## 2 Signal and System Model  
The signal at the input of a GNSS receiver, in a noisy environment with RFI, can be written as:  
\[y_{RF}(t)=\sum_{i=1}^{N_{s}}r_{RF,i}(t)+\eta_{RF}(t) \tag{1}\]  
_i.e._, the sum of \(N_{s}\) useful signals emitted by \(N_{s}\) different satellites and of a disturbing term \(\eta_{RF}(t)\) and \(N_{s}\) is the number of satellites in view. The expression of the signal in space (SIS) transmitted by the _i_-th satellite and received at the GNSS receiver antenna with a propagation delay \(\tau_{i}\) is usually assumed as the following structure:  
\[r_{RF,i}(t)=A_{i}c_{i}(t-\tau_{i})d_{i}(t-\tau_{i})\cos[2\pi(f_{RF}+f_{d,i})t+ \varphi_{RF,i}] \tag{2}\]  
where:  
* \(A_{i}\) is the amplitude of the _i_-th useful signal;
* \(\tau_{i}\) is the code phase delay introduced by the transmission channel;
* \(c_{i}(t-\tau_{i})\) is the pseudo random noise (PRN) code sequence, which is assumed to take a value in the set \(\{-1,1\}\);
* \(d_{i}(t-\tau_{i})\) is the bit stream of the navigation message, binary phase-shift keying (BPSK) modulated, including satellite data; and each binary unit is called a bit;
* \(f_{d,i}\) is the Doppler frequency shift affecting the _i_-th useful signal, and \(\varphi_{RF,i}\) is the initial carrier phase offset;
* \(f_{RF}\) is the carrier frequency, and it depends on the GNSS signal band under analysis; in the case of the GPS L1 signal, \(f_{RF}=f_{\rm L1}=1575.42\) MHz. In general, the disturbing term \(\eta_{RF}(t)\) can be expressed as:  
\[\eta_{RF}(t)=j_{RF}(t)+w_{RF}(t) \tag{3}\]  
where \(j_{RF}(t)\) is a non-stationary RFI and \(w_{RF}(t)\) is a zero-mean stationary AWGN process. The interfering signal \(j_{RF}(t)\) can assume different forms depending on the jammer that generates it. Several efforts have been devoted to the analysis and characterization of civilian GNSS jammers; despite significant differences, the transmitted jamming signal is usually frequency modulated with an almost constant amplitude. In this paper, the interference term \(j_{RF}(t)\) is assumed to be in the class of sweep interference (linear chirp). Sweep interference is one of the main classes of the interfering signals, and its corresponding time-domain function represented by sinusoids can be written as follows:  
\[j_{RF}(t)=A_{inst}(t)\text{cos}[2\pi\text{f}_{inst}(t)t+\varphi_{0}] \tag{4}\]  
where \(A_{inst}(t)\) is the interfering signal amplitude, \(f_{inst}(t)\) denotes its instantaneous frequency and \(\varphi_{0}\) is the initial phase of the interference (at time \(t=0\)), which can be assumed to be a random variable with a uniform distribution in the range \([-\pi,+\pi)\). In a linear chirp, the instantaneous frequency \(f_{inst}(t)\) of the interfering signal evolves linearly with time over the interval \([f_{RF}+\Delta f_{0},f_{RF}+\Delta f_{1}]\), where \(f_{RF}\) is the GNSS signal center frequency. Therefore, the instantaneous frequency \(f_{inst}(t)\) can be expressed as:  
\[f_{inst}(t)=f_{0}+kt\ \ \ \ \ 0\leq t\leq t_{j} \tag{5}\]  
where \(f_{0}\) is the starting frequency (at time \(t=0\)), \(f_{0}=f_{RF}+\Delta f_{0}\), \(t_{j}\) is the frequency sweep period of the jamming signal and \(k\) is the rate of frequency increase or chirp rate, written as:  
\[k=\frac{f_{1}-f_{0}}{t_{j}}=\frac{\Delta f_{1}-\Delta f_{0}}{t_{j}} \tag{6}\]  
where \(f_{1}\) is the final frequency during a specific frequency sweep period, \(f_{1}=f_{RF}+\Delta f_{1}\) and \(\Delta f_{1}-\Delta f_{0}\) stands for the frequency sweep. The input signal \(y_{RF}(t)\) defined in Equation (1) is received by the receiver antenna, down-converted and filtered by the receiver front-end. Then, the received signal before the analog to digital (A/D) conversion can be written as:  
\[ \begin{split} y(t)=&\sum_{i=1}^{N_{s}}r_{i}(t)+ \eta(t)\\ =&\sum_{i=1}^{N_{s}}A_{i}\tilde{c}_{i}(t-\tau_{i})d_ {i}(t-\tau_{i})\cos[2\pi(f_{IF}+f_{d,i})t+\varphi_{i}]+\eta(t)\end{split}  \tag{7}\]  
where \(f_{IF}\) is the receiver intermediate frequency (IF).