# A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform
## 2 GNSS Signal and Interference Models  
In an interfering environment, the model of the signal at the input of a GNSS receiver is expressed as [8; 9; 10; 11; 16; 25]:  
\[y_{RF}(t)=\sum_{i=1}^{N_{s}}r_{RF,i}(t)+\eta_{RF}(t) \tag{1}\]  
where \(r_{RF,i}(t)\) is the \(i\)th GNSS signal (\(i=1,2,\ \cdots,N_{s}\)), \(N_{s}\) denotes the number of satellites in view, and \(\eta_{RF}(t)\) is the disturbing term. When a single useful signal is considered, for example, the GNSS signal emitted by the \(i\)th satellite can be expressed as [8; 9; 10; 11; 16]:  
\[r_{RF,i}(t)=A_{i}c_{i}(t-\tau_{i})d_{i}(t-\tau_{i})\cos[2\pi(f_{RF}+f_{d,i})t+ \varphi_{RF,i}] \tag{2}\]  
where:  
* \(A_{i}\) is the amplitude of the \(i\)th useful GNSS satellite signal;
* \(\tau_{i}\) is the propagation delay for the \(i\)th satellite signal;* \(c_{i}(t-\tau_{i})\) denotes the Pseudo-Random Noise (PRN) code sequence extracted from a family of quasi-orthogonal codes modulated by rectangular pulses. For the simple case of the GPS L1-C/A signal, \(c_{i}(t)\) represents the C/A code;
* \(d_{i}(t-\tau_{i})\) represents the navigation data bit information;
* \(f_{RF}\) denotes the carrier center frequency of the GNSS signal, for the GPS L1-C/A signal, \(f_{RF}=f_{L1}=1575.42\) MHz;
* \(f_{d,i}\) denotes the Doppler frequency shift affecting the \(i\)th useful GNSS signal;
* \(\varphi_{RF,i}\) is the initial carrier phase offset for the \(i\)th GNSS useful signal. The disturbing term \(\eta_{RF}(t)\) can be written as [8; 9; 10; 11; 16; 25]:  
\[\eta_{RF}(t)=j_{RF}(t)+w_{RF}(t) \tag{3}\]  
where \(j_{RF}(t)\) is the nonstationary interfering signal, and \(w_{RF}(t)\) is the GNSS receiver thermal noise, usually in the form of a zero-mean stationary additive white Gaussian noise (AWGN) process. There are different forms of interfering signals that could be generated by potential sources of radio frequency interference. In this paper, without loss of generality, the interference term \(j_{RF}(t)\) can be considered as the sweep interference (i.e., linear chirp), which is usually linearly frequency modulated with an almost constant amplitude and typically regarded as a test bench in interference detection for GNSS applications. Sweep interference, one of the main types of the interfering signals, can be expressed by sinusoids in the time domain as [16]:  
\[j_{RF}(t)=A_{inst}(t)\cos\left[2\pi\int_{0}^{t}f_{inst}(t)\mathrm{d}t+\varphi_ {0}\right] \tag{4}\]  
where \(A_{inst}(t)\) is the carrier amplitude of the sweep interfering signal, \(f_{inst}(t)\) represents the instantaneous frequency of the sweep interference, and \(\varphi_{0}\) denotes the initial carrier phase of the sweep interference, which can be considered as a random variable presenting a uniform distribution in the range \([-\pi,+\pi)\). When considering a linear chirp, its instantaneous frequency \(f_{inst}(t)\) linearly evolves with time in the interval \([f_{RF}+\Delta f_{0},\ f_{RF}+\Delta f_{1}]\), where \(f_{RF}\) is the center frequency of the considered GNSS signal. In this way, the instantaneous frequency \(f_{inst}(t)\) of the linear chirp can be written as [16; 25]:  
\[f_{inst}(t)=f_{0}+kt\qquad 0\leq t\leq t_{j} \tag{5}\]  
where \(f_{0}\) is the initial frequency, i.e., \(f_{0}=f_{RF}+\Delta f_{0}\), \(t_{j}\) denotes the frequency sweep period of the interfering signal, and \(k\) is the chirp rate or the frequency increase rate, expressed as [16; 25]:  
\[k=\frac{f_{1}-f_{0}}{t_{j}}=\frac{\Delta f_{1}-\Delta f_{0}}{t_{j}} \tag{6}\]  
where \(f_{1}\) is the final frequency in the frequency sweep period \(t_{j}\), i.e., \(f_{1}=f_{RF}+\Delta f_{1}\), and \(f_{1}-f_{0}=\Delta f_{1}-\Delta f_{0}\) is the value of the frequency sweep. The input signal \(y_{RF}(t)\) defined in Equation (1) is filtered, and down-converted by the GNSS receiver front-end. Then, the received GNSS signal before the Analog to Digital (A/D) conversion can be written as [8; 9; 10; 11; 16; 25]:  
\[