# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 2 Tracking Error Estimation
### Relevant Baseband Signal Model
IF: intermediate frequency; NCO: numerically-controlled oscillator. bandwidth depends on the RF front-end filter bandwidth which is no less than double the PRN code frequency. When the GNSS receiver tracks a specified GNSS signal in one tracking channel, other GNSS signals are suppressed and become noise because cross-correlation of different PRN codes is very little. So, the signal model for a specified GNSS satellite can be written as:  
\[s(t)=aD(t)x(t)\cos[2\pi\cdot(f_{\rm IF}+f_{\rm d})t+\varphi_{0}]+n_{0}(t) \tag{2}\]  
After sampled by the analog-digital converter (ADC) of the RF front-end, the signal becomes:  
\[s[m]=aD(mT_{\rm s})x(mT_{\rm s})\cos[2\pi\cdot(f_{\rm IF}+f_{\rm d})mT_{\rm s} +\varphi_{0}]+n_{0}(mT_{\rm s}) \tag{3}\]  
where \(m\) is an integer and indicates the sampling point number, and \(T_{\rm s}\) is the sampling period (s) of the ADC. The digitalized IF signal will first be mixed with local generated \(\sin/\cos\) carrier in mixers and then be correlated with local generated E/P/L PRN codes in correlators. The coherent integration can be obtained from the six correlators, shown as follows:  
\[ \begin{array}{l}I_{\rm E}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}-\frac{d} {2})\sin{\rm c}(\delta f_{i}T_{\rm coh})\cos(\overline{\delta\varphi_{i}})+n _{\rm IE}\\ Q_{\rm E}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}-\frac{d}{2})\sin{\rm c}( \delta f_{i}T_{\rm coh})\sin(\overline{\delta\varphi_{i}})+n_{\rm QE}\\ I_{\rm P}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i})\sin{\rm c}(\delta f_{i}T _{\rm coh})\cos(\overline{\delta\varphi_{i}})+n_{\rm IP}\\ Q_{\rm P}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i})\sin{\rm c}(\delta f_{i}T _{\rm coh})\sin(\overline{\delta\varphi_{i}})+n_{\rm QP}\\ I_{\rm L}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}+\frac{d}{2})\sin{\rm c}( \delta f_{i}T_{\rm coh})\cos(\overline{\delta\varphi_{i}})+n_{\rm IL}\\ Q_{\rm L}(i)=A_{i}\cdot D_{i}\cdot R(\delta\tau_{i}+\frac{d}{2})\sin{\rm c}( \delta f_{i}T_{\rm coh})\sin(\overline{\delta\varphi_{i}})+n_{\rm QL}\end{array}  \tag{4}\]  
where \(i\) indicates the \(i\)-th coherent integration results, \(I_{\rm E}\), \(Q_{\rm E}\), \(I_{\rm P}\), \(Q_{\rm P}\), \(I_{\rm L}\), \(Q_{\rm L}\) are coherent integration in the six correlators, \(\sin{\rm c}(\delta fT_{\rm coh})=\sin(\pi\delta fT_{\rm coh})/\pi\delta fT_{ \rm coh}\), \(T_{\rm coh}\) is coherent integration time (s), \(\delta\tau\) denotes the code phase error (chip), \(\delta f\) denotes the carrier frequency error (Hz), \(\overline{\delta\varphi}\) denotes the mean carrier phase error (rad) in coherent integration time, \(R(\cdot)\) denotes the auto-correlation function of PRN code, \(d\) denotes the Early-Late correlator spacing (chip), \(n_{\rm IE}\), \(n_{\rm QE}\), \(n_{\rm IP}\), \(n_{\rm QP}\), \(n_{\rm IL}\), \(n_{\rm QL}\) represent the noise in coherent integration. It should be noted here that the distributions of \(n_{\rm IE}\), \(n_{\rm QE}\), \(n_{\rm IP}\), \(n_{\rm QP}\), \(n_{\rm IL}\), \(n_{\rm QL}\) almost fulfill the additive white Gaussian noise (AWGN) assumption (i.e., \(n_{\rm IE}\),..., \(n_{\rm QL}\sim N(0,\sigma_{\rm noise}{}^{2})\)). The value of \(A\) can be calculated by the following Equation (5) [9]:  
\[A=\sqrt{2(c/n_{0})T_{\rm coh}}\sigma_{\rm noise} \tag{5}\]  
where \(c/n_{0}\) is the carrier-to-noise density ratio (Hz) of the corresponding GNSS signal, \(\sigma_{\rm noise}\) is the standard deviation of \(n_{\rm IE}\), \(n_{\rm QE}\), \(n_{\rm IP}\), \(n_{\rm QP}\), \(n_{\rm IL}\), \(n_{\rm QL}\). After coherent integration is done, the auto-correlation power and auto-correlation amplitude can be defined as Equations (6) and (7), respectively:  
\[