# Study and Analysis of Interference Signals of the LTE System of the GNSS Receiver
## 3 Mathematical Model of Noise Removal from the LTE System  
In order to investigate effective methods of counteracting interference (unwanted LTE signals), at the very beginning, it is necessary to establish a mathematical model of the received signal, in which there are also other disturbances, including multipath, and to analyze their influence on the GNSS signal. In turn, to simplify the mathematical considerations, the GPS signal was used as an example, and the related conclusions can be generalized to other GNSS systems. Thus, the broadcast signal of the GNSS system can be mathematically represented as (2) [42]:  
\[s(t)=D(t)c(t)\cos(\omega_{c}t), \tag{2}\]  
where \(D(t)\) is the navigation message of the GNSS signal; \(c(t)\) represents \(C/A\) code; \(\omega_{c}\) is the carrier frequency. Assuming that the GPS receiver receives the unwanted signal from the LTE system combined with multipath interference from \(P\), the reflection paths, then the received signal can be represented as (3):  
\[\widetilde{\widetilde{x}}(t)=\sum_{p=0}^{p}\widetilde{\alpha}_{p}D\big{(}t- \tau_{p}\big{)}c\big{(}t-\tau_{p}\big{)}\cos\big{(}\omega_{c}\big{(}t-\tau_{p} \big{)}+\widetilde{\phi}_{p}\big{)}+\sum_{k=0}^{N-1}x_{k,l}\phi_{k}(t_{LTE}- IT), \tag{3}\]  
where \(\widetilde{\alpha}_{p}\), \(\tau_{p}\), \(\widetilde{\phi}_{p}\) represent the amplitude of the beacon signal, code delay, and start phase for the \(p\)-th multipath of the GNSS signal; \(p\)? 0 represents the LOS signal arriving directly on the unwanted signal of the LTE system, assuming that \(p=0\) represents the SLTE signal (unwanted signal of the LTE system) arriving on the direct path, being a useful signal (information) to noise floor ratio. In the above Equation (3), \(N\) denotes the number of subcarrier signals present in the signal reaching the GNSS receiver, while \(x_{k,l}\) denotes the number of constellation variables of the OFDM (Orthogonal Frequency Division Multiplexing) modulation signal, where \(k\) is a real number and \(l\) is an imaginary number, \(\phi_{k}\) denotes an OFDM modulating signal, \(t_{LTE}\) is an index of OFDM modulation prefix, \(T\) is the length of the symbol in the OFDM modulating signal, and \(l\) is the length of the block in which OFDM signals are transmitted. On the other hand, since the navigation message period is much longer than the \(C/A\) code period, the length of the data block is required in signal analysis by the GNSS receiver through a subsequent code delay, in which the estimation is relatively short (only a few code periods are needed \(C/A\)). During the duration of the data, the navigation message \(D(t-\tau_{p})=\pm 1\), and assuming no data bit change, can further combine the value \(D(t-\tau_{p})\) at (3) within an amplitude \(\widetilde{\alpha}_{p}\), and the new variable can be denoted as \(\alpha_{p}^{\prime}\). Then, (3) is transformed to the form (4):  
\[\widetilde{\widetilde{x}}(t)=\sum_{p=0}^{p}\alpha_{p}^{\prime}c\big{(}t-\tau_ {p}\big{)}\cos\big{(}\omega_{c}\big{(}t-\tau_{p}\big{)}+\widetilde{\varphi}_{p }\big{)}\cos(\omega_{LTE}(t_{LTE}-\tau_{LTE})+\widetilde{\varphi}_{k}), \tag{4}\]  
where \(\omega_{LTE}\), \(t_{LTE}\), and \(\tau_{LTE}\) mean the frequency, time, and delay LTE system signals, respectively, and \(\widetilde{\varphi}_{k}\) represents the phase for the \(p\)-th multipath signal of the LTE system. To determine the Doppler frequencies for signals arriving along different paths, consider the Doppler frequency of the SLTE signal.