## 3 Losnus
### Localization  
The _LOSNUS_ localization algorithm is based on TDoA and requires the calibrated transmitter positions, the time differences and the speed of sound as an input. For two transmitters \(\mathrm{T_{i}}\) and \(\mathrm{T_{j}}\), a receiver R and ToAs \(\mathrm{t_{i}}\) and \(\mathrm{t_{j}}\) the basic TDoA equation is given in Equation (3) where c is the speed of sound.  
\[\left(\mathrm{t_{i}}-\mathrm{t_{j}}\right)\mathrm{c}=\left\|\mathrm{T_{i}}- \mathrm{R}\right\|-\left\|\mathrm{T_{j}}-\mathrm{R}\right\| \tag{3}\]  
Having three such equations corresponding to four ToA measurements an analytical solution exists. An important parameter for a localization system is the dilution of precision (DOP) [27], which relates ToA uncertainty to locating uncertainty. For example if the DOP is 20, the standard deviation of ToA measurements is 700 ns, and the speed of sound is 343 m/s then the standard deviation of the position estimate is \(20\cdot 700\) ns \(\cdot 343\) m/s = 4.8 mm. The DOP is dependent on the placement of the beacons and the position of the receiver.  
As in a typical _LOSNUS_ system, more than four transmitters are available the redundancy in the data can be used for increasing the robustness of the locating procedure. The algorithm used by _LOSNUS_ is called _Grouping_. Having six (or more) transmitters and requiring four ToAs for localization fifteen different combinations are possible. If all ToA combinations are valid then all calculated positions should be approximately the same. If one of them is incorrect, only five over four possible combinations are approximately the same, and the remaining ten positions are different. Figure 7a shows that all fifteen solutions are close if no single ToA is an outlier. Figure 7b shows the solutions when a single ToA was changed by 1 cm. It can be seen that the positions calculated by the correct combinations, _i.e._, four over five, are still correct while all others a different.  
In this situation, neither the non-linear least squares, nor using the mean or median of all coordinates can deliver the correct result. The algorithm creates distinctive groups whereas the elements are determined by a similarity scale based on calculating the 2-norm of all positions differences. The group with the largest cardinality is selected. The final position is either calculated by taking the mean of all positions within this group, by selecting the element with the lowest dilution of precision (DOP) for the transmitter/receiver combination or by applying an NLS algorithm to the  
Figure 6: **(a)** Transmitter identification for n coding frequencies. Filter output values are only evaluated at discrete times as the starting time of the frame is known. **(b)** Example transmitter decoding on binary input data using \(f_{l}=40\) kHz \(+4.5\) kHz \(\cdot i,0\leq i\leq 5\). Identified transmitter is (40 kHz, 53.5 kHz). Red vertical lines mark sampling points.  
correct ToAs. A comparison on the performance of the different position calculation methods as well as a comparison to other robust position estimation algorithms like Least-Median of Squared (LMedS) and Robust Multilateration (RMult) is shown in [28]. Among the grouping algorithms, NLS performs best although the statistical differences are in the mm range. An important, but unproven aspect is that this algorithm requires that combinations including the wrong ToA do not form groups larger than the correct group. Simulation and practical results have shown us that this is indeed the case.