# Distributed Localization with Complemented RSS and AOA Measurements: Theory and Methods
## 4 Performance Analysis
### Analysis of Localization Accuracy  
In this section, a set of simulation results are presented in order to evaluate the performance of the existing algorithms in terms of localization accuracy. In the simulations presented here, all nodes were deployed completely randomly inside a square area with edge length \(B=30\) m in each Monte Carlo (\(M_{c}\)) run, forming connected networks. RSS and AOA measurements were generated according to (1) and (2). The reference distance was set to \(d_{0}=1\) m, the reference RSS to \(P_{0}=20\) dBm, and the PLE to \(\gamma=3\). The maximum number of steps in the bisection procedure was set to \(S_{\max}=30\). Moreover, \(\sigma_{n_{ij}}=10\) dB and \(\kappa_{ij}=16.676\) were adopted, where the latter parameter corresponds to the Normal standard deviation of \(\sigma_{n_{ij}}=10\) deg, since \(\sigma_{n_{ij}}^{2}=1-I_{1}(\kappa_{ij})/I_{0}(\kappa_{ij})\)[54, 55]. In this analysis, the following assumptions are made about the network:  
1. The network is connected and does not vary during the computational period;
2. All nodes are suitably equipped to measure RSS of the received radio signal;
3. Some nodes (anchors) or possibly all of them are conveniently equipped to measure AOA of the received radio signal;
4. Working order of the nodes is synchronized. The first part of Assumption (1) guarantees that there is a path between nodes \(i\) and \(j\), \(\forall i,j\in\mathcal{V}\), while the second part ensures that there is no topology variation (node/edge failure, node/edge addition or node movement) during the time required for information processing. Assumptions (2) and (3) are necessary in order for the nodes to acquire RSS and AOA measurements. While RSS measurements are _cheap_ (because they do not require specialized hardware) and are thus widely available Today in almost all devices, in order to measure AOA quantity, typically directional antennas, antenna arrays or video cameras are required at the receiving end. Assumption (4) is made for the sake of simplicity, since node synchronization is a difficult problem to tackle in distributed networks, and it does not represent the main concern of this work. One possible way to achieve node synchronization could be by using network coloring scheme, as explained in [40, 71, 72], or any other medium access protocol. Figure 12 depicts ARMSE (m) versus \(N\) comparison for fixed \(M=50\) and \(R=10\) m. It clearly shows that the performance of all methods improve as \(N\) grows. This behavior is anticipated, since the quantity of reliable information in the network increases when more anchor nodes are added into the network. The figure shows that the performance of LLS is the best one. This is not a surprise, since LLS assumes that all nodes can measure AOA and it assumes perfect knowledge about the noise powers, which might not be feasible in practical scenarios where only a single measurement between pair of nodes is used for localization. Hence, in order to study the robustness of this method to imperfect knowledge about the noise power, the results for LLS when instead of the true value of \(\sigma_{ij}\) (dB, deg), \(\sigma_{ij}=15\) (dB, deg) is used are also included. It can be seen that LLS is sensitive to imperfections about this information. Finally, Figure 12 shows that there is still considerable room for further improvements, since there is a margin between CRLB (assuming that all nodes can measure the AOA quantity) and the existing methods. Figure 13 depicts ARMSE (m) versus \(M\) comparison for fixed \(N=15\) and \(R=10\) m. It can be seen that the performance of LLS and MCMC-MH deteriorates and the performance of SOCP and GTRS meliorates slightly as \(M\) increases. This indicates that the latter two methods benefit from additional node cooperation achieved by introducing more target nodes into the network, while the latter ones are unable to take advantage of this additional cooperation. Figure 14 depicts ARMSE (m) versus \(R\) (m) comparison for fixed \(N=15\) and \(M=50\).