# Hybrid TOA/RSS Range-Based Localization with Self-Calibration in Asynchronous Wireless Networks
## 4 Performance Assessment  
In this section, we evaluate the performance of the proposed algorithm by means of Monte Carlo simulations. We consider, as a reference scenario, a WSN deployed over an area of \(20\ \times\ 20\) m; in particular, the \(N\) anchors are distributed in the space so as to provide a good coverage in terms of connectivity, while the blind node is randomly located within the considered area and its position varies in each simulation trial. We consider for the RSS measurements the parameter \(P_{0}=-30\) dB, while for the path-loss exponent the two cases \(\alpha=2\) and \(\alpha=4\) are evaluated. As to the synchronization offsets \(t_{i}\)s and \(t_{b}\), they are generated according to a uniform distribution in the interval \([b_{\text{min}}/c,\,b_{\text{max}}/c]\) where \(b_{\text{min}}=0.5\) m and \(b_{\text{max}}=3\) m represent the equivalent offsets expressed in terms of distances, respectively. In this analysis, we assume without loss of generality that both RSS and TOA measurements have the same accuracy, i.e., \(\sigma_{r}=c\sigma_{t}=\sigma\). Thus, all measurements have the same variance \(\sigma^{2}\), i.e., neither measurement has better \"quality\" than the other. This is not always the case, of course, but this choice has the advantage of reducing the parameter space for the simulations to the most interesting case: in fact, in [44] it is shown that if TOA and RSS variances are (significantly) different, then one of the two sources of information always dominates; hence, the hybrid estimator boils down to the TOA-only or RSS-only estimator, accordingly. For the assessment we consider values of \(\sigma\) ranging from 3 to 6, which are representative of several operational scenarios of WSNs, from outdoor open spaces to propagation environments full of obstructions (e.g., people, furniture,...). The proposed hybrid estimator is computed using \(A=-1\) and \(B=37\) as in [44], and it is compared against the most natural competitor algorithms based on Newton-Raphson iterations, given in [53] and in the following labeled as NR1 and NR2. The basic difference among the NR1 and NR2 methods consists in the number of Newton-Raphson iterations performed to compute the distance estimate. More precisely, the NR1 algorithm uses a single iteration, while the NR2 algorithm employs two iterations. Even though such algorithms were originally designed for range estimation, the ultimate blind node position can be easily retrieved by resorting to the ILS procedure in Equation (39) based on the \(N\) distance estimates obtained during the ranging phase. For comparison purposes, we also report the performance of the (range-free) joint ML position estimator in Equation (8), obtained via a two-dimensional grid search. We, however, remark that the latter cannot be taken as a direct competitor since it is not available in closed-form; hence, it will be considered only as a benchmark. Moreover, we also compare the performance of the hybrid estimators against the more traditional algorithms based only on RSS or TOA information, denoted by \"RSS-only\" and \"TOA-only\", respectively. To assess the ultimate algorithms performance, we consider the mean square error (MSE) metric, computed from the position error based on 1000 trials. We start the analysis by investigating the computational complexity of all the considered approaches. More specifically, we have recorded the runtimes of the estimators executed on the same hardware platform, for a network with \(N=8\) anchors. A grid of \(P=200\) evaluation points per dimension is considered for the joint ML estimator. A comparison between the average runtimes of the different estimators (normalized to the fastest one, that is the NR-based method,) is provided in Figure 3.  
As expected, the joint ML requires by far the longest runtime due to the two-dimensional search involved in the solution of Equation (8).