# Blind Spot Detection Radar System Design for Safe Driving of Smart Vehicles
## 2 FMCW Radar Principle
To interpret the chirp signal in Figure 2, the beat frequency of the up-chirp and down-chirp is given by Equation (3):  
\[f_{b,up}=\tau\cdot\frac{B}{T}-\frac{2v}{\lambda}=\frac{2 RB}{cT}-f_{d} \tag{3}\]  
\[f_{b,dn}=\tau\cdot\frac{B}{T}+\frac{2v}{\lambda}=\frac{2 RB}{cT}+f_{d},\left\{ \begin{array}{l}f_{d}=Doppler\ frequency\ [\mathrm{Hz}]\\ \lambda=wavelength\ [\mathrm{m}]\\ v=target\ velocity\ [\mathrm{m/s}]\end{array} \right. \tag{4}\]  
In a moving target environment, Range can be defined by Equation (5), and Velocity can be defined by Equation (6); additionally, the wavelength for interpreting the velocity can be defined by Equation (7):  
\[R=\frac{cT}{4B}\Big{(}f_{b,up}+f_{b,dn}\Big{)} \tag{5}\]  
\[v=\frac{\lambda}{4}(f_{b,up}-f_{b,dn}) \tag{6}\]  
Figure 2: Beat Frequency for Moving Target. \[\lambda=\frac{c}{f_{c}} \tag{7}\]  
In the design of a BSD radar system, the fundamental FMCW principle has been explained using Figures 1 and 2, and Equations (1)-(7). This paper provides a more detailed explanation of the fundamental principle of the FMCW radar signal used in the designed system. After setting arbitrary parameters for the Multiple Stationary Targets Case [14] and Multiple Moving Targets Case [15], by using MATLAB simulation tools [16; 17], this paper examines the form of the beat frequency based on the Tx signal in the time domain and the form of the signal's spectrum in the frequency domain. The results are presented below. Table 1 shows the arbitrary parameters used to represent Multiple Stationary Targets, while Table 2 shows the arbitrary parameters used to represent Multiple Moving Targets. Three targets were set, and the Time Delay, Doppler Frequency, Amplitude, and Beat Frequency for the simulation were defined in each table. Figure 3 shows the simulation results for multiple stationary targets, assuming the presence of three targets. As all three targets are stationary, the frequencies of \(f_{b,up}\) and \(f_{b,dn}\) are identical. The characteristics of the beat frequency can be observed in the frequency domain, with frequency spectra appearing at the positions of 5 [Hz], 10 [Hz], and 20 [Hz] defined in Table 1.  
Figure 4 shows the simulation results for multiple moving targets, assuming the presence of three targets. Since these targets are in motion, \(f_{b,up}\) and \(f_{b,dn}\) have different frequencies. In the frequency domain, the characteristics of \(f_{b,up}\) of Targets 1, 2, and 3 can be observed, with frequency spectra appearing at 4 [Hz], 7 [Hz], and 15 [Hz] as defined in Table 2. Likewise, the characteristics of \(f_{b,dn}\) can be observed with spectra appearing at 6 [Hz], 13 [Hz], and 25 [Hz]. Therefore, the equations proposed in this paper for the design of FWCM can be validated. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target** & **Parameter** & **Value** & **Unit** & **Beat Frequency (Cal.)** \\ \hline \multirow{3}{*}{Target 1} & Time Delay (\(\tau\)) & 0.1 & Sec & \(f_{b,up}=\left|\tau\left(B/T\right)+f_{d}\right|=5\) [Hz] \\  & Doppler Freq. (\(f_{d}\)) & 0 & Hz & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=5\) [Hz] \\  & Amplitude & 1 & Volt & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=5\) [Hz] \\ \hline \multirow{3}{*}{Target 2} & Time Delay (\(\tau\)) & 0.2 & Sec & \(f_{b,up}=\left|\tau\left(B/T\right)+f_{d}\right|=10\) [Hz] \\  & Doppler Freq. (\(f_{d}\)) & 0 & Hz & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=10\) [Hz] \\  & Amplitude & 0.5 & Volt & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=10\) [Hz] \\ \hline \multirow{3}{*}{Target 3} & Time Delay (\(\tau\)) & 0.4 & Sec & \(f_{b,up}=\left|\tau\left(B/T\right)+f_{d}\right|=20\) [Hz] \\  & Doppler Freq. (\(f_{d}\)) & 0 & Hz & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=20\) [Hz] \\ \cline{1-1}  & Amplitude & 0.3 & Volt & \(f_{b,dn}=\left|\tau\left(B/T\right)-f_{d}\right|=20\) [Hz] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for Multiple Stationary Targets.