## 4 Beamforming Based on INCM Reconstruction  
### Steering Vector Estimation  
The nominal SVs \(\bar{\mathbf{a}}\) are constituted by the directions of each signal obtained from the DOA estimation module. The actual SV can be expressed as the sum of the nominal SV and the error  
\[\mathbf{a}=\bar{\mathbf{a}}+\mathbf{e}, \tag{13}\]  
where \(\mathbf{e}=\mathbf{e}_{\perp}+\mathbf{e}_{\parallel}\). \(\mathbf{e}_{\perp}\) is the orthogonal component and \(\mathbf{e}_{\parallel}\) is the parallel component. The latter component does not affect the beamforming performance, so the corrected SV can be obtained as  
\[\mathbf{a}=\mathbf{a}+\mathbf{e}_{\perp}. \tag{14}\]  
A uniform division is performed within a small range around the nominal SV to obtain the SV neighborhood \(\overline{\mathbf{\Lambda}}\)  
\[\overline{\mathbf{\Lambda}}=[\bar{\mathbf{a}}(\theta-P\Delta\theta),\cdots,\bar{\mathbf{a} }(\theta),\cdots\bar{\mathbf{a}}(\theta+P\Delta\theta)], \tag{15}\]  
where \(\Delta\theta\) represents the interval of adjacent angles chosen and \(2P+1\) represents the number of selected SVs. Therefore, the dimension of the matrix \(\overline{\mathbf{\Lambda}}\) is \(M\times(2P+1)\). To ensure that the array manifold \(\overline{\mathbf{\Lambda}}\) is an underdetermined matrix, the size of \(P\) should satisfy \(2P+1<M\). Based on the properties of the matrix rank, it can be deduced that  
\[r\left(\overline{\mathbf{\Lambda}}\overline{\mathbf{\Lambda}}^{H}\right)\leq r\left( \overline{\mathbf{\Lambda}}\right)\leq 2P+1<M. \tag{16}\]  
Thus, an eigenvalue decomposition of \(\overline{\mathbf{\Lambda}}\overline{\mathbf{\Lambda}}^{H}\) can be obtained as  
\[\overline{\mathbf{\Lambda}}\overline{\mathbf{\Lambda}}^{H}=\mathbf{U}\mathbf{\Sigma}\mathbf{U}^{ H}=\sum_{\varepsilon=1}^{M}\sigma_{\varepsilon}\mathbf{u}_{\varepsilon}\mathbf{u}_{ \varepsilon}^{H}. \tag{17}\]  
By arranging the eigenvalues in descending order, since \(\overline{\mathbf{\Lambda}}\) is an underdetermined matrix, the eigenvectors corresponding to eigenvalues from \(2P+2\) to \(M\) are orthogonal to \(\overline{\mathbf{\Lambda}}\)  
\[\mathbf{u}=\sum_{\varepsilon=2P+2}^{M}\mathbf{u}_{\varepsilon}. \tag{18}\]  
That is, \(\mathbf{u}\) can be regarded as the vertical error vector \(\mathbf{e}_{\perp}\) orthogonal to the nominal SV \(\bar{\mathbf{a}}\). In practical calculations, although this part is not strictly zero, it can still be treated as zero. Therefore, the error neighborhood of the direction \(\theta\) can be expressed as follows  
\[\overset{-}{\mathbf{E}}=[\bar{\mathbf{a}}(\theta)-Q\eta\mathbf{u},\cdots,\bar{\mathbf{a}}( \theta),\cdots,\bar{\mathbf{a}}(\theta)+Q\eta\mathbf{u}], \tag{19}\]  
where \(\eta\) is the adjustment variable. The smaller its value is, the higher is the estimation accuracy. \(Q\) in Equation (19) is used to limit the search range. Combine \(\overline{\mathbf{\Lambda}}\) with \(\overset{-}{\mathbf{E}}\) to obtain the SV error neighborhood table [29]  
\[\mathbf{T}= \begin{bmatrix}\mathbf{a}^{(-P_{r}-Q)}(\theta)&\cdots&\mathbf{a}^{(0,-Q)}( \theta)&\cdots&\mathbf{a}^{(P_{r}-Q)}(\theta)\\ \vdots&&\vdots&&\vdots\\ \bar{\mathbf{a}}^{(-P,0)}(\theta)&\cdots&\bar{\mathbf{a}}^{(0,0)}(\theta)&\cdots&\bar{ \mathbf{a}}^{(P,0)}(\theta)\\ \vdots&&\vdots&&\vdots\\ \bar{\mathbf{a}}^{(-P,Q)}(\theta)&\cdots&\bar{\mathbf{a}}^{(0,Q)}(\theta)&\cdots&\bar{ \mathbf{a}}^{(P,Q)}(\theta)\\ \end{bmatrix} . \tag{20}\]The \((p\)th, \(q\)th) point in the SV error neighborhood can be expressed as:  
\[\bar{\mathbf{a}}^{(p,d)}(\theta)=\bar{\mathbf{a}}(\theta+p\Delta\theta)+q\eta\mathbf{u}. \tag{21}\]  
To ensure that the SV has the same norm as the actual SV [30], normalize each element in \(\mathbf{T}\) according to \(\sqrt{M}\), that is  
\[ \begin{array}{c}\frac{-(p,d)}{-}\\ \mathbf{a}\qquad(\theta)=\sqrt{M}\frac{\bar{\mathbf{a}}(\theta+p\Delta\theta)+q\eta \mathbf{u}}{\left\|\bar{\mathbf{a}}(\theta+p\Delta\theta)+q\eta\mathbf{u}\right\|_{2}}.\end{array}