# Benefits and Impact of Joint Metric of AOA/RSS/TOF on Indoor Localization Error
## 4 Simulation Results
### Impact of AP Number  
Based on the previous discussion, we can calculate that:  
\[V_{m}\left(\text{RSS}\right)-V_{m+1}\left(\text{RSS}\right)\geq\frac{ \left(\begin{array}{c}\sin\alpha_{i\left(m+1\right)}\\ \frac{\cos\alpha_{i\left(m+1\right)}}{d_{i\left(m+1\right)}}\sqrt{\eta_{im \left(RSS\right)}\sum\limits_{k=1}^{m}\left[\frac{\sin\alpha_{ik}}{d_{ik}} \right]^{2}-\omega_{im\left(RSS\right)}}\\ -\frac{\cos\alpha_{i\left(m+1\right)}}{d_{i\left(m+1\right)}}\sqrt{\eta_{im \left(RSS\right)}\sum\limits_{k=1}^{m}\left[\frac{\sin\alpha_{ik}}{d_{ik}} \right]^{2}-\omega_{im\left(RSS\right)}}\\ \end{array}\right)^{2}}{\rho_{1}\rho_{2}\omega_{im\left(RSS\right)}\cdot\omega_{ i\left(m+1\right)\left(RSS\right)}} \tag{39}\]  
where \(\left\{\begin{array}{c}\eta_{im\left(RSS\right)}=\sum\limits_{k=1}^{m}\left[ \frac{\cos\alpha_{ik}}{d_{ik}}\right]^{2}+\sum\limits_{k=1}^{m}\left[\frac{ \sin\alpha_{ik}}{d_{ik}}\right]^{2}\\ \omega_{im\left(RSS\right)}=\sum\limits_{k=1}^{m}\left[\frac{\sin\alpha_{ik}}{ d_{ik}}\right]^{2}\times\sum\limits_{k=1}^{m}\left[\frac{\cos\alpha_{ik}}{d_{ik}} \right]^{2}-\left[\sum\limits_{k=1}^{m}\frac{\sin\alpha_{ik}\cos\alpha_{ik}}{d _{ik}^{2}}\right]^{2}\\ \end{array}\right..\)  
\[V_{m}\left(\text{TOF}\right)-V_{m+1}\left(\text{TOF}\right)\geq\frac{ \left(\begin{array}{c}\sin\alpha_{i\left(m+1\right)}\sqrt{\eta_{im\left( \text{TOF}\right)}\sum\limits_{k=1}^{m}\left[\cos\alpha_{ik}\right]^{2}- \omega_{im\left(\text{TOF}\right)}}\\ -\cos\alpha_{i\left(m+1\right)}\sqrt{\eta_{im\left(\text{TOF}\right)}\sum \limits_{k=1}^{m}\left[\sin\alpha_{ik}\right]^{2}-\omega_{im\left(\text{TOF} \right)}}\\ \end{array}\right)^{2}}{\rho_{2}\omega_{im\left(\text{TOF}\right)}\cdot\omega_ {i\left(m+1\right)\left(\text{TOF}\right)}} \tag{40}\]  
where \(\left\{\begin{array}{c}\eta_{im\left(\text{TOF}\right)}=\sum\limits_{k=1}^{m }\left[\cos\alpha_{ik}\right]^{2}+\sum\limits_{k=1}^{m}\left[\sin\alpha_{ik} \right]^{2}\\ \omega_{im\left(\text{TOF}\right)}=\sum\limits_{k=1}^{m}\left[\sin\alpha_{ik} \right]^{2}\times\sum\limits_{k=1}^{m}\left[\cos\alpha_{ik}\right]^{2}-\left[ \sum\limits_{k=1}^{m}\sin\alpha_{ik}\cos\alpha_{ik}\right]^{2}\\ \end{array}\right)^{2}\cdot\eta_{im\left(\text{AOA}\right)}=\eta_{im\left(RSS \right)}\) and \(\omega_{im\left(\text{AOA}\right)}=\omega_{im\left(\text{RSS}\right)}\).  
\[V_{m}\left(\text{AOA}\right)-V_{m+1}\left(\text{AOA}\right)\geq\frac{\left( \begin{array}{c}\frac{\cos\alpha_{i\left(m+1\right)}}{d_{i\left(m+1\right)}} \sqrt{\eta_{im\left(\text{AOA}\right)}\sum\limits_{k=1}^{m}\left[\frac{\sin \alpha_{ik}}{d_{ik}}\right]^{2}-\omega_{im\left(\text{AOA}\right)}}\\ -\frac{\sin\alpha_{i\left(m+1\right)}}{d_{i\left(m+1\right)}}\sqrt{\eta_{im \left(\text{AOA}\right)}\sum\limits_{k=1}^{m}\left[\frac{\cos\alpha_{ik}}{d_{ ik}}\right]^{2}-\omega_{im\left(\text{AOA}\right)}}\\ \end{array}\right)^{2}}{\rho_{3}\omega_{im\left(\text{AOA}\right)}\cdot\omega_ {i\left(m+1\right)\left(\text{AOA}\right)}} \tag{41}\]  
where \(\eta_{im\left(\text{AOA}\right)}=\eta_{im\left(RSS\right)}\) and \(\omega_{im\left(\text{AOA}\right)}=\omega_{im\left(RSS\right)}\).  
As can be seen from (39)-(41), the increase of AP number decreases the error bound or keeps the error bound unchanged [26]. The relation \(V_{m}\left(\eta_{j}\right)=V_{m+1}\left(\eta_{j}\right)\) holds under the condition of \(\alpha_{i1}=\alpha_{i2}=\cdots=\alpha_{i\left(m+1\right)}\), which indicates that the APs are collinearly distributed. Similarly, we can also find that the increase of AP number decreases the error bound or keeps the error bound unchanged by using the metric of double measurements or JMART.