# Distributed Localization with Complemented RSS and AOA Measurements: Theory and Methods
## 2 RSS-AOA Distributed Localization Problem
### Problem Statement  
Let us consider a radio localization system consisting of \(M+N\) nodes (\(M\) of which are targets and \(N\) anchors, with preferably \(N\ll M\)), where their true locations are denoted by \(\mathbf{x}_{i}\in\mathbb{R}^{q}\), \(i=1,\ldots,M\), and \(\mathbf{a}_{j}\in\mathbb{R}^{q}\), \(j=M+1,\ldots,M+N\), respectively, with \(q=2\) or \(3\). One can view such a system as a connected graph \(\mathcal{G}(\mathcal{V},\mathcal{E})\), with \(|\mathcal{V}|=N+M\) vertices and a set of edges, \(\mathcal{E}\). Thus, we can also define the set of all targets, \(\mathcal{T}=\left\{i\colon i=1,\ldots,N\right\}\), as well as the set of all anchors, \(\mathcal{A}=\left\{j:j=N+1,\ldots,N+M\right\}\), where it follows that \(\mathcal{V}=\mathcal{T}\cup\mathcal{A}\). Due to limited energy constraints, it is assumed that nodes have constrained communication range. If two nodes, \(i\) and \(j\), are within the communication range of each other, they can interact, i.e., there is an edge between them, \(\mathcal{E}=\left\{(i,j):\|\mathbf{x}_{i}-\mathbf{s}_{j}\|\leq R,\,i,j\in\mathcal{V}\right\}\), with \(\mathbf{s}_{j}=\mathbf{x}_{j}\), if the \(j\in\mathcal{T}\) and \(\mathbf{s}_{j}=\mathbf{a}_{j}\), if the \(j\in\mathcal{A}\). Consequently, we can divide the set \(\mathcal{E}\) into two subsets, i.e., \(\mathcal{E}_{\mathcal{T}}=\left\{(i,j):\|\mathbf{x}_{i}-\mathbf{x}_{j}\|\leq R,\,i,j\in \mathcal{T}\right\}\) and \(\mathcal{E}_{\mathcal{A}}=\left\{(i,j):\|\mathbf{x}_{i}-\mathbf{a}_{j}\|\leq R,\,i\in \mathcal{T},\,j\in\mathcal{A}\right\}\). Lastly, let us define the neighborhood of the \(i\)-th target node as \(\mathcal{E}_{i}=\left\{j:(i,j)\in\mathcal{E}\right\}\). The localization system of interest in a 2-dimensional space is illustrated in Figure 4, where the distance and direction information are extracted from RSS and AOA measurements, respectively. RSS is defined as the voltage level measured by a receiver's RSS indicator circuit. It is often equivalent to measured power, i.e., the squared magnitude of the signal strength. In free-space, signal power decays proportionally to \(d^{-2}\), where \(d\) is the distance between a transmitter and a receiver. However, in practical applications the environment in which the signals propagate is not free-space, and an obstructed channel decays proportionally to \(d^{\gamma}\) instead, where \(\gamma\) is the path loss  
Figure 4: Illustration of the considered model in 2-dimensional space. Figure 3: Illustration of different network types based on the type of processing. CN = central node. exponent (PLE) which indicates the rate at which the received strength decreases with distance [41]. Typically, the RSS sensed by the \(i\)-th node related to the transmission of the \(j\)-th node [42] (Ch. 3), [43, 44], is defined as  
\[P_{ij}=P_{0}-10\gamma\log_{10}\frac{d_{ij}}{d_{0}}+n_{ij},\;\forall(i,j)\in \mathcal{E_{A}}\cup\mathcal{E_{T}},i\
eq j, \tag{1}\]  
where \(P_{0}\) is the reference RSS measured at a short reference distance \(d_{0}\) (\(d_{0}\leq d_{ij}\)), \(d_{ij}\) is the distance between nodes \(i\) and \(j\), and \(n_{ij}\) is the log-normal shadowing term modeled as a zero-mean Gaussian random variable with variance \(\sigma_{n_{ij}}^{2}\). Without loss of generality, it is assumed here that RSS observations between target nodes are symmetric, i.e., \(P_{ij}=P_{ji}\). Two common ways to measure AOA are by employing antenna arrays and by using directional antennas. The former method is based on the so-called array signal processing techniques at sensor nodes. In this method, the AOA is estimated from the differences in the arrival times of a received signal at each of the array elements. The latter method estimates AOA based on the RSS ratio between two (or more) directional antennas installed on a node, as illustrated in Figure 5. Usually, two directional antennas are pointed in different directions so that their main beams overlap, and the overlap is used to estimate the AOA from the ratio of their individual RSS values [41].