# Localization of GNSS Spoofing Interference Source Based on a Moving Array Antenna
## 3 Proposed Methods
### The CRB of the Method Proposed in This Article  
In this section, we present the CRB for our proposed method, taking into account the presence of measurement errors in carrier phase measurements and array antenna positioning. CRB is a widely recognized tool that establishes a lower bound on the accuracy of unbiased estimators. According to Equation (8), the error in spoofing location accuracy is influenced by two components: the measurement error of the carrier phase for each array element and the positioning error of the antenna array itself. To derive the CRB for the proposed method, we introduce an observation vector \(\hat{\mathbf{u}}=\left[\hat{a}_{11},\hat{a}_{12},\cdots,\hat{a}_{j\hat{\
u}}, \cdots,\hat{a}_{M6}\right]^{T}\), representing the carrier phase measurements. Here, \(\hat{a}_{ji}\) corresponds to \(u_{ji}\) in Equation (8) and encompasses the observation error. We define vector \(\hat{\mathbf{u}}\)'s true value as \(\mathbf{u}=\left[u_{11},u_{12},\cdots,u_{ji},\cdots,u_{M6}\right]^{T}\) and its measurement error vector as \(\mathbf{\Delta u}=\hat{\mathbf{u}}-\mathbf{u}\), where \(\mathbf{\Delta u}\) follows a zero-mean Gaussian distribution with a covariance matrix \(\mathbf{Q}_{\mathbf{u}}=E\left[\mathbf{\Delta u}\mathbf{\Delta u}^{T}\right]\). Similarly, we define the observation vector \(\hat{\mathbf{v}}=\left[\hat{\mathbf{v}}_{11}^{T},\hat{\mathbf{v}}_{12}^{T},\cdots,\hat{ \mathbf{v}}_{ji}^{T},\cdots,\hat{\mathbf{v}}_{M6}^{T}\right]^{T}\) related to the position of the antenna array, where \(\hat{\mathbf{v}}_{ji}\) corresponds to \(\mathbf{v}_{ji}\) in Equation (8) and contains the observation error. We define vector \(\hat{\mathbf{v}}\)'s true value as \(\mathbf{v}=\left[\mathbf{v_{11}}^{T},\mathbf{v_{12}}^{T},\cdots,\mathbf{v_{i\hat{\mathbf{\mu}}}}^{ T},\cdots,\mathbf{v_{M6}}^{T}\right]^{T}\) and its measurement error vector as \(\mathbf{\Delta v}=\hat{\mathbf{v}}-\mathbf{v}\), where \(\mathbf{\Delta v}\) follows a zero-mean Gaussian distribution with a covariance matrix \(\mathbf{Q}_{\mathbf{v}}=E\left[\mathbf{\Delta v}\mathbf{\Delta v}^{T}\right]\). According to the method in reference [36], we define vector \(\mathbf{\Phi}=\left[\mathbf{p_{t}}^{T},\mathbf{v}^{T}\right]\), where \(\mathbf{p_{t}}\) represents the exact location of the target spoofing interference source. Given that \(\mathbf{\Delta u}\) and \(\mathbf{\Delta v}\) are independent of each other and follow zero-mean Gaussian distributions, the logarithm of the joint density function of \(\hat{\mathbf{u}}\) and \(\hat{\mathbf{v}}\), parametrized on \(\mathbf{\Phi}\), can be expressed as follows:  
\[ \begin{array}{l}lnf(\hat{\mathbf{u}},\hat{\mathbf{v}},\mathbf{\Phi})=lnf(\hat{\mathbf{u}}, \mathbf{\Phi})+lnf(\hat{\mathbf{v}},\mathbf{\Phi})\\ =K-\frac{1}{2}[\hat{\mathbf{u}}-\mathbf{u}]^{T}\mathbf{Q_{u}}^{-1}[\hat{\mathbf{u}}-\mathbf{u}]- \frac{1}{2}[\hat{\mathbf{v}}-\mathbf{v}]^{T}\mathbf{Q_{v}}^{-1}[\hat{\mathbf{v}}-\mathbf{v}]\end{array}  \tag{24}\]  
in which \(f(\cdot)\) is the probability density function and the parameter \(K\) associated with \(\mathbf{Q}_{\mathbf{u}}\) and \(\mathbf{Q}_{\mathbf{v}}\) is calculated as follows. \[K=-\frac{1}{2}ln\Big{[}(2\pi)^{6M}|\mathbf{Q}_{\mathbf{u}}|\Big{]}-\frac{1}{2}ln\Big{[} (2\pi)^{6M}|\mathbf{Q}_{\mathbf{v}}|\Big{]} \tag{25}\]  
According to the definition of CRB, the CRB equation for \(\mathbf{\Phi}\) is as follows. \[CRB(\mathbf{\Phi})=-E\bigg{[}\bigg{(}\frac{\partial^{2}lnf(\hat{\mathbf{u}},\hat{\mathbf{v }},\mathbf{\Phi})}{\partial\mathbf{\Phi}\partial\mathbf{\Phi}^{T}}\bigg{)}\bigg{]}_{(6M+3) \times(6M+3)}^{-1} \tag{26}\]  
It can be seen from Equation (25) that \(CRB(\mathbf{\Phi})\) is a \(6M+3\) square matrix, in which only the upper left \(3\times 3\) submatrix corresponding to the CRB is of interest to us. For the convenience of expression, we convert Equation (26) into submatrix form shown as:\[CRB(\mathbf{\Phi})= \begin{bmatrix}\mathbf{R_{1}}&\mathbf{R_{2}}\\ \mathbf{R_{2}}^{T}&\mathbf{R_{3}}\end{bmatrix}