# Optical Boundaries for LED-Based Indoor Positioning System
## 5 Results and Discussions
### Effect of Optical Boundaries on Positioning Error  
Using Algorithm 1, the variation of positioning error for increasing beam radius and number of LEDs is presented in Figure 7. It is observed that the error in positioning is reduced by increasing the number of LEDs. For 1 LED, 2 LEDs, 8 LEDs, 16 LEDs, the minimum positioning error is 1907.2 mm, 1460.5 mm, 626.44 mm, and 230.99 mm, respectively. The characteristics plot in Figure 7 shows an optimal point for performance between regions of low beam radius and regions of high beam radius. This is because, at low beam radius, there are no overlaps between the LED beams and the probability that the receiver is outside the region of coverage of the beams is higher. As the low beam radius increases, this probability reduces, so the positioning error also reduces. As the overlap starts, the positioning error reduces further until the performance is optimal. However, as the beam radius continues to increase, the overlap regions also keep increasing and the non-overlapping regions reduce until every part on the room is identified as one single overlap region and the positioning error is high.  
The trend in Figure 8 shows that the minimum positioning error reduces as the number of LEDs represented as the LED exponent increases. It is deduced that the positioning error reduces to 27.6 mm at an LED exponent of 10 which corresponds to 1024 LEDs in the room. Perhaps in some scenarios, installing 1024 uniquely identifiable LEDs in a room is not feasible and will increase installation cost. This increased installation cost is prevented by choosing the desired accuracy based on specific applications. For instance, for human positioning, since the average shoulder breadth of a person is between 450 mm and 600 mm [87], a system with this range of positioning error will prove accurate  
Figure 6: Experimental setup for noise determination. **A**: Transmitter electronic module, **B**: Transmitter LED on stand, **C**: Power supply unit, **D**: Oscilloscope for measurement, **E**: Receiver electronics module, **F**: Receiver PD on stand.  
enough. Therefore, from Figure 8, the number of LEDs required for accurate human positioning is between 8 and 16, which is not only feasible but also keeps the system inexpensive.  
The information related to the number of LEDs and beam radius that provide the desired positioning accuracy, given in Figures 7 and 8, is used to estimate the correct threshold angle using (24) for minimal positioning delays. For practical purposes, this threshold angle value is used to determine the desired half angle for an LED using (13). In the design of an LED-based indoor positioning system, the available number of LEDs and desired positioning error can be maintained while the LED type is selected based on the desired threshold angle that prevents delays as presented in subsequent sections.  
Figure 8: Representation of minimal positioning error for increasing number of LEDs presented as the LED exponent factor as defined in (23)  
Figure 7: Positioning error as the beam radius and the number of LEDs in a room are increased.