# The UWB Radar Application in the Aviation Security Systems
## 2 Materials and Methods
### Localization of the Target by the Straight-Line Method for Two Independent Radar Sets
#### 2.1.4 Application Procedure for the Straight-Line Method Usage
\begin{array}{c}u_{0}=v_{2}v_{10}-v_{4}^{2},\\ u_{1}=v_{0}v_{10}+v_{2}(v_{7}+v_{9})-2v_{3}v_{4},\\ u_{2}=v_{0}(v_{7}+v_{9})+v_{2}(v_{6}-v_{8})-v_{3}^{2}-2v_{1}v_{4},\\ u_{3}=v_{0}(v_{6}-v_{8})+v_{2}v_{5}-2v_{1}v_{3},\\ u_{4}=v_{0}v_{5}-v_{1}^{2}.\end{array}  \tag{10}\]  
Calculation of coordinates-y:  
Calculation of polynomial roots \(R(y)=0\),  
\[y_{sur}=roots(u_{4},u_{3},u_{2},u_{1},u_{0}). \tag{11}\]  
Selection of real roots of the equation and elimination of multiple roots \(\mathbf{y}\)  
\[ \begin{array}{c}if\_imag(y\_sur(i))=0,\\ y\_sur\_real=y\_sur(i),\\ y\_sur\_real\_un=unique(y\_sur\_real),\\ y\_ru=y\_sur\_real\_un.\end{array}  \tag{12}\]  
Calculation of coordinates x:  
Calculation of coefficients of equations  
\[ \begin{array}{c}Q_{i}(x,y),i=1,2\\ z_{0}=a_{11}^{(i)}(y\_ru)^{2}+b_{1}^{(i)}(y\_ru)+c^{(i)},\\ z_{1}=2a_{01}^{(i)}(y\_ru)+b_{0}^{(i)},\ z_{2}=a_{00}^{(i)}.\end{array}  \tag{13}\]  
Calculation of roots of equations:  
\[ \begin{array}{c}Q_{i}(x,y),i=1,2\\ x_{sur}=roots(z_{2},z_{1},z_{0}).\end{array}  \tag{14}\]  
Selection of real roots of the equation and elimination of multiple roots x  
\[ \begin{array}{c}if\_imag(x\_sur(i))=0\\ x_{sur\_real}=x_{sur(i)}\\ x_{sur\_real}=unique(x_{sur\_real})\\ x\_ru=x\_sur\_real\_un.\end{array}  \tag{15}\]  
In the examined cases of calculation in each individual case (after elimination of imaginary and multiple), two real intersections remained. The position of lines No. 3, 4, 5, and 6 at the time of observation (\(\tau=100\)) is shown in Figure 9.  
Creating an Optimal Line  
n pairs of \([x_{i},y_{i}]\),\((i=1,2,\ldots n)\), values of equal accuracy (weight) \((n>2)\),  
which should satisfy the equation \(y=a*x+b\) of the alignment line [28; 29] were measured. The substitution of  
\[a*x+b-y=v_{i} \tag{16}\]  
after fulfilling the condition:  
\[[v*v\to min] \tag{17}\]  
coefficients \(a,b\) are determined from the equations:  
\[ \begin{array}{c}n*b+a[x]=[y]\\ b[x]+a[x*x]=[y*x].\end{array}  \tag{18}\]From the above relations, it is possible to create the optimal (straight) line shown in Figure 10. From the conclusions of the study of the course of pairs of ellipses, created by combinations of ellipses of different sets at selected moments (times) of observation (\(\tau\)), it is clear that a group of intersections was formed inside the observation area (partially visible in Figures 7-9), and the other group of intersections was formed outside the observation area. We will consider the first group as a realistic target. Since the distribution of intersections of ellipses around the alignment line must be (due to the nature of the line) in the whole observed area similar to Figure 11, to determine the target point, it is appropriate to use the center of gravity of combined, real intersections of ellipses). The intersections of ellipses created by a combination of two ellipses, each ellipse created by a different set RS, represent a potential target. I will mark the targets Tar1, Tar2... TarN. Calculation of the center of gravity of the real intersections of ellipses is listed in (28, 29):  
Line direction:  
\[k1=\frac{\mathit{TarN}-2(1,2)-\mathit{TarN}-1(1,2)}{\mathit{TarN}-2(1,1)- \mathit{TarN}-1(1,1))}. \tag{19}\]  
Line equations:  
\[xp1=\mathit{TarN}-1(1,1):0.1:\mathit{TarN}-2(1,1)), \tag{20}\] \[yp1=(k1*(xp1-(\mathit{TarN}-1(1,1))))+(\mathit{TarN}-1(1,2)).\]  
Line direction:  
\[kN-2=\frac{(\mathit{TarN}(1,2)-\mathit{Tar1}(1,2)}{\mathit{TarN}(1,1)- \mathit{Tar1}(1,1))}. \tag{21}\]  
Line equations:  
\[xpN-2=\mathit{Tar1}(1,1):0.1:\mathit{TarN}(1,1)), \tag{22}\] \[ypN-2=(kN-2*(xpN-2-(\mathit{Tar1}(1,1))))+(\mathit{Tar1}(1,2)).\]  
Calculation of intersections of lines:  
\[ \begin{array}{l}xgc=\frac{k1*\mathit{TarN}-1(1,1)+\mathit{Tar1}(1,2)- \mathit{TarN}-1(1,2)-kN-2*\mathit{Tar1}(1,1)}{-kN-2+k1},\\ ygc=(k1*(xgc-(\mathit{TarN}-1(1,1))))+(\mathit{TarN}-1(1,2)).\end{array}