# Adaptive Algorithm for Fast 3D Characterization of Magnetic Sensors
## 3 Results
### 1D Estimation, Control Schemes and Step Size Behavior  
First, we examined the algorithm without any influence of the orientation control--simply exciting one field direction in line with the assumed sensitive axis of the sensor under characterization. For that, we set the number of azimuth and elevation steps, \(N_{\theta}\) and \(N_{\theta}\), respectively, to 1 and the time thresholds to \(N_{\min}=N_{\max}=30\) s as shown in Table 3. As we wanted to investigate different control schemes, we performed four different runs of the experiment: two runs with fixed step sizes of \(\hat{\mu}=1\) and \(\hat{\mu}=0.1\), respectively, one run with a DT controlled step size, and one run with a CEA controlled step size. Our goal was a comparison of the resulting impulse response estimates and measuring the adaptation behavior with regards to the system distance and the step size. The resulting normalized impulse response estimations are visualized in Figure 6. The upper left depicts the impulse response estimated with a large fixed step size, which clearly leads to a noisy estimate. The smaller fixed step size (upper right) reduces this erratic nature in the long tail somewhat; it is, however, still less smooth than the estimation results from the adaptively controlled runs. Between these, the DT (lower left) and CEA (lower right) control do not show a visually distinguishable behavior and show only minuscule differences in the exact timing of the impulse peak. Next, we look at the smoothed system distance estimates calculated by our algorithm (cf. Figure 7). The DT-based system distance estimation \(\widehat{\Gamma}_{\text{DT}}(n)\) is performed for all four experiments, as it is only based on the current impulse response approximation (cf. (25)). The CEA system distance \(\widehat{\Gamma}_{\text{CEA}}(n)\), on the other hand, is only derived when this control scheme is used. The experiment with a fixed step size of 1 (green) shows a steep convergence at the start of adaptation but stabilizes around a relatively large system distance of \(-60\) dB. A smaller, fixed step size of 0.1 (purple) can achieve a lower remaining system distance  
Figure 6: Comparison of the normalized impulse response estimation results after complete 1D adaptation runs. The more polished and smooth nature of the estimates resulting from adaptive step size control schemes is clearly observable, while a clear distinction is not visible between the DT and CEA controlled results. after adaptation but it takes significantly longer for the first adaptation, observable from the much less steep convergence in the first second. Both adaptive control approaches achieve a similar convergence speed with the (fastest possible) fixed step size of 1; they are also able to surpass both fixed step size runs with their ongoing system distance estimations. The DT-controlled adaptation (blue) shows a higher fluctuation in its system distance estimate in comparison with the CEA control (red). The latter shows a deviation between the DT and CEA estimation for the system distance with the CEA estimate (dotted) achieving smaller modeled distances based on the initial behavior. Overall, both adaptive approaches led to significant improvement, with the model based on CEA showing potential in achieving lower remaining errors. The difference in the resulting step size behavior is visualized in Figure 8 with logarithmic scaling for better differentiation in small step size ranges. Both experiments with fixed step size obviously lead to constant behavior over the whole 30 s adaptation time frame. The adaptively controlled step sizes qualitatively resemble the different system distance estimates, which is expected due to their proportional calculation (cf. (30) and (31)). With the smaller step size at later adaptation times and an ever-decreasing modeled system distance estimate for CEA control, however, a smaller remaining error between system model and estimated physical sensor system is theoretically achievable [24]. This is of interest for the use case presented in this work.