# Benefits and Impact of Joint Metric of AOA/RSS/TOF on Indoor Localization Error
## 4 Simulation Results
### Impact of AP Locations  
In this section, we will continue to investigate the relations between the error bounds and AP locations. As the distance between the \(i\)-th real location and \(m\)-th AP, \(d_{im}\), decreases to \({d_{im}}^{\prime}=\ell d_{im}\)\((0~{}<\ell\leq 1)\), we have:  
\[V_{m}\left(\text{RSS}\right)-{V_{m}}^{\prime}\left(\text{RSS}\right)\geq\frac{ \left( \begin{array}{c}W_{1}(\text{RSS})\sqrt{\eta_{i\left(m-1\right)(\text{ RSS})}\sum\limits_{k=1}^{m-1}\left[\frac{\cos\alpha_{ik}}{d_{ik}}\right]^{2}- \omega_{i\left(m-1\right)(\text{RSS})}}\\ -W_{2}(\text{RSS})\sqrt{\eta_{i\left(m-1\right)(\text{RSS})}\sum\limits_{k=1}^{m -1}\left[\frac{\sin\alpha_{ik}}{d_{ik}}\right]^{2}-\omega_{i\left(m-1\right)( \text{RSS})}}\\ \hline\rho_{1}^{2}\cdot\omega_{im\left(\text{RSS}\right)}\cdot\omega^{\prime} _{im\left(\text{RSS}\right)}\end{array} \right)^{2}} \tag{42}\]  
where \(\left\{ \begin{array}{l}\cos^{\prime}\alpha_{im}=\frac{x_{i}-x_{m}}{d\hat{d}_{ im}}^{\prime};\sin^{\prime}\alpha_{im}=\frac{y_{i}-y_{m}}{d\hat{d}_{im}}^{\prime}\\ W_{1}(\text{RSS})=\frac{(\sin^{\prime}\alpha_{im})^{2}}{\ell^{2}d_{im}^{2}}- \frac{\sin^{2}\alpha_{im}}{\hat{d}_{im}}\\ W_{2}(\text{RSS})=\frac{(\cos^{\prime}\alpha_{im})^{2}}{\ell^{2}d_{im}^{2}}- \frac{\cos^{2}\alpha_{im}}{\hat{d}_{im}}\\ \end{array} \right.\) and \(\omega^{\prime}_{im\left(\text{RSS}\right)}=\sum\limits_{k=1}^{m-1}\left( \frac{\sin^{2}\alpha_{ik}}{d_{ik}}+\frac{\sin^{\prime 2}\alpha_{im}}{\ell^{2}d_{im}^{2}}\right)\)  
\(\sum\limits_{k=1}^{m-1}\left(\frac{\cos^{2}\alpha_{ik}}{d_{ik}}+\frac{\cos^{ \prime 2}\alpha_{im}}{\ell^{2}d_{im}^{2}}\right)-\left[\sum\limits_{k=1}^{m-1} \frac{\sin\alpha_{ik}\cos\alpha_{ik}}{d_{ik}^{2}}+\frac{\sin^{\prime}\alpha_{ im}\sin^{\prime}\alpha_{im}}{\ell^{2}d_{im}^{2}}\right]^{2}\). \[V_{m}\left(\text{TOF}\right)-{V_{m}}^{\prime}\left(\text{TOF}\right)\geq\frac{ \left( \begin{array}{c}W_{1}(\text{TOF})\sqrt{\eta_{i\left(m-1\right)(\text{ TOF})}\sum\limits_{k=1}^{m-1}\left[\cos\alpha_{ik}\right]^{2}-\omega_{i \left(m-1\right)(\text{TOF})}}\\ -W_{2}(\text{TOF})\sqrt{\eta_{i\left(m-1\right)(\text{TOF})}\sum\limits_{k=1} ^{m-1}\left[\sin\alpha_{ik}\right]^{2}-\omega_{i\left(m-1\right)(\text{TOF}) }}\\ \hline\rho_{2}^{2}\cdot\omega_{im\left(\text{TOF}\right)}\cdot\omega^{\prime} _{im\left(\text{TOF}\right)}\end{array} \right)^{2}} \tag{43}\]  
where \(\left\{ \begin{array}{l}W_{1}(\text{TOF})=\left(\sin^{\prime}\alpha_{im} \right)^{2}-\left(\sin\alpha_{im}\right)^{2}\\ W_{2}(\text{TOF})=\left(\cos^{\prime}\alpha_{im}\right)^{2}-\left(\cos\alpha_{im }\right)^{2}\end{array} \right.\), \(\omega^{\prime}_{im\left(\text{TOF}\right)}=\sum\limits_{k=1}^{m-1}\left( \sin^{2}\alpha_{ik}+\sin^{\prime 2}\alpha_{im}\right)\\ \sum\limits_{k=1}^{m-1}\left(\cos^{2}\alpha_{ik}+\cos^{\prime 2}\alpha_{im}\right) -\left[\sum\limits_{k=1}^{m-1}\sin\alpha_{ik}\cos\alpha_{ik}+\sin^{\prime} \alpha_{im}\sin^{\prime}\alpha_{im}\right]^{2}\text{. }\end{array}\right.\)  
\[V_{m}\left(\text{AOA}\right)-{V_{m}}^{\prime}\left(\text{AOA}\right)\geq\frac{ \left( \begin{array}{c}W_{2}(\text{AOA})\sqrt{\eta_{i\left(m-1\right)(\text{ AOA})}\sum\limits_{k=1}^{m-1}\frac{\sin^{2}\alpha_{ik}}{d_{ik}^{2}}-\omega_{i \left(m-1\right)(\text{AOA})}}\\ -W_{1}(\text{AOA})\sqrt{\eta_{i\left(m-1\right)(\text{AOA})}\sum\limits_{k=1}^{m -1}\frac{\cos^{2}\alpha_{ik}}{d_{ik}^{2}}-\omega_{i\left(m-1\right)(\text{AOA}) }}\\ \hline\rho_{3}^{2}\cdot\omega_{im\left(\text{AOA}\right)}\cdot\omega^{\prime} _{im\left(\text{AOA}\right)}\end{array} \right)^{2}} \tag{44}\]  
where \(\omega^{\prime}_{im\left(\text{AOA}\right)}=\omega^{\prime}_{im\left(\text{RSS}\right)}\). Similarly, we can also find that the decrease of the distance between the \(i\)-th real location and \(k\)-th AP decreases the error bound or keeps the error bound unchanged by using the metric of double measurements or JMART.