# 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  
Even in the case of the use of two radar systems, where each set separately has its antennas located on one line, see Figure 1, it is possible to calculate unknown coordinate values \((x,y)\) for the location of the target by the simple mathematical operations mentioned in [29] because part of the signal path \(\|\text{Tx}\rightarrow\text{T}\rightarrow\text{Rxi}\|\) is common for both receiving antennas of the proper set. The resulting relationships determined by this procedure allow separate calculations of the intersections of the ellipses located in the observation area for each of the radar sets. In the real case, these are two different values. Another possibility of calculating the intersections of ellipses is possible using the Bezout theorem. The principle of determining the intersections of ellipses and the algorithm for determining the intersections of ellipses using the Bezout theorem is described in detail in [29] and [36; 37; 38]. Complex roots are excluded from the estimated positions \((\overline{x},\overline{y})\) of the intersections of the ellipses. Some of the other values \((\overline{x},\overline{y})\) may be excluded if they are not in the possible target area. The above calculation of the intersections of two ellipses is derived for arbitrarily lying ellipses. The presented method of lines is a method of direct calculation. From the point of view of the complexity of the calculation, it is also advantageous to choose the local coordinate system so that the transmitter antenna of one of the sets is placed at its beginning (see the description of the experiment site, Figure 1). We assume the location of targets by two independent UWB radar systems. Each set of radars is made up of one transmitter and two receiver antennae, lying on two mutually perpendicular lines (e.g., Figure 1). In this case, it is possible to calculate the target coordinates based on the known coordinates of the antennae. The coordinates of the intersections of the ellipses \(X_{12 RS1}(t)\) and \(Y_{12 RS1}(t)\) for the radar set RS1 are determined using the method of finding the intersections of the ellipses, which is described in (28):  
\[X_{12 RS1}(t)=-((p_{11}\cdot d_{12})-(p_{12}\cdot d_{11}))/((X_{11}\cdot d_{12} )-(X_{12}\cdot d_{11})), \tag{3}\]  
\[Y_{12 RS1}(t)=-((p_{11}+X_{12 RS1}(t)\cdot X_{11})^{2}-X_{12 RS1}(t)^{2})^{0.5}, \tag{4}\]  
where \(d_{11}\) is the distance between the transmitter, the target, and the antenna of the first receiver of the RS1 kit; \(d_{12}\) is the distance between the transmitter, the target, and the antenna of the second receiver of the RS1 set; \(p_{11}=0.5\ (d^{2}{}_{11}-X^{2}{}_{11})\); \(p_{12}=0.5\ (d^{2}{}_{12}-X^{2}{}_{12})\); \(X_{11}\) and \(X_{12}\) are the coordinates of antennas of RS1 set receivers. Using the method of direct Equations (3) and (4) finding the intersections of ellipses, we determine the intersections of the ellipses from the TOA times measured by the given set. The results of determining the intersections of ellipses at the selected observation time \(\tau_{\rm k}\) are shown for the radar set RS1 in Figure 2 where the expression E1 RS 1 represents the ellipse No. 1 of the radar set RS1, with foci in the position of the transmitter TxA and in the position of the receiver RxA1 (see Figure 1). The analog expression E2 RS 1 represents the ellipse No. 2, the radar set RS1, with foci in the position of the transmitter TxA and in the position of the receiver RxA2. See Figure 2a. The dimensions of ellipses and location of UWB radars in Figures 2-10 are in line with the description in Figure 1. The line drawn through the intersections of the ellipses represents the 1 st line using the method of lines. See Figure 2b.