## 3 TDoA Method Experimental Validation
### Experimental Measurements  
In this section, we report the results of a massive campaign of measurements carried out putting the target emulated by the transmitting antenna on different heights and different distances; Figure 12 reports the entire set of target positions, with respect to the SBA anchor. An entire set of time observation was taken, for each target position and for a convenient number of SBA rotation angles, spanning the azimuthal domain \(\varphi\) with a step of 1 deg. Figure 10: Block diagram of the experiment setup. The transmitting antenna emulates the target. In Figure 13 are shown the typical measured calibrated data composing a steering vector, acquired by the instrumentation in terms of the magnitude of the transmission parameter S21. By observing the plots we can easily observe from the data associated to the SBA element 2, that the effective antenna bandwidth is between 4.75 GHz to 6.25 GHz, thus justifying its use as reduced bandwidth defined herein above. Note that in this example, antenna 2 is the one aligned with the transmitting antenna, while the others result directed elsewhere according to the SBA geometry shown in Figure 6; still according with the SBA geometry antenna 1 points upward, that is \(\theta=0\) degree and thus provides a low trace in the data plots. The second set of plots shown in Figure 13, provides instead the steering vector data at center frequency, as a function of the azimuthal angle. As expected, in this set of data the shapes of the curve resemble the one of the radiation diagram shown in Figure 6.  
Figure 11: Picture of the experimental setup with the target positioned on a pole and the SBA installed on a rotating platform. Figure 12: Schematic diagram of the SBA and transmitting antennas experimental relative positions. For each rotation the signal DoA and the related error estimations were estimated following the method introduced in Section 2.1, adopting the MUSIC ML estimator in its scalar implementation [21], dealing with phase-less steering vectors, and in its standard implementation [23; 24], with phase-reconstructed data. To assess the validity of the phase reconstruction process, the estimations were also carried out using steering vectors completed with pure phase information related to the center frequency \(f_{0}\). The results about all these cases are compared in the Figure 14 where are reported the CDFs for the obtained DoA estimation errors. In particular, in Figure 14a,b the distribution functions are calculated with respect to each angle subject to estimation, while Figure 14c shows the combined estimation error considering the euclidean distance between the estimated DoA and the real one as  
\[\varepsilon_{\text{DoA}}=\sqrt{\left(\hat{\vartheta}-\vartheta\right)^{2}+ \left(\hat{\varphi}-\varphi\right)^{2}} \tag{55}\]  
with \(\hat{\vartheta},\hat{\varphi}\) being the estimated DoA values and \(\vartheta,\varphi\) the actual ones. By observing Figure 14 it is clear how much the proposed phase reconstruction method is able to recover a coherent phase information. This is especially noticeable when looking at the accuracy improvement on azimuthal domain \(\varphi\) (Figure 14b), where the introduction of center frequency phase information actually worsened overall results. This is coherent with results in Figure 8 where severe phase wrapping phenomena were expected along the azimuthal domain, especially for lower azimuth angles. In the azimuthal direction the implementation of phase recovery did not lead to accuracy improvements but it prevented the disruption of steering vectors information. Conversely on elevation direction the re-introduction of phase allowed to overcome the lack of antenna elements due to SBA array geometry, giving an huge improvement on estimation accuracy, as shown in Figure 14a.