## 3 Proposed Mathematical Model of a Switched Beam Antenna with Beam Steering Abilities  
Focusing on optimizing the half power bandwidth (HPBW) and the first null bandwidth can improve the power concentration for the beam-to-beam management and V2V links. The concept of the HPBW and the FPBW can be illustrated in the accompanying Figure 2.  
The FNBW plays a role in the antenna radiation pattern, which can be considered as the angular breadth between the first side lobes or nulls. The resolution capacity of the antenna, which defines the system's ability to distinguish between two adjacent targets, is determined by the beam width. In the scenario of V2V communication, two targets are represented by two vehicles, and differentiation between them is important to prevent collisions. The factors that affect the beam width include:  
1. The shape of radiation patterns. 2. The antenna dimensions. 3. The wavelength. Furthermore, for power calculations of a fine beam, the HPBW and FNBW are both considered for concentrated beam power analysis. For V2V links, the SNR ratio should be maximized, followed by a high RSSI value. In this paper, a five-switched beam system with optimum and combined values of HPBW and FNBW is observed to enhance SNR and RSSI values. A comparative analysis will be conducted later on the amount of power transmission. The antenna gain function depends on beam directions, as reported in reference [29]. \(R(safe)\) indicates the distance the driver has left to prepare before entering a dangerous situation on the road. Additionally, the equation links the suggested system to the time to collision or safety time \((Tsafe)\). Therefore, the Effective Communication Range (RECR) of the antenna technology is crucial for the V2V communication link, and both \((Rsafe)\) and \((Tsafe)\) are essential for analysis. \[R=(Z-0.355)^{2}/99.8 \tag{1}\]  
\[Tsafe=(R(safe)-(R-D(ped))^{2}/Z \tag{2}\]  
\(D(ped)\), which refers to the pedestrian distance from a specific vehicle, was added. This is essential to address after the Uber fatality case [30]. \[Tsafe=(R(safe)-(R-D(ped))^{2}/Z \tag{3}\]  
Figure 2: Structural explanation of HPBW and FNBW lobes. The relationship between the transmitting and receiving end embedded by the beam switching antenna can be shown as:  
\[a=b(x1+x2+x3)+n \tag{4}\]  
where \(a\) represents the receiving antenna and \(b\) represents the wireless channel with different propagations. The power estimation of a specific beam is a crucial step and will be carried out by considering the transmitted data with respect to the channel propagation. \[Power=(X^{2}*B^{2}+N^{2})*HPBW+(X^{2}*B^{2}+N^{2})*FNBW \tag{5}\]  
Similarly, the received power with respect to beam switching is as follows:  
\[Power=Transmitted\ power*propagation*noise \tag{6}\]  
Beam communication is depicted in Figure 3.  
The channel coefficient can be determined by Friis transmission [31]. The angle of transmission with respect to the HPBW and FNBW is shown in Equations (7) and (8). \[Power(Angle) =g(i)\cdot 10^{0.1}(G(max)- \tag{7}\] \[R=(Z-0.355)^{2}/99.8 \tag{1}\]0 \[R=(Z-0.355)^{2}/99.8 \tag{1}\]1 (8) \[R=(Z-0.355)^{2}/99.8 \tag{1}\]2  
Similarly, the estimation of the probability of power with respect to the vehicle is as follows:  
\[R=(Z-0.355)^{2}/99.8 \tag{1}\]3  
For an appropriate beam switching antenna configuration, a Fourier analysis will be used:  
\[R=(Z-0.355)^{2}/99.8 \tag{1}\]4  
\[R=(Z-0.355)^{2}/99.8 \tag{1}\]5  
\[R=(Z-0.355)^{2}/99.8 \tag{1}\]6 Proposed beam-to-beam power connection. The Doppler frequency \(F(doppler)\) depends on the relative velocity \(V\) between the transmitter \(Tx\) and receiver \(Rx\), the speed of light \(Co\), and the carrier frequency \(Fo\) for fast-moving vehicles with all developed angles with respect to the power of the beam and fine beams \(FB\). \[ \begin{array}{ll}F(doppler)&=[(V/Co)*Fo*Cos(\alpha+\beta+\\ &+\gamma)*FB-P(beamswitch)]\end{array}  \tag{12}\]  
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