# SATCOM Earth Station Arrays Anti-Jamming Based on MVDR Algorithm
## 2 Satellite Communication Earth Station Array Model
To minimize the jamming and noise power of the array output under the condition that the output of the expected direction signal is constant, the cost function can be constrained as [22]  
\[\left\{ \begin{array}{ll}&\min_{w}\mathbf{w}^{\text{H}}\mathbf{R}\mathbf{w}\\ \text{s.t.}&\mathbf{w}^{\text{H}}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})=1\end{array} \right. \tag{18}\]  
where \(\mathbf{R}\) represents the covariance matrix of the array's received data. From Equation (18), it can be seen that to calculate the optimal weight vector of the MVDR algorithm, the covariance matrix of the array's received data must be obtained, but in practical situations, the array's sampling data is obtained instead. Therefore, under a certain sampling rate, the array's sampling covariance matrix can be used to replace the array's received data covariance matrix. \[ \begin{array}{ll}\mathbf{R}_{xx}&=\frac{1}{k}\sum\limits_{i=1}^{K}\mathbf{x}\mathbf{x}^{ \text{H}}\\ &=\sum_{i=0}^{N-1}\sigma_{i}^{2}\mathbf{Q}(\theta_{i})\mathbf{a}(\theta_{i})\mathbf{a}^{ \text{H}}(\theta_{i})\mathbf{Q}^{\text{H}}(\theta_{i})+\sigma_{\text{n}}^{2}\mathbf{I} \end{array}  \tag{19}\]where \(K\) represents the number of sampling data snapshots. Thus, Equation (18) can be transformed into  
\[\left\{ \begin{aligned} &\min_{w}\mathbf{w}^{\mathrm{H}}\mathbf{R}_{xx}\mathbf{w} \\ &\text{s.t.}\quad\mathbf{w}^{\mathrm{H}}\mathbf{Q}(\theta_{0})\mathbf{a}( \theta_{0})=1\end{aligned} \right. \tag{20}\]  
Using the Lagrange multiplier method, Equation (20) can be transformed into an unconstrained problem. Introducing the Lagrange factor \(\mu\) and constructing the function  
\[\mathrm{L}(w)=\frac{1}{2}\mathbf{w}^{\mathrm{H}}\mathbf{R}_{xx}\mathbf{w}-\mu\Big{[}\mathbf{w }^{\mathrm{H}}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})-1\Big{]} \tag{21}\]  
Differentiating Equation (21) with respect to \(\mathbf{w}\) and setting it to zero, the function with respect to the weight vector \(\mathbf{w}\) can be obtained  
\[\frac{\partial L}{\partial w}=\mathbf{R}_{xx}\mathbf{w}-\mu\mathbf{Q}(\theta_{0})\mathbf{a}( \theta_{0})=0 \tag{22}\]  
The optimal weight vector can be solved as  
\[\mathbf{w}=\mu\mathbf{R}_{xx}^{-1}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0}) \tag{23}\]  
Then, using the constraint condition \(\mathbf{w}^{\mathrm{H}}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})=1\), the Lagrange factor can be obtained as  
\[\mu=\frac{1}{\mathbf{a}^{\mathrm{H}}(\theta_{0})\mathbf{Q}^{\mathrm{H}}(\theta_{0}) \mathbf{R}_{xx}^{-1}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})} \tag{24}\]  
Substituting Equation (24) into Equation (23), the optimal weight vector can be obtained as  
\[\mathbf{w}=\frac{\mathbf{R}_{xx}^{-1}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})}{\mathbf{a}^{ \mathrm{H}}(\theta_{0})\mathbf{Q}^{\mathrm{H}}(\theta_{0})\mathbf{R}_{xx}^{-1}\mathbf{Q}( \theta_{0})\mathbf{a}(\theta_{0})} \tag{25}\]  
In practical communication environments, the performance of the traditional MVDR algorithm can sharply decrease due to factors such as external environment, antenna array, and estimation error of the sampling covariance matrix [23]. To address this, the diagonal loading MVDR algorithm can be used to improve the algorithm's robustness [24]. Let \(\mathbf{R}_{x}=\mathbf{R}_{xx}+\delta\mathbf{I}\), the optimal weight vector of the diagonal loading MVDR algorithm can be represented as  
\[\mathbf{w}=\frac{(\mathbf{R}_{xx}+\delta\mathbf{I})^{-1}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{ 0})}{\mathbf{a}^{\mathrm{H}}(\theta_{0})\mathbf{Q}^{\mathrm{H}}(\theta_{0})(\mathbf{R}_{ xx}+\delta\mathbf{I})^{-1}\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})} \tag{26}\]  
The recommended range of the loading factor \(\delta\) in the above equation is \(\delta_{S}\leq\delta\leq\delta_{L}\), and the maximum loading level \(\delta_{L}\) depends on the specific application, while the minimum loading level \(\delta_{S}\) is generally not less than the noise power. Due to the inclusion of matrix inversion algorithms in this article, its algorithmic complexity is at most cubic.