A method for locating a scanning radiation source based on semi-positive relaxation

Abstract

The application discloses a scanning radiation source positioning method based on semi-positive relaxation, and comprises the following steps: firstly, a positioning geometric model based on a scanning angle and a time difference in a multi-station scene is constructed; secondly, an auxiliary variable is introduced to construct a convex optimization positioning solving framework, and an original non-convex observation model is equivalently converted into a convex optimization problem; finally, a convergence domain of the positioning problem after semi-positive relaxation is geometrically analyzed, the feasible domain expansion caused by the semi-positive relaxation is reduced, and the convex optimization problem is efficiently solved by using an optimization toolbox, so that accurate positioning of a static scanning radiation source is realized.

Description

Technical Field

[0001] This invention belongs to the field of positioning technology, specifically relating to a scanning radiation source positioning method based on semidefinite relaxation. Background Technology

[0002] Currently, positioning systems can be divided into active and passive positioning systems based on their positioning methods. Active positioning systems actively transmit signals towards a target radiation source and use the echo signals to obtain location information. They have advantages such as all-weather operation, high accuracy, and resistance to external environmental influences. However, these systems rely on their own equipment to transmit high-power signals, resulting in poor concealment and susceptibility to interference, making them unsuitable for missions requiring high levels of countermeasures and stealth. Passive positioning technology, also known as passive location technology, determines the location of a radiation source by receiving and measuring parameters containing radiation source location information in non-cooperative signals. It has received widespread attention and research in fields such as reconnaissance and surveillance.

[0003] Among typical radiation source types, scanning emitters (SEs) achieve active target detection over a 360° area through the periodic movement of their antennas. These emitters typically feature high main lobe signal-to-noise ratios, narrow beamwidths, and stable scanning periods, making them widely used in electromagnetic monitoring. For the passive localization problem of scanning emitters, traditional location methods based on Angle of Arrival (AOA) become increasingly complex in practical engineering applications due to coordinate system transformations and the attitude of the observation station, introducing new error terms. For non-cooperative scanning emitters, the carrier frequency and modulation parameters of their transmitted signals are difficult to obtain accurately from prior information, making location methods based on Frequency of Arrival (FOA) unsuitable for scanning emitter scenarios.

[0004] The most in-depth research on scanning radiation sources focuses on location methods based on the Time of Interception (TOI) of the main lobe signal. The TOI observation is the moment when the main lobe peak of the signal reaches the observation station, encompassing the relative positional relationship between the scanning radiation source and the observation station. This method converts a set of TOI observations from multiple observation stations within a single scanning cycle into a Time Difference of Interception (TDOI), and combines this with the known scanning cycle to calculate the scanning angle (SA), thereby achieving the location determination of the radiation source. This method reduces the accuracy requirement for time measurement to the nanosecond level while still ensuring high accuracy for the scanning angle, thus lowering the accuracy requirements for parameter measurement in multi-station scenarios.

[0005] Existing methods for locating scanning radiation sources mostly combine AOA and SA, but both are essentially angular measurement information, resulting in some information redundancy and limited improvement in positioning accuracy. In contrast, in multi-station reconnaissance scenarios, a hyperbola can be obtained by extracting the time difference information of the scanning radiation source arriving at different receiving stations. This hyperbola can then be combined with the obtained equiangular circles of the scanning time difference to construct a TDOA-SA heterogeneous measurement positioning system, thereby solving for the target location. Moreover, the intercepted time difference and time difference information are simple and readily available, and have high measurement accuracy. Therefore, this invention improves the positioning accuracy of scanning radiation sources by combining intra-pulse TDOA and inter-pulse TDOI for joint positioning. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention provides a scanning radiation source localization method based on semi-definite relaxation. First, a localization geometric model based on scanning angle and time difference is constructed in a multi-station scenario. Second, by introducing auxiliary variables, a convex optimization localization solution framework is constructed, which transforms the original non-convex observation model into a convex optimization problem. Finally, geometric analysis is performed on the convergence domain of the localization problem after semi-definite relaxation to reduce the expansion of the feasible region caused by relaxation. The convex optimization problem is then solved efficiently using an optimization toolbox, achieving accurate localization of the stationary scanning radiation source.

[0007] The technical solution adopted by this invention to solve its technical problem is as follows: Step 1: Construct a geometric positioning model based on scanning angle and time difference for a multi-observation station system; Step 2: Introduce auxiliary variables to transform the original non-convex observation model into a convex optimization problem, and construct a convex optimization localization solution framework; Step 3: Introduce bounded constraints to narrow the feasible region of the semidefinite relaxed problem, making the relaxed problem approximate the original localization problem.

[0008] Preferably, step 1 specifically comprises: Consider a two-dimensional passive localization problem, assuming that both the scanning radiation source and the observation station remain stationary, where the position of the scanning radiation source is denoted as... The location of the observation station is , Let be the number of observation stations; taking the first observation station as the reference station, based on the time difference positioning principle, the time difference positioning equation is obtained, namely: (1) In the formula, Represents the speed of light; For L2 norm operations, the distance between the observation station and the scanning radiation source is represented in equation (1); This indicates that the signal has arrived at the observation station. The time difference with observation station 1; The time-measured noise follows a zero mean and a variance of . Gaussian distribution; Using the first observation station as the reference station, and based on the geometric principle of scanning time difference positioning, the scanning angle difference is expressed as: (2) In the formula, This indicates that the main lobe of the scanned radiation source is from the observation station. The angle between observation station 1 and observation station 2. Represents the arctangent function; The noise is represented by the scanning angle difference measurement, which has zero mean and variance. Gaussian distribution; Construct a weighted least squares function based on time difference and scan angle residuals: (3) In the formula, , Represents a weighted matrix. This represents the process of diagonalizing a vector into matrix form; , This indicates the measurement deviation between multiple observation stations.

[0009] Preferably, step 2 specifically comprises: Step 2-1: Use Schur complement to transform the objective function into a positive semidefinite constraint; Considering that both the objective function and the constraint function in formula (3) are nonlinear and nonconvex, the objective function can be expressed in quadratic form through matrix constraints, i.e.: (4) In the formula, For the introduced auxiliary error variables, Indicates the joint measurement error; and They represent and Square root factorization, satisfying , ; Using Schur complement, the objective function (4) is transformed into a positive semidefinite constraint, i.e.: (5) In the formula, Represents a matrix with 1s on the diagonal; Step 2-2: Introduce auxiliary variables to relax the time difference positioning equation and the scanning angle positioning equation into a linear expression. First, expand the distance term in the TDOA positioning equation (1) by square, and obtain the expression: (6) The scanning time difference localization equation can be further transformed using the positive tangent formula, resulting in: (7) In the formula, Represents the tangent function; Introducing lifting vectors This converts a vector into a matrix variable, i.e.: (8) In the formula, Representing three-dimensional real space, This represents the element in the 3rd row and 3rd column of the matrix; at this time For real pairwise matrices, satisfy the positive semidefinite matrix constraint, i.e. And at this time , To calculate the rank of a matrix; Define the auxiliary matrix: (9) Therefore, the nonlinear variables in equation (6) are expressed as: (10) (11) In the formula, Represents the Frobenius inner product. This indicates finding the trace of a matrix; Therefore, the original nonlinear time difference positioning equation is convexly relaxed into a linear form, i.e.: (12) (13) Similarly, the localization equation (7) based on the scanning time difference is obtained by cross-multiplication: (14) In the formula, and All , , , One combination, namely: (15) (16) In the formula, and This is a constant matrix related to the location of the observation station, namely: (17) (18) Therefore, the original nonlinear scanning angle positioning equation (2) is convexly relaxed into a linear expression: (19) In the formula, is a coefficient.

[0010] Preferably, step 3 specifically comprises: By imposing distance constraints, the difficulty of the solution is reduced, i.e.: (20) At this point, constraint (20) is non-convex. According to Schur's theorem, the original norm constraint is transformed into a second-order cone constraint SOC, thus transforming the non-convex problem into a convex problem, i.e.: (twenty one) in, Represents a two-dimensional identity matrix; Add hard constraints to the scanning angle difference Based on the geometric model of an equal-angled circle with a scanning angle, the formula for calculating the radius is: , Indicates the observation station The distance between observation station 1 and the center of the circle is calculated using the following formula: , Describes an orthogonal vector. Let be the orthogonal distance from the center of the circle to the observation station forming the baseline; therefore, the following constraint is established: (twenty two) Using Schur's theorem, the hard angular constraint is transformed into a convex constraint, i.e.: (twenty three) (twenty four) The final enhanced positive semidefinite localization model is expressed as: (25) By employing an optimization toolbox, the precise location of the stationary scanning radiation source can be achieved; if the estimation results... Satisfying the rank-one constraint, i.e. Therefore, the estimated location of the scanned radiation source is as follows: ,like If it does not meet the requirements, then its principal eigenvectors are used. Projection, i.e. .

[0011] An electronic device includes a processor and a memory; the memory stores a computer program, and the processor executes the computer program stored in the memory to enable the electronic device to perform the above-described method for locating scanning radiation sources.

[0012] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method for locating scanning radiation sources.

[0013] A chip includes a processor for retrieving and running a computer program from a memory, causing a device equipped with the chip to perform the above-described method for locating a scanning radiation source.

[0014] A computer program product includes a computer storage medium storing a computer program, the computer program including instructions executable by at least one processor, which, when executed by the at least one processor, implement the above-described scanning radiation source localization method.

[0015] The beneficial effects of this invention are as follows: This invention can be applied to the field of passive electronic reconnaissance. By fully utilizing the periodic characteristics of scanning radiation sources, it provides a two-step localization method for stationary scanning radiation sources that combines scanning angle information and time difference information. Compared with traditional single-measurement localization methods, this method can effectively improve localization accuracy in multi-observation station systems. Secondly, by introducing a semi-definite relaxation technique, this invention transforms the original non-convex observation model into an equivalent convex optimization problem, thereby obtaining a stable global optimal solution under relaxation conditions. Compared with the maximum likelihood method, it can achieve accurate localization of scanning radiation sources without being affected by initial values. Attached Figure Description

[0016] Figure 1 This is a geometric schematic diagram of the combined time difference and scanning angle measurement positioning of the present invention.

[0017] Figure 2 This is a flowchart of the method of the present invention.

[0018] Figure 3 This is an example of the error ellipse distribution diagram of the present invention.

[0019] Figure 4 This is a comparison chart of the errors between the example of this invention and the traditional positioning method.

[0020] Figure 5 This is a comparison chart of the time measurement error performance curves in the examples of this invention. Detailed Implementation

[0021] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0022] Because the localization problem based on TDOA and SA measurements is highly nonlinear and nonconvex, the maximum likelihood criterion can be used to solve the target localization problem, but this method is sensitive to initial values ​​and is prone to getting trapped in local optima. The purpose of this invention is to fully utilize scanning angle and time difference information to provide a high-precision localization algorithm that is not affected by initial values. By introducing the concept of semidefinite relaxation (SDR), a localization solution framework based on convex optimization is constructed. By introducing auxiliary variables, the original nonconvex observation model is equivalently transformed into a convex optimization problem, thereby obtaining a stable global optimal solution under relaxation conditions and achieving accurate localization of the scanning radiation source.

[0023] The basic idea behind this invention is as follows: First, a positioning geometric model based on scanning angle and time difference is constructed for multi-station scenarios. Second, by introducing auxiliary variables, a convex optimization positioning solution framework is constructed, which transforms the original non-convex observation model into a convex optimization problem. Finally, geometric analysis is performed on the convergence domain of the semi-positive definite relaxation positioning problem to reduce the expansion of the feasible region caused by relaxation. The convex optimization problem is then solved efficiently using an optimization toolbox to achieve accurate positioning of the stationary scanning radiation source.

[0024] To achieve the above-mentioned objectives, the specific steps for implementing the invention are as follows: like Figure 2 As shown, step 1: Construct a geometric positioning model based on scanning angle and time difference under a multi-observation station system.

[0025] Consider a two-dimensional passive localization problem, assuming that both the scanning radiation source and the observation station remain stationary, where the position of the scanning radiation source is denoted as... The location of the observation station is , Let be the number of observation stations. Taking the first observation station as the reference station, based on the principle of time difference positioning, the time difference positioning equation can be obtained, namely: (1) In the formula, Represents the speed of light. The 2-norm operation represents the distance between the observation station and the scanning radiation source in this invention. This indicates that the signal has arrived at the observation station. The time difference with observation station 1 The time-measured noise follows a zero mean and a variance of . The Gaussian distribution.

[0026] Using the first observation station as the reference station, and based on the geometric principle of scanning time difference positioning, the scanning angle difference can be expressed as: (2) In the formula, This indicates that the main lobe of the scanned radiation source is from the observation station. The angle between observation station 1 and observation station 2. This represents the arctangent function. The noise is represented by the scanning angle difference measurement, which has zero mean and variance. The Gaussian distribution.

[0027] To achieve precise target positioning, a weighted least squares function based on time difference and scanning angle residuals is constructed: (3) In the formula, , Represents a weighted matrix. This indicates that the vector is diagonalized into matrix form. , This indicates the measurement deviation between multiple observation stations.

[0028] Step 2: Introduce auxiliary variables to transform the original non-convex observation model into an equivalent convex optimization problem, and construct a convex optimization localization solution framework.

[0029] Step 2.1: Use Schur complement to transform the objective function into a positive semidefinite constraint. Considering that both the objective function and the constraint function in formula (3) are nonlinear and nonconvex, the objective function can be expressed in quadratic form through matrix constraints, i.e.: (4) In the formula, For the introduced auxiliary error variables, Indicates the joint measurement error, where, and They represent and Square root factorization, satisfying , Decomposition conditions.

[0030] Using Schur complement, the objective function (4) is transformed into a positive semidefinite constraint, i.e.: (5) In the formula, Represents a unit array.

[0031] Step 2.2: Introduce auxiliary variables to relax the time difference positioning equation and the scanning angle positioning equation into a linear expression.

[0032] To transform the nonlinear, non-convex constraint function into a linear, convex constraint, the distance term in the TDOA positioning equation (1) is first expanded by square, yielding the classical expression: (6) The scanning time difference localization equation can be further transformed using the positive tangent formula, resulting in: (7) In the formula, This represents the tangent function.

[0033] At this time, the positioning equation contains , , Since there are nonlinear variables, a lifting vector is introduced. This converts a vector into a matrix variable, i.e.: (8) In the formula, Representing three-dimensional real space, This represents the element in the 3rd row and 3rd column of the matrix.

[0034] at this time For real pairwise matrices, satisfy the positive semidefinite matrix constraint, i.e. And at this time , This indicates the rank of the matrix being calculated.

[0035] To further relax the nonlinear variables, an auxiliary matrix is ​​defined: (9) Therefore, the nonlinear variables in equation (6) can be expressed as: (10) (11) In the formula, Represents the Frobenius inner product. This indicates finding the trace of a matrix.

[0036] Therefore, the original nonlinear time difference positioning equation is convexly relaxed into a linear form, i.e.: (12) (13) Similarly, the positioning equation (7) based on the scanning time difference can be obtained by cross-multiplication: (14) In the formula, and All , , , One combination, namely: (15) (16) In the formula, and This is a constant matrix related to the location of the observation station, namely: (17) (18) Therefore, the original nonlinear scanning angle positioning equation (2) is convexly relaxed into a linear expression: (19) In the formula, is a coefficient.

[0037] Step 3: Introduce bounded constraints to narrow the feasible region of the semidefinite relaxed problem, making the relaxed problem approximate the original localization problem.

[0038] Since the distance between each observation station and the radiation source is unknown, a distance constraint is considered to reduce the difficulty of the optimization solution, i.e.: (20) At this point, constraint (20) is non-convex. According to Schur's theorem, the original norm constraint is transformed into a second-order cone constraint (SOC), thus transforming the non-convex problem into a convex problem, i.e.: (twenty one) In addition, a hard constraint is added for the scanning angle difference. Based on the geometric model of an equal-angled circle with a scanning angle, the formula for calculating the radius is: , Indicates the observation station The distance between observation station 1 and the center of the circle is calculated using the following formula: , Describes an orthogonal vector. Let be the orthogonal distance from the center of the circle to the observation station forming the baseline. Therefore, the following constraint can be established: (twenty two) Using Schur's theorem, the hard angular constraint is transformed into a convex constraint, i.e.: (twenty three) (twenty four) By introducing distance second-order cone constraints and angle hard constraints, the final constraints are transformed from a large finite region to a finite region, significantly shortening the feasible region of the variables. This makes the relaxed semidefinite programming problem closer to the original problem. Therefore, the final enhanced semidefinite positioning model can be expressed as: (25) By employing an optimization toolbox to efficiently solve this problem, precise localization of the static scanning radiation source can be achieved. If the estimation results... Satisfying the rank-one constraint, i.e. Therefore, the estimated location of the scanned radiation source is as follows: ,like If it does not meet the requirements, then its principal eigenvectors are used. Projection, i.e. .

[0039] Example: Step 1: Construct a geometric positioning model based on scanning angle and time difference for a multi-observation station system.

[0040] Consider the two-dimensional passive localization problem. Assume that both the scanning radiation source and the observation station remain stationary. The location of the scanning radiation source is represented as [50, 50] km, and the number of observation stations is set to 4. The locations of the observation stations are shown in Table 1 below. This represents the number of observation stations.

[0041] Table 1. Location of Observation Stations

[0042] Using the first observation station as the reference station, based on the principle of time difference positioning, the time difference positioning equation can be obtained, namely: (1) In the formula, Represents the speed of light. The 2-norm operation represents the distance between the observation station and the scanning radiation source in this invention. This indicates that the signal has arrived at the observation station. The time difference with observation station 1 The time measurement noise follows a Gaussian distribution with zero mean and a root mean square error of 100 ns.

[0043] Using the first observation station as the reference station, and based on the geometric principle of scanning time difference positioning, the scanning angle difference can be expressed as: (2) In the formula, This indicates that the main lobe of the scanned radiation source is from the observation station. The angle between observation station 1 and observation station 2. This represents the arctangent function. It is represented as the scanning angle difference measurement noise, which follows a Gaussian distribution with zero mean and root mean square error of 0.5°.

[0044] Geometric diagrams of the isoangular circles and hyperbolas formed by any two observation stations are attached. Figure 1As shown, the geometric intersection of the hyperbola and the circle with equal angles is the true location of the radiation source.

[0045] To achieve precise target positioning, a weighted least squares function based on time difference and scanning angle residuals is constructed: (3) In the formula, , Represents a weighted matrix. This indicates that the vector is diagonalized into matrix form. , This indicates the measurement deviation of the scanning angle of the three observation stations relative to observation station 1.

[0046] Step 2: Introduce auxiliary variables to transform the original non-convex observation model into an equivalent convex optimization problem, and construct a convex optimization localization solution framework.

[0047] Step 2.1: Use Schur complement to transform the objective function into a positive semidefinite constraint. Considering that both the objective function and the constraint function in formula (3) are nonlinear and nonconvex, a positive semidefinite matrix is ​​introduced, and the objective function is expressed in quadratic form through matrix constraints: (4) In the formula, For the introduced auxiliary error variables, Indicates the joint measurement error, where, , .

[0048] Using Schur complement, the objective function is transformed into a positive semidefinite constraint, i.e.: (5) In the formula, This represents a square matrix with a diagonal of 1 and a dimension of 6.

[0049] Step 2.2: Introduce auxiliary variables to relax the time difference positioning equation and the scanning angle positioning equation into a linear expression.

[0050] To transform the nonlinear, non-convex constraint function into a linear, convex constraint, we first expand the distance term in the TDOA positioning equation by square, obtaining the classical expression: (6) The scanning time difference localization equation can be further transformed using the positive tangent formula, resulting in: (7) In the formula, This represents the tangent function.

[0051] At this time, the positioning equation contains , , Since there are nonlinear variables, a lifting vector is introduced. This converts a vector into a matrix variable, i.e.: (8) In the formula, Representing three-dimensional real space, This represents the element in the 3rd row and 3rd column of the matrix.

[0052] at this time For real pairwise matrices, satisfy the positive semidefinite matrix constraint, i.e. And at this time , This indicates the rank of the matrix being calculated.

[0053] To further relax the nonlinear variables, an auxiliary matrix is ​​defined: (9) Therefore, the nonlinear variables in the time difference positioning equation can be expressed as: (10) (11) In the formula, Represents the Frobenius inner product. This indicates finding the trace of a matrix.

[0054] Therefore, the original nonlinear time difference positioning equation is convexly relaxed into a linear form, i.e.: (12) (13) Similarly, the positioning equation (7) based on the scanning time difference can be obtained by cross-multiplication: (14) In the formula, and All , , , One combination, namely: (15) (16) In the formula, and This is a constant matrix related to the location of the observation station, namely: (17) (18) Therefore, the original nonlinear scanning angle positioning equation is convexly relaxed into a linear expression: (19) In the formula, is a coefficient.

[0055] Step 3: Introduce bounded constraints to narrow the feasible region of the semidefinite relaxed problem, making the relaxed problem approximate the original localization problem.

[0056] Since the distance between each observation station and the radiation source is unknown, a distance constraint is considered to reduce the difficulty of the optimization solution, i.e.: (20) At this point, the above constraints are non-convex. According to Schur's theorem, the original norm constraints are transformed into second-order cone constraints, thus transforming the non-convex problem into a convex problem, i.e.: (twenty one) In addition, a hard constraint is added for the scanning angle difference. Based on the geometric model of an equal-angled circle with a scanning angle, the formula for calculating the radius is: , Indicates the observation station The distance between observation station 1 and the center of the circle is calculated using the following formula: , Describes an orthogonal vector. Let be the orthogonal distance from the center of the circle to the observation station forming the baseline. Therefore, the following constraint can be established: (twenty two) Using Schur's theorem, the hard angular constraint is transformed into a convex constraint, i.e.: (twenty three) (twenty four) By introducing distance second-order cone constraints and angle hard constraints, the final constraints are transformed from a large finite region to a finite region, significantly shortening the feasible region of the variables. This makes the relaxed semidefinite programming problem closer to the original problem. Therefore, the final enhanced semidefinite positioning model can be expressed as: (25) This problem is solved efficiently using the CVX optimization toolbox, thus achieving precise localization of the static scanning radiation source. If the estimation results... Satisfying the rank-one constraint, i.e. Therefore, the estimated location of the scanned radiation source is as follows: ,like If it does not meet the requirements, then its principal eigenvectors are used. Projection, i.e. .

[0057] Based on the above steps, the Monte Carlo independent repeated experiment was performed 100 times, and the 90% error ellipse distribution was plotted. The results are attached. Figure 3 As shown. The experimental positioning results of the method of this invention are basically located near the actual location, and the major and minor axes of the 90% confidence error ellipse are evenly distributed.

[0058] The positioning error was statistically analyzed based on the actual location, and compared with traditional single-measurement SA algorithms such as SA-LS, SA-TLS, and SA+TDOA-ML. The results are shown in the appendix. Figure 4 As shown in the figure. The results show that the positioning accuracy is significantly improved by combining the scanning angle and time difference. Compared with the SA+TDOA-ML method, the algorithm proposed in this invention is an approximation of the original problem, and its performance is slightly worse than the maximum likelihood method. However, it does not require iterative solutions based on initial values, thus avoiding positioning failures caused by inaccurate initial positions in traditional methods.

[0059] The positioning accuracy under different time measurement errors was further verified through 500 Monte Carlo experiments. The root mean square error of the time difference measurement was increased exponentially from 100 ns to 100 µs, and the scanning angle measurement error was set to 0.5°. The simulation results are attached. Figure 5 As shown. The results indicate that when the time measurement error is less than 50µs, the proposed algorithm approximates CRLB. However, when the error is greater than 50µs, the enhanced positive semidefinite RMSE curve still deviates, but the trend is more stable compared to the ordinary positive semidefinite method.

Claims

1. A method for locating scanning radiation sources based on semi-definite relaxation, characterized in that, Includes the following steps: Step 1: Construct a geometric positioning model based on scanning angle and time difference for a multi-observation station system; Step 2: Introduce auxiliary variables to transform the original non-convex observation model into a convex optimization problem, and construct a convex optimization localization solution framework; Step 3: Introduce bounded constraints to narrow the feasible region of the semidefinite relaxed problem, making the relaxed problem approximate the original localization problem.

2. The scanning radiation source localization method based on semi-definite relaxation according to claim 1, characterized in that, Step 1 specifically involves: Consider a two-dimensional passive localization problem, assuming that both the scanning radiation source and the observation station remain stationary, where the position of the scanning radiation source is denoted as... The location of the observation station is , Let be the number of observation stations; taking the first observation station as the reference station, based on the time difference positioning principle, the time difference positioning equation is obtained, namely: (1) In the formula, Represents the speed of light; For L2 norm operations, the distance between the observation station and the scanning radiation source is represented in equation (1); This indicates that the signal has arrived at the observation station. The time difference with observation station 1; The time-measured noise follows a zero mean and a variance of . Gaussian distribution; Using the first observation station as the reference station, and based on the geometric principle of scanning time difference positioning, the scanning angle difference is expressed as: (2) In the formula, This indicates that the main lobe of the scanned radiation source is from the observation station. The angle between observation station 1 and observation station 2. Represents the arctangent function; The noise is represented by the scanning angle difference measurement, which has zero mean and variance. Gaussian distribution; Construct a weighted least squares function based on time difference and scan angle residuals: (3) In the formula, , Represents a weighted matrix. This represents the process of diagonalizing a vector into matrix form; , This indicates the measurement deviation between multiple observation stations.

3. The scanning radiation source localization method based on semi-definite relaxation according to claim 2, characterized in that, Step 2 specifically involves: Step 2-1: Use Schur complement to transform the objective function into a positive semidefinite constraint; Considering that both the objective function and the constraint function in formula (3) are nonlinear and nonconvex, the objective function can be expressed in quadratic form through matrix constraints, i.e.: (4) In the formula, For the introduced auxiliary error variables, Indicates the joint measurement error; and They represent and Square root factorization, satisfying , ; Using Schur complement, the objective function (4) is transformed into a positive semidefinite constraint, i.e.: (5) In the formula, Represents a matrix with 1s on the diagonal; Step 2-2: Introduce auxiliary variables to relax the time difference positioning equation and the scanning angle positioning equation into a linear expression. First, expand the distance term in the TDOA positioning equation (1) by square, and obtain the expression: (6) The scanning time difference localization equation can be further transformed using the positive tangent formula, resulting in: (7) In the formula, Represents the tangent function; Introducing lifting vectors This converts a vector into a matrix variable, i.e.: (8) In the formula, Representing three-dimensional real space, This represents the element in the 3rd row and 3rd column of the matrix; at this time For real pairwise matrices, satisfy the positive semidefinite matrix constraint, i.e. And at this time , To calculate the rank of a matrix; Define the auxiliary matrix: (9) Therefore, the nonlinear variables in equation (6) are expressed as: (10) (11) In the formula, Represents the Frobenius inner product. This indicates finding the trace of a matrix; Therefore, the original nonlinear time difference positioning equation is convexly relaxed into a linear form, i.e.: (12) (13) Similarly, the localization equation (7) based on the scanning time difference is obtained by cross-multiplication: (14) In the formula, and All , , , One combination, namely: (15) (16) In the formula, and This is a constant matrix related to the location of the observation station, namely: (17) (18) Therefore, the original nonlinear scanning angle positioning equation (2) is convexly relaxed into a linear expression: (19) In the formula, is a coefficient.

4. The scanning radiation source localization method based on semi-definite relaxation according to claim 3, characterized in that, Step 3 specifically involves: By imposing distance constraints, the difficulty of the solution is reduced, i.e.: (20) At this point, constraint (20) is non-convex. According to Schur's theorem, the original norm constraint is transformed into a second-order cone constraint SOC, thus transforming the non-convex problem into a convex problem, i.e.: (21) in, Represents a two-dimensional identity matrix; Add hard constraints to the scanning angle difference Based on the geometric model of an equal-angled circle with a scanning angle, the formula for calculating the radius is: , Indicates the observation station The distance between observation station 1 and the center of the circle is calculated using the following formula: , Describes an orthogonal vector. Let be the orthogonal distance from the center of the circle to the observation station forming the baseline; therefore, the following constraint is established: (22) Using Schur's theorem, the hard angular constraint is transformed into a convex constraint, i.e.: (23) (24) The final enhanced positive semidefinite localization model is expressed as: (25) By employing an optimization toolbox, the precise location of the stationary scanning radiation source can be achieved; if the estimation results... Satisfying the rank-one constraint, i.e. Therefore, the estimated location of the scanned radiation source is as follows: ,like If it does not meet the requirements, then its principal eigenvectors are used. Projection, i.e. .

5. An electronic device, characterized in that, include: Processor and memory; The memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory to cause the electronic device to perform the method as described in any one of claims 1 to 4.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 4.

7. A chip, characterized in that, include: A processor for retrieving and running a computer program from memory, causing a device on which the chip is mounted to perform the method as described in any one of claims 1 to 4.

8. A computer program product, characterized in that, The computer program product includes a computer storage medium storing a computer program, the computer program including instructions executable by at least one processor, which, when executed by the at least one processor, implement the method as described in any one of claims 1 to 4.

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