# Numerical Integration-Based Performance Analysis of Cross-Eye Jamming Algorithm
## 2 Cross-Eye Gain
Since cross-eye gain from Equation (3), GC is a function of \(a\) and \(\phi\), GC\({}_{\text{real}}\) and GC\({}_{\text{nominal}}\) can be expressed in terms of terms \(a_{\text{real}}\), \(\phi_{\text{real}}\) and \(a_{\text{nominal}}\) and \(\phi_{\text{nominal}}\):  
\[GC_{\text{nominal}}(a_{\text{nominal}},\phi_{\text{nominal}})=\frac{1-a_{ \text{nominal}}^{2}}{1+a_{\text{nominal}}^{2}+2a_{\text{nominal}}\cos\phi_{ \text{nominal}}} \tag{8}\]  
\[GC_{\text{real}}(a_{\text{real}},\phi_{\text{real}})=\frac{1-a_{\text{real}}^{2} }{1+a_{\text{real}}^{2}+2a_{\text{real}}\cos\phi_{\text{real}}}. \tag{9}\]  
The square of the difference between Equations (8) and (9) can be expressed as  
\[(GC_{\text{real}}-GC_{\text{nominal}})^{2}=\Bigg{(}\frac{1-a_{\text{real}}^{2} }{1+a_{\text{real}}^{2}+2a_{\text{real}}\cos\phi_{\text{real}}}-\frac{1-a_{ \text{nominal}}^{2}}{1+a_{\text{nominal}}^{2}+2a_{\text{nominal}}\cos\phi_{ \text{nominal}}}\Bigg{)}^{2}. \tag{10}\]  
Figure 4: Miss angle with respect to amplitude ratio and phase difference. Figure 5: Contour of the cross-eye gain. Let the MSD denote the expectation of Equation (10):  
\[ \begin{split}\text{MSD}&=\text{E}\Big{[}(GC_{\text{real }}-GC_{\text{nominal}})^{2}\Big{]}\\ &=\text{E}\Bigg{[}\Bigg{(}\frac{1-a_{\text{real}}^{2}}{1+a_{\text{ real}}^{2}+2a_{\text{real}}\cos\phi_{\text{real}}}-\frac{1-a_{\text{nominal}}^{2}}{1+a_{ \text{nominal}}^{2}+2a_{\text{nominal}}\cos\phi_{\text{nominal}}}\Bigg{)}^{2} \Bigg{]}.\end{split}  \tag{11}\]  
MSD from the Monte-Carlo simulation can be obtained from:  
\[ \begin{split}\text{MSD}&=\frac{1}{N}\sum\limits_{i= 1}^{N}\Big{(}(GC_{\text{real}}(a_{\text{real}},\phi_{\text{real}})_{i}-GC_{ \text{nominal}}(a_{\text{nominal}},\phi_{\text{nominal}}))^{2}\Big{)}\\ &=\text{SimulationE}\Big{[}(GC_{\text{real}}(a_{\text{real}},\phi_{ \text{real}})_{i}-GC_{\text{nominal}}(a_{\text{nominal}},\phi_{\text{nominal}} ))^{2}\Big{]},\end{split}  \tag{12}\]  
where the \(N\) is the number of Monte-Carlo repetitions. The simulation-based MSD of the first-order [PERSON] approximation is given as  
\[\text{E}\bigg{[}\big{(}GC_{\text{real}}^{\text{(approxi=1)}}-GC_{\text{nominal} }\big{)}^{2}\bigg{]}=\text{E}\Big{[}\big{(}\alpha_{a}(a_{\text{real}}-a_{0})+ \alpha_{\phi}(\phi_{\text{real}}-\phi_{0})\big{)}^{2}\Big{]}, \tag{13}\]  
where \(\alpha_{a}\) and \(\alpha_{\phi}\) are coefficients of the first-order [PERSON] approximation, respectively. The MSD of the second-order Taylor approximation is given as  
\[ \begin{split}&\text{E}\bigg{[}\big{(}GC_{\text{real}}^{\text{(approxi =2)}}-GC_{\text{nominal}}\big{)}^{2}\bigg{]}\\ &=\text{E}\Bigg{[}\bigg{(}\begin{array}{c}\big{(}a_{a}(a_{ \text{real}}-a_{0})+\alpha_{\phi}(\phi_{\text{real}}-\phi_{0})\big{)}+\\ \big{[}\beta_{a}(a_{\text{real}}-a_{0})^{2}+\beta_{a\phi}(a_{\text{real}}-a_{0 })(\phi_{\text{real}}-\phi_{0})+\beta_{\phi}(\phi_{\text{real}}-\phi_{0})^{2} \big{]}\end{array}\bigg{)}^{2}\Bigg{]},\end{split}  \tag{14}\]  
where \(\beta_{a}\), \(\beta_{a\phi}\) and \(\beta_{\phi}\) are coefficients of the second-order [PERSON] approximation, respectively. The analytic MSD based on the first-order [PERSON] approximation and the second-order [PERSON] approximation can be expressed as  
\[\text{E}\Big{[}(GC_{\text{real}}^{\text{approxi=1}}-GC_{\text{nominal}})^{2} \Big{]}=\alpha_{a}^{2}\sigma_{a}^{2}+\alpha_{\phi}^{2}\sigma_{\phi}^{2} \tag{15}\]  
\[\text{E}\Big{[}(GC_{\text{real}}^{\text{approxi=2}}-GC_{\text{nominal}})^{2} \Big{]}=\alpha_{a}^{2}\sigma_{a}^{2}+\alpha_{\phi}^{2}\sigma_{\phi}^{2}+\frac {3}{4}\beta_{\phi}^{2}\sigma_{a}^{4}+\frac{3}{4}\beta_{\phi}^{2}\sigma_{\phi}^ {4}+\beta_{a\phi}^{2}\sigma_{a}^{2}\sigma_{\phi}^{2}+\frac{1}{2}\beta_{a}\beta _{\phi}\sigma_{a}^{2}\sigma_{\phi}^{2}. \tag{16}\]  
Each coefficient of the first-order Taylor approximation and the second-order Taylor approximation \(\alpha_{a}\), \(\alpha_{\phi}\), \(\beta_{a}\), \(\beta_{a\phi}\) and \(\beta_{\phi}\) are derived in the Appendix A.