# Benefits and Impact of Joint Metric of AOA/RSS/TOF on Indoor Localization Error
## 3 Error Bound Analysis  
In this section, we will derive the error bounds of localization by using the single measurement of RSS, TOF or AOA, double measurements of RSS/AOA, RSS/TOF or AOA/TOF and the JMART based on the FIM.  
Let \(\widehat{\theta}_{i}=\left(\widehat{x_{i}}\,\widehat{y_{i}}\right)^{\text{T}}\) be the unbiased estimate of the \(i\)-th real location \(\theta_{i}=\left(x_{i}\,y_{i}\right)^{\text{T}}\). Then, the covariance matrix with respect to \(\widehat{\theta}_{i}\) is calculated by:  
\[\text{E}\left[\left(\widehat{\theta}_{i}-\theta_{i}\right)\left(\widehat{ \theta}_{i}-\theta_{i}\right)^{\text{T}}\right]=\left[\begin{array}{cc} \sigma_{\widehat{x}_{i}}^{2}&\sigma_{\widehat{x}_{i}\widehat{y}_{i}}^{2}\\ \sigma_{\widehat{y}_{i}\widehat{x}_{i}}^{2}&\sigma_{\widehat{y}_{i}}^{2} \end{array}\right] \tag{1}\]  
where \(\sigma_{\widehat{x}_{i}}^{2}\) and \(\sigma_{\widehat{y}_{i}}^{2}\) are the Mean Square Errors (MSEs) with respect to \(\widehat{x_{i}}\) and \(\widehat{y_{i}}\). \(\sigma_{\widehat{x}_{i}\widehat{y_{i}}}^{2}\) and \(\sigma_{\widehat{y}_{i}\widehat{x}_{i}}^{2}\) are the covariance between \(\widehat{x_{i}}\) and \(\widehat{y_{i}}\) and between \(\widehat{y_{i}}\) and \(\widehat{x_{i}}\), respectively. The notations \"var\", \"E\" and \"T\" stand for the variance, Expectation and Transpose operations. If \(\widehat{\theta}_{i}\) is calculated from the unbiased estimate of the measurement (e.g., RSS, TOF or AOA) at the \(i\)-th real location \(\mathbf{m}_{i}\), we can use the FIM to construct the relations below to present the lower bound of the variance of \(\widehat{\theta}_{i}\).  
\[\text{var}[\widehat{\theta}_{i}]\geq\text{E}\left\{-\frac{\partial^{2}\ln \theta_{i}(\mathbf{m}_{i})}{\partial\Theta_{i}{}^{2}}\right\}^{-1}=J(\theta_ {i})^{-1} \tag{2}\]  
where \(f_{  
Based on (2), since:  
\[J(\theta_{i})^{-1}=\frac{1}{\left|J\left(\theta_{i}\right)\right|}\cdot\left[ \begin{array}{cc}J_{y_{y_{i}}}\left(\mathbf{m}_{i}\right)&-J_{y_{i}x_{i}} \left(\mathbf{m}_{i}\right)\\ -J_{x_{i}y_{i}}\left(\mathbf{m}_{i}\right)&J_{x_{i}x_{i}}\left(\mathbf{m}_{i} \right)\end{array}\right], \tag{6}\]  
where \(\left|J\left(\theta_{i}\right)\right|=\)\(J_{x_{i}x_{i}}\left(\mathbf{m}_{i}\right)\)\(J_{y_{i}y_{i}}\left(\mathbf{m}_{i}\right)-J_{x_{i}y_{i}}\left(\mathbf{m}_{i} \right)^{2}\), we calculate that:  
\[\left\{\begin{array}{c}\sigma_{\widehat{x}_{i}}^{2}=\mathrm{E}\left(\widehat {x}_{i}-x_{i}\right)^{2}\geq\frac{J_{y_{y_{i}y_{i}}}\left(\mathbf{m}_{i} \right)}{\left|J\left(\theta_{i}\right)\right|}\\ \sigma_{\widehat{y}_{i}}^{2}=\mathrm{E}\left(\widehat{y}_{i}-y_{i}\right)^{2} \geq\frac{J_{x_{i}x_{i}}\left(\mathbf{m}_{i}\right)}{\left|J\left(\theta_{i} \right)\right|}\end{array}\right. \tag{7}\]  
Then, by selecting the RSS as the metric for the localization, the CRLB with respect to \(\theta_{i}\) under \(m\) APs condition equals:  
\[V_{m}\left(\mathbf{m}_{i}\right)=\frac{J_{x_{i}x_{i}}\left(\mathbf{m}_{i} \right)+J_{y_{i}y_{i}}\left(\mathbf{m}_{i}\right)}{\left|J\left(\theta_{i} \right)\right|} \tag{8}\]  
Therefore, the average localization error bound for the target environment, \(V_{\mathrm{ave}}\left(\mathbf{m}_{i}\right)\), is:  
\[V_{\mathrm{ave}}\left(\mathbf{m}_{i}\right)= \frac{1}{n}\sum\limits_{i=1}^{n}V_{m}\left(\mathbf{m}_{i}\right) \tag{9}\]  
where \(n\) is the number of real locations in the target environment.