## 3 Analytical Model
The index \(u\), in terms of \(\varphi\) can be elaborated as  
\[u=-ln(2)/ln(cos(\varphi)) \tag{3}\]  
Even for signal with no interference, the effect of atmospheric attenuation coefficient (\(\alpha\)) and field of view (FOV) \(\theta_{FOV}\) on the received power (\(P_{r}\)) has to be considered. The general equation for the \(P_{r}\) becomes  
\[P_{r}=\frac{e^{-\alpha D}P}{\left(D\theta_{FOV}/2\pi\right)^{2}} \tag{4}\]  
where several factors still need to be considered for VLC transmission. The accuracy in the estimation of received power can be increased by a detailed model [36] for the LOS signal, given by  
\[P_{r,\ eff}=P_{r}\frac{G(\varphi)\left(A_{eff}cos\varphi\right)}{D^{2}2\pi \int_{0}^{\pi}G(\varphi)sin\varphi d\varphi} \tag{5}\]  
In Equation (5), \(G(\varphi)\) denotes beam pattern for transmitter and \(A_{eff}\) is the effective area of the photodetector. The \(P_{r,eff}\) is the improved version of the estimated power and will be used in the simulation analysis. The signal-to-noise ratio (SNR) for the SISO transmission, for a noise spectral density of \(\sigma_{N}^{2}\), can be estimated as  
\[SNR=\frac{\frac{P}{m}R^{2}\left|\sum_{i=1}^{m}H_{i}\right|^{2}}{\sigma_{N}^{2}} \tag{6}\]  
Given the condition in Equation (6), the data rate (\(\Upsilon\)) is directly related to the bandwidth of the transmitter (\(B\)) for the case of on-off-keying (OOK). To increase the data rate by using an advanced modulation scheme such as Pulse Amplitude Modulation (PAM), in which the intensity of the carrier pulses can be changed with high speed, the tradeoff exists between the PAM level (\(L\)) and the power penalty caused due to the increase in intensity-modulated levels. For using L-PAM, which encodes \(b_{sym}\) number of bits to a symbol, the penalty in terms of power (\(P_{p}\)) can be given as  
\[P_{p}=\frac{L-1}{\sqrt{b_{sym}}} \tag{7}\]  
This shows that increasing the spectral efficiency by a factor of two, by using a higher level PAM scheme, will require approximately 6.5 dB more power. This does not include the power penalty caused by the non-linearities due to the higher order PAM. Due to this reason, only OOK has been used in this work to present a low-complex and highly linear system for ITS. To consider a reflected version of the signal which exists due to the reflection from the road surface, street pole or a sign-board, a diffused signal with a modified road surface reflection profile [37] can be used to model the reflected (non-LOS) signal as  
\[X_{ref}=Xcos(\Phi)\frac{\mu}{\pi} \tag{8}\]  
In Equation (8), \(\Phi\) denotes the angle of diffused reflection and \(\mu\) denotes the amount of reflectivity. The H represents a \(4\times 4\) channel matrix which is evaluated using a training sequence (TS) before the actual transmission. The received signal \(Y\) and the pre-known TS can be used at the receiver end to estimate the channel using a simple zero-forcing equalizer performing the \(Y_{i}\)/TS operation (where i represents the number of transmitting units). The inverse of the estimated channel matrix can be therefore used to recover the data from the channel effects. The channel matrix is estimated using a training sequence before the actual transmission. To achieve \(4\times 1\) MISO, the diversity is implemented on the transmitter side by coding the information signal into four different versions. Each version is transmitted through an individual LED unit. Even if the complexity of the system is reduced by transmitting same signal through each LED unit, array gain [38] can be achieved using the channel time response \(h(t)\) (which can be estimated from Equation (2) for individual units) for all \(m\) units as  
\[Gain=\frac{1}{\sqrt{m\sum_{i=1}^{m}\left\|h_{i}(t)\right\|^{2}}} \tag{9}\]  
With MISO processing, the Equation (6) can be modified as  
Figure 1:0  
where \(W\) represents the coding matrix which is applied to the information signal at the transmitter side.