# Anti-Jamming Method and Implementation for GNSS Receiver Based on Array Antenna Rotation
## 2 Array Signal Reception and Anti-Jamming Model  
Signals received by each array element have different propagation delays and the delays appear in the form of phase differences. The steering vector of the signal is composed of the phase differences in a column containing signal DOA (direction of arrival) information in the spatial domain. Changes in the signal DOA might lead to changes in the steering vector. Matters such as element coupling and the inconsistency of radio frequency (RF) channels affect the measurement of phase difference, which worsens the anti-jamming performance. To simplify the analyses, this study ignores the non-ideal factors above and adopts the narrowband array signal model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Explanation** \\ \hline \([\cdot]^{\text{H}}\) & “H” means Hermitian Transpose \\ \([\cdot]^{\text{T}}\) & “T” means transpose \\ min\([\cdot]\) & “min” means minimum value \\ \(||\cdot||\) & means norm of vector \\ \(\lambda_{\text{min}}[\cdot]\) & means the minimum eigen value of matrix \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequently used symbols. Figure 1: Block diagram of the work. Assume that the navigation signal and interference signal are received by the \(N\)-element antenna array. The navigation signal, signal power, and steering vector are denoted as \(s_{i}(t)\), \(p_{s_{i}}\), \(\mathbf{a_{s_{i}}}\), respectively. The interference signal, interference power, and steering vector are denoted as \(j_{k}(t)\), \(p_{j_{k}}\), \(\mathbf{a_{j_{k}}}\), respectively. The thermal noise and noise power are denoted as \(\mathbf{n_{N}}(t)\) and \(p_{n}\), respectively. \"\(t\)\" is the sampling time and the steering vector is a \(N\times 1\)-dimensional column vector. The received signal of \(N\)-element array is  
\[\mathbf{x}(t)=\sum\limits_{i=1}^{I}\sqrt{2p_{s_{i}}}s_{i}(t)\mathbf{a_{s_{i}}} +\sum\limits_{k=1}^{K}\sqrt{2p_{j_{k}}}j_{k}(t)\mathbf{a_{j_{k}}}+\sqrt{p_{n}} \mathbf{n_{N}}(t) \tag{1}\]  
where \(i\) represents the \(i\)th navigation signal, \(k\) represents the \(k\)th interference signal, and \(I\) and \(K\) denote the number of satellite signals and the number of jammers, respectively. \(\mathbf{n_{N}}(\mathbf{t})\) represents the noise signal column vector of the \(N\)-element array. This work concentrates on the condition that \(K\geq N\). The steering vector is the function of array geometry, antenna attitude \(a_{antenna}\), and the signal's incident DOA (\(\theta\) in elevation, \(\varphi\) in azimuth). Assuming each element is isotropic and the array geometry is settled, the steering vector has the form  
\[\mathbf{a}= \begin{bmatrix}1&a_{1}(\theta,\varphi,a_{antenna})&\cdots&a_{N-1}( \theta,\varphi,a_{antenna})\end{bmatrix} ^{T} \tag{2}\]  
where \(\mathbf{a}\) represents steering vector. Denote the normalized coefficient of the spatial anti-jamming filter as \(\overline{\mathbf{w}}\). It is an \(N\times 1\)-dimensional column vector. The output signal of spatial anti-jamming processing is  
\[y(t)=\overline{\mathbf{w}}^{\mathrm{H}}\mathbf{x}(t) \tag{3}\]  
Since the intensity of interference and noise is much greater than that of navigation signals, the output residual interference-plus-noise power is approximately  
\[p_{inres}=\overline{\mathbf{w}}^{\mathrm{H}}\mathbf{R_{xx}}\overline{\mathbf{ w}} \tag{4}\]  
where \(\mathbf{R_{xx}}\) is the covariance matrix of the array signal. The computing of \(\mathbf{R_{xx}}\) is given later. Correspondingly, the \(ICR\) (interference cancellation ratio) is defined to help evaluate the interference cancellation performance. The \(ICR\) is the ratio of input interference-plus-noise power and residual interference-plus-noise power. \[ICR=\frac{\sum\limits_{k=1}^{K}p_{j_{k}}+p_{n}}{p_{inres}} \tag{5}\]  
The residual interference-plus-noise power is composed of two parts: residual interference power and noise power. Both of them have an impact on the signal \(CNR\) after the anti-jamming process.