# Optical Boundaries for LED-Based Indoor Positioning System
## 4 Investigations Showing the Effect of Defined Optical Boundaries
### Quantifying Effect of Full Angle Positioning Boundary  
In this section, the process of examining the effect of the conventional full angle positioning boundary on an IPS with single and overlapping LED beams is explained. In the full angle positioning boundary, a receiver in the boundary region takes a longer time to determine its position due to the low SNR in the region. This is because low SNR causes a higher BER which leads to reduced PDR. Since packets with error are discarded, the receiver waits for a longer time to receiver errorless packets. This wait increases positioning time. Consequently, analysis to show the effect of the full angle boundary on positioning is done by determining the average positioning time (APT) when the full angle is used as the LED beam region and repeating the process using the threshold angle. Considering an untilted receiver at an incidence angle \(\varphi=\phi\) from the transmitter, if the BER at this point is BER\({}_{\phi}\) for a single LED transmitting \(N_{p}\) bits in a packet, given the pulse duration \(L\) and the PDR from (11) PDR\({}_{\phi}\), the positioning time is computed as  
\[t_{\phi}=\frac{2N_{p}L}{\text{PDR}_{\phi}}. \tag{14}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Scheme** & **Modifier** & **Symbol 1** & **Symbol 0** \\ \hline BPC & - & \(\sqrt{\frac{L}{2}}\theta_{1}(t)\) & \(\sqrt{\frac{L}{2}}\theta_{2}(t)\) \\ PWM & Additive & \(\sqrt{L}\theta_{1}(t)\) & \(\sqrt{\frac{L-1}{L}}\theta_{2}(t)+\theta_{1}(t)\) \\ PWM & Gain & \(\sqrt{L}\theta_{1}(t)\) & \(\sqrt{\frac{L}{2}}\theta_{2}(t)+\theta_{1}(t)\) \\ PWM & Power & \(\sqrt{L}\theta_{1}(t)\) & \((\sqrt{L-1})\theta_{2}(t)+\theta_{1}(t)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Signal space parameters for encoding schemesFor a single LED positioning system illustrated in Figure 5a with the radius of beam being \(R\) at the full angle of LED\({}_{2}\), if the positioning time \(t_{\phi}\) at a point with incidence angle \(\phi\) is \(t_{1\phi}\), the positioning time of all points on a circle at the radius \(r\) is given as \(2\pi rt_{1\phi}\). By geometry, \(r=h\tan\phi\). Therefore, the positioning time for all points in the LED beam is given as  
\[t_{1}=2\pi h\int_{0}^{\Phi}t_{1\phi}\tan\phi d\phi. \tag{15}\]  
The APT is the ratio of the total positioning time to the total number of points given by the area of the beam. Therefore the APT is  
\[\bar{t}_{1}=\frac{2h}{R^{2}}\int_{0}^{\Phi}t_{1\phi}\tan\phi d\phi. \tag{16}\]  
Given that \(R=h\tan\Phi\), \(\bar{t}_{1}\) can be written as  
\[\bar{t}_{1}=\frac{2}{h\tan^{2}\Phi}\int_{0}^{\Phi}t_{1\phi}\tan\phi d\phi. \tag{17}\]  
For the system with two overlapping LED beams, a probabilistic PDM process is introduced in Reference [66; 75] to handle collisions. In the region where two LED beams meet, the positioning time is taken as the time to receive packets from one of the LEDs twice. As a result of the stochastic nature of PDM, packet collision may or may not occur. If there are no collisions in transmitted packets, the positioning time at \(\phi\), \(t_{n\phi}\) varies between \(\frac{t_{p\phi}(t_{y}+t_{p})}{t_{p}}\) and \(\frac{2t_{p\phi}t_{y}}{t_{p}}\), where \(t_{y}\) is the PDM-based transmission cycle time and \(t_{p}\) is the encoded packet duration. By taking the average, the positioning time when no collision occurs is estimated as  
\[\bar{t}_{n\phi}=\frac{3t_{p\phi}(t_{y}+t_{p})}{2t_{p}}. \tag{18}\]  
If collisions occur, the positioning time can be written  
\[\bar{t}_{c\phi}=\bar{n}\bar{t}_{n\phi} \tag{19}\]  
where n is the number of cycles required to guarantee that a packet is received without collision and is given as \(\text{n}=\log_{2D}\left(1-0.9999\right)\) to guarantee a 99.99% chance that a packet is received given the probability of collision for two LEDs in the overlap region is \(2D\), where \(D<0.5\) is the transmission duty cycle given as \(\frac{t_{p}}{t_{y}}\).