## 3 Analytical Model  
The considered VLC system in this work is based on commercially available LEDs which can be used in the vehicles as head lights. The typical bandwidth of 20 MHz from an LED is continuously improving with the development of new techniques in LED manufacturing and advancements in fabrication processes. Figure 1 shows a typical application of VLC in an ITS where the head light of a rear vehicle is used to transmit information towards the tail light of a vehicle in front. Such a scenario is considered here to make a realistic model for the ITS because the concept of vehicle lights-based VLC for road safety is already in place for commercial implementation. The associated benefits include prevention of backside accidents and detection of the road surface by receiving data from the vehicles ahead. An LED unit can be composed of a single high-power LED or a group of multiple small LED units in a matrix from which all are operated using the same information signal. Transmission from a single LED unit in the form of SISO is accomplished by modulating the data signal \(x\) and transporting it through the channel with \(h\) time response. The received signal \(y\) at the photodetector can be composed of noise-affected transmitted signal, a reflected version and an interference signal from another light source. For improved channel estimation, multiple LED units can be considered, as shown in Figure 1, to transmit different versions of the same signal with different encoding. For a case of four LED units transmitting to a single receiver, MISO transmission can be performed to obtain a set of four different channel response elements. The MISO transmission can either be realized with transmission of same signal from multiple LED units, or by encoding each signal to de-correlate each transmitting unit from the other. In the former case, array gain is achieved at the transmitter, which compensates the propagation loss and misalignment of the VLC modules. Such arrangement is equally useful to implement selection diversity and the receiver can select the best available signal from the multiple transmitters. Encoding each signal increases the complexity of the system on both the transmitter and receiver end, but is really useful in the mitigation of interference signal and noise. For SISO transmission, where only one LED unit transmits, the received signal contains only one trace of information signal and retrieves channel state information from it. However, in the case of MISO where the four transmitted copies of the same signal experience different channel conditions, the estimation is much improved and results in optimized channel equalization. For \(m\) number of LED units at the transmitter end, where each transmitted signal \(X_{i}\) experiences a distinct frequency channel response \(H_{i}\), the received signal \(Y\) at the Photodetector (PD) (having a responsivity of \(R\)) can be given as  
\[Y=\sum_{i=1}^{m}\sqrt{\frac{P}{m}}RX_{i}H_{i}+A_{int} \tag{1}\]  
Figure 1: A \(4\times 1\) MISO configuration for the VLC-based ITS using the head and tail light of the vehicles. where \(P\) represents the total power drawn from the vehicle for the transmitter, \(A_{int}\) represents the interference signal and \(N\) denotes background noise. For a separation of \(D\) between the transmitter and receiver, the channel response (\(H\)) for the \(i_{\text{th}}\) LED will have a strong LoS component given by  
\[H_{i}=cos^{u}\theta cos\ \varphi\frac{u+1}{2\pi D^{2}} \tag{2}\]  
where \(\theta\) represents the angle for irradiance which depends on the orientation of the LED and \(\varphi\) denotes the angle of incident between the axes of transmitter and receiver. The value of \(\varphi\) remains between 0 and the receiver's semi-angle.