# Hybrid TOA/RSS Range-Based Localization with Self-Calibration in Asynchronous Wireless Networks
## 4 Performance Assessment
These results are reasonable since for higher values of \(N\) the increasing amount of information collected within the network enables a much more accurate estimation of the unknown models parameters (through the self-calibration procedure proposed in Section 3.1); conversely, as \(N\) decreases, the algorithms performance experience a rapid degradation until achieving unacceptable results for a number of anchors \(N<6\). In addition to a higher amount of information in the network (anchor-to-blind measurements scale linearly with \(N\), anchor-to-anchor measurements scale quadratically with \(N\)), increasing the number of anchors \(N\) also leads to a better coverage of the considered area, resulting in turn into more favorable geometric conditions for the estimation of the unknown blind position. To quantify the impact of a better distribution of the anchors in the space, in Figure 8 we evaluate the horizontal dilution of precision (HDOP) for each different number of anchors \(N\). Figure 6: Estimation errors obtained during the calibration phase for \(\sigma\in\{3,6\}\) with \(\alpha=2\) and \(N=8\). As it can be seen, increasing the density of anchors from 8 to 14 yields an accuracy improvement in terms of HDOP of about 40%. As concerns the poor performance of the NR-based methods, it can be attributed to the fact that the hybrid TOA/RSS ranging problem is solved via a suboptimal approach based on Newton-Raphson iterations. More precisely, from the closed-form expressions of the estimators provided in [53], it can be noticed that the Newton-Raphson procedure is initialized with a coarse range estimate obtained using the sole TOA measurement; hence, it is reasonable to expect that the performance of both NR1 and NR2 approaches heavily depend on the accuracy of such data. Such theoretical considerations are in agreement with the performance of the NR1 method, whose performance curves are superimposed to the ones provided by the TOA-only estimator, hence are not visible in the figures, meaning that in the considered scenario one iteration of the Newton-Raphson method is indeed not sufficient to find a good approximate solution to the joint TOA/RSS ML problem. To complete the analysis, in Figure 9 we show the empirical cumulative distribution functions (ECDFs) of the algorithms localization errors for the case \(\alpha=2\), \(\sigma=4\) and assuming a number of anchors \(N=8\). As it can be noticed, a perfect knowledge of the models parameters would lead to positioning errors that are always below 7 m for all the hybrid estimators; conversely, in case of a realistic self-calibration, all the algorithms suffer evident impairments in the achieved performance, with errors that can even reach 20 m (tails of the ECDFs). Remarkably, the proposed hybrid estimator closely follows the joint ML benchmark, with a position error that is lower than 5 m in 98% of the cases. As a confirmation, in Figure 10, we report the ECDFs of the localization errors for the challenging case of \(\sigma=6\). Figure 8: Horizontal dilution of precision (HDOP) as function of the number of anchors \(N\). Figure 7: MSEs of the considered algorithms as function of the number of anchors \(N\) for \(\alpha=4\) and \(\sigma=4\). **(left)** Ideal TOA/RSS parameters and **(right)** estimated TOA/RSS parameters. As it could be expected, the obtained performance are generally worse than the ones in the previous case, especially for the RSS-only and TOA-only estimators which cannot mitigate the negative effects of an increased noise. Nevertheless, the proposed hybrid estimator still outperforms all the competitors, achieving localization performance that remains very close to the one provided by the joint ML. Notice also that for some percentile the joint ML has inferior performance; this is partially due to the fact that it is obtained by a numerical optimization of a non-linear function, which may not always terminate at the absolute maximum.