## 4 Impact of the Constellation Mesh Area on Localization Accuracy
### Simulation Results
\begin{array}{c}\varphi_{i}=\text{atan}\Big{(}\frac{\varphi-y_{i}}{x- x_{i}}\Big{)}\\ \mathfrak{s}_{i}=\frac{\pi}{2}-\text{atan}\Bigg{(}\frac{z_{i}-z}{\sqrt{\left(x -x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}}}\Bigg{)}\end{array} \right., \tag{10}\]  
while the noise contribution \(\hat{N}\) is described by:  
\[\hat{N}=N_{MN}(\mu,\sigma)=N_{MN}(0,\varepsilon_{RSSIMAX}). \tag{11}\]  
Here it is assumed an RSSI noise characterized by a Gaussian model with null mean and variance \(\varepsilon_{RSSIMAX}\), [16]. Moving on to the expected RSSI data, the measurement noise term embeds both the effective RSSI detection error and second order effects (e.g., fast fading). It is possible to get a reliable characterization of the RSSI variance, removing fast-fading by the rejection techniques described in [12]. Considering an example of the typical recorded RSSI data by the anchor SBA elements 1-7, reported in Figure 10, the worst cases show a maximum RSSI drift of about 5 dB during a single steering vector acquisition. Given this observation, we can set for the simulations the value \(\sigma=\varepsilon_{RSSIMAX}=5\). Note that in a real implementation, the RSSI variance can be reduced by applying a mobile mean along the acquired steering vectors. The simulation results across the entire area for the three cases of study are reported in Figure 11, in terms of Cumulative Error Distribution Function (CDF) for the area within the single rectangular mesh and across the entire room, respectively. In this set of simulations, we generated 5 RSSI randomly distributed according the noise model described above at each of the 299 points distributed as in Figure 9a. From the figure we can observe that, due to the reduced dimension, the tiny mesh is capable to estimate with comparably smaller error the position within the area defined by the mesh itself, but it is not suitable for the outer part, as it determines high errors at the border of the room; this is already predicted by the investigation reported in Section 3.2 and specifically Figure 7a. For the other mesh sizes, the data of Figure 11 show a convergence between the outer and inner mesh results, thus confirming that the large mesh provides almost the same results within and across the entire area, as the two results almost the same. For a better comparison of the data, the results are summarized in Table 2. They clearly show that a trade-off involved in the anchor distribution exists: a wider mesh increases the sub-metrical coverage on the full area, while smaller configurations increase the precision inside the mesh yet degrade significantly the overall area performance. With this conclusion we have an objective way to evaluate the dimension of the mesh according to the accuracy needed and the dimensions of the room. The next Section deals with the experimental verification of these results. Figure 11: Simulated localization error CDFs, with the three anchor networks. Data acquired at the 299 points in the room of Figure 9a, assuming the mobile tag pointing upward, 5 averaged RSSI data for point and each antenna element, mobile tag height = 1.1 m.  
Figure 10: Typical Received Signal Strength Indicator (RSSI) acquisition trace: case C, anchor installed at room up-right corner (5.75, 5.00) m.