# Flexible Null Broadening Robust Beamforming Based on JADE
## 3 Proposed Algorithm
### Flexible Null Broadening Method  
According to the structure of CMT in reference [15], the covariance matrix taper can be expressed as  
\[T_{m,n}(\Delta\theta,Q)=\sum_{q=-Q}^{Q}\exp(-\text{j}(m-n)\pi\sin(q\Delta \theta)) \tag{18}\]  
Here, \(Q\) virtual interferences are added on both sides of the \(i\)-th real interference with \(\Delta\theta\) interval. The novel covariance matrix taper can be written by  
\[\overline{T}_{m,n}\left(\Delta\theta,Q,P_{q}\right)=\sum_{q=-Q}^{Q}P_{q}\exp( -\text{j}(m-n)\pi\sin(q\Delta\theta)) \tag{19}\]  
where \(P_{q}\) is the null deepening coefficient, and it can control the depth of the null. In addition, the deeper null with a lager \(P_{q}\) can suppress the interference more toughly, proved by Proposition 1. **Proposition 1**. : _The deepening of the beamforming null is beneficial to the suppression of interference, \(\eta_{i}\triangleq 1-\sigma_{io}^{2}/\sigma_{i}^{2}\) is defined as the i-th interference suppression ratio and the null deepening coefficient \(P_{q}\) is proportional to \(\eta_{i}\)._  
Proof of Proposition 1.: Detailed proof is shown in Appendix A. \(\square\)  
When \(\eta_{i}\to 0\), the interference suppression ability is poor; when \(\eta_{i}\to 1\), the interference suppression performance is close to the optimal. From Proposition 1, we can see that a larger \(P_{q}\) can make the higher interference suppression ratio \(\eta_{i}\). Therefore, the proposed novel covariance matrix taper in this paper can improve the interference suppression ability by setting a larger null deepening coefficient \(P_{q}\). To avoid the waste of DOFs and to broaden the null more flexibly, the interference mismatch matrix can be obtained by  
\[ \begin{array}{l}\mathbf{R}_{di}=\left[\hat{\sigma}_{i}^{2}\mathbf{a}_{\Gamma}(\hat{ \theta}_{i})\mathbf{a}_{\Gamma}^{\text{H}}(\hat{\theta}_{i})\right]\odot\overline {T}\\ =\left[P_{q}\hat{\sigma}_{i}^{2}\mathbf{a}_{\Gamma}(\hat{\theta}_{i})\mathbf{a}_{ \Gamma}^{\text{H}}(\hat{\theta}_{i})\right]\odot\mathbf{T}\end{array}  \tag{20}\]  
From (20), the operation of the new array taper and the interference covariance matrix is equivalent to the Hadamard product of the original one and the weighted interference covariance matrix. The null deepening coefficient \(P_{q}\) in the weighted interference covariance matrix can adjust the power of original interference, resulting in an adjustable null depth. Then, the interference-plus-noise covariance matrix can be expressed as  
\[\hat{\mathbf{R}}_{in}=\sum_{i=1}^{L_{d}}\mathbf{R}_{di}+\sum_{i=1}^{L_{nd}}\mathbf{R}_{i}+ \hat{\sigma}_{n}^{2}\mathbf{I} \tag{21}\]where \(L_{d}\) and \(L_{nd}\) are the number of dynamic interference and non-dynamic interference, respectively, and \(L=L_{d}+L_{nd}\). In addition, \(\mathbf{R}_{i}=\hat{\sigma}_{i}^{2}\mathbf{a}_{\Gamma}(\hat{\theta}_{i})\mathbf{a}_{\Gamma}^ {\text{II}}(\hat{\theta}_{i})\) is the non-dynamic interference covariance matrix, and \(\mathbf{I}\) is an N \(\times\) N identity matrix. The \(i\)-th interference power is estimated by Capon spectrum [20]  
\[\hat{\sigma}_{i}^{2}=\frac{1}{\mathbf{a}_{\Gamma}^{\text{H}}(\hat{\theta}_{i})\hat{ \mathbf{R}}^{-1}\mathbf{a}_{\Gamma}(\hat{\theta}_{i})} \tag{22}\]  
The noise power \(\hat{\sigma}_{n}^{2}\) is estimated by small eigenvalue of matrix \(\hat{\mathbf{R}}\)[21]  
\[\hat{\sigma}_{n}^{2}=\frac{1}{N-L+1}\sum_{i=L}^{N}\lambda_{i} \tag{23}\]  
where \(\lambda_{i}\) is the \(i\)-th eigenvalue of the sample covariance matrix and \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{L}\geq\cdots\geq\lambda_{N} \approx\sigma_{n}^{2}\).