## 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, \(f_{RF}=f_{\text{L1}}=1,575.42\) 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:  
\[j_{RF}(t)=A_{inst}(t)\text{cos}[2\pi f_{inst}(t)t+\varphi_{0}] \tag{4}\]  
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).