## 3 TDoA Method Experimental Validation
### Experimental Measurements
The experiment was repeated by considering the reduced bandwidth, 4.75-6.25 GHz, where the SBA antenna provides a maximally gain according to Table 1. The DoA estimation error in the reduced bandwidth is reported in the Figure 14c, in comparison with those carried out in the nominal bandwidth. From the data we can recognize that reducing the bandwidth, to that where the antenna performs better, the noise bandwidth reduces and the overall SNR increases thus leading to slightly better DoA accuracy. In the Figure 14c  
Figure 13: Example of measured \(S21\) parameter for entire set of 6 SBA elements, labeled ‘data\(\{n\}\)’, with transmitting antenna positioned at height: 280 cm, range 220 cm. Data acquired across the nominal frequency bandwidth (**left**) at \(\phi=0\) deg, and across the entire azimuthal domain at center frequency band (**right**). we can estimate that the CDF improves of 1.5% at 20 degree, while it remains constant at 10 degrees. In addition to pure DoA estimations, the involvement of wide-band signals enables to calculate range estimations. Applying the method already proposed in [15], measuring the peak delay with respect to the transmission time range estimation is possible by simply calculating  
\[R=c_{0}\cdot\tau_{\mathrm{PE}d}^{2}=c_{0}(\tau_{\mathrm{PE}d}+t_{0}), \tag{56}\]  
Figure 14: Cumulative Distribution Functions (CDFs) of experimental DoA estimations error. where \(d\) is the index of the antenna showing the maximum absolute peak value for its related time observation, \(\tau_{\text{PK},d}\) is the time of arrival of peak, and \(\mathfrak{t}_{0}\) is the signal transmission beginning time. Note that here the knowledge of \(\mathfrak{t}_{0}\) parameter is mandatory, thus range estimations require an absolute synchronization between transmitter and receiver achievable through the VNA or applying specific signal coding schemes [42; 43; 44]. The range resolution is related to timing resolution, and the latter is approximable to inverse of signal bandwidth following signal condition shown in (40), so  
\[\Delta R=c_{0}\cdot\Delta\tau\bar{\text{P}}\bar{\text{K}}_{d}\approx\frac{c_{ 0}}{\text{B}} \tag{57}\]  
where with a bandwidth \(\text{B}=3\:\text{GHz}\) it results \(\Delta R\approx 10\) cm. Furthermore dealing with a one-way transmission there is not any issue related to range ambiguity. Position estimation is calculated as  
\[\overline{\text{x}}=R\cdot\left( \begin{array}{c}\sin\hat{\vartheta}\cos\hat {\phi}\\ \sin\hat{\vartheta}\sin\hat{\phi}\\ \cos\hat{\vartheta}\end{array} \right), \tag{58}\]  
where \((\hat{\vartheta},\hat{\phi})\) is the estimated DoA for each transmitting position respect the SBA reference system. The positioning error is calculated as the euclidean distance between estimation and real position. Figure 15 shows the CDF for both the range estimation error and the positioning error, being the latter a combination of the error due to range and angle estimations. As shown by plots, a correct time offset compensation let the range estimation error to be comparable with range resolution, i.e., \(\approx\)10 cm. The CDFs of final position estimations follow the trends of Figure 14c, upholding the efficacy of the proposed phase reconstruction method. Furthermore, in this case we report the positioning estimation carried out by considering the reduced operation bandwidth of 4.75-6.25 GHz. We see from the curves, that the range estimation exhibits a slight reduction of the accuracy, nominally at 0.1 m the CDF reduces of 3.7%, as result of the analysis bandwidth reduction. Nevertheless, the combined range--DoA positioning error maintains the same slight advantage from reduced noise bandwidth.