## 3 The Proposed Method
### Impact on the Acquisition Stage  
In the case of a single satellite and a single chirp interference, the input signal \(X_{IF}(t)\) in Equation (1) enters the ADC without considering the quantization effect and can be rewritten as  
\[X_{IF}(n)=\sqrt{2A}C(n-\tau)D(n-\tau)\cos[2\pi(f_{IF}+f_{d})nT_{s}+\varphi_{IF} ]+\sqrt{2A_{j}}\cos\!\left[2\pi(f_{ins}(t))nT_{s}+\theta_{f}\right]+N(n) \tag{23}\]  
Then, the signal \(X_{IF}(n)\) is multiplied by two orthogonal sinusoids as well as a local signal replica and is integrated; the results are as follows [27]  
\[\begin{array}{c}S_{I}(\tau,f_{D})=\frac{1}{N}\sum\limits_{n=0}^{N-1}r_{I}(n )c(n-\tau)=r_{I}(\tau)*h_{c}(\tau)\\ S_{Q}(\tau,f_{D})=\frac{1}{N}\sum\limits_{n=0}^{N-1}r_{Q}(n)c(n-\tau)=r_{Q}( \tau)*h_{c}(\tau)\end{array} \tag{24}\]  
where \(r_{I}(n)=X_{IF}(n)\cos(2\pi f_{D}n)\), \(r_{Q}(n)=X_{IF}(n)\sin(2\pi f_{D}n)\), \(f_{D}=(f_{IF}+f_{d})T_{s}\), \((\star)\) denotes integral operation, \(N\) is the length of local code and \(h_{c}(\tau)\) is an equivalent filter. The Cross Ambiguity Function (CAF) is obtained [27]:  
\[S(\tau,f_{D})=\sqrt{S_{I}(\tau,f_{D})^{2}+S_{Q}(\tau,f_{D})^{2}} \tag{25}\]  
When the Doppler shift \(f_{d}\) and the code delay \(\tau\) are correctly recovered, the detection probability that the statistical variable \(S(\tau,f_{D})\) surpasses a fixed threshold \(\beta\) in the GNSS acquisition stage can be defined:  
\[P_{\rm det}(\beta)=P(S(\tau,f_{D})>\beta) \tag{26}\]  
Figure 3: Block diagram of the joint method based on the Hough transform of RSPWVD to detect chirp interferences.  
In order to assess the impact of chirp interference on the detection probability, it is supposed that the GNSS signal and the chirp interference are known. The integration results of GNSS signals in Equation (24) can be written [27]:  
\[S_{y}\approx\sqrt{C/2}\exp\{-j\varphi_{IF}\} \tag{27}\]  
Similarly, the integration results of chirp interference in Equation (24) can be written [27]:  
\[S_{f}=k_{1}\sqrt{\frac{A_{f}}{2}}\exp\{j2\pi(f_{\text{ins}}+f_{D})\tau+j\theta _{J}+j\theta_{1}\}+k_{2}\sqrt{\frac{A_{J}}{2}}\exp\{-j2\pi(f_{\text{ins}}-f_{D })\tau-j\theta_{J}+j\theta_{2}\} \tag{28}\]  
where \(\theta_{1}=\angle\{H_{c}(f_{\text{ins}}+f_{D})\}\) and \(\theta_{2}=\angle\{H_{c}(-f_{\text{ins}}+f_{D})\}\). \(H_{c}(f)\) is the Fourier transform of \(h_{c}(n)\). \(k_{1}\) and \(k_{2}\) can be written as  
\[k_{i}^{2}=\big{|}H_{c}(\pm f_{\text{ins}}+f_{D})\big{|}^{2}=\int_{-\infty}^{ \infty}\big{|}H_{c}(f)\big{|}^{2}\delta(f-(\pm f_{\text{ins}}+f_{D}))df=\int_{ -\infty}^{\infty}G_{s}(f)G_{i}(f)df \tag{29}\]  
where \(\delta(\cdot)\) is the delta Dirac, \(G_{s}(f)=\big{|}H_{c}(f+f_{D})\big{|}^{2}\) and \(G_{i}(f)=\delta(f\pm f_{\text{ins}})\). When the chirp interference appears, the CAF in Equation (25) follows a Rice distribution [27]  
\[S(\tau,f_{D})=\frac{x}{\sigma^{2}}\exp\{-\frac{x^{2}+\alpha^{2}}{2\sigma^{2}} \}I_{0}\Big{(}\frac{x\alpha}{\sigma^{2}}\Big{)},\ x>0 \tag{30}\]  
where \(\alpha^{2}=\big{|}S_{y}+S_{f}\big{|}^{2}\), \(\sigma^{2}=\frac{\sigma_{out}^{2}}{2\sigma}\), \(\sigma_{out}^{2}\) is the noise variance of output of equivalent filter \(h_{c}(n)\) and \(\sigma_{out}^{2}=\frac{1}{N}N_{0}B_{IF}\), \(I_{0}\) is the modified Bessel function with first kind and zero order. The detection probability can be defined:  
\[P_{d}(\beta)=\int_{\beta}^{\infty}\frac{x}{\sigma^{2}}\exp\{-\frac{x^{2}+ \alpha^{2}}{2\sigma^{2}}\}I_{0}\Big{(}\frac{x\alpha}{\sigma^{2}}\Big{)}dx=Q \Big{(}\frac{\alpha}{\sigma};\frac{\beta}{\sigma}\Big{)} \tag{31}\]  
where \(Q(\cdot;\cdot)\) denotes the Marcum Q function. From Equation (31), the chirp interference seriously affects the detection probability in the GNSS acquisition stage.