# A Simple and Efficient Method for RSS-AOA-Based Localization with Heterogeneous Anchor Nodes
## 2 Problem Formulation and Cramer-Rao Lower Bound
### System Model and Problem Formulation  
Consider a three-dimensional (3D) WSN consisting of \(N\) anchor nodes whose locations are known, denoted by \(\mathbf{a}_{i}=[a_{i1},a_{i2},a_{i3}]^{T},i=1,\cdots,N\), and a target node whose location is unknown, denoted by \(\mathbf{x}=[x_{1},x_{2},x_{3}]^{T}\). As shown in Figure 1, the true value of the azimuth angle and elevation angle between the target node and the \(i\)-th anchor node is referred to as \(\phi_{i}\) and \(\dot{a}_{i}\), which can be written as  
\[\dot{\phi}_{i}=\arctan(\frac{x_{2}-a_{i2}}{x_{1}-a_{i1}}),\quad i=1,\cdots,N, \tag{1}\]  
\[\dot{a}_{i}=\arccos(\frac{x_{3}-a_{i3}}{\|\mathbf{x}-\mathbf{a}_{i}\|}),\quad i=1, \cdots,N, \tag{2}\]  
where \(\phi_{i}\in(-\pi,\pi)\) and \(\dot{a}_{i}\in(-\frac{\pi}{2},\frac{\pi}{2})\). Figure 1: Illustration of an anchor node and the target node in 3D space. The RSS measurements without noise are denoted by  
\[\dot{P}_{i}=P_{0}-10\gamma\log_{10}\frac{\|\mathbf{x}-\mathbf{a}_{i}\|}{d_{0}},\quad i=1, \cdots,N, \tag{3}\]  
where \(P_{0}\) represents the reference received signal power when the reference distance is \(d_{0}\), and \(\gamma\) is referred to as the path loss exponent which is environment-related. Taking measurement noise into consideration, the hybrid RSS-AOA time series model can be represented as  
\[\phi_{it}=\dot{\phi}_{i}+m_{it},\quad t=1,\cdots,T, \tag{4}\]  
\[\alpha_{it}=\dot{a}_{i}+v_{it},\quad t=1,\cdots,T, \tag{5}\]  
\[P_{it}=\dot{P}_{i}+n_{it},\quad t=1,\cdots,T, \tag{6}\]  
where \(T\) denotes the number of time steps, which means the total number of discrete time intervals or steps in the time series. \(m_{it}\), \(v_{it}\), and \(n_{it}\) refer to measurement noise of the \(t\)-th time step of the azimuth angle, elevation angle and received signal strength, respectively, between the target node and the \(i\)-th anchor node. Generally, it is assumed that \(\{m_{it}\}\) follows independent identical distributions, where \(i=1,\ldots,N\) and \(t=1,\ldots,T\). \(\{v_{it}\}\) and \(\{n_{it}\}\) also follow independent identical distributions. However, AOA and RSS measurements are subject to measurement noise that is related to many factors, such as the quality of the hardware components, the location of the nodes, the orientation of the antennas, and the characteristics of the signal propagation environment. Therefore, it is possible that the variances of measurement noises from different anchor nodes are different. So, in this paper, the measurement noise from different anchor nodes follows different distributions, which can be modeled as a zero-mean Gaussian random variables, i.e., \(m_{it}\sim\mathcal{N}\left(0,\sigma_{m_{i}}^{2}\right)\), \(v_{it}\sim\mathcal{N}\left(0,\sigma_{v_{i}}^{2}\right)\), \(n_{it}\sim\mathcal{N}\left(0,\sigma_{n_{i}}^{2}\right)\), where \(\sigma_{m_{i}}\), \(\sigma_{v_{i}}\) and \(\sigma_{n_{i}}\) are the standard deviation of \(m_{it}\), \(v_{it}\) and \(n_{it}\), respectively. To make the standard deviation of measurement noise of different anchor nodes different, such that \(m_{p}\) is not equal to \(m_{q}\), \(p\
eq q\), \(p\), \(q=1,\cdots,N\), the standard deviations can be assumed to follow exponential distributions with the means of \(m_{m}\), \(\mu_{v}\), and \(\mu_{n}\). It can be expressed as \(\sigma_{m_{i}}\sim\text{Exp}(\mu_{m})\), \(\sigma_{v_{i}}\sim\text{Exp}(\mu_{v})\), and \(\sigma_{n_{i}}\sim\text{Exp}(\mu_{n})\). Then, the conditional probability density function (PDF) of the hybrid RSS-AOA measurement \(\mathbf{\theta}\) that \(\mathbf{x}\) is given can be expressed as  
\[