## 4 Modeling the Target Tracking Process Using Kalman Filters
Let us consider the following modeling parameters: interference intensity, \(\lambda\), time intervals, \(T\), and other input parameters of the model. Based on the analytical expression for the probability density function of the radar's interaction time with the target \(p(t)\)calculations were carried out for various values of interference intensity. The results are presented in the form of probability density function (PDF) and cumulative distribution function (CDF) graphs. To model radar performance under interference conditions, parameters characteristic of modern radar systems were used. The interference intensity, \(\lambda\), was chosen in the range of 0.1 to 1, corresponding to different scenarios, from weak natural interference to conditions of active electronic countermeasures. The radar's pulse repetition frequency (\(f\)) is 10 Hz, which is typical for medium-range detection and tracking systems. The target velocity (\(v\)) was set at 200 m/s, representing the parameters of highly maneuverable airborne objects. The radar's operational range (\(R\)) is 10 km, which is typical for ground-based short-range systems. These parameters allow the evaluation of the effectiveness of the proposed model in scenarios close to real-world operating conditions. The modeling and calculations were performed using a Python 3.11-based simulation tool developed for this study. This software allows for the flexible implementation of radar parameters and interference conditions, as well as the dynamic visualization of results. The tool utilizes NumPy for numerical computations and Matplotlib for plotting graphs, ensuring precise and reliable results. Additionally, the interactive features of the software enable the real-time adjustment of key parameters such as \(\lambda\), radar frequency (\(f\)), and target velocity (\(v\)) to evaluate their influence on radar performance. Figure 5 shows the graph of the dependence of the radar's mean response time on different interference intensities. As seen from the graph, at low interference intensity (\(\lambda\approx 0.1\)), the mean response time reaches its maximum value (\(t^{-}\approx 10\) s). With an increase in interference intensity (\(\lambda>1.00\)), the mean response time decreases sharply and asymptotically approaches a value of approximately 1.5 s.  
The mean response time is an important parameter that reflects the radar's performance dependence on interference levels. The reduction in response time with increasing \(\lambda\) is due to the fact that, under challenging conditions, the system is forced to complete data processing more quickly. Figure 6 presents the probability graphs of interaction completion within specified time intervals. The graph demonstrates the radar's probability of completing interaction with a target within given time intervals (\(T=10\) s, 20 s, 30 s) at various values of \(\lambda\). As can  
Figure 5: The graph of the dependence of the radar’s mean response time on different interference intensities. be seen, for a short interval (\(T=10\) s), the probability of interaction completion increases rapidly with growing \(\lambda\), reaching its maximum at \(\lambda\approx 0.5\). For longer intervals (\(T=20\) s), the probability reaches 1 for nearly all values of \(\lambda\), but at lower intensities (\(\lambda\approx 0.1\)), this growth occurs more slowly. The probability of interaction completion is higher for longer time intervals, as expected, since the radar has more time to complete its tasks. These results demonstrate the influence of interference intensity and time constraints on radar performance and illustrate the applicability of the proposed model in analyzing radar operations under interference conditions.