## 3 Construction of Multi-Correlators and Maximum Likelihood Rstimation of Dignal Parameters  
In order to achieve the mitigation of spoofing, we employ the idea of multipath estimation. Firstly, in the presence of spoofing, the composite signals for the in-phase (I) and quadrature (Q) branches of any channel of the target receiver can be modeled as  
\[i_{\Sigma}(t)=\sum_{n=0}^{N}a_{n}(t)A_{0}D[t-\tau_{n}(t)]c[t-\tau_{n}(t)]\cos [\phi_{n}(t)]+v_{\text{q}}(t), \tag{1}\]  
\[q_{\Sigma}(t)=\sum_{n=0}^{N}a_{n}(t)A_{0}D[t-\tau_{n}(t)]c[t-\tau_{n}(t)]\sin [\phi_{n}(t)]+v_{\text{q}}(t), \tag{2}\]  
Figure 1: The structure of a repeater. Figure 2: Spoofing rebroadcasted by a repeater. respectively, where \(A_{0}\) is the known amplitude of the authentic signal, \(D(\cdot)\) is the navigation data bit, \(c(\cdot)\) is the CA code, \(N\) is the number of counterfeit signals in the current channel, \(n=0,1,2,\cdots,N,v_{\rm i}(t)\) and \(v_{\rm q}(t)\) denote the noises in the I and Q branches, respectively. \(a_{n}(t)\), \(\tau_{n}(t)\), \(\phi_{n}(t)\) are the amplification coefficient of the amplitude, code delay (time delay) and carrier phase corresponding to the \(n\)-th signal, respectively. If \(n=0\), they correspond to the authentic signal and \(a_{0}(t)=1,\ \tau_{0}(t)=0,\ \phi_{0}(t)=0\). Otherwise, they correspond to the counterfeit signals and \(a_{n}(t)\geq\sqrt{1.1}\). In reality, the repeater delays and amplifies the authentic signal, and then rebroadcasts it. This principle determines that there is usually only one counterfeit signal corresponding to each satellite, namely, \(N=1\). Although the counterfeit signals for spoofing can be ahead of the authentic signal, we assume \(\tau_{n}(t)\geq 0\) by symmetry in order to facilitate the discussion. As shown in Figure 3, a series of correlators whose time delays are \(\beta_{0},\beta_{1},\beta_{2},\cdots,\beta_{M}\) is deployed in the channel. Take the I branch for example, the composite signal is correlated with these correlators. It is assumed that the variations of \(a_{n}(t)\), \(\tau_{n}(t)\) and \(\phi_{n}(t)\) are negligible and \(D(\cdot)\) does not flip in the coherent integration time of \(T_{\rm coh}\). Then the result of this coherent integration can be expressed as  
\[ \begin{array}{rl}I_{m,k}&=\sum\limits_{n=0}^{N}\frac{x_{n}}{c_{n,k}}f_{t_{k} }^{t_{k}+T_{\rm coh}}\ c(t-\tau_{n})c(t-\beta_{m}){\rm d}t+I_{m,k}^{\rm v}\\ &=\sum\limits_{n=0}^{N}x_{n}R_{\rm C}(\beta_{m}-\tau_{n})+I_{m,k}^{\rm v}, \end{array}  \tag{3}\]  
where \(k\) is the index of the discretized sampling time \(t_{k}\), \(R_{\rm C}(\cdot)\) is the autocorrelation function of the CA code, \(m=0,1,2,\cdots,M\), and  
\[x_{n}=a_{n}A_{0}\cos(\phi_{n}), \tag{4}\]  
\[I_{m,k}^{\rm v}=\frac{1}{T_{\rm coh}}\int_{t_{k}}^{t_{k}+T_{\rm coh}}v_{\rm q }(t)c(t-\beta_{m}){\rm d}t. \tag{5}\]  
Similarly, the expressions for the quadrature branch are written as  
\[Q_{m,k}=\sum\limits_{n=0}^{N}y_{n}R_{\rm C}(\beta_{m}-\tau_{n})+Q_{m,k}^{\rm v}, \tag{6}\]  
\[y_{n}=a_{n}A_{0}\sin(\phi_{n}), \tag{7}\]  
\[Q_{m,k}^{\rm v}=\frac{1}{T_{\rm coh}}\int_{t_{k}}^{t_{k}+T_{\rm coh}}v_{\rm q }(t)c(t-\beta_{m}){\rm d}t. \tag{8}\]  
Assume that \(v_{\rm i}(t)\) and \(v_{\rm q}(t)\) are white Gaussian noises with zero mean and variance \(\sigma_{\rm v}^{2}\), then \(I_{m,k}^{\rm v}\) and \(Q_{m,k}^{\rm v}\) are white Gaussian noises with  
\[\left\{ \begin{array}{l}\mathbb{E}\Big{(}I_{m,k}^{\rm v}\Big{)}=\mathbb{E} \Big{(}Q_{m,k}^{\rm v}\Big{)}=0\\ E\Big{[}I_{m_{1},k}^{\rm v}I_{m_{2},k}^{\rm v}\Big{)}=\mathbb{E}\Big{[}Q_{m_{1 },k}^{\rm v}\mathcal{Q}_{m_{2},k}^{\rm v}\Big{]}=\frac{\sigma_{\rm v}^{2}}{N_ {\rm coh}}R_{\rm C}(\beta_{m_{1}}-\beta_{m_{2}}),\end{array} \right. \tag{9}\]  
where \(m_{1}=0,1,2,\cdots,M\), \(m_{2}=0,1,2,\cdots,M\). Figure 3: Multi-correlator and maximum likelihood estimator. When the time delay of a counterfeit signal equals \(\beta_{n}\), only the sum of squares of \(\hat{x}_{n}\) and \(\hat{y}_{n}\) is not obviously close to 0 regardless of the authentic signal.