# Constant-Beamwidth LCMV Beamformer with Rectangular Arrays
## 5 Experimental Results  
In this section, we present the simulations performed to verify the proposed method's implementation and to compare its performance to that of existing methods. The code used for these simulations is available at [[https://github.com/VCurtarelli/py-cb-lcmv-rect](https://github.com/VCurtarelli/py-cb-lcmv-rect)]([https://github.com/VCurtarelli/py-cb-lcmv-rect](https://github.com/VCurtarelli/py-cb-lcmv-rect)) (accessed on 27 July 2023). We test different combinations of the number of sensors for each subarray, as established in Table 1. The table presents the dimensions for each subarray, for the full array of sensors, and the virtual number of sensors being employed. [A, B] are obtained through the RCKP, [C, D] through the LCKP, and [E, F] through the KP + LCKP, with the KP and LCKP methods being as defined in Section 3.4. [A, C, E] utilize the SD and DS beamformers, while [B, D, F] \"disable\" them in favor of the CB beamformer. For [E, F], the LCMV beamformer is more spaced out. In all situations, \(S\) has a total of \(M=36\) sensors. The sensors are assumed to be ideal omnidirectional sensors with a plain frequency response over the spectrum. The intersensor distances are \(\delta_{\mathrm{x}}=0.5\) cm and \(\delta_{\mathrm{y}}=3.0\) cm. We assume that \(\theta_{\mathrm{d}}=0^{\mathrm{o}}\) and \(\theta_{\mathrm{B}}=40^{\mathrm{o}}\). Since the LCMV has four sensors, we use three interfering sources (for all situations), with directions \(\theta_{i}\in[-90^{\mathrm{o}},60^{\mathrm{o}},130^{\mathrm{o}}]\), each with \(\phi_{\mathbf{v}_{i}}=1\), and we also assume \(\mathbf{v}^{\prime}\) to be uncorrelated Gaussian white noise with unit variance (that is, \(\mathbf{\Phi}_{\mathbf{v}^{\prime}}\) is the identity matrix). The variance of the desired signal is \(\phi_{\mathrm{X}}=\mathsf{E}\Big{[}|X|^{2}\Big{]}=5\). The simulations are performed for the range \(f\in[4,8]\) kHz. This range is chosen to satisfy the conditions from ([22], Equation (29)) for condition [A]. Figures 1 and 2a-c show the simulation results for \(\mathcal{B}\), measured \(\theta_{\mathrm{B}}\), \(\mathcal{W}\), and \(\mathcal{D}\). \(\theta_{\mathrm{B}}\) was measured as the first angle in which \(\mathcal{B}\big{(}\mathbf{h},\mathbf{d}_{\theta_{\mathrm{B}}/2}\big{)}\leq 0.05\). All methods manage to achieve the nulls in the desired directions from the LCMV. Observing Figure 2a, all but [E] are capable of maintaining a reasonable beamwidth across all frequencies, with [E] not being able to maintain the beamwidth because of too few sensors in the CB beamformer. The \"naive\" combination of KP and LCKP [E, F] leads to worse performance for all metrics; thus, their results are not further compared. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cond.** & **LCMV** & **SD** & **DS** & **CB** & **FA** & \(\boldsymbol{M^{\prime}}\) \\ \hline A & \(2\times 2\) & \(2\times 1\) & \(2\times 1\) & \(1\times 8\) & \(4\times 9\) & 128 \\ B & \(2\times 2\) & \(1\times 1\) & \(1\times 1\) & \(1\times 17\) & \(2\times 18\) & 68 \\ C & \(1\times 4\) & \(1\times 2\) & \(1\times 2\) & \(1\times 31\) & \(1\times 36\) & 496 \\ D & \(1\times 4\) & \(1\times 1\) & \(1\times 1\) & \(1\times 33\) & \(1\times 36\) & 132 \\ E & \(2\times 2\) & \(2\times 1\) & \(2\times 1\) & \(1\times 3\) & \(6\times 6\) & 48 \\ F & \(2\times 2\) & \(1\times 1\) & \(1\times 1\) & \(1\times 9\) & \(2\times 18\) & 36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of sensors for each subarray, dimension of the full array (FA), and size of the virtual array in each simulation. Both methods (the RCKP and LCKP) lead to very akin results for \(\theta_{\rm B}\), even though the LCKP simulations [C, D] have many more sensors in the CB beamformer than the RCKP ones [A, B]. This indicates that increasing the CB array size does not result in a better performance in either FNBW or directivity. However, in Figure 1, it can be seen that it generates a more focused beam. Both the RCKP and LCKP have similar WNG results of around 10 dB.