# Flexible Null Broadening Robust Beamforming Based on JADE
## 2 Signal Model and MVDR Beamformer  
Without loss of generality, the uniform linear array is considered here to build the signal model. In this paper, the array is composed of N sensors, as shown in Figure 1.  
Considering that one desired signal impinges on the array from a known direction along with L jammer signals from unknown directions, respectively, the array-receiving signal model can be expressed as  
\[\mathbf{x}(t)=\mathbf{\Gamma}\mathbf{As}(t)+\mathbf{n}(t) \tag{1}\]  
where \(\mathbf{As}(t)\) is the signal model, \(\mathbf{As}(t  
where \(\mathbf{x}(t)\) is the snapshot data received by the array, \(\mathbf{\Gamma}\) is the array gain-phase errors matrix. \(\mathbf{A}\) is the array flow matrix, \(\mathbf{s}(t)\) is the incident signal vector, \(\mathbf{n}(t)\) is the Gaussian noise. The array flow matrix is further expressed as  
\[\mathbf{A}=[\mathbf{a}(\theta_{0}),\ldots,\mathbf{a}(\theta_{L})] \tag{2}\]  
\[\mathbf{s}(t)=[s_{0}(t),\cdots,s_{L}(t)]^{\mathrm{T}} \tag{3}\]  
\[\mathbf{n}(t)=[n_{0}(t),\cdots,n_{N-1}(t)]^{\mathrm{T}} \tag{4}\]  
where \(\theta\) is the pitch angle, and \([\cdot]^{\mathrm{T}}\) denotes transpose operation.  
The array gain-phase errors matrix \(\mathbf{\Gamma}\) is defined as  
\[\mathbf{\Gamma}\triangleq\mathrm{diag}\Big{(}\Big{[}1,\rho_{2}e^{j\phi_{2}},\rho_ {3}e^{j\phi_{3}},\ldots,\rho_{N}e^{j\phi_{N}}\Big{]}\Big{)} \tag{5}\]  
where \(\rho_{i}\) and \(\phi_{i}\) are the gain and phase errors of the \(i\)-th array, \(\mathrm{diag}(\cdot)\) denotes matrix diagonalization. The steering vector \(\mathbf{a}(\theta)\) is written as  
\[\mathbf{a}(\theta)=[1,e^{j\phi_{1}},\cdots,e^{j\phi_{N-1}}]^{\mathrm{T}} \tag{6}\]  
where the phase difference between sensors is \(\phi_{n}=-\frac{2\pi nd}{\lambda}\sin\theta,0\leq n\leq N-1\), \(\lambda\) is wavelength.  
Then, the actual steering vector with array gain-phase errors is  
\[\mathbf{a}_{\mathbf{\Gamma}}(\theta)=\mathbf{\Gamma}\mathbf{a}(\theta) \tag{7}\]  
Under the maximum signal-to-interference-plus-noise ratio (SINR) criterion, the optimal weights of the MVDR beamformer can be transformed into optimization problem  
\[\begin{split}\underset{w}{\text{min}}&\mathbf{w}^{ \mathrm{H}}\mathbf{R}_{in}\mathbf{w}\\ s.t.&\mathbf{w}^{\mathrm{H}}\mathbf{a}_{0}=1\end{split} \tag{8}\]  
From (8), the mathematical model of beamforming can be interpreted as minimizing the array output power while ensuring that the response at the desired direction is \(1\). Therefore, it can be realized that the main lobe is formed at the desired position and the null is formed at the interference position.  
Figure 1: Moving target model based on uniform linear array.  
According to the Lagrange multipliers method, the optimal weight vector \(\mathbf{w}_{opt}\) is obtained by  
\[\mathbf{w}_{opt}=\frac{R_{in}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H}R_{in}^{-1}\mathbf{a}_{0}} \tag{9}\]  
where \(\mathbf{a}_{0}\) is the desired signal steering vector, and \(R_{in}\) is the interference-plus-noise covariance matrix (INCM).  
The interference-plus-noise covariance matrix \(\mathbf{R}_{in}\) is defined by  
\[\mathbf{R}_{in}=\sum_{l=1}^{L}\sigma_{l}^{2}\mathbf{a}(\theta_{l})\mathbf{a}^{\mathrm{H}}( \theta_{l})+\sigma_{n}^{2}\mathbf{I} \tag{10}\]  
where \(\sigma_{l}^{2}\) is the power of the l-th interference, \(\sigma_{n}^{2}\) is the noise power.  
Furthermore, the sample covariance matrix \(\hat{\mathbf{R}}\) is defined by  
\[\hat{\mathbf{R}}=\frac{1}{K}\sum_{t=1}^{K}x(t)x^{\mathrm{H}}(t) \tag{11}\]  
where, \(K\) is the snapshot number.