# Hybrid TOA/RSS Range-Based Localization with Self-Calibration in Asynchronous Wireless Networks
## 4 Performance Assessment
Remarkably, the proposed hybrid TOA/RSS approach has only slightly higher complexity compared to NR, while it is significantly less complex than the optimal joint ML. In Figure 4, we report the MSEs of all the considered algorithms as function of the measurement error \(\sigma\) for \(\alpha=2\) and \(N=8\). The solid curves represent the ideal estimation performance obtained by assuming that the TOA/RSS model parameters \(\theta\) and \(\beta\) are perfectly known a priori, while the dashed curves show the corresponding versions with the unknown parameters replaced by their estimates obtained from the preliminary calibration phase presented in Section 3.1. The obtained results clearly show that the proposed hybrid estimator outperforms the state-of-the-art competitors for all the considered range of \(\sigma\) and, remarkably, can achieve almost the same accuracy of the joint  
Figure 3: Averaged runtimes of the considered localization approaches. ML benchmark, but at a significantly reduced computational cost. To better highlight the obtained results, in Figure 5 we depict the considered \(20\) m \(\times\)\(20\) m network with \(N=8\) anchors for a given position of the target, together with a graphical representation of the estimated circular error probable (CEP) (the CEP is usually defined as the radius of the circle, centered in the average estimated position, which is expected to include 50% of the estimated positions) resulting from the application of the proposed hybrid TOA/RSS approach for the two specific cases of \(\sigma=3\) and \(\sigma=6\). Figure 4: Mean square error (MSEs) of the considered estimators as function of the measurement error \(\sigma\) for \(\alpha=2\) and \(N=8\): ideal vs. estimated parameters. Figure 5: Graphical representation of the considered network for \(N=8\) and for a fixed target position. Shaded circles represent the circular error probable (CEP) of the proposed two-step localization approach for \(\sigma\in\{3,6\}\). As expected, the actual performance in the more realistic case of self-calibrated algorithms are much worse compared to that obtained under ideal knowledge of the models parameters, especially for high values of \(\sigma\). To corroborate such results, we investigate the performance of the self-calibration procedure adopted to estimate the unknown TOA/RSS model parameters. In Figure 6, we depict the estimation errors obtained during the calibration phase for the two values of the measurement error \(\sigma\in\{3,6\}\), which are representative of a \"low-noise\" and \"high-noise\" scenarios, respectively. The obtained performance is in agreement with the findings in [43] (which however addressed the specific problem of RSS-only ranging): in fact, higher values of \(\sigma\) amplify the errors in the model parameters estimation and, in turn, introduce severe non-linear effects that are detrimental to the final positioning task. From this analysis, it clearly emerges that in practical WSN deployments--where the true values of the environmental parameters are unknown--the ultimate localization accuracy is also affected by the quality of the parameter estimation task. To further investigate how a realistic calibration affects the localization process, in Figure 7 we report the algorithms performance as a function of the number of anchors \(N\), now for the case \(\alpha=4\) and \(\sigma=4\). As it can be noticed, for a sufficient number of anchors \(N\), almost all the algorithms exhibit a satisfactory localization accuracy, in spite of the presence of a quite high noise level \(\sigma\). Interestingly, for \(N\geq 12\) the dashed-curves (estimated parameters) closely attain the ideal ones (known parameters). Again, the proposed hybrid TOA/RSS position estimator offers the best performance, with an MSE below \(10\) m for almost all the considered span of \(N\).