta\tau_{i}-\tfrac{d}{2})\sin\text{c}(\delta f_{i}T_{\text{coh }})\cos(\overline{\delta\varphi})&+\\ &2n_{\text{QP}}A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}-\tfrac{d}{2})\sin\text{c }(\delta f_{i}T_{\text{coh}})\sin(\overline{\delta\varphi})&\\ &\approx n_{\text{IP}}{}^{2}+n_{\text{QP}}{}^{2}+n_{\text{IP}}\\ n_{\text{L}^{2}}&=n_{\text{IL}}{}^{2}+n_{\text{QL}}{}^{2}+2n_{\text{IL}}A_{i} \cdot D_{i}\cdot R(\delta\tau_{i}+\tfrac{d}{2})\sin\text{c}(\delta f_{i}T_{ \text{coh}})\cos(\overline{\delta\varphi})&+\\ &2n_{\text{QL}}A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}+\tfrac{d}{2})\sin\text{c }(\delta f_{i}T_{\text{coh}})\sin(\overline{\delta\varphi})&\\ &\approx n_{\text{IL}}{}^{2}+n_{\text{QL}}{}^{2}+n_{\text{IL}}\\ \end{array}  \tag{8}\]  
\[ \begin{array}{rl}n_{\text{E}}&=\sqrt{A_{i}{}^{2}R^{2}(\delta\tau_{i}-\tfrac{ d}{2})\sin\text{c}^{2}(\delta f_{i}T_{\text{coh}})+n_{\text{E}^{2}}}-A\cdot R( \delta\tau_{i}-\tfrac{d}{2})|\sin\text{c}(\delta f_{i}T_{\text{coh}})|\\ &\approx\sqrt{A_{i}{}^{2}(1-\tfrac{d}{2})^{2}+n_{\text{E}^{2}}-A_{i}(1-\tfrac {d}{2})}\\ n_{\text{P}}&=\sqrt{A_{i}{}^{2}R^{2}(\delta\tau_{i})\sin\text{c}^{2}(\delta f _{i}T_{\text{coh}})+n_{\text{P}^{2}}}-A\cdot R(\delta\tau_{i})|\sin\text{c}( \delta f_{i}T_{\text{coh}})|\\ &\approx\sqrt{A_{i}{}^{2}+n_{\text{P}^{2}}}-A_{i}\\ n_{\text{L}}&=\sqrt{A_{i}{}^{2}R^{2}(\delta\tau_{i}+\tfrac{d}{2})\sin\text{c }^{2}(\delta f_{i}T_{\text{coh}})+n_{\text{L}^{2}}}-A_{i}\cdot R(\delta\tau_{ i}+\tfrac{d}{2})|\sin\text{c}(\delta f_{i}T_{\text{coh}})|\\ &\approx\sqrt{A_{i}{}^{2}(1-\tfrac{d}{2})^{2}+n_{\text{L}^{2}}-A_{i}(1-\tfrac {d}{2})}\\ \end{array}  \tag{9}\]  
It should be noted that the tracking loops are assumed to be in locking state when we derive the Equations (8) and (9). That is, tracking error is assumed to be zero. After that, the non-coherent integration can be obtained by accumulating the auto-correlation amplitude of size \(N_{\text{nc}}\), as shown in Equation (10). Another form of non-coherent integration is calculated by accumulating the auto-correlation power of size \(N_{\text{nc}}\), as shown in Equation (11). The carrier frequency error and signal amplitude are assumed to be constant when we derive Equations (10) and (11). \[