# Study and Analysis of Interference Signals of the LTE System of the GNSS Receiver
## 3 Mathematical Model of Noise Removal from the LTE System
Hence, the following Equation (5) can be obtained:  
\[\omega_{c}\tau_{0}=\omega_{c}\bigg{(}\frac{R_{0}(t)}{c}+\frac{L_{0}(t)}{c} \bigg{)}, \tag{5}\]  
where in Formula (5), the following symbols have been adopted: the speed of light is denoted by \(c\), and the expression \(R_{0}(t)=r_{0}+v_{0}t\) defines the distance between the GNSS system receiver and the signal \(n\) received from the satellite in the time sequence \(t\), in which the distance of the measurement made in the initial stage of observation received by the receiver is defined by the signal from the satellite \(r_{0}\), and \(v_{0}\) defines the relative speed occurring during the transmission of the signal from the satellite. Furthermore, \(L_{0}(t)=l_{0}+v_{LTE}t\) is the distance between the GNSS receiver and the LTE (4G) system base station; \(t\) are \(l_{0}\) represent the distance between the receiver and the base station at the initial observation time, and \(v_{LTE}\) is the relative rate of signal transmission between the receiver and the base station. The above Formula (5) can be further expressed as (6):  
\[\omega_{c}\tau_{0}=\omega_{c}\bigg{(}\frac{r_{0}+v_{0}t}{c}+\frac{l_{0}+v_{ LTE}t}{c}\bigg{)}=2\pi\bigg{(}\frac{r_{0}}{\lambda}+\frac{v_{0}t}{\lambda}+ \frac{l_{0}}{\lambda}+\frac{v_{LTE}t}{\lambda}\bigg{)}. \tag{6}\]  
In the case where the relative velocity \(v_{0}\) and \(l_{0}\) are constant, the Doppler frequency is (7):  
\[\omega_{d0}=2\pi\frac{v_{0}+l_{0}}{\lambda}. \tag{7}\]  
The propagation distance of the LTE system signal can be expressed as (8):  
\[R_{p}(t)=r_{0}+v_{0}t+l_{0}+v_{LTE}t+\Delta R_{p}(t) \tag{8}\]  
where \(\Delta R_{p}(t)\) is the propagation distance difference between the multipath interference and the LTE system signal. In a situation where the distance between the reflection point and the receiver is large, the code delay between the multipath interference and the SLTE signal is greater than 1.5 times the chip; then, the effect of the multipath interference on the receiver can be neglected. Since the satellite is far from the receiver on Earth, assuming the multi-way reflection points are not too far from the receiver, it can be assumed that \(\Delta R_{p}(t)\) does not change in a short time. Then, \(\omega_{c},\ \tau_{p}\) can be further expressed as (9):  
\[\omega_{c}\tau_{p}=\omega_{c}\frac{r_{0}+v_{0}t+l_{0}+v_{LTE}t+\Delta R_{p}}{ c}=2\pi\bigg{(}\frac{r_{0}}{\lambda}+\frac{v_{0}t}{\lambda}+\frac{l_{0}}{ \lambda}+\frac{v_{LTE}t}{\lambda}+\frac{\Delta R_{p}}{\lambda}\bigg{)}. \tag{9}\]  
It can be seen that the \(p\)-th Doppler multipath interference \(\omega_{dp}=\frac{2\pi v_{LTE}}{\lambda}\), i.e., the SLTE signal and the multipath interference approximately have the same Doppler frequency,and they can be determined using a unified representation \(\omega_{d0}\). Hence, (4) can be further simplified as (10):  
\[\overline{x}(t)=\sum\limits_{p=0}^{p}a^{\prime}_{p}e\big{(}t-\tau_{p}\big{)}\cos \Big{[}(\omega_{c}+\omega_{d0})t+\varphi^{\prime}_{p}+\varphi^{\prime}_{k}\Big{]}, \tag{10}\]  
where \(\phi^{\prime}_{0}=\varphi^{\prime}_{0}+\frac{2\pi\tau_{0}}{\lambda}\) is the phase of the SLTE signal, \(\varphi^{\prime}_{p}=\varphi^{\prime}_{0}+\Delta\varphi_{p}\), \((p=1,2,\ldots,P)\) is the phase of the \(p\)-th multipath signal, \(\Delta\varphi_{p}=\frac{2\pi R_{0}}{\lambda}\) is the additional phase difference between the \(p\)-th multipath signal and the GNSS signal, and \(\varphi^{\prime}_{k}\) represents the estimated carrier phase of the SLTE signal.