# Distributed Localization with Complemented RSS and AOA Measurements: Theory and Methods
## 2 RSS-AOA Distributed Localization Problem
### Problem Statement
The AOA can also be estimated from RSS measurements by employing electronically steerable parasitic array radiator antennas [45] or by rotating a directional antenna at the receiving end [46], or even by using video cameras [47]. It is also worth mentioning that some recent works estimate the angle by exploiting the known network topology, rather than installing any additional hardware at the receiving end [48]. Furthermore, phase of the received signal can play a significant role in the direction of arrival estimation [49, 50]. From geometry, azimuth angle measurements (at anchor nodes) [15, 46, 51, 52, 53] can be modeled as  
\[\varphi_{ij}=\arctan\left(\frac{x_{i2}-a_{j2}}{x_{i1}-a_{j1}}\right)+m_{ij},\; \forall(i,j)\in\mathcal{E_{A}}, \tag{2}\]  
where \(x_{kl}\) and \(a_{kl}\) denote, respectively, the \(l\)-th coordinate of \(k\)-th target and anchor node, and \(m_{ij}\) is the measurement error of the azimuth angle, modeled as a zero-mean von Mises random variable with the concentration parameter \(\kappa_{m_{ij}}\in[0,\infty)\), i.e., \(m_{ij}\sim VM(0,\kappa_{m_{ij}})\). The von Mises distribution can be seen as a circular analogue of the Normal distribution. There is a close connection between the mean direction and the concentration parameters of the von Mises distribution and the mean and variance of the Normal one [54, 55]. Figure 5: Illustration of angle of arrival (AOA) estimation process based on the received signal strength (RSS) ratio, \(\text{RSS}_{1}/\text{RSS}_{2}\), between two directional antennas. In order to simplify the notation, we define \(\mathbf{P}=[P_{ij}]^{T}\), \(\mathbf{\varphi}=[\varphi_{ij}^{A}]^{T}\), \(\mathbf{X}=[\mathbf{x}_{1},\ \ldots,\mathbf{x}_{M}]\), \((\mathbf{X}\in\mathbb{R}^{2\times M})\), and \(\mathcal{E}=\mathcal{E}_{\mathcal{A}}\cup\mathcal{E}_{T}\). Following the model in (1), the conditional probability density function (PDF) of RSS measurements is given by  
\[f_{\mathbf{P}}(\mathbf{P}|\mathbf{X})=\prod_{(i,j)\in\mathcal{E}}\frac{1}{\sqrt{2\pi\sigma _{\mathbf{n}ij}^{2}}}\,\mathrm{e}^{-\frac{\left(p_{ij}-p_{0}+10\gamma_{\mathbf{n}ij} \log\frac{d_{ij}}{d_{0}}\right)^{2}}{2\pi_{\mathbf{n}ij}^{2}}}\,. \tag{3}\]  
In a similar manner, according to the model in (2), the conditional PDF of AOA measurements can be written as  
\[f_{\mathbf{\varphi}}\left(\mathbf{\varphi}|\mathbf{X}\right)=\prod_{(i,j)\in\mathcal{E}_{ \mathcal{A}}}\frac{1}{2\pi I_{0}(\kappa_{\mathbf{n}ij})}\,\mathrm{e}^{\kappa_{\mathbf{ n}ij}\mathbf{n}_{ij}^{T}\frac{\mathbf{a}_{j}-\mathbf{x}_{i}}{|\mathbf{a}_{i}-\mathbf{a}_{j}|}}\,, \tag{4}\]  
where \(I_{k}(\bullet)\) is the modified Bessel function of first kind of order \(k\)[54; 55], and \(\mathbf{u}_{ij}=\left[\cos\left(\varphi_{ij}\right),\sin\left(\varphi_{ij}\right) \right]^{T}\) is a unit vector. If one would maximize (3),(4), one would obtain the maximum likelihood (ML) estimator [56] (Chapter 7) of \(\mathbf{X}\). In distributed networks, the ML problem is actually split into \(M\) local ML estimators, which is then solved individually by each one of the target nodes, resorting only to measurements acquired from the target's direct neighbors. Nonetheless, the ML estimator obviously implies dealing with highly non-convex problem, in which a closed-form solution is not directly attainable; hence, directly solving the RSS and AOA ML estimator might not be feasible in practice. Therefore, many researchers opt to tackle an approximation of the ML estimator instead.