# Constant-Beamwidth LCMV Beamformer with Rectangular Arrays
## 4 Constant-Beamwidth LCMV Beamformer with Rectangular Arrays
### CB-LCMV Beamformer with RCKP  
Here, we propose the use of the RCKP method as derived previously to construct a CB-LCMV beamformer with an increase in white noise gain and directivity measures. For this, full array \(S\) is separated into four subarrays: \(S_{1}\), \(S_{2,\times}^{\prime}\), \(S_{2,\times}^{\prime\prime}\), and \(S_{2,\times}\). Each subarray is used to design one of the desired beamformers presented in Sections 3.1-3.3.  
\(S_{1}\) is used to design the LCMV beamformer, following the steps detailed in Section 3.1. Since the LCMV beamformer does not require the array to be linear, we use \(S_{1}\) as a rectangular array of size \(M_{1,\times}\times M_{1,y}\). As explained previously, this choice has the advantage of the rectangular array's lesser symmetry compared to a linear array. We have \(M_{1}=M_{1,\times}M_{1,y}\) virtual sensors and, at most, \(M_{1}-1\) nulls to be placed. If \(N=0\), this same array is used for the MVDR beamformer instead.  
The subarray \(S_{2,y}\) is used to construct the CB beamformer. Given that the CB is achieved through the window technique (as per Section 3.2), this subarray must be a ULA, and the desired source direction should be on its broadside direction. Here, this implies that \(\theta_{\mathrm{d}}=0^{\mathrm{o}}\).  
The SD and DS beamformers are built from the \(S_{2,\times}^{\prime}\) and \(S_{2,\times}^{\prime\prime}\) subarrays, respectively, based on Section 3.3. We assume that they are constructed from linear arrays, but this condition is flexible and can be changed in other implementations.  
Once all the subarray beamformers are designed, we use the RCKP method to combine these beamformers into a full-array beamformer. Thus, we construct a beamformer with many desired features (null placement \(+\) constant beamwidth \(+\) white noise gain \(+\) directivity factor gain) that exploit the symmetry of the rectangular array, which implies more spatial information and more performance. Algebraically, the full-array beamformer is given by  
\[\mathbf{h}=\mathcal{V}\Big{(}\mathcal{V}^{-1}\big{(}\mathbf{h}_{\mathrm{LCMV }};\,M_{1,y}\big{)}\mathbin{\raisebox{-1.29 pt}{\scalebox{0.8}{$\circ$}}} \mathcal{V}^{-1}(\mathbf{h}_{\mathrm{SD}};\,1)\mathbin{\raisebox{-1.29 pt}{ \scalebox{0.8}{$\circ$}}}\mathcal{V}^{-1}(\mathbf{h}_{\mathrm{DS}};\,1) \mathbin{\raisebox{-1.29 pt}{\scalebox{0.8}{$\circ$}}}\mathcal{V}^{-1} \big{(}\mathbf{h}_{\mathrm{CB}};\,M_{2,y}\big{)}\Big{)}. \tag{19}\]An implementation of the proposed CB-LCMV beamformer is shown in Algorithm A2 in Appendix B. The procedure calculates beamformer \(\mathbf{h}\) for a single frequency.  
For the CB beamformer to be effective, its condition must be valid for all beamformers. That is, \(|\mathcal{B}(\mathbf{h},\mathbf{d})|>0\) if \(|\theta-\theta_{\mathrm{d}}|<\theta_{\mathrm{B}}/2\), where \(\mathbf{h}\in[\mathbf{h}_{\mathrm{LCMV}},\ \mathbf{h}_{\mathrm{CB}},\ \mathbf{h}_{ \mathrm{SD}},\ \mathbf{h}_{\mathrm{DS}}]\). This is true by definition for the CB beamformer, and by setting \(N=M_{1}-1\) for the LCMV, no nulls are free to end up inside the main beam. For the SD and DS beamformers, we assume that their FNBWs are sufficient to satisfy the condition.