TDOA / FDOA passive positioning method based on alternating direction multiplier method

Abstract

The invention discloses a TDOA / FDOA passive positioning method based on an alternating direction multiplier method. The method comprises the following steps: firstly, carrying out initial estimation on a target position and speed by utilizing a weighted least square method; then, performing first-order Taylor expansion on the nonlinear geometric constraint at the initial point, and reconstructing an original non-convex optimization problem into a constraint weighted least square model with linear coupling constraint; finally, an ADMM framework is introduced, and efficient decoupling solution of position and speed variables is achieved by alternately updating the target position, the speed and the Lagrange multiplier. The system, the device and the medium are used for realizing the TDOA / FDOA passive positioning method based on the alternating direction multiplier method. According to the method, the problem that traditional nonlinear optimization is prone to falling into local optimum is effectively solved, the positioning precision is remarkably improved, meanwhile, the calculation complexity is greatly reduced, and the robustness and the real-time performance of the algorithm under the conditions of low signal-to-noise ratio and complex observation are enhanced.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, specifically relating to a passive positioning method for TDOA / FDOA based on the alternating direction multiplier method. Background Technology

[0002] The Alternating Direction Multiplier Method (ADMM) combines the parallelizability of dual decomposition with the good convergence performance of the augmented Lagrange multiplier method. This framework utilizes the separable structure of the objective function to decompose the complex primal problem into a series of easily tractable subproblems for alternating iteration, effectively alleviating the computational burden of large-scale, complex constrained optimization problems. It has shown great application potential in the field of optimization algorithms and has been attempted to be introduced into localization problems. For example, patent application CN120595232A, entitled "Localization Method and System Based on Differential Time Delay and Alternating Direction Multiplier Method," proposes a passive target localization method based on ADMM, using differential time delay (DTD) to jointly estimate the positions of a stationary target and an unknown transmitter. This method demonstrates the effectiveness of ADMM in solving localization problems. However, this method and the existing ADMM localization schemes it represents have fundamental limitations: their models are designed only for stationary targets, completely lacking the utilization and modeling of FDOA observations, thus making it theoretically impossible to estimate target velocity; the constraints they handle are only static geometric constraints, failing to characterize the dynamic coupling relationship between position and velocity. This means that the existing ADMM positioning framework cannot be directly applied to the more challenging dynamic scenario of joint positioning and velocity measurement of mobile radiation sources.

[0003] To address the positioning needs of moving targets, existing technologies mainly explore two paths, but both have significant shortcomings: The first type of approach employs closed-form solutions or two-step optimization frameworks based on the traditional Lagrange multiplier method. For example, the patent application "Improved Constrained Weighted Least Squares TDOA / FDOA Passive Positioning Method Based on Lagrange Multipliers" (CN119471565A) proposes a two-step solution strategy: first, an initial closed-form solution is obtained through pseudo-linearization and constraint simplification; then, a nonlinear equation system is established based on the original constraints for iterative optimization. Although this method can obtain an analytical preliminary solution, its "step-by-step solution and iterative correction" process still has significant drawbacks: model errors introduced by constraint simplification in the first step are propagated to the second step, affecting convergence accuracy and stability; secondly, the nonlinear iterative optimization in the second step may still be affected by the quality of the initial value, and the overall computational process still has room for improvement in terms of balancing efficiency and accuracy. The second type of approach shifts to heuristic global optimization algorithms based on random search. For example, the patent application "A Passive Localization Method Based on Adaptive Step-Size Cuckoo Search Algorithm" (CN121299583A) uses an improved heuristic search to handle non-convex optimization problems. However, such methods generally suffer from slow convergence speed, high computational cost, and the quality of the solution is greatly affected by parameter settings and randomness. In scenarios with high noise or strict real-time requirements, their robustness and engineering practicality are often insufficient.

[0004] In summary, existing technologies exhibit significant "capability gaps" and "performance bottlenecks" in the joint localization of mobile radiation sources TDOA / FDOA: On the one hand, the efficient and convergent ADMM framework is only applicable to static scenarios and lacks comprehensive functionality; on the other hand, closed-form solutions and heuristic algorithms capable of handling dynamic problems have limitations in error propagation, computational efficiency, or convergence reliability, respectively. Specifically: 1. Functional limitations and model gaps: Existing ADMM localization frameworks (such as CN120595232A) are only designed for stationary targets and lack modeling for FDOA observations and velocity estimation, which cannot meet the need for estimating the complete motion state (position and velocity) of targets in dynamic scenes.

[0005] 2. Insufficient solution efficiency and robustness: Existing methods capable of handling dynamic localization, such as algorithms based on two-step closed solutions or heuristic search, often suffer from problems such as error accumulation, sensitivity to initial values, high computational complexity, or poor convergence, making it difficult to balance real-time performance and estimation accuracy in high-noise environments.

[0006] In summary, although the ADMM algorithm has shown potential in static localization, existing technologies (including the latest ADMM localization patents) have not solved the core challenge of successfully applying it to "localization and velocity measurement of mobile radiation sources based on TDOA / FDOA joint observations." Existing solutions all have inherent and insurmountable shortcomings in terms of problem modeling, observation utilization, constraint handling, and performance robustness. Summary of the Invention

[0007] To overcome the problems existing in the prior art, the present invention discloses a passive localization method for TDOA / FDOA based on the alternating direction multiplier method. First, the weighted least squares method is used to initially estimate the target position and velocity. Then, at this initial point, a first-order Taylor expansion is performed on the nonlinear geometric constraints, reconstructing the original non-convex optimization problem into a constrained weighted least squares model with linear coupling constraints. Finally, the ADMM framework is introduced, and by alternately updating the target position, velocity, and Lagrange multipliers, efficient decoupling of the position and velocity variables is achieved. This method has the advantages of high positioning accuracy and low computational complexity, enhancing the robustness and real-time performance of the algorithm under low signal-to-noise ratio and complex observation conditions.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: A passive localization method for TDOA / FDOA based on the alternating direction multiplier method includes the following steps: Step 1: Arrange at least 5 observation stations in three-dimensional space and initialize them to determine the position coordinates, motion velocity vector, signal propagation speed and iteration termination threshold of each observation station; Step 2: All observation stations synchronously receive the target signal and acquire the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements between each observation station and the reference station. Step 3: Introduce auxiliary variables of distance and rate of change of distance to transform the nonlinear TDOA / FDOA equation into a pseudo-linear system of equations; use the weighted least squares method to calculate the initial position and velocity estimates of the target as the starting point for subsequent iterations; Step 4: At the initial position and velocity estimate, perform a first-order Taylor series expansion on the nonlinear geometric constraints to reconstruct the original nonconvex optimization problem into a constrained weighted least squares model with linear coupling constraints; Step 5: Solve the constrained weighted least squares model using the Alternating Direction Multiplier Method (ADMM) framework. By alternately updating the target position, target velocity, and Lagrange multipliers until the convergence condition is met, the final output is the accurate estimation results of the target position coordinates and velocity vector.

[0009] The number of observation stations set in step 1 is , No. The location coordinates of each observation station are The velocity vector is The target location to be estimated is The velocity vector is The true distance between the target and the observation station is The rate of change of distance .

[0010] In step 2, zero-mean Gaussian white noise is added to the TDOA and FDOA measurements to simulate the actual measurement environment. in, and Given Gaussian measurement noise with a mean of zero, squaring both sides and neglecting the second-order noise term, we obtain the following TDOA / FDOA localization equation: .

[0011] In step 3, the pseudo-linear equation system is: The estimated quantity to be measured is ;matrix and Its structure is as follows: The weighting matrix used in weighted least squares The noise covariance matrix is ​​determined by the time difference of arrival (TDOA) measurement and the frequency difference of arrival (FDOA) measurement. The initial solution is obtained using the weighted least squares method: Due to intermediate variables and and unknown quantities and There is a nonlinear relationship between them, thus we obtain the following constrained weighted least squares problem: .

[0012] In step 4, the linear coupling constraint is obtained by defining the Jacobian matrix with respect to the target position and velocity and performing a first-order Taylor expansion at the initial estimation point; At the initial point Expand get: definition as well as The corresponding constraints are expressed as follows: in, , Therefore, the reconstructed constrained weighted least squares problem is restated as follows: in, .

[0013] In step 5, the solution using the ADMM framework specifically includes: Construct the augmented Lagrangian function; in, To correspond to the constraints Lagrange multipliers, This is the penalty coefficient; Using the ADMM method, the following three iterative update formulas are derived: Given a fixed target velocity and Lagrange multipliers, solve for a closed-form solution regarding the target position: First, with variables fixed... and Under the premise that the constraints are transformed into ,definition as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is ; Given a fixed target position and Lagrange multipliers, solve for the closed-form solution regarding the target velocity. Update the Lagrange multipliers; Similarly, in a fixed as well as Under the condition that the constraint is transformed into ,in as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form: Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is in, After completing the alternating updates of position and velocity, update the Lagrange multipliers to... Repeat the iterative process until the convergence condition is met, and finally output the target position. and target speed .

[0014] The convergence condition is that the change in the estimated target position and velocity values ​​obtained from two consecutive iterations is less than a preset iteration termination threshold.

[0015] A passive TDOA / FDOA localization system based on the alternating direction multiplier method, implementing the passive TDOA / FDOA localization method based on the alternating direction multiplier method described in steps 1 to 5, includes: The data acquisition module is used to control at least 5 observation stations arranged in three-dimensional space to synchronously receive the target signal and acquire the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements between the target signal and the reference station. The initial estimation module is used to introduce auxiliary variables of distance and rate of change of distance, transform the nonlinear TDOA / FDOA equation into a pseudo-linear system of equations, and use the weighted least squares method to calculate the initial position and velocity estimates of the target. The model reconstruction module is used to perform a first-order Taylor series expansion on the nonlinear geometric constraints at the initial position and velocity estimates, and reconstruct the original non-convex optimization problem into a constraint-weighted least squares model with linear coupling constraints. The iterative solution module is used to solve the constrained weighted least squares model using the Alternating Direction Multiplier Method (ADMM) framework. By alternately updating the target position, target velocity, and Lagrange multipliers, the module continues until the convergence condition is met, and finally outputs the accurate estimation results of the target position coordinates and velocity vector.

[0016] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the TDOA / FDOA passive positioning method based on the alternating direction multiplier method as described in steps 1 to 5.

[0017] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the TDOA / FDOA passive localization method based on the alternating direction multiplier method as described in steps 1 to 5.

[0018] Compared with the prior art, the present invention has the following advantages: 1. Effectively overcomes the local convergence problem of traditional nonlinear optimization methods, significantly improving positioning accuracy. Traditional TDOA / FDOA positioning methods typically solve highly nonconvex and nonlinear original observation equations directly. These problems are extremely sensitive to initial values ​​and are prone to getting trapped in local optima, especially under low signal-to-noise ratio or poor observation geometry conditions, leading to large positioning deviations and poor stability.

[0019] Step 5 of this invention obtains a robust initial estimate using the weighted least squares method. Then, a first-order Taylor series expansion is performed on the complex nonlinear geometric constraints near this initial point, transforming it into an optimization model with linearly coupled constraints. This linearization process greatly simplifies the complexity of the problem. Based on this, the ADMM framework is used for solving the problem, which can more reliably approximate the global optimum, thereby significantly improving the estimation accuracy of the target position and velocity, especially in harsh environments.

[0020] 2. The joint estimation of position and velocity was successfully decoupled, which significantly reduced the computational complexity.

[0021] TDOA information is primarily associated with the target's position, while FDOA information couples both the target's position and velocity. When the two are solved together, the position and velocity variables are deeply intertwined, forming a high-dimensional coupled optimization problem. The computational complexity of the solution process is enormous, making it difficult to meet real-time requirements.

[0022] The core innovation of this invention lies in the ingenious application of the ADMM decomposition concept. Step 5 decomposes the original coupled problem into two independent subproblems by constructing an augmented Lagrangian function: one concerning only the target position and the other concerning only the target velocity. In each iteration, the system can obtain closed-form solutions for these two subproblems separately. This divide-and-conquer strategy completely decouples the position and velocity estimation processes, transforming a complex joint optimization problem into a series of simple, parallelizable subtasks, thereby greatly reducing the computational complexity of the algorithm, improving computational efficiency, and laying the foundation for real-time engineering applications.

[0023] 3. The robustness and adaptability of the algorithm have been enhanced.

[0024] Many existing algorithms are highly sensitive to external conditions such as measurement noise and observation station layout; their performance drops sharply once these conditions change.

[0025] The two-stage design of this invention—initial estimation → ADMM fine-tuning iteration—is inherently robust. The weighted least squares estimation in step 3 effectively suppresses the influence of noise; while the ADMM iteration process in step 5, through dynamic updates of the Lagrange multipliers, adaptively adjusts the degree of constraint satisfaction, achieving stable convergence even in the presence of measurement errors. Furthermore, the framework itself possesses good flexibility, easily incorporating different prior information or weighting strategies, thus exhibiting stronger adaptability and stability to different battlefield environments, observation geometries, and noise levels.

[0026] In summary, this invention, by introducing advanced ADMM optimization theory into the field of TDOA / FDOA passive positioning, not only fundamentally solves the problem of traditional methods easily getting trapped in local optima and achieves high-precision positioning, but also achieves a balance between low complexity and high efficiency through ingenious variable decoupling. These beneficial effects collectively make this invention stand out with significant substantive features and remarkable progress compared to existing technologies in application scenarios with stringent requirements for positioning accuracy and real-time performance, such as electronic reconnaissance, target tracking, and emergency rescue. Attached Figure Description

[0027] Figure 1 This is a flowchart of the positioning algorithm of the present invention.

[0028] Figure 2 This is a schematic diagram of the positioning scenario of the present invention.

[0029] Figure 3 This is a position estimation deviation diagram under different noise levels according to the present invention.

[0030] Figure 4 This is a velocity estimation deviation diagram under different noise levels according to the present invention.

[0031] Figure 5 This is a root mean square error diagram of position estimation under different noise levels according to the present invention.

[0032] Figure 6 This is a root mean square error diagram of velocity estimation under different noise levels according to the present invention.

[0033] Figure 7 This is the cumulative density function for location estimation in this invention.

[0034] Figure 8 This is the velocity estimation cumulative density function of the present invention. Detailed Implementation

[0035] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0036] A passive localization method for TDOA / FDOA based on the alternating direction multiplier method includes the following steps: Step 1: Arrange at least 5 observation stations in three-dimensional space and initialize them to determine the position coordinates, motion velocity vector, signal propagation speed and iteration termination threshold of each observation station; The number of observation stations set in step 1 is , No. The location coordinates of each observation station are The velocity vector is The target location to be estimated is The velocity vector is The true distance between the target and the observation station is The rate of change of distance .

[0037] Step 2: All observation stations synchronously receive the target signal and acquire the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements between each observation station and the reference station. In step 2, zero-mean Gaussian white noise is added to the TDOA and FDOA measurements to simulate the actual measurement environment. in, and Given Gaussian measurement noise with a mean of zero, squaring both sides and neglecting the second-order noise term, we obtain the following TDOA / FDOA localization equation: .

[0038] Step 3: Introduce auxiliary variables of distance and rate of change of distance to transform the nonlinear TDOA / FDOA equation into a pseudo-linear system of equations; use the weighted least squares method to calculate the initial position and velocity estimates of the target as the starting point for subsequent iterations; In step 3, the pseudo-linear equation system is: The estimated quantity to be measured is ;matrix and Its structure is as follows: The weighting matrix used in weighted least squares The noise covariance matrix is ​​determined by the time difference of arrival (TDOA) measurement and the frequency difference of arrival (FDOA) measurement. The initial solution is obtained using the weighted least squares method: Due to intermediate variables and and unknown quantities and There is a nonlinear relationship between them, thus we obtain the following constrained weighted least squares problem: .

[0039] Step 4: At the initial position and velocity estimate, perform a first-order Taylor series expansion on the nonlinear geometric constraints to reconstruct the original nonconvex optimization problem into a constrained weighted least squares model with linear coupling constraints; In step 4, the linear coupling constraint is obtained by defining the Jacobian matrix with respect to the target position and velocity and performing a first-order Taylor expansion at the initial estimation point; At the initial point Expand get: definition as well as The corresponding constraints are expressed as follows: in, , Therefore, the reconstructed constrained weighted least squares problem is restated as follows: in, .

[0040] Step 5: Solve the constrained weighted least squares model using the Alternating Direction Multiplier Method (ADMM) framework. By alternately updating the target position, target velocity, and Lagrange multipliers until the convergence condition is met, the final output is the accurate estimation results of the target position coordinates and velocity vector.

[0041] In step 5, the solution using the ADMM framework specifically includes: Construct the augmented Lagrangian function; in, To correspond to the constraints Lagrange multipliers, This is the penalty coefficient; Using the ADMM method, the following three iterative update formulas are derived: Given a fixed target velocity and Lagrange multipliers, solve for a closed-form solution regarding the target position: First, with variables fixed... and Under the premise that the constraints are transformed into ,definition as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is ; Given a fixed target position and Lagrange multipliers, solve for the closed-form solution regarding the target velocity. Update the Lagrange multipliers; Similarly, in a fixed as well as Under the condition that the constraint is transformed into ,in as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form: Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is in, After completing the alternating updates of position and velocity, update the Lagrange multipliers to... Repeat the iterative process until the convergence condition is met, and finally output the target position. and target speed .

[0042] Example A passive localization method for TDOA / FDOA based on the alternating direction multiplier method includes the following steps: Step 1: Set the observation station to be located in three-dimensional space, such as... Figure 2 There are 5 observation stations. The location coordinates of each observation station are as follows: With the velocity vector As shown in Table 1: Table 1 The true position of the target to be estimated is (500, 200, 100) m, and the velocity vector is (-20, 15, 40) m / s. An iteration termination threshold is set. .

[0043] Step 2: Synchronously receive the target signal through the observation station to acquire TDOA and FDOA measurement values. In the simulation, zero-mean Gaussian white noise is added to the measurement values, with noise power... The variation range is set to -10dB to 25dB.

[0044] Step 3: Construct a pseudo-linear matrix and observation vector The initial position of the target is calculated using the weighted least squares method. and speed This serves as the starting point for the iteration of the ADMM algorithm.

[0045] Step 4: Using the initial values ​​obtained in Step 3, perform a first-order Taylor series expansion on the nonlinear distance constraint at the initial point to construct the augmented Lagrange equation.

[0046] Step 5: Substitute the data into the augmented Lagrangian function and set the initial penalty coefficient. Alternately update the target position, velocity, and Lagrange multipliers until the convergence threshold is met.

[0047] This embodiment comprehensively evaluates the performance of the algorithm proposed in this invention through detailed simulation experiments. 10,000 Monte Carlo simulations were conducted to obtain statistically stable results. The system's estimation accuracy of target position and velocity was verified under different levels of measurement noise power. The algorithm's evaluation criteria used were estimation bias and root mean square error (RMSE), which were used to measure the average deviation and overall dispersion level of the estimation results, respectively. To further objectively evaluate the algorithm's performance, the Cramero lower bound (CRLB) was also introduced in the simulation as a theoretically achievable benchmark for the best estimation accuracy, thus clearly revealing the gap between each algorithm and the theoretical limit.

[0048] In this scenario, the proposed method is compared with the classic TSWLS algorithm, the improved TSWLS algorithm, the iterative constrained weighted least squares (ICWLS) algorithm, and the bias reduction (BiasRed) algorithm, using CRLB as the benchmark. Figures 3 to 6 As shown, the method of this invention exhibits estimation accuracy close to or even reaching CRLB under all test noise conditions, with particularly significant advantages in high-noise environments. Specifically, the method of this invention achieves position estimation performance (e.g., at 25 dB noise power) of [missing information - likely a specific accuracy value]. Figure 3 ) and speed estimation performance (e.g. Figure 4 The proposed algorithm still achieves CRLB accuracy. Other comparative algorithms exhibit varying degrees of performance degradation with increasing noise. TSWLS and BiasRed algorithms deviate from CRLB first, followed by ICWLS and Improved TSWLS at 20dB noise power. Notably, because the ICWLS method relies on the previous stage's estimation results to approximate the distance and distance change rate in its constraints, this approximation error accumulates significantly in high-noise environments, causing its estimation accuracy to deviate severely from the theoretical lower bound of CRLB. Under 25dB noise power, the improved TSWLS algorithm has a root mean square error (RMSE) of 118.15 meters, while the proposed method significantly reduces it to 66.23 meters, improving positioning accuracy by 43.9%. Figure 5 Regarding velocity estimation, the improved TSWLS algorithm has a root mean square error (RMSE) of 1809.3 m / s, while the method of this invention has a RMSE of only 25.45 m / s, representing a two-order-of-magnitude improvement in accuracy. Figure 6This highlights the core advantage of the ADMM algorithm in motion parameter estimation.

[0049] This superior performance primarily stems from the ADMM framework and problem reconstruction strategy employed in this invention: by decomposing the original non-convex problem into two subproblems—position and velocity—with closed-form analytical solutions, and by dynamically harmonizing constraints using augmented Lagrange multipliers during iteration, the model error accumulated due to linearization approximation in traditional methods is effectively suppressed. Furthermore, ADMM's alternating iteration mechanism enhances robustness to initial value selection, avoiding the problem of conventional algorithms easily getting trapped in local optima or diverging, thus ensuring the numerical stability and convergence reliability of the algorithm under high-noise environments at the system level.

[0050] To further verify the robustness and engineering applicability of the method under non-ideal observation conditions, a test scenario with randomly distributed sensor locations was designed in the specific implementation. In this scenario, the spatial coordinates of the five observation stations were randomly generated within a three-dimensional cube region with a side length of 1000 meters, and their velocity vectors remained consistent with the parameters in Table 1. The actual position coordinates of the fixed moving radiation source were also considered. With velocity vector Under set noise power conditions, a large number of independent experimental samples were obtained through Monte Carlo simulation, and the cumulative distribution function curves of position estimation error and velocity estimation error were plotted. The position (e.g., ...) obtained by the method of this invention... Figure 7 ) and speed (e.g.) Figure 8 The cumulative distribution function curve of the estimation error is superior to that of the comparative algorithm, indicating that the estimation results of this invention are more reliable at the same error level. This proves that the method of this invention can maintain a high-precision joint estimation capability for the position and velocity of a moving radiation source even in complex and uncontrollable station deployment environments, and its performance advantages are statistically significant and environmentally adaptable. From a geometric robustness perspective, this result strongly supports the effectiveness and reliability of the technical solution of this invention in real-world complex scenarios.

[0051] In summary, the simulation results not only verify the high accuracy and robustness of the present invention in dynamic target positioning and velocity measurement, but also highlight its important application potential in complex electromagnetic environments and real-time processing scenarios, especially suitable for electronic reconnaissance and target tracking systems with extremely high requirements for concealment, real-time performance and reliability.

Claims

1. A passive localization method for TDOA / FDOA based on the alternating direction multiplier method, characterized in that, Includes the following steps: Step 1: Arrange at least 5 observation stations in three-dimensional space and initialize them to determine the position coordinates, motion velocity vector, signal propagation speed and iteration termination threshold of each observation station; Step 2: All observation stations synchronously receive the target signal and acquire the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements between each observation station and the reference station. Step 3: Introduce auxiliary variables of distance and rate of change of distance to transform the nonlinear TDOA / FDOA equation into a pseudo-linear system of equations; use the weighted least squares method to calculate the initial position and velocity estimates of the target as the starting point for subsequent iterations; Step 4: At the initial position and velocity estimate, perform a first-order Taylor series expansion on the nonlinear geometric constraints to reconstruct the original nonconvex optimization problem into a constrained weighted least squares model with linear coupling constraints; Step 5: Solve the constrained weighted least squares model using the Alternating Direction Multiplier Method (ADMM) framework. By alternately updating the target position, target velocity, and Lagrange multipliers until the convergence condition is met, the final output is the accurate estimation results of the target position coordinates and velocity vector.

2. The passive positioning method for TDOA / FDOA according to claim 1, characterized in that, The number of observation stations set in step 1 is , No. The location coordinates of each observation station are The velocity vector is The target location to be estimated is The velocity vector is ; The true distance between the target and the observation station is The rate of change of distance .

3. The passive positioning method for TDOA / FDOA according to claim 1, characterized in that, In step 2, zero-mean Gaussian white noise is added to the TDOA and FDOA measurements to simulate the actual measurement environment. Without loss of generality, the first observation station is set as the reference station. and These are Gaussian measurement noises with a mean of zero. and Each is an observation station Squaring both sides of the distance difference and the rate of change of distance between the reference station 1 and the reference station 1, and neglecting the second-order noise term, we obtain the following TDOA / FDOA positioning equation: 。 4. The passive positioning method for TDOA / FDOA according to claim 1, characterized in that, In step 3, the pseudo-linear equation system is: The estimated quantity to be measured is ;matrix and Its structure is as follows: The weighting matrix used in weighted least squares The noise covariance matrix is ​​determined by the time difference of arrival (TDOA) measurement and the frequency difference of arrival (FDOA) measurement. The initial solution is obtained using the weighted least squares method: Due to intermediate variables and and unknown quantities and There is a nonlinear relationship between them, thus we obtain the following constrained weighted least squares problem: 。 5. The passive positioning method for TDOA / FDOA according to claim 1, characterized in that, In step 4, the linear coupling constraint is obtained by defining the Jacobian matrix with respect to the target position and velocity and performing a first-order Taylor expansion at the initial estimation point; At the initial point Expand get: definition as well as The corresponding constraints are expressed as follows: in, , Therefore, the reconstructed constrained weighted least squares problem is restated as follows: in, 。 6. The passive positioning method for TDOA / FDOA according to claim 1, characterized in that, In step 5, the solution using the ADMM framework specifically includes: Construct the augmented Lagrangian function; in, To correspond to the constraints Lagrange multipliers, This is the penalty coefficient; Using the ADMM method, the following three iterative update formulas are derived: Given a fixed target velocity and Lagrange multipliers, solve for a closed-form solution regarding the target position: First, with variables fixed... and Under the premise that the constraints are transformed into ,definition as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is ； Given a fixed target position and Lagrange multipliers, solve for the closed-form solution regarding the target velocity. Update the Lagrange multipliers; Similarly, in a fixed as well as Under the condition that the constraint is transformed into ,in as well as To augment the Lagrange equation and retain the contents The term is: Organized into a list of things to do Standard convex quadratic form: Among them, matrix and The specific forms are as follows: Therefore, we can obtain The weighted least squares estimate is in, After completing the alternating updates of position and velocity, update the Lagrange multipliers to... Repeat the iterative process until the convergence condition is met, and finally output the target position. and target speed .

7. The passive positioning method for TDOA / FDOA according to claim 6, characterized in that, The convergence condition is that the change in the estimated target position and velocity values ​​obtained from two consecutive iterations is less than a preset iteration termination threshold.

8. A passive TDOA / FDOA positioning system based on the alternating direction multiplier method, implementing the passive TDOA / FDOA positioning method based on the alternating direction multiplier method as described in any one of claims 1 to 7, characterized in that, include: The data acquisition module is used to control at least 5 observation stations arranged in three-dimensional space to synchronously receive the target signal and acquire the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements between the target signal and the reference station. The initial estimation module is used to introduce auxiliary variables of distance and rate of change of distance, transform the nonlinear TDOA / FDOA equation into a pseudo-linear system of equations, and use the weighted least squares method to calculate the initial position and velocity estimates of the target. The model reconstruction module is used to perform a first-order Taylor series expansion on the nonlinear geometric constraints at the initial position and velocity estimates, and reconstruct the original non-convex optimization problem into a constraint-weighted least squares model with linear coupling constraints. The iterative solution module is used to solve the constrained weighted least squares model using the Alternating Direction Multiplier Method (ADMM) framework. By alternately updating the target position, target velocity, and Lagrange multipliers, the module continues until the convergence condition is met, and finally outputs the accurate estimation results of the target position coordinates and velocity vector.

9. An electronic device, characterized in that, The method includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the TDOA / FDOA passive localization method based on the alternating direction multiplier method as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the TDOA / FDOA passive localization method based on the alternating direction multiplier method as described in any one of claims 1 to 7.

Images