# Simulating Signal Aberration and Ranging Error for Ultrasonic Indoor Positioning
## 4 Discussion

Table 1: Low error cone angle and range maximum error as a function of the aperture D.  
Immediately outside this low error area, on the other hand, the abrupt appearance of a higher error, often greater than 11 mm, is observed, which is ultimately produced by the bifurcation of the cross-correlation values, as also shown in Figure 4.  
Instead, in Figure 6e,f, no conical region is seen, but a fairly uniform error appears, which is about one order of magnitude lower than that seen in Figure 6a-d. This is the numerical error due to the numerical approximations and the sampling frequency chosen for the simulation, everywhere less than about 3.3 mm. The absence of the low error conical region is due to the fact that, for aperture values D = 8.5 mm and D = 6 mm, the low error region includes all the half-space in front of the transducer. The isolated points of yellow color (relatively large ranging errors) in Figure 6c,d refer to positions where the peak detection error is large due to the similarity in the height of adjacent peaks of the cross-correlation (see also peaks of almost equal height for \(\vartheta\) = 45\({}^{\circ}\) and \(\vartheta\) = 90\({}^{\circ}\) in Figure 5a). The same phenomenon is also observed in Figure 7d. In Figure 7a-d, for D from 25 mm to 13.1 mm, it is possible to clearly recognize increasing low error circular areas, which are the circular sections of the low error cones already seen in Figure 6a-d. Using such aperture values, therefore, it is not possible to cover the whole room for that height, and things go even worse for \(z\) higher than 1.5 m, which, however, is certainly of interest for a three-dimensional internal positioning system. On the contrary, the volume of the room is completely covered by the last two apertures D = 8.5 mm and D = 6 mm, as can also be seen from the vertical sections of Figure 6e,f, which show ranging errors everywhere lower than about 3.09 mm. The ranging error is mainly due to the numerical approximations and the sampling frequency chosen for the simulation. As a significant result, for apertures D from 25 mm down to 13.1 mm, it is possible to clearly recognize the cone-shaped favorable zone. With these apertures, however, it is not possible to cover the whole room. In fact, the room can only be covered up to a height of less than 1 meter from the floor in the most favorable case. This unfortunately prevents, in many cases, from reaching the coverage required by three-dimensional indoor positioning systems. The room, on the other hand, is completely covered by the last two apertures, as can also be seen from the vertical Section A shown in Figure 6, where, in fact, the conical region is no longer recognized, since the low error area is now extended to the whole volume of the room. The simulations presented demonstrate that, using Field II in the design phase, by varying the transducer aperture and the others parameters, it is therefore possible to check whether the acoustic coverage required by a specific application is reached, i.e., whether the region of interest for that application is within the region where the ranging error is sufficiently low or not.