# Simulating Signal Aberration and Ranging Error for Ultrasonic Indoor Positioning
## 3 Simulations Results
In order to observe in detail the extent and shape of the spatial regions within which it is possible to obtain the typical accuracy of the technique based on the cross-correlation, the ranging error on two rectangular grids of points (see Section A and B of the room volume, Figure 1) was evaluated as a function of D. The grid pitch is 5 cm in the \(x\) and \(z\) directions. For each point, the lag of the correlation peak (\(\tau_{\rm MAX}\)) and, from these, the estimates of the distance from the emitter through (3) were obtained. Finally, the competent ground-truth value at each point of the simulation grid was subtracted from the values just obtained, thus generating a grid of estimates of the ranging error. Figure 6 shows the ranging error along a rectangular vertical section (Section of the room volume, Figure 1) of 3-m height and 4-m base passing through the center of the transducer, equal to the vertical section of the typical office room taken as a reference in some positioning works [27; 33; 34], for each aperture D of the set defined above. The grid pitch is 5 cm in the \(x\) and \(y\) directions. Figure 5: Cross-correlations along a semicircular path at distance R = 1 m from the emitting transducer for two different transducer apertures: (**a**) D = 25 mm and (**b**) D = 8.5 mm. For D = 25 mm, it is possible to see that the single unique peak of the cross-correlation for \(\vartheta\) = 0\({}^{\circ}\) is no longer present at the 45\({}^{\circ}\) and 90\({}^{\circ}\) angles, while, for D = 8.5 mm, it is possible to appreciate the single correlation peak at all angles. The cross-correlation values are normalized to their maximum value for each aperture: for D = 8.5 mm, the amplitude relative reduction with respect to the increasing angle is much lower than for D = 25 mm, due to the much wider emission of the smaller aperture. Finally, Figure 7 shows the behavior of the ranging error on a horizontal section of 4 m \(\times\) 4 m at z = 1.5 m, or halfway between the ground and the ceiling (Figure 1b), for all the apertures considered. Figure 6: Computed range error at different transducer apertures D = {25, 20, 16, 13.1, 8.5, 6} mm in a dense grid of points (horizontal and vertical step = 0.05 m) belonging to the vertical 4 m \(\times\) 3 m Section A (see Figure 1); it is possible to appreciate the progressive widening of the cone of the minimum ranging error going from D = 25 mm to D = 6 mm. 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. For D from 25 mm to 13.1 mm, Figure 7a-d, 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 apertures, therefore, it is not possible to cover the whole room for that height, and things go  
Figure 7: Computed range error at different transducer apertures D = {25, 20, 16, 13.1, 8.5, 6} mm in a dense grid of points (horizontal and vertical step = 0.05 m) belonging to the horizontal 4 m \(\times\) 4 m Section B (see Figure 1); it is possible to appreciate the progressive widening of the disk of the minimum ranging error going from D = 25 mm to D = 6 mm. 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. even worse for high heights, which, however, are 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 due to the numerical approximations and the sampling frequency chosen for the simulation.