# A Millimeter-Wave Radar Tunnel Obstacle Detection Method Based on Invalid Target Filtering
## 2 Materials and Methods
### Theoretical Framework  
#### 2.2.1 Principle of Millimeter-Wave Radar Target Detection  
The millimeter-wave radar used in this study adopts FMCW (frequency-modulated continuous wave) technology, which can measure both distance and speed and has obvious advantages in short-range measurement [21; 22]. The radar transmit signal is a solid blue line, the radar received signal of a moving target is a red dashed line, the signal of a stationary target is a yellow dashed line, \(tc\) is half of the sweep period, \(fc\) is the sweep bandwidth, and \(\tau\) is the time elapsed from the transmission of the signal to the reception of the return wave. Let \(f_{S}(t)\) and \(f_{E}(t)\) be the frequency variation functions of the transmitted signal and the echo signal, respectively, then the frequency of the differential signal \(f_{b}(t)\) is  
\[f_{b}(t)=f_{S}(t)-f_{E}(t) \tag{1}\]  
Based on the geometric relationships in Figure 1, using the similarity triangle theorem, it follows that:  
\[\frac{fb}{\tau}=\frac{fc}{tc} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Long Distance** & **Long Distance** & **Short Range** & **Short Range** \\  & **Narrow Vision** & **Narrow Vision** & **Wide View** & **Wide View** \\  & **Angle (\(\pm\)9\({}^{\circ}\))** & **Angle (\(\pm\)4\({}^{\circ}\))** & **Angle (\(\pm\)60\({}^{\circ}\))** & **Angle (\(\pm\)40\({}^{\circ}\))** \\ \hline Detection distance/m & 0.2–150 & 0.2–250 & 0.2–20 & 0.2–70 \\ Horizontal angle resolution/(\({}^{\circ}\)) & 3.2 & 1.6 & 12.3 & 4.5 \\ Horizontal angle accuracy/(\({}^{\circ}\)) & \(\pm\)0.3 & \(\pm\)0.1 & \(\pm\)5 & \(\pm\)1 \\ Detection range resolution/m & 1.79 & 1.79 & 0.39 & 0.39 \\ Detection range accuracy/m & \(\pm\)0.4 & \(\pm\)0.4 & \(\pm\)0.1 & \(\pm\)0.1 \\ Speed resolution/(km.h\({}^{-1}\)) & 0.37 & 0.37 & 0.43 & 0.43 \\ Speed accuracy/(km.h\({}^{-1}\)) & & \(\pm\)0.1 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of ARS 408-21 XX millimeter-wave radar. The relative distance \(R\) is calculated as  
\[R=\frac{\tau c_{0}}{2} \tag{3}\]  
where \(c_{0}\)--electromagnetic wave propagation speed in free space, about \(3\times 10^{8}\) m/s. The relationship between the relative distance and the frequency of the differential signal can be obtained from Equations (2) and (3):  
\[R=\frac{c_{0}t_{c}f_{b}}{2f_{c}} \tag{4}\]  
The relative distance \(R\) of the target is proportional to the frequency \(f_{b}\) of the differential signal. Therefore, the relative distance \(R\) of the target can be determined by measuring the frequency of the differential frequency signal. When there is relative motion between the target obstacle and the radar, the Doppler effect occurs, and the frequency of the reflected signal when the relative motion occurs is the Doppler frequency [23]. The Doppler frequency \(f_{D}^{(2)}\) is calculated as  
\[f_{D}^{(2)}=\frac{2f_{S}v_{r}}{c_{0}} \tag{5}\]  
The \(f_{b1}\) and \(f_{b2}\) in Figure 1 are the difference frequency functions obtained when the sent signal and the received signal are at the rising and falling edges, respectively, and the difference frequency functions at this time can be found by combining with Equations (4) and (5)  
\[ \begin{cases}f_{b1}=\frac{f_{c}}{f_{c}}\cdot\frac{2R}{c_{0}}-\frac{2f_{S}v_{r }}{c_{0}}\\ f_{b2}=\frac{f_{c}}{f_{c}}\cdot\frac{2R}{c_{0}}+\frac{2f_{S}v_{r}}{c_{0}}\\ \end{cases}  \tag{6}\]  
The following equations can be solved to obtain specific values for both \(v_{r}\) and \(R\). \[ \begin{cases}v_{r}=\frac{f_{c}}{f_{c}\cdot f_{S}}\cdot R-\frac{1}{2}\cdot\frac {c_{0}}{f_{S}}\cdot f_{b1}\\ v_{r}=\frac{1}{2}\cdot\frac{c_{0}}{f_{S}}\cdot f_{b2}-\frac{f_{c}}{f_{c}\cdot f _{S}}\cdot R\\ \end{cases}  \tag{7}\]  
When measuring the relative distance and speed of multiple targets, the frequency of the transmitted signal needs to be processed using a moving target detection technique [24].