# A Novel Localization Technique Using Luminous Flux
## 3 Power Measurement for Both LoS and NLoS Models  
Considering LEDs as emitters, modeled as Lambertian radiation for a collection of wavelengths with axial symmetry, a model number n for each lobe of radiation is expressed as follows.  
\[R_{E}(\varphi,n,\lambda)=\frac{n+1}{2\pi}P_{E}(\lambda)\cos^{n}(\varphi) \tag{16}\]  
In the above equation, the nominal power \(P_{E}\) is obtained by integrating \(P_{E}(\lambda)\) over the wavelength intervals. The value of \(\varphi\) is \(-\frac{\pi}{2}\leq\varphi\leq\frac{\pi}{2}\). However, regarding power estimation based on the rays, which propagate around the room and bounce off the walls with a probability density function in a particular direction, his is summed up to obtain the impulse response. The total impulse response is composed of the primary and secondary reflections, with the amplitude varying upon reflections received in X, Y, and Z planes. Figure 1a,b show the spread of energy or the received power in the Y direction and the surface map of the reflections.  
Figure 1: Distribution of received power in each plane.  
Figure 1a has no reflection state and the corresponding RSS is measured in the (X,Y) plane. The Z-direction in the RSSI measurement and (X,Y) are given for interpolation of points, as shown in Figure 1b. Figure 1c expresses the 2D presentation of the optical power at the receiver. Typically, the positions of LEDs are parallel to the receiver plane and share a common clock, which transmits their three-dimensional coordinate information. The impetus for this research comes from the analysis of positional error using PDoP impulse response. It can be empirically shown that the signal-to-noise ratio (SNR) at the center of the room is more pronounced against the reflections. Hence, the effect of the reflections is small and weak at the center and does not contribute to the positioning accuracy.  
Moreover, the light reflections of LEDs are taken into account (Single, 10 reflections). Figure 2 below shows the reflected or bounced signals measured after a single reflection and 10 reflections, respectively. As seen, the reflections cause changes in the received power, demonstrating a positional error.  
The 2D views of the single reflection and ten reflections for the power spread are shown in Figure 2b,d. The power is focused in the first quadrant, as seen in Figure 2d while Figure 2b illustrates a more uniform spread. In order to visualize a uniform spread rather than a cross-sectional view we scaled the room dimensions from \(6\times 6\) to \(24\times 24\). This represents a scaled version of the room dimensions for visualization purposes.  
Figure 2: Analysis of a single reflection and 10 reflections.  
From Figure 2, it can be seen that the power intensity distribution becomes more uniform at the edges and corners. As reflections increase the received power is reduced, creating an error in positioning. It can be seen that positioning error across the room is low, especially in the corner. The error is pronounced because of the low SNR. Comparing Figure 2a,b, we find that the reflections have a lower SNR, as seen in the concentration between (0,1) values. The receiver plane is closer to the floor, hence the improvement in determination accuracy. Comparing the single reflection versus ten reflections shows the spread in energy in the (X,Y) plane, as seen in Figure 2a,b. The spread of energy is non-uniform in the case of higher reflections, and hence is useful for accuracy calculation.