# Numerical Integration-Based Performance Analysis of Cross-Eye Jamming Algorithm
## 1 Introduction
Because the cross-eye gain is a non-linear function expressed in fractional form, it is difficult to obtain closed-form analytic expression of the MSD. Therefore, after approximating the cross-eye gain as a polynomial function for \(a\) and \(\phi\) using the first-order and the second-order Taylor approximation, the analytical expansion of the expectation operator can be achieved. The Monte-Carlo simulation method is used to obtain the simulation MSD without approximation, the first-order [PERSON] approximation, and the second-order Taylor approximation cross-eye gain. The analytic expressions of MSD are given as explicit expressions in terms of the variance of the amplitude ratio, \(a\), and phase difference, \(\phi\). Numerical integration-based MSD of cross-eye gain was obtained from numerical integration. Using joint PDF of two random variables of \(a\) and \(\phi\), the mean of functions consisting of two random variables, \(a\) and \(\phi\), is calculated. In order to analyze the accuracy of the cross-eye jamming effect under various mechanical defects the performance degradation of the cross-eye jamming was evaluated for several values of standard deviations. In the previous study [14], the analytic expression of MSD is obtained via the first order and the second order Taylor series, and the validity of the derived expression is shown by comparing the analytic results and the Monte-Carlo simulation-based results. In order to improve the accuracy of the MSD calculation, we propose a numerical integration-based MSD of cross-eye gain. In the previous study [14], the authors derived a close form expression of the analytic MSDs associated with the simulation based on the first-order Taylor expansion and with the simulation based on the second-order Taylor expansion. Note that the simulation based on the first-order Taylor expansion and the second-order Taylor expansion is slightly different from the simulation with non-approximation. In [14], the authors proposed how the closed-form expression of the analytic MSD associated with the simulation based on the first-order Taylor expansion and the second-order Taylor expansion can be derived. In this paper, it is shown how the closed-form expression of the analytic MSD associated with the simulation with non-approximation can be expressed. The computation time of MSD using an analytical method is much less than that of MSD calculated by the Monte-Carlo method with a large number of repetitions. In other words, the scheme in [14] has demonstrated that analytical MSDs have significantly less computational complexity than the Monte-Carlo simulation-based MSD. However, accuracy is reduced because of approximation. There is a difference between the non-approximated simulation-based MSD and the analytic-based approximated MSD. From that point of view, the accuracy is reduced. Therefore, we improve the MSD calculation accuracy of the cross-eye gain by considering the perturbation through numerical integration-based method. It does not mean that the integral method's estimated performance is better than the analytic method's one. The important point is that it is closer to the non-approximated simulation-based MSD than the analytic method. In this paper, how close various MSDs are to the MSD with non-approximation is the concern. In Section 2, the cross-eye jamming method is briefly introduced, and why the cross-eye gain affects the cross-eye jamming performance. Further, MSD of approximated cross-eye gain, and non-approximated cross-eye gain are derived, respectively. In Section 3, numerical integration-based MSD, which uses joint PDF of two Gaussian random variables is derived. In Section 4, the result of each MSD is shown, and the performance of analytic MSDs, numerical integration-based MSD and simulation MSD are analyzed. Finally, in the conclusion, we summarize the overall contents of this paper and emphasize the advantages of the numerical integration-based MSD method compared to Monte-Carlo simulation-based MSD and analytic-based approximated MSD.