## 2 Effect of Spoofing Signals on Acquisition  
### Noise Floor Estimation  
When the target receiver receives both spoofing signal and authentic signal, the complex signal model can be expressed as [10]:  
\[ \begin{array}{l}r(nT_{s})=\sum\limits_{h=l}\sqrt{P_{h}^{u}}D_{h}^{a}(nT_{s}- \tau_{h}^{a})c_{h}^{a}(nT_{s}-\tau_{h}^{a})e^{j\varphi_{h}^{a}+j2\pi\tau_{h}^{ f_{n}nT_{s}}}+\\ \sum\limits_{m=l}\sqrt{P_{m}^{u}}D_{m}^{s}(nT_{s}-\tau_{m}^{s})c_{m}^{s}(nT_{ s}-\tau_{m}^{s})e^{j\varphi_{m}^{a}+j2\pi\tau_{m}^{f_{n}nT_{s}}}+\eta(nT_{s})\end{array}  \tag{1}\]  
where subscripts \(h\) and \(m\) represent the received authentic satellite signals and spoofing signals, respectively, \(l^{a}\) and \(l^{s}\) are the set of authentic and spoofing signals, respectively, and the superscripts \(a\) and \(s\) represent the received authentic satellite signals and spoofing signals, respectively. \(T_{s}\) is the sampling interval, \(P\) is the power of received signal, \(c\) is the pseudorandom noise (PRN) code sequence, \(D\) is the navigation message, \(\varphi,f\), and \(\tau\) are the carrier phase of received signal, carrier frequency Doppler shift, and code phase delay, and \(\eta(nT_{s})\) is an additive white Gaussian noise with a mean of zero and a variance of \(\sigma_{n}^{2}\). The coherent integrated output value of the \(l\)th signal can be expressed as:  
\[ \begin{array}{l}u_{l}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right] =\frac{1}{N}\sum\limits_{n=(k-1)N+1}^{kN}r(nT_{s})c_{l}(nT_{s}- \widetilde{\tau}_{l})e^{-j2\pi\widetilde{f}_{l}nT_{s}}\end{array}  \tag{2}\]  
where \(\widetilde{f}_{l}\), \(\widetilde{\tau}_{l}\) represent the estimated Doppler shift and code phase delay, respectively, and \(k\) represents the number of integration interval related outputs. Referring to Gaussian Sum Theory [22], the \(\hat{\sigma}^{2}\) calculated using the averaging method can be expressed as:  
\[\hat{\sigma}^{2}=\frac{1}{2}\left[\sum\limits_{h=j^{a}}P_{h}^{a}\text{var}\left[ \psi_{hl}^{a}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right]\right]+\sum \limits_{m=j^{a}}P_{m}^{s}\text{var}\left[\psi_{ml}^{s}\left[\widetilde{f}_{l },\widetilde{\tau}_{l},k\right]\right]+\text{var}[\overline{\eta}[k]]\right] \tag{3}\]  
The noise estimator comprises three parts: 1 related interference generated by authentic signal; 2 related interference generated by spoofing signal; 3 related interference generated by Gaussian noise. \(\psi_{hl}^{a}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right]\) and \(\psi_{ml}^{s}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right]\) represent the parameters of one channel authentic and one channel spoofing signal with the local code and carrier, respectively. Because this is a complex signal, both contain \(I\) and \(Q\) branches that are orthogonal to each other, and both the \(I\) and \(Q\) branches are subject to a zero-mean Gaussian distribution. Then, \(\psi_{hl}^{a}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right]\) is represented by a two-dimensional covariance matrix with the \(I\) and \(Q\) branches as random variables [23]:  
\[\left\{ \begin{array}{l}\psi_{hl}^{a}\left[\widetilde{f}_{l},\widetilde{\tau }_{l},k\right]\sim N\left(\left[\begin{array}{cc}0\\ 0\end{array}\right],\left[\begin{array}{cc}\overline{\sigma}_{\psi,normalization}^{2}&0\\ 0&\overline{\sigma}_{\psi,normalization}^{2}\end{array}\right]\right)\\ \overline{\sigma}_{\psi,normalization}^{2}=\overline{\sigma}_{\psi}^{2}/\text{max} \left(\overline{\sigma}_{\psi,normalization}^{2}\right)\overline{\sigma}_{\psi}^{2} \end{array} \right\} \tag{4}\]  
The cross-correlation variance \(\overline{\sigma}_{\psi,normalization}^{2}\) of the normalized spreading codes of the \(I\) and \(Q\) branches is statistically averaged as 0.00033. \(\psi_{ml}^{s}\left[\widetilde{f}_{l},\widetilde{\tau}_{l},k\right]\) obeys the same distribution. The correlation value generated by noise is still a zero-mean Gaussian noise, and \(\overline{\eta}[k]\) can be expressed by the same distribution as  
\[\overline{\eta}[k]\sim N\bigg{(}\left[