# SATCOM Earth Station Arrays Anti-Jamming Based on MVDR Algorithm
## 2 Satellite Communication Earth Station Array Model
The received data output of the M reflector antennas at time \(t\) is arranged into a column vector [18], which can be obtained as follows  
\[\mathbf{x}(t)=\mathbf{A}\mathbf{s}(t)+\mathbf{n}(t) \tag{8}\]  
In the formula, \(\mathbf{x}(t)=[x_{1}(t)\)\(x_{2}(t)\)\(\cdots\)\(x_{M}(t)]^{\rm T}\) is the M \(\times\) 1-dimensional snapshot data vector of the array, \(\mathbf{n}(t)=[n_{1}(t)\)\(n_{2}(t)\)\(\cdots\)\(n_{M}(t)]^{\rm T}\) is the M \(\times\) 1-dimensional noise data vector of the array, and \(s(t)=[s_{0}(t)\)\(s_{1}(t)\)\(\cdots\)\(s_{N-1}(t)]^{\rm T}\) is the N \(\times\) 1-dimensional vector of spatial signals and jammings, where \(s_{0}(t)\) and \(s_{i}(t)\) are the complex envelopes of the signal and the i-th jamming at time \(t\), respectively, \((\bullet)^{\rm T}\) represents the transpose operation, and \(\mathbf{A}\) is the array manifold matrix, which can be expressed as  
\[\mathbf{A}= \begin{bmatrix}\mathbf{\hat{a}}(\theta_{0})&\mathbf{\hat{a}}(\theta_{1})&\cdots &\mathbf{\hat{a}}(\theta_{N-1})\end{bmatrix}  \tag{9}\]  
In the formula, \(\mathbf{\hat{a}}(\theta_{i})\) is the directional vector of the signal/jamming, which can be expressed as  
\[\mathbf{\hat{a}}(\theta_{i})=\mathbf{Q}(\theta_{i})\mathbf{a}(\theta_{i})=\begin{bmatrix} Q_{1}(\theta_{i})\\  
power are minimized, achieving suppression of jamming and noise. In this article, the constrained target is the satellite signal received by the earth station, and the goal is to minimize the total power of jamming and noise signals received by the earth station, thereby maximizing the array output SINR. Assuming that the satellite signal is incident from direction \(\theta_{0}\) and N \(-\) 1 jammings are incident from directions \(\theta_{i}\) (\(i\) = 1, 2,..., N \(-\) 1), the output of the linear array at time \(t\) is [19]  
\[\mathbf{x}(t)=\mathbf{x}_{\text{S}}(t)+\mathbf{x}_{\text{I}}(t)+\mathbf{n}(t) \tag{13}\]  
where \(\mathbf{x}_{\text{S}}(t)=\hat{\mathbf{a}}(\theta_{0})\mathbf{x}_{0}(t)\), \(\mathbf{x}_{\text{I}}(t)=\sum_{i=1}^{N}\hat{\mathbf{a}}(\theta_{i})\mathbf{x}_{i}(t)\), and \(n(t)\) represent the signal, jamming, and noise, respectively, and \(\hat{\mathbf{a}}(\theta_{0})\) and \(\hat{\mathbf{a}}(\theta_{i})\) are the steering vectors of the signal and jamming, respectively. The covariance matrix of jamming and noise can be derived as [20]  
\[ \begin{array}{ll}\mathbf{R}_{\text{i+n}}&=E\Big{\{}(\mathbf{x}_{\text{I}}(t)+\mathbf{n} (t))(\mathbf{x}_{\text{I}}(t)+\mathbf{n}(t))^{\text{H}}\Big{\}}\\ &=\sum_{i=1}^{N-1}\sigma_{i}^{2}\mathbf{Q}(\theta_{i})\mathbf{a}(\theta_{i})\mathbf{a}^{ \text{H}}(\theta_{i})\mathbf{Q}^{\text{H}}(\theta_{i})+\sigma_{\text{n}}^{2}\mathbf{I} \end{array}  \tag{14}\]  
where \(\sigma_{i}^{2}\) represents the power of the i-th jamming, \(\sigma_{\text{n}}^{2}\) represents the power of the noise, \(I\) represents the M \(\times\) M dimensional identity matrix, E[\(\bullet\)] represents the expectation operation, and (\(\bullet\))\({}^{\text{H}}\) represents the conjugate transpose operation. Assuming that the weight of the \(l\)-th element is \(w_{l}\), the weight vector of the M elements can be arranged into a column vector [21]  
\[\mathbf{w}= \begin{bmatrix}w_{1}&w_{2}&\cdots&w_{M}\end{bmatrix} ^{\text{T}} \tag{15}\]  
The output of the array is the product of its weight vector and the received data  
\[y(t)=\mathbf{w}^{\text{H}}\mathbf{x}(t)=\sum_{l=1}^{M}w_{l}^{*}\mathbf{x}_{\text{I}}(t) \tag{16}\]  
where (\(\bullet\))* represents the conjugate operation. From the above equations, the signal to interference plus noise ratio (SINR) of the array output can be derived as  
\[\Psi=\frac{\sigma_{s}^{2}\big{|}\mathbf{w}^{\text{H}}\mathbf{Q}(\theta_{0})\mathbf{a}( \theta_{0})\big{|}^{2}}{\mathbf{w}^{\text{H}}\mathbf{R}_{\text{i+n}}\mathbf{w}} \tag{17}\]  
where \(\mathbf{Q}(\theta_{0})\mathbf{a}(\theta_{0})\) is the steering vector of the signal, and \(\sigma_{\text{s}}^{2}\) is the power of the signal.