# Benefits and Impact of Joint Metric of AOA/RSS/TOF on Indoor Localization Error
## 3 Error Bound Analysis
### Jmart  
In this section, we will continue to investigate the benefits and impact of the JMART on indoor localization error. Based on the previous discussion, the PDF of the JMART with respect to \(\theta_{i}\) can be calculated by:  
\[\left\{ \begin{array}{l}f_{\theta_{i}}\left(\mathrm{JMART}\right)=\frac{1}{ \sqrt{\left(2\pi\right)^{3}\left|\mathbf{\Sigma}\right|^{1/2}}}\exp\left\{- \frac{1}{2}\mathbf{x}^{\mathrm{T}}\mathbf{\Sigma}^{-1}\mathbf{x}\right\}\\ \\ \mathbf{\Sigma}\ =\ \mathrm{var}(\mathbf{x})=\left[\begin{array}{ccc}\sigma_{1 }^{2}&\rho_{12}\sigma_{1}\sigma_{2}&\rho_{13}\sigma_{1}\sigma_{3}\\ \rho_{12}\sigma_{1}\sigma_{2}&\sigma_{2}{}^{2}&\rho_{23}\sigma_{2}\sigma_{3}\\ \rho_{13}\sigma_{1}\sigma_{3}&\rho_{23}\sigma_{2}\sigma_{3}&\sigma_{3}{}^{2} \end{array}\right]\end{array} \right. \tag{31}\]  
where \(\mathbf{x}\ =\left(\xi_{i1},\xi_{i2},\xi_{i3}\right)\). Then, we have:  
\[J\left(\theta_{i}\right)=\frac{\partial\mathbf{x}^{\mathrm{T}}}{\partial\theta _{i}}\mathbf{\Sigma}^{-1}\frac{\partial\mathbf{x}}{\partial\theta_{i}}+ \frac{1}{2}\mathrm{tr}(\mathbf{\Sigma}^{-1}\frac{\partial\mathbf{\Sigma}}{ \partial\theta_{i}}\mathbf{\Sigma}^{-1}\frac{\partial\mathbf{\Sigma}}{\partial \theta_{i}}) \tag{32}\]  
Since the variances of \(\widehat{P_{ik}}\), \(\widehat{d_{ik}}\), and \(\widehat{\alpha_{ik}}\) are constant, we convert (32) into:  
\[\left\{ \begin{array}{l}J\left(\theta_{i}\right)=\frac{\partial\mathbf{x}^{ \mathrm{T}}}{\partial\theta_{i}}\mathbf{\Sigma}^{-1}\frac{\partial\mathbf{x}^{ \mathrm{T}}}{\partial\theta_{i}^{2}}\\ \\ \mathbf{\Sigma}^{-1}\ =\ \frac{1}{r}\left[\begin{array}{ccc}\frac{1-\rho_{23} }{\sigma_{1}}^{2}&\frac{\rho_{13}\rho_{23}-\rho_{12}}{\sigma_{1}\sigma_{2}}& \frac{\rho_{12}\rho_{23}-\rho_{13}}{\sigma_{1}\sigma_{3}}\\ \\ \frac{\rho_{13}\rho_{23}-\rho_{12}}{\sigma_{1}\sigma_{2}}&\frac{-\rho_{13}\rho_{2 }}{\sigma_{2}}&\frac{\rho_{12}\rho_{13}-\rho_{23}}{\sigma_{2}\sigma_{3}}\\ \\ \frac{\rho_{12}\rho_{23}-\rho_{13}}{\sigma_{1}\sigma_{3}}&\frac{\rho_{12}\rho_{13} -\rho_{23}}{\sigma_{2}\sigma_{3}}&\frac{-\rho_{12}\rho^{2}}{\sigma_{3}}\end{array}\right]\end{array} \right. \tag{33}\]  
where \(r\ =\ 1\ +\ 2\rho_{12}\rho_{13}\rho_{23}-{\rho_{12}}^{2}-{\rho_{13}}^{2}-{\rho_{23}}^{2}\). Then, we can obtain:  
\[J\left(\theta_{i}\right)=\left[ \begin{array}{ccc}\frac{\partial\xi_{i1}}{ \partial x_{i}}&\frac{\partial\xi_{i2}}{\partial x_{i}}&\frac{\partial\xi_{i3}}{ \partial x_{i}}\\ \frac{\partial\xi_{i1}}{\partial y_{i}}&\frac{\partial\xi_{i2}}{\partial y_{i}}& \frac{\partial\xi_{i3}}{\partial y_{i}}\end{array} \right]\cdot\mathbf{\Sigma}^{-1} \cdot\left[ \begin{array}{ccc}\frac{\partial\xi_{i1}}{\partial x_{i}}&\frac{ \partial\xi_{i1}}{\partial y_{i}}\\ \frac{\partial\xi_{i2}}{\partial x_{i}}&\frac{\partial\xi_{i2}}{\partial y_{i}} \\ \frac{\partial\xi_{i2}}{\partial x_{i}}&\frac{\partial\xi_{i3}}{\partial y_{i}} \end{array} \right]=\!\frac{1}{r}\left[ \begin{array}{ccc}M_{11}&M_{12}\\ M_{21}&M_{22}\end{array} \right] \tag{34}\]  
where:  
\[\left\{ \begin{array}{l}M_{11}\ =\ \left(1-{\rho_{23}}^{2}\right)[PERSON]_{xx}\left( \mathrm{RSS}\right)+\left(1-{\rho_{13}}^{2}\right)[PERSON]_{xx}\left(\mathrm{AOA}\right)+ \left(1-{\rho_{12}}^{2}\right)[PERSON]_{xx}\left(\mathrm{TOF}\right)-2A\\ M_{12}\ =\ \left(1-{\rho_{23}}^{2}\right)[PERSON]_{xy}\left(\mathrm{RSS}\right)+\left(1-{\rho_{13}}^{2} \right)[PERSON]_{xy}\left(\mathrm{AOA}\right)+\left(1-{\rho_{12}}^{2}\right)[PERSON]_{xy} \left(\mathrm{TOF}\right)-B_{1}-B_{2}\\ M_{21}\ =\ \left(1-{\rho_{23}}^{2}\right)[PERSON]_{yx}\left(\mathrm{RSS}\right)+\left(1-{\rho_{13}}^{2} \right)[PERSON]_{yx}\left(\mathrm{AOA}\right)+\left(1-{\rho_{12}}^{2}\right)[PERSON]_{yx} \left(\mathrm{TOF}\right)-C_{1}-C_{2}\\ M_{22}\ =\ \left(1-{\rho_{23}}^{2}\right)[PERSON]_{yy}\left(\mathrm{RSS}\right)+\left(1-{\rho_{13}}^{2} \right)[PERSON]_{yy}\left(\mathrm{AOA}\right)+\left(1-{\rho_{12}}^{2}\right)[PERSON]_{yy} \left(\mathrm{TOF}\right)-2D\end{array} \right. \tag{35}\]\[\left\{