# Numerical Integration-Based Performance Analysis of Cross-Eye Jamming Algorithm
## 5 Conclusions  
It can be seen in Equation (3) that the jamming effect of cross-eye changes significantly due to the amplitude ratio and phase difference. That is, the cross-eye gain is the function of amplitude ratio and phase difference. Due to mechanical defects, the amplitude ratio and phase difference can change differently from the nominal values; thus, changing the gain of the cross-eye. Degraded cross-eye gain is denoted by GCreal and the original cross-eye gain is denoted by GCnominal. GCreal is a cross-eye gain obtained from \(a_{\text{real}}\) and \(\phi_{\text{real}}\), the values of zero mean Gaussian random variables added to the amplitude ratio and phase difference, respectively. MSD is the mean of the squared differences between GCreal and GCnominal. MSD is one of the methods used to measure the accuracy of an estimate. As a way to obtain these MSDs, previous papers introduced the simulation and analytic methods obtained through the first-order Taylor approximation and the second-order Taylor approximation. In the case of the simulation method, a number of repetitions were calculated using the Monte-Carlo method to obtain an MSD. Using this method, it can be shown that computational costs increase as the number of repetitions increases. On the contrary, in the case of the analytic method, it can be confirmed that the calculation cost is independent on the number of repetitions. This confirms that the analytic method has a much lower time cost than the time cost of the simulation. However, cross-eye gain is a nonlinear function shown in a fractional form, it is difficult to develop the mean operation analytically. Thus, the approximations are applied, which leads to poor accuracy. Since approximation has been applied, it can be confirmed that the MSD of the analytical method is close to the MSD of non-approximated simulation method, but not completely matched. This paper proposed a method for calculating MSD through the numerical integral method. This is the method of calculating the MSD by using numerical integration and a joint PDF. Because \(a\) and \(\phi\) are uncorrelated, it can be calculated using each PDF, \(f_{a}\) and \(f_{\phi}\) without using joint PDF, \(f_{a,\phi}\). Equation (21) shows that a double integral is used. Table 1 shows that numerical integration-based MSD time cost is higher than analytic MSD because it uses dual integrals. However, Figures 6 and 13 show the accuracy of numerical integration-based MSD is higher than the accuracy of the analytic MSD. The numerical integration-based MSD is closer to the non-approximation simulation MSD than \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**MSD Method**} & **Operation Time (sec)** & **Operation Time (sec)** \\  & **(Number of Repetitions: 30,000)** & **(Number of Repetitions: 3,000,000)** \\ \hline Simulation MSD & 0.09 & 8.35 \\ Numerical integration-based MSD & 0.03 & 0.03 \\ Analytic MSD & 0.002 & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of operation time. the analytic MSD. In other words, this means that MSD can be calculated using numerical integration without performing numerous repetitive calculations. The higher the order, the closer the non-approximation MSD, but the higher the order, the higher the complexity and computational time. Therefore, the method calculating an MSD using the numerical integral proposed in this paper can be said to be superior to the analytic method proposed in the previous paper. A quantitative description on performance degradation, due to the imperfect amplitude ratio and phase difference between two jammer antennas, has been presented. More specifically, imperfections in amplitude ratio and phase difference is modeled with zero-mean Gaussian random variables, and the validity of the proposed scheme is shown using numerical results. Future work includes an extension to the non-zero mean Gaussian random variable and an improvement to the accuracy of the numerical integration by adopting more sophisticated numerical integration schemes.