# 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
#### 2.1.3 Principle of the Straight-Line Method
The coordinates of the intersections of the ellipses \(X_{12 RS2}(t)\) and \(Y_{12 RS2}(t)\) for the radar set RS2 are determined using the method of finding the intersections of the ellipses as follows:  
\[Y_{12 RS2}(t)=((p_{22}\cdot d_{21}-p_{21}\cdot d_{22}))/(d_{22}(Y_{TR2}-Y_{21}) +d_{21}(Y_{22}-Y_{TR2})), \tag{5}\]  
\[X_{12 RS2}(t)=(((p_{21}+Y_{12 RS2}(t))\cdot(Y_{TR2}-Y_{21}))^{2}/d_{21}-(Y_{12 RS2 }(t)-Y_{TR2})^{2})^{0.5}+Y_{TR2}, \tag{6}\]  
where \(d_{22}\) is the distance between the transmitter, the target, and the antenna of the second receiver of the RS2 kit; \(d_{21}\) is the distance between the transmitter, the target, and the antenna of the first receiver of the RS2 set; \(p_{22}=0.5\). \((Y_{22}^{\ 2}-Y^{\ 2}{}_{TR2}-d^{\ 2}{}_{22})\); \(p_{21}=0.5\). \((Y^{2}{}_{21}-Y^{\ 2}{}_{TR2}-d^{\ 2}{}_{21})\); \(Y_{21}\) and \(Y_{22}\) are the coordinates of antennas of RS2 set receivers; \(Y_{TR2}\) is the coordinate of antenna of RS2 set transmitter. The results of determining the intersections of the ellipses at the selected observation time \(\tau_{\rm K}\) are shown for the radar set RS2 in Figure 3, where the expression E\({}_{3}\)R\({}^{\rm 82}\) represents ellipse No. 3 of the radar set RS2, with foci in the position of the transmitter Tx\({}_{\rm B}\) and in the position of the receiver RxB1. The analog expression \(\text{E}_{4}\)RS2 represents ellipse No. 4 in radar set RS2, with foci in the position of the transmitter \(\text{Tx}_{\text{B}}\) and in the position of the receiver \(\text{Rx}_{\text{B2}}\). The line drawn through the intersections of the ellipses represents the 2 nd line using the method of lines. The number of real intersections of ellipses in each of the investigated points was = 4. Other ellipse intersections can be found in this way. Finding the intersections of ellipses created by combinations of ellipses of different sets and their use is described in [29]. Figure 4: Position of lines of RS1 and RS2 sets in the points (**a**) \(\tau\) = 55 and (**b**) \(\tau\) = 100. Figure 5: Position of lines of RS1 and RS2 sets in the points (**a**) \(\tau\) = 150 and (**b**) \(\tau\) = 225. Figure 3: Intersections of RS2 ellipses. Figure 8: Combined intersections of ellipses: (**a**) E2 sets RS1 and E3 sets RS2 and (**b**) E2 sets RS1 and E4 sets RS2. Figure 6: The intersection of the lines joining the intersections of the ellipse RS1 and RS2 sets in the point \(\tau=150\). \[Y_{12 RS1}(t)=-((p_{11}+X_{12 RS1}(t)\cdot X_{11})^{2}-X_{12 RS1}(t)^{2})^{0.5}, \tag{4}\]0 Combined intersections of ellipses: (**a**) E1 sets RS1 and E3 sets RS2 and (**b**) E1 sets RS1 and E4 sets RS2. Let us investigate the mutual position of lines No. 1 and line No. 2, which we stated in previous paragraphs, at different times of observation (\(\tau\)). As can also be seen in Figures 4 and 5, in the observation area (room according to the above scenario), there is a straight line. Vertical and horizontal lines in most positions (\(\tau\)) intersect. The intersection of the lines connecting the intersections of the ellipses of the sets RS1 and RS2 is obtained by the least squares method [29; 36]. Using this method, we estimate the vector \(\hat{p}=\left[ \begin{array}{cc}\hat{x}&\hat{y}\end{array} \right]^{T}\), which represents the unknown coordinates of the target according to the relation:  
\[\hat{p}=\left(\mathrm{A}^{T}\cdot A\right)^{-1}\cdot\mathrm{A}^{T}\cdot b \tag{7}\]  
where  
\[A=\left[ \begin{array}{cc}X_{11}\cdot d_{12}-X_{12}\cdot d_{11}&Y_{11}\cdot d _{12}-Y_{12}\cdot d_{11}\\ d_{22}\cdot\left(X_{TR2}-X_{12}\right)+d_{21}\cdot\left(X_{22}-X_{TR2}\right)& d_{22}\cdot\left(Y_{TR2}-Y_{12}\right)+d_{21}\cdot\left(Y_{22}-Y_{TR2} \right)\end{array} \right],\]  
\[b=\left[ \begin{array}{cc}-p_{11}\cdot d_{12}+p_{12}\cdot d_{11}\\ p_{22}\cdot d_{21}-p_{12}\cdot d_{22}\end{array} \right],\]  
where \(X_{TR2}\) represents the coordinates of antennas of the RS2 set transmitter.