# Study and Analysis of Interference Signals of the LTE System of the GNSS Receiver
## 3 Mathematical Model of Noise Removal from the LTE System
### Influence of Other Undesirable Signals  
Assuming that the carrier wave recovery is completely accurate, the signal after PLL (Phase-Locked Loop) demodulation can be represented as (11):  
\[\overline{x}(t)=\sum\limits_{p=0}^{p}\alpha^{\prime}_{p}c\big{(}t-\tau_{p} \big{)}\cos\Big{(}\varphi^{\prime}_{p}-\widehat{\varphi}^{\prime}_{0}-\varphi ^{\prime}_{k}\Big{)}, \tag{11}\]  
where \(\widehat{\varphi}^{\prime}_{0}\) is the phase shift of the carrier signal of the GNSS system, and \(\varphi^{\prime}_{k}\) represents the estimated carrier phase of the SLTE signal. If the time interval between the early code correlator and the late code correlator is \(d\), then the locally generated early and late codes can be represented as (12), respectively [43]:  
\[ \begin{array}{l}s_{E}(t)=c(t-\widehat{\tau}_{0}-d/2)\\ s_{L}(t)=c(t-\widehat{\tau}_{0}+d/2)\end{array}  \tag{12}\]  
where \(\widehat{\tau}_{0}\) represents the estimated code delay of the SLTE signal. After demodulation, correlations between signal and locally generated early/late codes \(R_{E}(\varepsilon)\) and \(R_{L}(\varepsilon)\) can be represented as (13), respectively:  
\[ \begin{array}{l}R_{E}(\varepsilon)=\sum\limits_{p=0}^{p}\alpha^{\prime}_{p} R\big{(}\varepsilon+\Delta\tau_{p}-d/2\big{)}\cos(\varphi^{\prime}_{p}- \varphi^{\prime}_{k})\\ R_{L}(\varepsilon)=\sum\limits_{p=0}^{p}\alpha^{\prime}_{p}R\big{(} \varepsilon+\Delta\tau_{p}+d/2\big{)}\cos(\varphi^{\prime}_{p}-\varphi^{ \prime}_{k})\end{array}  \tag{13}\]  
where \(R(\cdot)\) represents the autocorrelation function of the \(C/A\) code; \(\varepsilon=\tau_{0}-\widehat{\tau}_{0}\) is an error in estimating the code delay of the SLTE signal; \(\Delta\tau_{p}=\tau_{p}-\tau_{0}\) is the relative code delay between the \(p\)-th multipath interference and the SLTE signal. The code-tracking loop discrimination function of the GNSS receiver may be represented as (14). \[ \begin{array}{l}f(\varepsilon)\ \ \ =R_{E}(\varepsilon)-R_{L}(\varepsilon)\\ \ \ \ =\sum\limits_{p=0}^{p}\alpha^{\prime}_{p}\big{[}R\Big{(}\varepsilon+ \Delta\tau_{p}-\frac{d}{2}\Big{)}-R\Big{(}\varepsilon+\Delta\tau_{p}+\frac{d}{ 2}\Big{)}\big{]}\cos\Big{(}\varphi^{\prime}_{p}-\varphi^{\prime}_{k}\Big{)}, \end{array}  \tag{14}\]  
Then, at the stage of implementing the PRN code tracking loop, in the absence of a disturbing signal (coming from the LTE system) and when there is no multipath signal influx, i.e., when the parameter \(\varepsilon=0\) occurs, tracking errors appear in zero values code in the discrimination function. It should be noted that if the value of the discriminant function is zero, then the correlation function derived from the PRN code association values (early and late) are identical, except that the correlation function obtained from the quick code at any intermediate point obtains its maximum value. Assuming that the locally produced \(C/A\) code in the code tracking process is synchronized with the received code from the satellite by the GNSS system receiver, the stage of local \(C/A\) code phase tuning is performed by analyzing the zero values of the functions obtained in the discriminator. This situation provides the same correlation value for early and late codes, which means that the tracing error zeros correspond to the discrimination function zeros. When considering the situation related to the occurrence of undesirable signals, one should take into account their influence in the process of determining the correlation function. At this stage, mainly multipath signals arriving at the input of the GNSS receiver should be taken into account. Thus, when a condition occurs where the value of the function obtained in the discriminant is zero, and the correlation functions obtained from the early and late codes have the same waveforms, then the correlation function obtained from the quick codes is located on intermediate values and has the same waveform as the correlation function for the codes early and late.