# Research on the Compound Jamming Method against FDA-MIMO Beamforming
## 3 MVDR Adaptive Beamforming Algorithm  
The MVDR adaptive beamforming algorithm adopts the sampling matrix inversion algorithm in adaptive beamforming, which minimizes the output power of the array in the desired direction by adding a weight coefficient at each array element, and it has a fast convergence speed under SINR.  
The desired target signal in space is \(s(t)\), the desired target position information is \((r_{0},\theta_{0})\), the jamming signal is \(i_{j}\), \((j=1,2,\cdots,J)\), the jamming position is \((r_{j},\theta_{j})\), and the array element noise is \(n(t)\). Then the \(n\)-th array element signal at the receiving side is  
\[\mathbf{x}_{n}(t)=\mathbf{a}_{n}(\theta_{d})s(t)+\sum_{j=1}^{J}\mathbf{a}_{n}(\theta_{j}) i_{j}(t)+\mathbf{n}_{n}(t) \tag{20}\]  
Then the matrix form of the whole array is  
\[\mathbf{x}(t)=\mathbf{a}(\theta_{d})s(t)+\sum_{j=1}^{J}\mathbf{a}(\theta_{j})i_{j}(t)+\mathbf{ n}(t) \tag{21}\]  
where \(\mathbf{a}(\theta)\) is the receiving steering vector from the direction of \((r,\theta)\), which can be expressed as  
\[\mathbf{a}(\theta)\!=[1,\text{e}^{-j(2\pi f_{0}d\sin(\theta)/c)},\cdots,\text{e} ^{-j(N-1)(2\pi f_{0}d\sin(\theta)/c)}] \tag{22}\]The array output is  
\[\mathbf{y}(t)=\mathbf{w}^{H}\mathbf{a}(\theta_{d})s(t)+\sum_{j=1}^{J}\mathbf{w}^{H}\mathbf{a}(\theta_{ j})i_{j}(t)+\mathbf{n}(t) \tag{23}\]  
When \(J\) interferences are known, to maximize the power of the beamforming output signal and accurately receive the desired signal in the \(\theta_{d}\) direction, according to Equation (23), the constraint condition of the weight coefficient can be obtained as  
\[\left\{\begin{array}{l}\mathbf{w}^{H}a(\theta_{d})=1\\ \mathbf{w}^{H}a(\theta_{j})=0\end{array}\right. \tag{24}\]  
This constraint causes the zero point of each beam pattern of the receive array to point to J interfering signals. To minimize the noise when the constraints of Equation (24) are satisfied at the same time, the objective function is optimized as  
\[\underset{\omega}{\min}E\Big{[}\Big{|}\mathbf{y}(t)^{2}\Big{|}\Big{]}=\underset {\omega}{\min}w^{H}\mathbf{R}\mathbf{w} \tag{25}\]  
Then the MVDR weight optimization problem can be expressed as  
\[\underset{\omega}{\min}\mathbf{w}^{H}\mathbf{R}\mathbf{w} \tag{26}\] \[s.t.\left\{\begin{array}{l}\mathbf{w}^{H}\mathbf{a}(\theta_{d})=1\\ \mathbf{w}^{H}\mathbf{a}(\theta_{ij})=0\end{array}\right.\]  
The essence of the MVDR beamformer is to solve the weight coefficient of each array element. Then the Lagrange multiplier method can be used to obtain  
\[L(\mathbf{w})=\mathbf{w}^{H}\mathbf{R}\mathbf{w}+\lambda[\mathbf{w}^{H}\mathbf{a}(\theta_{d})-1] \tag{27}\]  
After deriving Equation (27) and making it zero, we can obtain  
\[\frac{\partial L(\mathbf{w})}{\partial\mathbf{w}}=2\mathbf{R}\mathbf{w}+\lambda\mathbf{a}(\theta_ {d})=0 \tag{28}\]  
Solving Equation (28), we can obtain  
\[\mathbf{w}=\mu\mathbf{R}^{-1}\mathbf{a}(\theta_{d}) \tag{29}\]  
Expressing the constraint condition \(\mathbf{w}^{H}\mathbf{a}(\theta_{d})=1\) as \(\mathbf{a}(\theta_{d})^{H}\mathbf{w}=1\) and then multiplying both sides of Equation (29) by \(\mathbf{a}^{H}(\theta_{d})\) at the same time to obtain  
\[\mu=\frac{1}{\mathbf{a}^{H}(\theta_{d})\mathbf{R}^{-1}\mathbf{a}(\theta_{d})} \tag{30}\]  
Finally, the optimal value of the array weight is obtained according to the MVDR criterion as  
\[\mathbf{w}_{MVDR}=\frac{\mathbf{R}^{-1}\mathbf{a}(\theta_{d})}{\mathbf{a}^{H}(\theta_{d})\bm {R}^{-1}\mathbf{a}(\theta_{d})} \tag{31}\]  
where \(\mathbf{R}=E[\mathbf{x}_{I}(t),\mathbf{x}_{I}^{H}(t)]\mathbf{x}_{I}(t)=\mathbf{J}(t)+\mathbf{n}(t)\). The output SINR is  
\[R_{SINR}=\frac{\big{|}\mathbf{w}^{H}\mathbf{a}(\theta_{d})\big{|}^{2}}{\mathbf{w}^{H}\mathbf{R} ^{-1}\mathbf{w}-\big{|}\mathbf{w}^{H}\mathbf{a}(\theta_{d})\big{|}^{2}} \tag{32}\]