The passive detection system can realize the positioning and tracking of multiple targets. It can fully play the advantages of passive detection, such as good concealment and extraction of target attribute information. It can also enhance the system's survivability and effective working ability in electronic warfare [1]. In the passive positioning method, multi-station direction-finding cross positioning is one of the more used ones. It uses high-precision direction-finding equipment to find the direction of the target at more than two observation points. The intersection of each direction finding line is the position of the target. Currently, in the world's existing passive detection systems, Russia's BEÃA85B6-A three-coordinate electronic intelligence station and Israel's EL/L8300G electronic support system adopt a three-station direction-finding cross positioning system. The multi-station direction-finding cross positioning system does not have very high requirements for time synchronization and data transmission between stations. The feature that allows working in a narrow beam scanning model improves the sensitivity of the system. And compared with single-station passive positioning, multi-station passive positioning has the advantages of omnidirectional and fast [2]. Therefore, although multi-station direction-finding cross positioning has shortcomings such as large positioning errors, it still has important research and application value.

In a complex environment, two or more observing stations perform direction finding and cross positioning of multiple targets. The intersection of different direction-finding lines will produce many false positioning points [3]. The number of false positioning points increases sharply with the number of observation stations and targets. At present, people have developed a variety of methods to solve this problem. For example, the minimum distance method, the maximum likelihood algorithm, and the Lagrangian relaxation algorithm. Among them, the minimum distance method has a simple algorithm and a small amount of calculation. Still, in a complex environment, the correct correlation rate of the data is low, and the probability of misjudgment is greater in practical use. The maximum likelihood algorithm and the Lagrangian relaxation algorithm consider correlation from the measurement domain. They require a general trial and comparison of the total effect of each segmentation, so the correct correlation rate is relatively high. But in the case of a large number of sensors and targets, the amount of calculation is relatively large, and it is not suitable for real-time processing. This article adopts a new method to carry on the position association to the target. This method first constructs a test statistic and compares the test statistic with a certain threshold to make a rough correlation. In this way, some false positioning points are eliminated, and the amount of calculation in the future is reduced. On this basis, the maximum likelihood algorithm is used to carry out the good correlation of orientations to find out the most likely combination of orientations from the same target [4]. This can ensure a certain azimuth correct correlation rate. The simulation results show that the method proposed in this paper can quickly and accurately eliminate false positioning points to realize the positioning and tracking of multiple targets.

Assume that _{ki}_{sj}_{k}_{k}_{s}_{s}

Through simple mathematical operations, the coordinate (

By deriving equations (3) and (4), the positioning error _{x}_{y}_{xy}_{θk}_{θs}

Bearing data association is a key issue in multi-sensor multi-target direction-finding cross-location technology [6]. This problem is relatively simple when there is only a single target in the surveillance area. This only involves positioning issues. The target's position can be obtained by formulas (3) and (4) through simple mathematical operations. However, if there are multiple targets in the monitoring area, the situation becomes more complicated. At this time, it is necessary to distinguish which position measurement data of each sensor comes from the same target. At the same time, we need to combine the position data belonging to the same target to locate the target [7]. This process is also the process of location data association. Here, a two-level azimuth correlation algorithm is used for azimuth data correlation. First, use certain criteria to perform a rough correlation of azimuths and eliminate some false azimuth combinations to reduce the amount of calculation in the future. Then use the Maximum Likelihood Algorithm to carry out the good correlation of azimuths and find the combination of azimuths most likely to come from the same target. This realizes the positioning and tracking of multiple targets.

Assume that 3 sensors each perform direction finding on _{s}_{s}_{1}, _{s}_{2}, _{sN}_{1}, _{2}, _{3} can constitute a possible association combination. For example, _{i1i2i3} = {_{1i1}, _{2i2}, _{3i3}} is a combination of orientations [8]. The combination may be the correct combination from the same target, or it may be wrong. We combine these 3 azimuth angles _{1i1}, _{2i2} and _{3i3} in pairs. From formula (3) to formula (5), the coordinates and positioning error of the intersection of the direction-finding line can be obtained. The following assumptions are made here.

(1) (_{1}, _{1}) is the anchor point determined by the azimuth angles _{1i1} and _{2i2}, measured by sensor 1 and sensor 2.
_{x1x1} represents the corresponding positioning error and cross-covariance. (2) (_{2}, _{2}) is the anchor point determined by the azimuth angles _{1i1} and _{2i2} measured by sensor 1 and sensor 3.
_{x2x2} represents the corresponding positioning error and cross-covariance [9]. Since the azimuth angle measurement errors of different sensors are independent of each other,

In the formula,
_{x1x2}, _{y1y2} can be obtained. _{y1x2} And _{x1y2} satisfy the following formula

Define the distance difference between anchor point (_{1}, _{1}) and (_{2}, _{2}) as

Then the mean and variance of the distance difference [^{T} can be obtained, which are respectively

Where

In the formula,
_{xiyi}_{x1x2}, _{y1y2}, _{y1x2}, and _{x1y2}, can be obtained by formula (7). We use the distance difference obtained by equation (8) and the variance obtained by equation (10). We can construct the following test statistics

Where

The test statistic obtained by formula (14) can be regarded as approximately following the ^{2} distribution. When the test statistic _{i1i2i3} is lower than a certain detection threshold obtained from the ^{2} distribution, it is considered that the azimuth angles _{1i1}, _{2i2} and _{3i2} corresponding to the positioning points (_{1}, _{1}) and (_{2}, _{2}) may belong to the same target. This combination of orientation may be correct, so we keep it; otherwise, it is considered a wrong combination, and we delete it.

Assume that the combination of the positions retained after the rough correlation of the positions is represented by the set _{T}_{T}_{i1i2i3} = {_{1i1}, _{2i2}, _{3i3}} is any combination of azimuths inset _{T}

We combine the set positions _{T}

Where

The above formula is the position combination corresponding to

Then the likelihood function of feasible partition

In the same way, other feasible partitions inset _{T}

Sensors have measurement errors in actual detection. The direction-finding lines corresponding to the same target may not intersect at one point, so at this time, it is necessary to perform position fusion on these intersection positioning points. This minimizes the random error of the system and obtains an estimate closer to the true position of the target [11]. At present, the more commonly used target location fusion algorithms include an average method, least-squares algorithm, iterative method, etc. In the multi-station direction finding and positioning, the bearing data association is the key. We use different methods for position fusion to have a small impact on the tracking results, so here we use the average method with a small amount of calculation for position fusion.

Suppose that after the azimuth data is associated, it is determined that the azimuth angle measurement values correspond to the same target measured by the three sensors. We combine these 3 azimuth angles in pairs by formulas (3) and (4) to obtain the positions of 3 different intersections. Then take the average of the positions of these 3 intersections to get the estimated position of the target. After obtaining the target's location information, it is equivalent to using a single active sensor to track multiple targets. We can track multiple targets using conventional sensor tracking methods.

This article simulates the algorithms described in the previous sections. In the simulation, it is assumed that the angle measurement errors of the three observation stations are the same, and all obey the Gaussian distribution with a mean value of zero. The target is assumed to be a ship [12]. The speed is 22 knots. The target distance is 5km. The longest distance between the target and the sensor is 71.25km. As shown in picture 2. Detection probability Pd=0.95, false alarm rate Pf=0.05. After 30 Monte-Carlo experiments, the azimuth correct correlation rate for different targets with different angle measurement errors is shown in Table 1. Figure 3 shows the estimated trajectory of the target when the angle measurement error is 0.1°.

Correlation rate of azimuth under different angle measurement errors

Angle measurement error | 0.1 degree | 0.8 degrees | 1.5 degrees | 2.5 degrees |
---|---|---|---|---|

Goal 1 | 0.9802 | 0.9703 | 0.8812 | 0.6713 |

Goal 2 | 1 | 0.9802 | 0.901 | 0.6931 |

Goal 3 | 0.9901 | 0.9802 | 0.8218 | 0.6535 |

Goal 4 | 0.9604 | 0.8218 | 0.7723 | 0.6634 |

It can be seen from Table 1 and Figure 3 that the angle measurement error has a greater influence on the correct correlation rate of the azimuth. When the angle measurement error is small, the correlation rate of position correctness is high, and the positioning error is small. In this way, more accurate positioning and tracking of multiple targets can be achieved. When the angle measurement error is large, there will be more wrong positioning. The location of the target cannot be accurately given at this time.

Multi-station direction-finding cross positioning is one of the most used passive positioning methods. However, this method is prone to produce many false positioning points in a complex environment. The rapid and accurate elimination of false positioning points has always been a difficult problem to solve. Based on the existing methods, this paper adopts a new method to eliminate false positioning points. This method first uses certain criteria to carry out a rough correlation of azimuths. It eliminates a part of false positioning points to reduce the amount of calculation in the future. On this basis, the azimuth is finely correlated to ensure a certain correct correlation rate. The simulation analysis shows that the method proposed in this paper has a high correlation rate of correct azimuth and has a moderate amount of calculation.

#### Correlation rate of azimuth under different angle measurement errors

Angle measurement error | 0.1 degree | 0.8 degrees | 1.5 degrees | 2.5 degrees |
---|---|---|---|---|

Goal 1 | 0.9802 | 0.9703 | 0.8812 | 0.6713 |

Goal 2 | 1 | 0.9802 | 0.901 | 0.6931 |

Goal 3 | 0.9901 | 0.9802 | 0.8218 | 0.6535 |

Goal 4 | 0.9604 | 0.8218 | 0.7723 | 0.6634 |

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Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model