## 4 The Geometries of the Proposed Transmitters
### \(3\times 3\) Square Array  
As depicted in Figure 4, a square LED array of size \(3\times 3\) with a spacing of \(s\) is proposed, where \(s\) is initially set to \(1\) cm. This value was chosen arbitrarily, and it will be shown later that variations in \(s\) do not significantly impact system performance. Each LED element, denoted as \(e_{i,j}\), has a field of view of \(\phi_{\frac{1}{2}}^{\prime}\), and is aligned along an orientation vector \(\mathbf{O}_{i,j}\), with each vector intersecting the \(y=d_{0}\) plane at point \(e_{i,j}^{\prime}\), where \(d_{0}\) is a design parameter. The \(y=d_{0}\) plane is defined as the plane on which the two arrays formed by the points \(e_{i,j}^{\prime}\) from the two transmitters are tangent, as shown in Figure 5. In other words, the outermost points of the two arrays intersect each other. For this condition to be met, the elements \(e_{i,j}^{\prime}\) should be separated by \(D/2\) on this plane. The naming convention for the array elements is illustrated in the matrix below.  
\[\begin{bmatrix}e_{1,1}&e_{1,0}&e_{1,-1}\\ e_{0,1}&e_{0,0}&e_{0,-1}\\ e_{-1,1}&e_{-1,0}&e_{-1,-1}\end{bmatrix} \tag{19}\]  
The location of the element \(e_{i,j}\) relative to the center element \(e_{0,0}\) is given by \(\mathrm{P}(sj,0,si)\). To achieve the geometry illustrated in Figure 4, the orientation vector \(\mathbf{O}_{i,j}\) with starting point \(\mathrm{P}(sj,0,si)\) (location of \(e_{i,j}\)) should intersect the \(y=d_{0}\) plane at \(\mathrm{P}(\frac{D}{2}j,d_{0},\frac{D}{2}i)\) (location of \(e_{i,j}^{\prime}\)). Thus, the required vector can be found using  
\[\mathbf{O}_{i,j}=\langle(\frac{D}{2}-s)j,d_{0},(\frac{D}{2}-s)i\rangle \tag{20}\]  
Figure 3: The geometry for the single light-emitting diode (LED) case.  
The reason why the distance \(d_{0}\) and half FOV \(\phi^{\prime}_{\frac{1}{2}}\) are determined as the design parameters is that a narrower half FOV \(\phi^{\prime}_{\frac{1}{2}}\) yields a larger SNR, and the value of \(d_{0}\) conveniently adjusts the orientation vectors (refer to Equation (20)) of the LEDs in the array. The orientation vectors are crucial since the taillights of a vehicle are essentially used for lighting purposes, not as transmitters in the communication link. Therefore, we need to ensure that they can illuminate a certain area.  
We assume that each of the two LEDs, one for Tx1 and one for Tx2, in the single-LED case has a power \(P_{0}\) and an intensity \(I_{0}\). To keep the power dissipation of both proposed transmitter configurations at the same level, the power of each LED in the square array transmitter is determined as \(P_{0}/9\). Since the intensity is proportional to the power level divided by the squared tangent of the half FOV, the intensity \(I_{1}\) of each LED of the \(3\times 3\) square array is set as follows:  
\[I_{1}=\frac{1}{9}\frac{\tan^{2}\phi_{\frac{1}{2}}}{\tan^{2}\phi^{\prime}_{ \frac{1}{2}}}I_{0} \tag{21}\]  
The initial values of the design parameters for the two transmitters are listed in Table 2. The design parameter \(d_{0}\) is set at 130 cm, which is equal to the y-distance where the  
Figure 4: The \(3\times 3\) square-shaped LED array geometry.  
Figure 5: The \(3\times 3\) square arrays formed by the points \(e^{\prime}_{i,j}\) on the \(y=d_{0}\) plane.  
two circles illustrated in Figure 3 become tangent (found to be 130 cm using Equation (18)). Using Equations (2) and (3), and the values given in Table 2, the irradiance distributions in the plane \(y=1\) m are plotted using MATLAB software for the two proposed configurations. The results are given in Figure 6a,b.