# Study and Analysis of Interference Signals of the LTE System of the GNSS Receiver
## 4 Simulation Studies
In turn, \(\theta_{\rm{inter}}\) and \(\theta_{\rm{LTE}}\) denote the phase of the interference signal and phase of the LTE signal. In fact, the authentic signal and the signal of the LTE system will be mixed upstream of the receiving antenna of the GNSS receiver. However, in the presented simulation, the real signal of the GPS system is a digital IF (Intermediate Frequency) signal, so the LTE system signals will be self-processed in the RF (Radio Frequency) front-end and then added to the real GNSS signal. In the receiver of the radio part R-front end, the complex IF signal can be expressed as follows (19):  
\[ \begin{array}{ll}r_{IF}(t)=&\sum\limits_{m=1}^{M}\sqrt{2P_{m}}d_{m}(t-\tau_ {m})c_{m}(t-\tau_{m})\cos\left(2\pi\Big{(}f_{IF}+f_{\rm{dopp}}\Big{)}t+\theta _{m}\right)\\ &+Ac_{\rm{inter}}(t-\tau_{\rm{inter}}-\tau_{\rm{LTE}})\cos(2\pi(f_{IF}+f_{\rm{ inter}}+f_{LTE})t+\theta_{\rm{inter}}+\theta_{\rm{LTE}})+n(t).\end{array}  \tag{19}\]  
where \(\tau_{\rm{inter}}\) and \(\tau_{\rm{LTE}}\) mean the time shift of the interference signal and LTE system, respectively. Then, the IF signal is passed on for secondary processing: acquisition, tracking, and computing. At a later stage, one satellite signal, e.g., from a satellite \(k\), was considered. Then, when the local PRN code replica signal is aligned with the incoming IF signal, the output is described by the mathematical relationship (20):  
\[ \begin{array}{ll}r_{k}(t)=&\sqrt{2P_{k}}d_{k}(nT)\cos\theta_{k}\\ &+\int_{jT}^{(j+1)T}Ac_{k}(t-\tau_{\rm inter}-\tau_{\rm LTE})c_{k}(t-\tau_{k}) \cos(\Delta ft+\theta_{\rm inter}+\theta_{\rm LTE})dt,\end{array}  \tag{20}\]  
where \(r_{k}(t)\) is the acquired satellite signal \(k_{c}(t-\tau_{\rm inter})\) is the PRN code of the LTE system signal; \(c_{k}(t-\tau_{k})\) is the PRN code of the local replica; \(\Delta f\) is the difference between the carrier frequency of the LTE system signal and the carrier frequency of the local replica. The quantities appearing in the first part are information from the navigation message, while the remaining part is the result of determining the correlation function determined from the locally generated replica code and the LTE system signal. Thus, the acquisition process performed in the GNSS system receiver depends on the second signal (LTE). It should be taken into account that the cross-correlation is characterized by low values between different PRN codes. For example, in a situation where one of the PRN codes has a high amplitude, this results in a new correlation function. Thus, in the acquisition phase, the GNSS receiver makes the decision to acquire one satellite signal by comparing the correlation peak to one predetermined threshold; hence, in the presence of an LTE system signal, it may not acquire the correct satellites due to this amplified PRN code. After the acquisition step, the IF signal will be processed in the tracking step based on the obtained coarse carrier frequency and the phase of the PRN code. As mentioned before, the tracking process includes carrier wave tracking and PRN code phase tracking. In this case, carrier tracking with PLL loop technology for a given satellite \(k\) will be analyzed:  
\[I_{k}(j)=\sqrt{P_{k}}d_{k}(nT)\cos\phi_{k}+\int_{jT}^{(j+1)T}\frac{A}{2T}c_{k} (t-\tau_{\rm inter}-\tau_{\rm LTE})c_{k}(t-\tau_{k})\cos(\Delta ft+\Delta\phi)dt \tag{21}\]  
or  
\[Q_{k}(j)=\sqrt{P_{k}}d_{k}(nT)\sin\phi_{k}+\int_{jT}^{(j+1)T}\frac{A}{2T}c_{k} (t-\tau_{\rm inter}-\tau_{\rm LTE})c_{k}(t-\tau_{k})\sin(\Delta ft+\Delta\phi)dt. \tag{22}\]  
As discussed previously, the discriminator used in the carrier tracking loop is based on the form of the function \(\phi_{k}=\tan^{-1}(Q_{k}\ /I_{k})\); in the design algorithm, the phase error is minimized when \(Q_{k}\) is equal to zero and \(I_{k}\) is maximum.