## 3 TDoA Method Experimental Validation
### Anchor Switched Beam Antenna Design  
The SBA-based anchor device is capable to enable the Space Division Multiplexing Access (SDMA) approach, whose structure is shown in Figure 5. The structure is engineered to be well suited for DoA estimations for phase-less observations, ensuring a complete coverage of hemispherical domain surrounding to the device, according to [37; 38]. The six antennas that compose the anchor system are designed to operate in the nominal bandwidth 4-7 GHz. The individual antenna element that composes the SBA, is a 2 \(\times\) 2 array of patch antennas, whose design was proposed in [39; 40]. They are and arranged to cover a semi-spherical range, following the design technique proposed in [41]. Each antenna element works in circular polarization (CP), increasing the robustness of the radio link and thus contrasting the multi-path impairment, regardless of the relative orientation of the target. In Figure 6, the radiation pattern of the six patch antennas implemented in the SBA at the center frequency of 5.5 GHz is reported; for the purpose of the localization purpose, these can be considered to be identical. The Half-Power Beamwidth is reasonably maintained constant across the entire working frequency band. A detailed antenna performance is provided in Table 1, where a distinction between nominal and operative frequency bandwidth is made, with the latter being the frequency range where the realized antenna gain is 8 \(\pm\) 2 dB. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Nominal BW** & **Operative BW** & **Gain** & **HPBW** & **Cross-Pol.** \\
**(GHz)** & **(GHz)** & **(dBi)** & **(deg)** & **(dB)** \\ \hline
4–7 & 4.75–6.25 & \(8\pm 2\) & \(40\pm 10\) & \textgreater{}15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance summary of the circularly polarized antenna element composing the SBA; experimental data. Figure 5: The assembled SBA of the anchor device adopted in the experiments **(left)**, detail of the single array element **(right)**. The implemented positioning procedures take advantage of the capability to steer the radiation pattern to different direction and thus areas in the space domain. The beam control is achieved through the sequential activation of the six antenna elements. Indeed, the SBA rotation is performed by means of a microwave six-pole single-through digitally controlled by a microcontroller, similarly to the system definition of Figure 4.  
To provide the needed directivity throughout the frequency band, the antenna element itself is based on a 2 \(\times\) 2 array, so increasing the overall element aperture, it reaches a dimension of 77 mm \(\times\) 77 mm. Antenna elements are arranged side by side and are tilted 34 deg with respect to the vertical direction, and each one covers an angular aperture of 72 deg on \(\varphi\) angle. Table 2 lists the center points of each antenna element, considering the reference system as in Figure 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Antenna** & \(\mathbf{\phi_{0}}\) (deg) & \(\mathbf{\varphi_{0}}\) (deg) & **x (cm)** & **y (cm)** & **z (cm)** \\ \hline \#1 & 0 & - & 0.00 & 0.00 & 0.00 \\ \#2 & 34 & 0 & 4.95 & 0.00 & \(-\)3.20 \\ \#3 & 34 & 72 & 1.53 & \(-\)4.70 & \(-\)3.20 \\ \#4 & 34 & 144 & \(-\)4.00 & \(-\)2.91 & \(-\)3.20 \\ \#5 & 34 & 216 & \(-\)4.00 & 2.91 & \(-\)3.20 \\ \#6 & 34 & 288 & 1.53 & 4.70 & \(-\)3.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Facing direction and center points of SBA antenna elements. Figure 6: Radiation pattern of each SBA element. Figure 7: Pictorial representation of the observation model for the experiment. Observing Table 2 and Figure 7, the directions leading to the worse phase impairments should be the ones with \(\vartheta=90\) deg. In this regard, Figure 8 (top) shows the ideal expected phase differences respect to the reference antenna 1 for center frequency \(f_{0}=5.5\) GHz, while on the bottom the same are represented considering the phase wrapping phenomena.