# A Simple and Efficient Method for RSS-AOA-Based Localization with Heterogeneous Anchor Nodes
## 2 Problem Formulation and Cramer-Rao Lower Bound
### Cramer-Rao Lower Bound  
The Cramer-Rao lower bound (CRLB) is a theoretical lower bound on the variance of any unbiased estimator of a parameter in a statistical model. In other words, it provides a lower limit on how well we can estimate a parameter, given the statistical properties of the data and the model used to describe them. In the hybrid RSS-AOA measurement sequences, the CRLB of location estimation depends on the locations of the anchor nodes, the true location of the target node, and the measurement noise variances. The CRLB can be calculated based on the Fisher information matrix (FIM). The FIM is a mathematical tool that measures the amount of information that an observable random variable carries about an unknown parameter in a statistical model. In the localization system model, it can be defined as  
\[\left[\text{FIM}(\mathbf{x})\right]_{ij}=-\mathrm{E}\Bigg{[}\frac{\partial^{2 }\mathrm{In}P(\theta|\mathbf{x})}{\partial x_{i}\partial x_{j}}\Bigg{]}, \tag{10}\]  
where \(1\leq i\), \(j\leq 3\). For the RSS-AOA time series model, the FIM is calculated using the PDF that have been given in (7). Specifically, each term has been derived as follows  
\[\left[\text{FIM}(\mathbf{x})\right]_{11} = T\sum_{i=1}^{N}\Bigg{(}\frac{(x_{2}-a_{2i})^{2}}{\sigma_{m_{i}} ^{2}d_{2i}^{4}}+\frac{(x_{1}-a_{i1})^{2}(x_{3}-a_{i3})^{2}}{\sigma_{v_{i}}^{2 }d_{2i}^{2}d_{i}^{4}}-\frac{\eta_{i}(x_{1}-a_{i1})^{2}}{\left\|\mathbf{x}- \mathbf{a}_{i}\right\|^{4}}\Bigg{)},\] \[\left[\text{FIM}(\mathbf{x})\right]_{22} = T\sum_{i=1}^{N}\Bigg{(}\frac{(x_{1}-a_{i1})}{\sigma_{m_{i}}^{2} d_{2i}^{4}}+\frac{(x_{2}-a_{2i})^{2}(x_{3}-a_{i3})^{2}}{\sigma_{v_{i}}^{2}d_{2i}^{ 4}d_{i}^{4}}-\frac{\eta_{i}(x_{2}-a_{2i})^{2}}{\left\|\mathbf{x}-\mathbf{a}_{i }\right\|^{4}}\Bigg{)},\] \[\left[\text{FIM}(\mathbf{x})\right]_{33} = T\sum_{i=1}^{N}\Bigg{(}\frac{d_{2i}^{2}}{\sigma_{v_{i}}^{2}d_{i}^{4 }}-\frac{\eta_{i}(x_{3}-a_{i3})^{2}}{\left\|\mathbf{x}-\mathbf{a}_{i}\right\|^ {4}}\Bigg{)},\] \[\left[\text{FIM}(\mathbf{x})\right]_{13} = T\sum_{i=1}^{N}\Bigg{(}-\frac{(x_{1}-a_{i1})(x_{3}-a_{i3})}{ \sigma_{v_{i}}^{2}d_{i}^{4}}-\frac{\eta_{i}(x_{1}-a_{i1})(x_{3}-a_{i3})}{ \left\|\mathbf{x}-\mathbf{a}_{i}\right\|^{4}}\Bigg{)}, \tag{11}\] \[\left[\text{FIM}(\mathbf{x})\right]_{23} = T\sum_{i=1}^{N}\Bigg{(}\frac{(x_{2}-a_{i2})(x_{3}-a_{i3})}{ \sigma_{v_{i}}^{2}d_{i}^{4}}-\frac{\eta_{i}(x_{2}-a_{i2})(x_{3}-a_{i3})}{ \left\|\mathbf{x}-\mathbf{a}_{i}\right\|^{4}}\Bigg{)},\] \[\left[\text{FIM}(\mathbf{x})\right]_{12} = T\sum_{i=1}^{N}\Bigg{(}-\frac{(x_{1}-a_{i1})(x_{2}-a_{i2})}{ \sigma_{m_{i}}^{2}d_{2i}^{4}}+\frac{(x_{1}-a_{i1})(x_{2}-a_{i2})(x_{3}-a_{i3} )^{2}}{\sigma_{v_{i}}^{2}d_{2i}^{2}d_{i}^{4}}-\frac{\eta_{i}(x_{1}-a_{i1})(x_{ 2}-a_{i2})}{\left\|\mathbf{x}-\mathbf{a}_{i}\right\|^{4}}\Bigg{)}.\]where \(\mathbf{FIM}(x)\) is a symmetric matrix, e.g., \(\left[\mathbf{FIM}(x)\right]_{ij}=\left[\mathbf{FIM}(x)\right]_{ji}\), \(i,j=1,2,3\), \(\eta_{i}=\left(\frac{10\gamma}{\sigma_{n_{i}}\log_{10}10}\right)^{2}\), \(d_{2i}=\sqrt{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}}\), and \(d_{i}=\sqrt{(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+(x_{3}-a_{3})^{2}}\). Then, the CRLB of the hybrid RSS-AOA time series model is obtained as  
\[\text{CRLB}=\sqrt{\text{trace}(\mathbf{FIM}^{-1})}. \tag{12}\]  
In order to provide a more intuitive illustration of the CRLB for the RSS-AOA time series, we simulated a specific scenario. Figure 2 illustrates a simple wireless localization scenario with randomly located target nodes and fixed anchor node locations. In this scenario, four anchor nodes are positioned at \([0,10,10]^{T}\), \([10,30,15]^{T}\), \([30,10,20]^{T}\), and \([30,30,25]^{T}\), with the noise variances of \(\sigma_{m_{1}}=5\deg\), \(\sigma_{v_{1}}=5\deg\), and \(\sigma_{n_{1}}=1\deg\) for the first anchor node, and \(\sigma_{m_{i}}=10\deg\), \(\sigma_{v_{i}}=10\deg\), and \(\sigma_{n_{i}}=2\deg\) for the rest anchor nodes, where \(i=2,3,4\). The target nodes are randomly placed within a square with sides ranging from 1 to 40 m for \(x_{1}\) and \(x_{2}\), and at \(x_{3}=0\).