## 7 Experimental Results  
In order to prove the validity of the DoA estimation concept using the M\({}^{3}\)PA under investigation, two experiments are conducted. In the first experiment, the receiving M\({}^{3}\)PA is assumed to be positioned in the origin of the Cartesian coordinate system, see Figure 2a. The test signals generated by the LO are transmitted using the quad-ridged horn antenna from multiple known directions \([\mathbf{\phi},\mathbf{\theta}]\) in the far field and received using the considered cuboid M\({}^{3}\)PA. As explained in Section 6, the received signals are fed into the SDR and the DoAs are estimated using the MUSIC algorithm presented in Section 4. In order to demonstrate the impact of transmit power on the accuracy of the estimation, five different transmit power levels \(\mathbf{P}_{\text{T}}=\) [0 dBm, -10 dBm, -20 dBm, -30 dBm, -40 dBm, -50 dBm] are used for each of the tested directions \([\mathbf{\phi},\mathbf{\theta}]\). The performance of the estimation is evaluated by means of the errors \(\varepsilon_{\phi}\) and \(\varepsilon_{\theta}\) for the azimuth and co-elevation angles, respectively. These errors are calculated as the absolute difference between the known angle and the estimated angle, i.e., \(\varepsilon_{\phi}=|\phi-\hat{\phi}|\) and \(\varepsilon_{\theta}=|\theta-\hat{\theta}|\). Furthermore, as a performance metric, the angle \(\psi\) between the position vector pointing to the known direction \([\hat{\phi},\theta]\) and the position vector pointing to the estimated direction \([\hat{\phi},\theta]\) is considered. This angle is a known measure, utilized for determining the orthodrome, which is the shortest distance between two distinctive points on the surface of a sphere [33]. For each conducted measurement, the angle \(\psi\) is calculated according to  
\[\psi=\arccos\big{(}\sin(\theta)\sin(\hat{\theta})\cos(\phi-\hat{\phi})+\cos( \theta)\cos(\hat{\theta})\big{)}. \tag{9}\]  
In the remainder of this paper, the angle \(\psi\) is called the error angle. The results of the experiment are represented in Table 1 for single-shot measurements. As can be seen in the table, with the exception of the lowest transmit power \(P_{\text{T}}=\) -50 dBm, the direction estimation error is less than 5\({}^{\circ}\) for signals arriving from directions with \(\theta\) = 90\({}^{\circ}\). The estimation error for signals arriving from directions with \(\theta\) = 90\({}^{\circ}\) is larger than 5\({}^{\circ}\) in some cases. However, this is considered to be a very good estimation accuracy, since DF errors can reach up to 30\({}^{\circ}\) in ACAS II [5]. By means of standard signal processing techniques like smoothing and tracking, the estimation error can be further reduced. Only for signals arriving from directions with \(\theta\) > 90\({}^{\circ}\) does the estimation accuracy degrade due to the antenna's geometry. Figure 6: M\({}^{3}\)PA antenna and coordinate system in antenna measurement chamber. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{(**a**) Transmit power \(P_{\rm T}=0\) dBm and \(P_{\rm T}=-10\) dBm.} \\ \hline  & \multicolumn{2}{c}{\(P_{\rm T}=0\) dBm} & \multicolumn{2}{c}{\(P_{\rm T}=-10\) dBm} \\ \hline \((\phi,\,\theta)\) & \((\hat{\phi  
As is evident in Figure 2b-d, only a small gain in the vertically polarized components is realizable due to the existence of the feed network and the ground plane. Hence, a performance degradation in DoA estimation is expected for signals arriving from directions underneath the horizon. However, these signals will be received by the M\({}^{3}\)PA mounted on the bottom of the UAV or manned aircraft, see Section 3, allowing an accurate estimation of their DoA. In order to obtain a more thorough overview of the results, the mean (\(\mu_{\text{MU}}\)) and the standard deviation (\(\sigma_{\text{MU}}\)) of the error angle \(\psi\) are depicted in Figure 7. Furthermore, an ML estimator is applied to the experimental data to estimate the DoAs.