# Distributed Localization with Complemented RSS and AOA Measurements: Theory and Methods
## Appendix A CRLB Derivation for RSS-AOA Localization  
In estimation theory and statistics, the CRLB gives a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter [56] (Chapter 3). Loosely speaking, the bound states that the variance of any unbiased estimator is at least as high as the inverse of the FIM. Thus, the CRLB is a reference point against which the performance of unbiased estimators can be analyzed.  
According to the definition [56] (Chapter 3), the variance of any unbiased localization estimator is lower bounded by \(\operatorname{var}(\hat{\mathbf{X}})\geq\operatorname{trace}\left(\mathbf{J}^{-1}( \mathbf{X})\right)\), where \(\mathbf{J}(\mathbf{X})\) is the \(2M\times 2M\) FIM. The elements of the FIM are defined as \(\left[\mathbf{J}(\mathbf{X})\right]_{i,j}=-\mathbb{E}\left[\frac{\partial^{2}\ln p( \mathbf{\theta}|\mathbf{X})}{\partial\mathbf{x}_{j}}\right]\), where \(i,j=1,\ldots,2M\), and \(p(\mathbf{\theta}|\mathbf{X})\) is the joint conditional probability density function of the observation vector \(\mathbf{\theta}=[\mathbf{P}^{T},\mathbf{\varphi}^{T}]^{T}\), given \(\mathbf{X}\).  
Therefore, it follows that the FIM can be computed as:  
\[\mathbf{J}(\mathbf{X})=\frac{1}{\sigma_{\tilde{n}_{ij}}^{2}}\sum_{(i,j):j\in\mathcal{E}_{ \mathcal{A}_{i}}}\mathbf{h}_{ij}\mathbf{h}_{ij}^{T}+\frac{1}{\sigma_{\tilde{m}_{ij}}^{2 }}\sum_{(i,j):j\in\mathcal{E}_{\mathcal{A}_{i}}}\mathbf{u}_{ij}\mathbf{u}_{ij}^{T}+ \frac{1}{\sigma_{\tilde{n}_{ik}}^{2}}\sum_{(i,k):k\in\mathcal{E}_{\mathcal{T}_ {i}}}\mathbf{h}_{ik}\mathbf{h}_{ik}^{T}+\frac{1}{\sigma_{\tilde{m}_{ik}}^{2}}\sum_{(i,k ):k\in\mathcal{E}_{\mathcal{T}_{i}}}\mathbf{u}_{ik}\mathbf{u}_{ik}^{T},\]  
where \(\mathcal{E}_{\mathcal{A}_{i}}=\{j:(i,j)\in\mathcal{E}_{\mathcal{A}}\}\) and \(\mathcal{E}_{\mathcal{T}_{i}}=\{j:(i,j)\in\mathcal{E}_{\mathcal{T}}\}\) are the anchor and target neighborhoods of the \(i\)-th target,  
\[\mathbf{h}_{ij}=\mathbf{\rho}-\frac{10\gamma d_{0}}{\ln(10)}\frac{E_{i}(E_{i}^{T}\mathbf{y }-\mathbf{a}_{j})}{\|E_{i}^{T}\mathbf{y}-\mathbf{a}_{j}\|^{2}},\]  
\[\mathbf{u}_{ij}=\frac{E_{i}\mathbf{e}_{2}(\mathbf{e}_{1}^{T}\mathbf{E}_{i}^{T}\mathbf{y}-\mathbf{e}_{1 }^{T}\mathbf{a}_{j})-E_{i}\mathbf{e}_{1}(\mathbf{e}_{2}^{T}\mathbf{E}_{i}^{T}\mathbf{y}-\mathbf{e}_{2}^ {T}\mathbf{a}_{j})}{(\mathbf{e}_{1}^{T}\mathbf{E}_{i}^{T}\mathbf{y}-\mathbf{e}_{1}^{T}\mathbf{a}_{j}) ^{2}+(\mathbf{e}_{2}^{T}\mathbf{E}_{i}^{T}\mathbf{y}-\mathbf{e}_{2}^{T}\mathbf{a}_{j})^{2}},\]  
\[\mathbf{h}_{ik}=\mathbf{\rho}-\frac{10\gamma d_{0}}{\ln(10)}\frac{(E_{i}-E_{k})(E_{i}^ {T}\mathbf{y}-\mathbf{E}_{k}^{T}\mathbf{y})}{\|E_{i}^{T}\mathbf{y}-\mathbf{E}_{k}^{T}\mathbf{y}\|^{2}},\]  
\[\mathbf{u}_{ik}=\frac{(\mathbf{E}_{i}-\mathbf{E}_{k})(\mathbf{e}_{2}(\mathbf{e}_{1}^{T}\mathbf{E}_{i}^ {T}\mathbf{y}-\mathbf{e}_{1}^{T}\mathbf{E}_{k}^{T}\mathbf{y})-\mathbf{e}_{1}(\mathbf{e}_{2}^{T}\mathbf{E}_ {i}^{T}\mathbf{y}-\mathbf{e}_{2}^{T}\mathbf{E}_{k}^{T}\mathbf{y}))}{(\mathbf{e}_{1}^{T}\mathbf{E}_{i}^ {T}\mathbf{y}-\mathbf{e}_{1}^{T}\mathbf{E}_{k}^{T}\mathbf{y})^{2}+(\mathbf{e}_{2}^{T}\mathbf{E}_{i}^ {T}\mathbf{y}-\mathbf{e}_{2}^{T}\mathbf{E}_{k}^{T}\mathbf{y})^{2}},\]  
\(\mathbf{E}_{i}=[\mathbf{e}_{2i-1},\mathbf{e}_{2i}]\), with \(\mathbf{e}_{i}\) representing the \(i\)-th column of the identity matrix \(\mathbf{I}_{2M}\), and \(\mathbf{e}_{1}=[1,\,0]^{T}\), \(\mathbf{e}_{2}=[0,\,1]^{T}\).  
Therefore, the CRLB for the estimate of the target locations is computed as:  
\[\text{CRLB}=\text{trace}\left(\mathbf{J}^{-1}(\mathbf{X})\right).\]