# Numerical Integration-Based Performance Analysis of Cross-Eye Jamming Algorithm
## 1 Introduction  
The radar receives information on the target's direction, distance, and velocity by using electromagnetic waves. The radar uses the received information to search and track the targets. Amplitude-comparison monopulse (ACM) radar is one of the tracking radars. The ACM uses only one pulse to measure the tracking error of a target accurately. ACM's antenna consists of four beam patterns, which have the same squint angle. ACM can estimate the angle of the target by comparing the sum and difference of the received signals from each antenna for each pulse of the reflected signal [1]. In [2], multiple antennas in the ACM are used to estimate the angle of the target. In this way, the radar estimates the distance, velocity, and angle of the target and provides information to the guided weapon. This information poses a great threat to our ships and planes, so it is essential to disrupt the radar. In order to avoid ACM tracking, the target needs to disturb the ACM. A way to protect our ships from ACM is jamming, among which angle disturbances are effective. However, ACM calculates the sum pattern and the difference pattern from a single receiving pulse to estimate the angle, so the effect of the amplitude change between pulses is small. Cross eye jamming can effectively interfere with ACM's angular tracking by changing the value of the received sum and difference of the ACM sensor [3; 4; 5; 6; 7]. Cross-eye is a method that causes the ACM to track the wrong angle. Cross-eye jamming causes angle errors, allowing the ACM to measure the false position of a target rather than the true position of the target through induced angle errors. The cross-eye jamming radiates jamming signals using two separate transmitters to ACM. At this time, because the phase difference between the two transmitters is \(180^{\circ}\), propagation is distorted when the ACM receives a signal. The jamming signal generated by the cross-eye's jammer antenna changes the sum of reception of the ACM and the value of the difference, causing an angle error [3; 4; 5; 6; 7; 8; 9; 10]. The jammer antenna in the cross-eye that generates the deceiving signal is located on board ship at a distance of \(L\). The disturbed angle caused by the signal from the jamming antenna will deceive the actual target's position. The cross-eye gain is obtained by usingthe two jammer antenna's jamming signals which have the amplitude ratio, \(a\), and phase difference, \(\phi\). Because the cross-eye gain is the function of two variables, cross-eye jamming performance is greatly influenced by \(a\) and \(\phi\). It is the most effective when variable \(a\) is a unity and variable \(\phi\) is \(\pi\)[11; 12]. Variables \(a\) and \(\phi\) may differ from the nominal values of \(a_{0}\) and \(\phi_{0}\) due to various reasons. The difference between \(a\) and \(a_{0}\) can be modeled as a zero-mean Gaussian random variable. Similarly, the difference between \(\phi\) and \(\phi_{0}\) can be modeled as a zero-mean Gaussian random variable. The amount of degradation is dependent on the deviation of actual values of \(a\) and \(\phi\) from nominal values of \(a\) and \(\phi\). Many studies on the performance analysis of cross-eye gain have been conducted [11; 12; 13]. Previous studies focused on how to maximize cross-eye gain to prevent monopulse radars from accurately tracking angles by increasing the cross-eye gain. In contrast, this paper does not focus on increasing the value of cross-eye gain by controlling \(a\) and \(\phi\). In this paper, we are concerned with how the perturbations in \(a\) value and \(\phi\) value degrade the performance of the cross-eye jammer. More specifically, we are concerned the quantitative description of the performance degradation due to the difference between the nominal values and the true values of \(a\) and \(\phi\). To get an analytic expression of the mean square difference (MSD), the Taylor approximation should be applied to the cross-eye gain expression.