# Round-Trip Time Ranging to Wi-Fi Access Points Beats GNSS Localization
## 8 Wi-Fi Installations Used for AP Localization  
We determined the positions of APs in three installations using the \"inverse\" Bayesian method described above. In each case, the APs where inside buildings in condo developments or hotels (see Figure 3). We report here specifically on one large complex (\(210\times 120\) m) with 108 APs concentrated in three main buildings (with another 5 APs in two auxiliary buildings). The APs in this case were from Ruckus Wireless according to their Organizational Unique Identifier (OUI) (94:B3:4F). These APs do not advertise support for IEEE 802.11 mc in the Wi-Fi beacon, but do respond to two-sided RTT ranging requests. The RTT measurements in this case were determined to have an offset or bias of 2.8 m (to be subtracted from the RTT measurement) [53]--and a standard deviation of about 3 m (note that the standard deviation reported in the RTT measurement itself is typically smaller because it is based on multiple measurements in rapid succession under near identical conditions and from basically the same position). We experimented with several smartphones including Pixel 5, Pixel 6 Pro and Pixel 7a. The data reported here was obtained using a Google Pixel 5. The ground truth is based on coordinates from Google Maps of landmarks--such as intersections of paths--along with interpolation over short distances based on step counting and headings obtained using the smartphone's acceleration and magnetic sensors. Coordinates determined from Google Maps are claimed to be accurate to about 1 m in some regions of the developed world (with a world-wide average of about 4.38 m) [54]. Because many RTT measurements were available for locating each AP, the accuracy of the derived AP position is better than that of the raw RTT measurement. If the probability distribution was Gaussian, and the directional geometry ideal, we would expect  
\[\sigma_{\text{AP}}=\frac{1}{\sqrt{N/D}}\sigma_{\text{RTT}} \tag{9}\]  
where \(\sigma_{\text{RTT}}\) is the standard deviation of the RTT measurements, \(\sigma_{\text{AP}}\) is the standard deviation of the computed position of the AP, \(N\) is the number of different UE positions, and \(D=2\) in the 2D case, and \(D=3\) in the 3D case. Based on measurements on three different days, the error in AP position was estimated to be around 1-2 m (this uncertainty in positions of the APs naturally induces some small error later when ranging to these APs in order to determine the position of the UE). Figure 2 shows four sample Bayesian grids for locating APs in the first building. In each subfigure, the red cross marks the mode of the probability distribution, while the outlines of the buildings are shown in blue. The grid cells here are \(1\times 1\) m. To accommodate the large dynamic range of the probabilities, the grey level in the figure is made proportional to the square root of the logarithm of the probability. RTT measurements often have outliers far from the correct distance (an example may be seen in the lower left of the bottom right image). Some of these are easy to reject (e.g., very large RTT standard deviations or negative RTT distances) but many masquerade as legitimate measurements (with reasonable looking standard deviations) even when they are dozens of meters off the correct value. These outliers do not present much of problem for the BGU method, but can potentially be problematic for LSQ-based multilateration because of the large contributions to the least-squares loss function (an effect that can be ameliorated to some extent by weighting the contributions with the inverse of the standard  
Figure 2: Inverse Bayesian grid distributions for four different Wi-Fi access points. The red crosses mark the peak values in each probability distribution. Outlines of the buildings containing the APs are shown in blue. deviation of the RTT measurement, and by dropping the worst of a group of related range measurements). The above figures are based on data collected during a single \"walk-through\" of the property.