## 3 System Description
### System Model
#### 3.2.4 Photometric and Radiometric Quantities  
Luminous intensity is an intrinsic property of LEDs which is governed by the irradiance angle (\(\psi\)). It falls off as the cosine of this angle such that [5]:  
\[I(\phi)=I_{0}cos(\phi) \tag{11}\]  
where \(I_{0}\) is the center luminous intensity of the LED.  
Illuminance is derived from the luminous intensity. It is defined as the photometric flux incident on a surface per unit area and is measured in lux. Illuminance can have vertical as well as horizontal components. However, it is mainly characterized by the latter.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Room dimensions & \(5\times 5\times 3\) m\({}^{3}\) \\ \hline Reflection coefficients (all walls) & 0.8 \\ \hline Area of differential element & 0.0017 m\({}^{2}\) \\ \hline \multicolumn{2}{c}{**For Transmitter**} \\ \hline Center luminous intensity per LED chip & 0.73 Cd \\ \hline Semi-angle at half-power & 60\({}^{\circ}\) \\ \hline Lambert’s order & 1 \\ \hline LED luminous efficacy & 90 lumens/watt \\ \hline Number of LEDs per array & 2500 (\(50\times 50\)) \\ \hline Azimuth angle & 0 \({}^{\circ}\) \\ \hline Elevation angle & \(-90\)\({}^{\circ}\) \\ \hline \multicolumn{2}{c}{**For Receiver**} \\ \hline Field-of-View & 90\({}^{\circ}\) \\ \hline Physical Area & 1 cm\({}^{2}\) \\ \hline Gain of an optical filter & 1 \\ \hline Refractive index of lens & 1.5 \\ \hline Height (from the floor) & 0.85 m \\ \hline Bandwidth & 20 MHz \\ \hline Photodiode responsivity & 0.54 A/W \\ \hline Fixed Capacitance & 112 pF/cm\({}^{2}\) \\ \hline Open loop gain & 10 \\ \hline Temperature & 300 K \\ \hline Noise Bandwidth factor (I\({}_{2}\)) & 0.562 \\ \hline Noise Bandwidth factor (I\({}_{3}\)) & 0.0868 \\ \hline FET transconductance & 30 mS \\ \hline FET noise factor & 1.5 \\ \hline Dark current (I\({}_{d}\)) & 2 pA \\ \hline N\({}_{x}\) & 100 \\ \hline N\({}_{y}\) & 100 \\ \hline N\({}_{z}\) & 90 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of Simulation Parameters [5; 11].  
Mathematically, for any point (x,y,z) in 3D space, horizontal illuminance for a single source in an array of transmitters \(\{S=S_{1},S_{2}\ldots S_{n}\}\) can be expressed as [5]:  
\[E_{r}^{n}(x,y,z)=(I_{o}\cos^{m}\phi\cos\psi)/d_{n}^{2} \tag{12}\]  
where \(\psi\) is the angle with which the ray is incident on the receiver, \(d\) represents the distance from the light source to the PD, and \(m\) is Lambert's order, which defines the directivity of the source, given by:  
\[m=\frac{ln2}{ln\cos\phi_{\frac{1}{2}}} \tag{13}\]  
Here, \(\phi_{\frac{1}{2}}\) is the semi-angle at half power. Note that, for a transmitter and receiver lying perpendicular to the ceiling, \(\phi=\psi\)[34] and \(cos\phi=h/d\). For multiple transmitters, the total horizontal illuminance, \(E_{T}\), is given by:  
\[E_{T}(x,y,z)=\sum_{n=1}^{N_{LED}}E_{r}^{n} \tag{14}\]  
where \(N_{LED}\) denotes the total number of LEDs. Interestingly, most of the studies reported in the literature, including the pioneering work by [PERSON] et al. [11; 13; 16], have considered luminous intensity \(I_{0}\) (0.73 cd / LED) and optical power \(P_{t}\) (20 mW/ LED) as independent quantities to perform photometric and radiometric computations for illuminance and received power, respectively. However, in this study, instead of assuming the arbitrary value of 20 mW, a simplified relationship between \(P_{t}\) and \(I_{0}\) has been employed to establish a correspondence between these quantities for more realistic calculations. \(P_{t}\) is related to \(I_{0}\), as follows:  
\[P_{t}=I_{0}\Omega/\eta \tag{15}\]  
where \(\eta\) is the luminous efficacy of LEDs in lumens per watt and \(\Omega\) is the solid angle in steradian, given by:  
\[\Omega=2\pi(1-\cos(\theta/2)) \tag{16}\]  
Here, \(\theta\) is the apex angle of the cone and it is given by \(\theta=2\phi\). The power obtained from this relationship against 0.73 cd is approximately 63 mW per LED.