# Numerical Integration-Based Performance Analysis of Cross-Eye Jamming Algorithm
## 2 Cross-Eye Gain  
In this section, cross-eye gain is defined, and MSD is defined as Equations (1)-(5). Cross-eye jamming can deceive the monopulse's angle tracking. Cross-eye jamming is implemented by placing two jamming antennas on board ship. The jamming antennas radiate the jamming signal. Figure 1 is a brief description of the cross-eye jamming technique. A false target is created by two jammer antennas on the board ship separated by distance \(L\). Figure 1 shows that \(L\) is similar to the length of the ship, and is drawn to explain that cross-eye jammer antennas are on the board ship. Miss distance and miss angle are given by [1]. \[r_{\text{miss}}=\frac{L\cos\psi}{2}\frac{1-a^{2}}{1+2a\cos\phi+a^{2}} \tag{1}\]  
\[\theta_{\text{miss}}=\frac{\Delta\theta}{2}\frac{1-a^{2}}{1+2a\cos\phi+a^{2}} \tag{2}\]  
where \(\psi\) is the angle between the track axis and the line that is vertically bisected between the two jammers and \(\Delta\theta\) is the angle between the two jammers. \(r_{\text{miss}}\) and \(\theta_{\text{miss}}\) have a common term, \(\frac{1-a^{2}}{1+2a\cos\phi+a^{2}}\), which is the cross-eye gain term  
\[GC(a,\phi)=\frac{1-a^{2}}{1+2a\cos\phi+a^{2}}. \tag{3}\]  
Thus, \(r_{\text{miss}}\) and \(\theta_{\text{miss}}\) can be simplified as  
\[r_{\text{miss}}=\frac{L\cos\psi}{2}GC \tag{4}\] \[\theta_{\text{miss}}=\frac{\Delta\theta}{2}GC. \tag{5}\]  
Figure 1: Cross-eye jamming. Since \(r_{\text{miss}}\) and \(\theta_{\text{miss}}\) are functions of \(a\) and \(\phi\), it is greatly affected by the \(a\) and \(\phi\). Figure 2 shows that jammer antenna A and jammer antenna B produce a disturbance signal. The cross-eye jamming induces an angular error by recreation the angular error in the jammed radar. The cross-eye jamming effect is sensitive to changing the amplitude ratio and phase difference values of the two jamming signals. Figure 3 shows the miss distance with respect to \(a\), and Figure 4 shows the angle error with respect to amplitude ratio and phase difference. In [15; 16; 17; 18], the authors have made tolerance analysis based on the contour plot. In Figure 5, contour plot of the cross-eye gain values, which is similar to the contour plots in [15; 16; 17; 18], are presented. The contour plot in Figure 5 displays how much degradation in cross-eye gain occurs due to a deviation in amplitude ratio and phase difference between two jammer antennas. Accordingly, cross-eye jamming performance is highly dependent on the amplitude ratio and phase difference. Suppose you can model the instability of amplitude ratios and phase differences as zero mean Gaussian random variables:  
\[ \begin{array}{rcl}\delta_{a}&\sim&N(0,\sigma_{a})\\ \delta_{\phi}&\sim&N(0,\sigma_{\phi}),\end{array}  \tag{6}\]  
where \(\sigma_{a}\) and \(\sigma_{\phi}\) are the standard deviation of amplitude ratio and standard deviation of phase difference, respectively. Therefore, \(a_{\text{real}}\) and \(\phi_{\text{real}}\), which occur, the perturbation can be written as  
\[ \begin{array}{rcl}a_{\text{real}}&=&a_{\text{nominal}}+\delta_{a}\\ \phi_{\text{real}}&=&\phi_{\text{nominal}}+\delta_{\phi}.\end{array}  \tag{7}\]  
Figure 3: Miss distance with respect to amplitude ratio. Figure 2: Two jammer antenna’s signal.