# Analysis of Multi-Antenna GNSS Receiver Performance under Jamming Attacks
## 2 Methodology
### Signal and System Model  
A rectangular planar array planar is used herein. Consider the case of an \(M\times N\) element uniform rectangular array. The elements are lying in the \(x\)-\(y\) plane and are equally spaced by \(d_{m}\) in the \(x\)-direction and \(d_{n}\) in the \(y\)-direction as shown in Figure 1.  
After down-conversion and sampling, the digitized signal received at the (\(m\), \(n\))th antenna element can be expressed as  
\[\begin{array}{rl}\mathrm{x}_{m,n}\left(t\right)=&\sum\limits_{k=1}^{K}{{s_{ k}}\left(t\right){e^{j\frac{2\pi}{\lambda}\left[\left(m-1\right)d_{m}\sin( \theta_{k})\sin(\phi_{k})+\left(n-1\right)d_{n}\sin(\theta_{k})\cos(\phi_{k}) \right]}}}\\ &+\sum\limits_{j=1}^{I}{{j_{j}}\left(t\right){e^{j\frac{2\pi}{\lambda}\left[ \left(m-1\right)d_{m}\sin(\theta_{j})\sin(\phi_{j})+\left(n-1\right)d_{n}\sin( \theta_{j})\cos(\phi_{j})\right]}}}+\eta_{m,n}\left(t\right)\end{array} \tag{1}\]  
where \({s_{k}}\left(t\right)\) is the \(k\)th satellite signal component, \({j_{j}}\left(t\right)\) is the \(j\)th electronic interference, \(\lambda\) refers to the wavelength of the signal, \((\theta_{k},\phi_{k})\) is the elevation and azimuth angle of the \(k\)th satellite, \(\left(\theta_{j},\phi_{j}\right)\) is the elevation and azimuth angle of the electronic interference signal, \(K\) is the number of Pseudo Random Numbers (PRNs), \(J\) is the number of electronic interference signals, and \(\eta_{m,n}\left(t\right)\) is the additive spatially white noise of the (\(m\), \(n\))th antenna element.  
Figure 1: Rectangular array structure with \(M\times N\) elements.  
The impinged signals from all the antenna elements can be represented in matrix form as  
\[\mathbf{x}=\mathbf{As}+\mathbf{Bj}+\boldsymbol{\eta} \tag{2}\]  
where \(\mathbf{x}\) is the \(M\!N\times 1\) received signal vector, \(\boldsymbol{\eta}\) is the \(M\!N\times 1\) noise vector, \(\mathbf{s}\) is the \(K\times 1\) satellite signal vector, and \(\mathbf{j}\) is the \(J\times 1\) signal vector affected by electronic interference; these vectors are given by  
\[\mathbf{x}=\left[x_{1,1}\left(t\right),x_{2,1}\left(t\right),\ldots,\ x_{M,1} \left(t\right),x_{1,2}\left(t\right),\ldots,\ x_{M,N}\left(t\right)\right]^{T} \tag{3}\]  
\[\mathbf{s}=\left[s_{1}\left(t\right),s_{2}\left(t\right),\ldots,s_{K}\left(t \right)\right]^{T} \tag{4}\]  
\[\mathbf{j}=\left[j_{1}\left(t\right),j_{2}\left(t\right),\ldots,j_{l}\left(t \right)\right]^{T} \tag{5}\]  
\[\boldsymbol{\eta}=\left[\eta_{1,1}\left(t\right),\eta_{2,1}\left(t\right), \ldots\eta_{M,1}\left(t\right),\eta_{1,2}\left(t\right),\ldots\eta_{M,N}\left( t\right)\right]^{T} \tag{6}\]  
The steering matrices \(\mathbf{A}\) and \(\mathbf{B}\) are of dimensions \(M\!N\times K\) and \(M\!N\times J\) respectively and are given by  
\[\mathbf{A}=\left[\mathbf{a}_{1},\mathbf{a}_{2},\ \ldots,\ \mathbf{a}_{k}\right]^{T} \tag{7}\]  
\[\mathbf{B}=\left[\mathbf{b}_{1},\mathbf{b}_{2},\ \ldots,\ \mathbf{b}_{j}\right]^{T} \tag{8}\]  
where \(\mathbf{a}_{k}\) is the \(M\!N\times 1\) steering vector of the \(k\)th satellite signal component coming from direction \(\left(\theta_{k},\phi_{k}\right)\) and \(\mathbf{b}_{j}\) is the \(M\!N\times 1\) steering vector of the \(j\)th electronic interference signal coming from direction \(\left(\theta_{j},\phi_{j}\right)\). Both \(\mathbf{a}_{k}\) and \(\mathbf{b}_{j}\) have a similar structure, hence only \(\mathbf{a}_{k}\) is defined and is given by  
\[\mathbf{a}_{k}=\left[\mathbf{c}_{k}^{T},\gamma_{k}\mathbf{c}_{k}^{T},\ \ldots,\ \gamma_{k}^{N-1}\mathbf{c}_{k}^{T}\right]^{T} \tag{9}\]  
\[\mathbf{c}_{k}=\left[1,\beta_{k},\ \ldots,\ \beta_{k}^{M-1}\right]^{T} \tag{10}\]  
\[\gamma_{k}=e^{j\frac{2\pi}{\lambda}\left[d_{m}\sin\left(\theta_{k}\right)\cos \left(\phi_{k}\right)\right]} \tag{11}\]  
\[\beta_{k}=e^{j\frac{2\pi}{\lambda}\left[d_{m}\sin\left(\theta_{k}\right)\sin \left(\phi_{k}\right)\right]} \tag{12}\]