## 5 Numerical Results  
To validate the applicability of the cuboid M\({}^{3}\)PA for DF, Monte Carlo simulations using the algorithm presented in Section 4 were conducted before measurements were made. To that end, test signals \(\mathbf{x}(k)\) that impinge on the cuboid M\({}^{3}\)PA from directions \([\mathbf{\phi}_{s},\mathbf{\theta}_{s}]\) were simulated. These directions \([\mathbf{\phi}_{s},\mathbf{\theta}_{s}]\) are the points in the far field, at which the patterns are measured in an anechoic chamber, as presented in [40]. Consequently, the term \(\mathbf{A}(\mathbf{\phi},\mathbf{\theta})\) representing the antenna response in (1) becomes \(\mathbf{A}(\mathbf{\phi}_{s},\mathbf{\theta}_{s})\). They cover the whole 3D space with a 5\({}^{\circ}\) step size in both azimuth and co-elevation, i.e., \(\mathbf{\phi}_{s}=[0^{\circ},5^{\circ},10^{\circ},...,355^{\circ}]\) and \(\mathbf{\theta}_{s}=[0^{\circ},5^{\circ},10^{\circ},...,180^{\circ}]\). This results in \(n_{\phi_{s}}=72\) azimuth angles and \(n_{\theta_{s}}=37\) co-elevation angles. Given that both poles, i.e., \(\theta=0^{\circ}\) and \(\theta=180^{\circ}\), represent the same point for any azimuth angle \(\phi\) in the 3D space, the considered angles yield a total number of 2522 directions \([\mathbf{\phi}_{s},\mathbf{\theta}_{s}]\) covering the whole 3D space. For each simulation run, signals impinging from all of these directions were simulated. However, in order for the estimation error not to be limited by the 5\({}^{\circ}\) measurement step size, \(\mathbf{a}(\phi,\theta)\) in (6) and hence \(P_{\text{MU}}(\phi,\theta)\) need to be calculated at any arbitrary direction \((\phi,\theta)\). Therefore, the wavefield modeling (WM) technique [43; 44; 45] was applied to the measured antenna response \(\mathbf{A}(\mathbf{\phi}_{s},\mathbf{\theta}_{s})\). The application of WM on M\({}^{3}\)PAs was thoroughly investigated in [31]. For a range of SNR values, the root mean square error (RMSE) for each estimated \(\hat{\phi}\) and \(\hat{\theta}\) is calculated according to  
\[\text{RMSE}(\phi_{s})=\sqrt{\frac{1}{N_{\text{MC}}}}\sum_{n_{\text{MC}}=1}^{N _{\text{MC}}}(\hat{\phi}_{n_{\text{MC}}}-\phi_{s})^{2}\hskip 28.452756 pt \text{and} \tag{7}\]  
Figure 3: MUSIC-based 3D DoA estimation adopted to multi-mode multi-port antennas.  
\[\text{RMSE}(\theta_{s})=\sqrt{\frac{1}{N_{\text{MCr}}}\sum_{n_{\text{MCr}}=1}^{N_{ \text{MCr}}}(\hat{\theta}_{n_{\text{MCr}}}-\theta_{s})^{2}}, \tag{8}\]  
respectively. \(N_{\text{MCr}}\) represents the number of simulation runs. As a benchmark for the quality of direction estimation, the Cramer-Rao bound (CRB) is used, as it is known to be the lower bound of any unbiased estimator [46,47]. Figure 4 shows the results of the numerical simulations for \(N_{\text{MCr}}=1000\) runs. In Figure 3(a), for each SNR value, the RMSE values are averaged over all values of \(\phi_{s}\). Similarly, Figure 3(a) depicts the averaged RMSE values over all values of \(\theta_{s}\) for each SNR. It can be observed in Figure 4 that the estimation error performs close to the CRB over a large range of SNR values and converges asymptotically to the CRB at high SNRs. This demonstrates the validity of the cuboid M\({}^{3}\)PA under investigation in combination with the MUSIC algorithm for the purpose of direction finding.