Positioning method and system based on differential time delay and alternating direction multiplier method

Abstract

The present invention belongs to the field of communications technology and discloses a positioning method and system based on differential time delay and alternating direction multiplier method. The method comprises: obtaining the position information and differential time delay of a receiver; and jointly estimating the position of a target object and the position of a transmitter based on the alternating direction multiplier method. During the joint estimation, a first variable and a second variable are introduced, and during iterative calculations during the joint estimation process, the original optimization problem is decomposed into two sub-problems related to the first variable, the second variable, and the Lagrange multiplier coefficient. During each iteration, analytical solutions are obtained for the two sub-problems, and the first variable, the second variable, and the Lagrange multiplier coefficient are updated. The present invention can significantly reduce computational complexity while improving the accuracy of target position estimation.

Description

Technical Field

[0001] The present invention relates to the field of wireless communication technology, and in particular to a positioning method and system based on differential time delay and alternating direction multiplier method. Background Art

[0002] Passive target positioning technology plays a vital role in modern wireless communications, sonar, and radar systems. With the continuous advancement of technology, the demand for high-precision and efficient positioning methods is increasing. Traditional distance-based positioning methods, such as measuring direct distance, distance difference, and distance sum, can meet positioning needs to a certain extent. However, these methods often rely on precise clock synchronization, which is often difficult to achieve in practical applications. Furthermore, these methods often suffer from insufficient accuracy and high computational complexity when dealing with positioning problems in complex environments.

[0003] In recent years, positioning methods based on differential time delay (DTD) have attracted considerable attention due to their ability to avoid clock synchronization issues. DTD methods determine the location of a target by measuring the time difference between the signal's transmission from the transmitter to the target and then to the receiver. While this method theoretically offers high accuracy and robustness, its performance is often limited in practice by measurement noise and environmental factors. To address these issues, researchers have proposed various improved methods, such as maximum likelihood estimation (MLE), weighted least squares (WLS), and convex relaxation techniques.

[0004] Maximum likelihood estimation is a well-known method for solving measurement noise that follows a Gaussian distribution. However, because the MLE problem is inherently nonlinear, its solution is often complex and prone to falling into local optima. To address this issue, the weighted least squares method (WLS) simplifies the computation by transforming the nonlinear problem into a pseudo-linear expression. However, the WLS method requires the introduction of a large number of auxiliary variables when generating the pseudo-linear expression, which not only increases computational complexity but also leads to performance degradation when insufficient measurement data is available.

[0005] In summary, how to improve the accuracy of target position estimation while reducing computational complexity is one of the important issues that need to be solved urgently in this field. Summary of the Invention

[0006] The purpose of the present invention is to provide a positioning method and system based on differential time delay and alternating direction multiplier method to address the deficiencies in the prior art, which can significantly reduce the computational complexity while improving the accuracy of target position estimation.

[0007] The present invention provides a positioning method based on differential time delay and alternating direction multiplier method, which includes:

[0008] Obtain receiver position information and differential time delay;

[0009] The position of the target object and the position of the transmitter are jointly estimated based on the alternating direction multiplier method. During the joint estimation, the first variable and the second variable are introduced, and during the iterative calculation in the joint estimation process, the original optimization problem is decomposed into two sub-problems related to the first variable, the second variable and the Lagrange multiplier coefficient. In each iteration, the analytical solutions of the two sub-problems are solved respectively, and the first variable, the second variable and the Lagrange multiplier coefficient are updated.

[0010] As described above, the positioning method based on differential time delay and alternating direction multiplier method, wherein, optionally, in the standard double-block alternating multiplier method, when the first variable and the second variable have linear constraints, the objective function of the original optimization problem is expressed in the form of a two-part sum, and the augmented Lagrangian function of the objective function of the original optimization problem is determined.

[0011] As described above, the positioning method based on differential time delay and alternating direction multiplier method, wherein, optionally, in the process of jointly estimating the target object position and the transmitter position based on the alternating direction multiplier method, the iterative stopping condition is that the original residual is less than the convergence tolerance.

[0012] As described above, the positioning method based on differential time delay and alternating direction multiplier method, wherein, optionally, the original optimization problem is,

[0013] ;

[0014] Among them, x is the first variable and z is the second variable;

[0015] ;

[0016] ;

[0017] is the location coordinate of the receiver, M is the number of receivers; K is the spatial dimension.

[0018] As described above, in the positioning method based on differential time delay and alternating direction multiplier method, optionally, the augmented Lagrangian function of the objective function of the original optimization problem is:

[0019] ;

[0020] in, are the Lagrange multiplier coefficients, is the penalty parameter.

[0021] The positioning method based on differential time delay and alternating direction multiplier method as described above, wherein, optionally, in the first During the iterative update of the step, the first of the two sub-problems after decomposition is:

[0022] ;

[0023] The second of the two sub-problems after decomposition is:

[0024] ;

[0025] The update formula of the Lagrange multiplier coefficient is:

[0026] .

[0027] As described above, in the positioning method based on differential time delay and alternating direction multiplier method, optionally, the analytical solution of the first subproblem is:

[0028] ;in,

[0029] ;

[0030] The analytical solution to the second subproblem is:

[0031] ;

[0032] Among them, G and is a constant,

[0033] ;

[0034] in, , and For The three vectors extracted from ;

[0035] in, .

[0036] The present invention also proposes a positioning system based on differential time delay and alternating direction multiplier method, comprising:

[0037] at least one transmitter for transmitting a measurement signal;

[0038] a plurality of receivers for receiving measurement signals;

[0039] A processor is communicatively connected to the receiver, and is used to obtain the position information of each receiver and the measurement result of each receiver, and estimate the position of the target object; wherein,

[0040] The processor uses any of the above methods to solve the problem of locating the target object by converting the problem into two sub-problems with analytical solutions.

[0041] As described above, in the positioning system based on differential time delay and alternating direction multiplier method, optionally, the measurement result of each receiver includes the differential time delay corresponding to each receiver.

[0042] As described above, in the positioning system based on differential time delay and alternating direction multiplier method, optionally, the number of the transmitters is 1-20 and they are deployed separately; the number of the receivers is 2 to 20 and they are deployed separately.

[0043] Compared to existing technologies, this invention utilizes a standard two-stage ADMM method based on DTD to achieve efficient and accurate estimation of target and transmitter positions, improving computational accuracy. Within the ADMM method, this invention effectively simplifies the computational process by decomposing the complex optimization problem into two subproblems with analytical solutions. As a result, the ADMM solution demonstrates rapid processing speed and excellent convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flowchart of the steps of the algorithm of embodiment 1 of the present invention;

[0045] Figure 2 It is a schematic diagram of the working principle of the target positioning system proposed by the present invention;

[0046] Figure 3 This is a flowchart of the algorithm steps of the second embodiment of the present invention;

[0047] Figure 4 This is a structural block diagram of the system proposed in Example 3 of the present invention. DETAILED DESCRIPTION

[0048] The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and are not to be construed as limiting the present invention.

[0049] To address the issues raised in the background, convex relaxation techniques improve positioning accuracy by relaxing the nonlinear positioning problem into a semi-definite programming (SDP) problem. However, the SDP method also requires the introduction of a large number of auxiliary variables, which limits its applicability in practical applications. Furthermore, the SDP method has high computational complexity when dealing with large-scale distributed optimization problems, making it difficult to meet real-time requirements.

[0050] To overcome the limitations of these methods, the Alternating Direction Method of Multipliers (ADMM) was introduced to the localization problem. By decomposing the complex optimization problem into multiple subproblems, ADMM effectively simplifies the computational process, demonstrating fast processing and good convergence performance. However, existing ADMM methods often require additional proof of global convergence when addressing DTD localization, which limits their application to a certain extent.

[0051] The main reasons for the above problems are: first, the introduction of a large number of auxiliary variables will lead to increased computational complexity and decreased performance when measurement data is insufficient; second, the convergence of the problem solution is the key to obtaining the final estimation result and finding the position of the target object; third, there is no guarantee that the sub-problem has an analytical solution, which increases the complexity of the calculation process.

[0052] To further illustrate the innovations and underlying principles of the present invention, some basic concepts are first introduced: a transmitter transmits signals, whose location is unknown and can be single or multiple; a receiver receives signals, whose location is known and generally contains at least two data points. The signals received by the receiver include unreflected signals and signals reflected by a target; a target is an object capable of reflecting radar wave signals, which can be single or multiple, and whose position coordinates are the result to be calculated; differential time delay (DTD): a transmitter sends a radar wave signal, which is reflected by a target and then transmitted to the receiver. The signal propagation time from transmitter to target to receiver is called time delay, which is the difference between the time the signal travels directly from the transmitter to the receiver. The time difference between the signal reaching different receivers is calculated as differential time delay; positioning calculation refers to using differential time delay and the receiver's position information to calculate the position of the signal source through geometric methods.

[0053] Using DTD measurements between the transmitter, target, and receiver, and introducing auxiliary variables to establish a pseudo-linear equation, the ADMM algorithm is used to accurately estimate the target's position. Because a large number of auxiliary variables are generated in this process, the use of WLS and SDP solutions is inefficient when there are insufficient receivers. This invention simplifies the solution process by applying a standard two-block ADMM method, achieving efficient and accurate estimation. The process is as follows:

[0054] 1. System Deployment: Deploy N transmitters and M receivers in a K-dimensional monitored area to locate a target, where M ≥ K, and N*M ≥ (N+1)K, N ≥ 2, so as to jointly locate the target and the transmitter; the value of K is usually 2 or 3.

[0055] 2. DTD measurement: A transmitter sends a signal, which reflects off a target and is received by a receiver. The transmitter records all signal propagation time delays from the transmitter to the target and back to the receiver, and the measurement results are sent to a server or centralized processor. This means obtaining the receiver's location information and differential time delay (DTD). The receiver's location information includes the receiver's ID and coordinates.

[0056] 3. Estimation of target and transmitter positions: The ADMM method is used on a server or processor to jointly estimate the target and transmitter positions, decomposing the positioning problem into two sub-problems and solving their analytical solutions. Decomposing the positioning problem into two sub-problems and solving their analytical solutions is a key improvement in the present invention. That is, during the joint estimation, the first and second variables are introduced, and during the iterative calculation during the joint estimation process, the original optimization problem is decomposed into two sub-problems related to the first and second variables and the Lagrange multiplier coefficients. During each iteration, the analytical solutions of the two sub-problems are solved separately, and the first and second variables and the Lagrange multiplier coefficients are updated.

[0057] 4. Estimation error compensation: Perform position error compensation on the target object position estimation result in step 3.

[0058] Assume that a passive target positioning system is deployed in a K-dimensional space (K is usually equal to 2 or 3). The system consists of N The transmitter and M located , =1, 2, ..., M receivers. Figure 1 The embodiment shown is a schematic diagram of a separate radar wave target positioning system including two transmitters and two receivers in a K-dimensional space. The position coordinates of the two receivers are known and are represented as 、 The position coordinates of the two transmitters are unknown and are expressed as 、 There is also a target object to be located in the monitoring area, and its position coordinates are expressed as .like Figure 1 As shown, the radar wave signal has a direct path from the transmitter to the receiver. , the indirect path from the transmitter to the receiver after being reflected by the target , for single transmitter (ST) positioning, n=1, there is a relationship:

[0059] (1)

[0060] Since the transmitter's position information is also unknown, it needs to be estimated together with the target's position p, so the actual distance It cannot be measured directly, and its observable measurement value is expressed as:

[0061]

[0062] in, To include noise Therefore, the vector form is expressed as:

[0063]

[0064] Need to be jointly determined and the value of p, denoted as The maximum likelihood estimation (MLE) estimate for this problem is expressed as the following minimization problem:

[0065] ;

[0066] Among them, ∑ noise covariance is a known value.

[0067] (5).

[0068] Through the above process, the positioning problem of the target object can be converted into the minimization problem in formula (4).

[0069] According to the number of transmitters, there are two types of problems: single transmitter (ADMM-ST) and multiple transmitter (ADMM-MT). The following describes the single transmitter and multiple transmitters in the first and second embodiments.

[0070] Example 1

[0071] For the ADMM-ST (single transmitter) problem, in the standard dual-block alternating direction multiplier method (ADMM), the first variable x and the second variable z are solved independently under linear constraints by decomposing the optimization problem into two subproblems. The original optimization problem can be expressed as:

[0072] ;

[0073] Among them, the objective function is expressed in the form of the sum of two parts, namely and Therefore, the augmented Lagrangian function of formula (6) is:

[0074] ;

[0075] in, is the Lagrange multiplier coefficient, A is the coefficient matrix related to the receiver position, is the penalty parameter. According to the ADMM method, the iterative update at step k is:

[0076]

[0077] The analytical solution is modified to reduce estimation bias caused by noise by iteratively updating the first variable x, the second variable z, and the Lagrange multiplier. During this iterative process, the constraints between the subproblems are coordinated using the augmented Lagrange multiplier method until a preset convergence criterion is met. During this process, the position variable (i.e., the first variable x) and the geometric relationship variable (i.e., the second variable z) are alternately optimized.

[0078] The following uses formulas (9a) to (14) to explain the theoretical basis for decomposing formula (4) into two sub-problems, formula (8a) and formula (8b).

[0079] In order to express the problem (4) as a dual-block ADMM form, as shown in formula (6), the intermediate variable is introduced , and , which is defined as follows:

[0080]

[0081] The objective function is decomposed into two parts related to the position variable and the norm constraint through the intermediate variables. Second variable for:

[0082]

[0083] Formula (4) is equivalent to the following minimization problem:

[0084]

[0085] in is a function of the second variable z,

[0086]

[0087] Therefore, problem (11) can be rewritten as formula (6), where

[0088]

[0089] in addition, and is defined as:

[0090]

[0091] Therefore, the augmented Lagrangian function is expressed as Equation (7), and then the problem is solved by calculating Equations (8a)-(8c) through iterative steps.

[0092] The following will explain how to solve the two sub-problems corresponding to formula (8a) and formula (8b). Specifically, for the sub-problem corresponding to formula (8a), since , by ignoring irrelevant terms, the minimization problem (8a) is transformed into:

[0093]

[0094] in

[0095] Obviously, the analytical solution of formula (15) is:

[0096]

[0097] According to the definition of A, the expression of formula (16) is simplified to:

[0098]

[0099] in,

[0100]

[0101] That is, for the subproblem corresponding to formula (8a), since there is an analytical solution, the solution process of the subproblem corresponding to formula (8a) can be greatly simplified during the estimation process, which is conducive to simplifying the calculation.

[0102] For the subproblem corresponding to formula (8b), since is a function of the norm of the intermediate variable. Ignoring the constant term, the subproblem corresponding to formula (8b) can be reorganized into formula (19).

[0103]

[0104] in , Indicates assignment.

[0105] set up , and For The three vectors extracted from

[0106]

[0107] To achieve the minimization goal, the vector needs to satisfy the following conditions:

[0108]

[0109] In addition, update the value for:

[0110]

[0111] in, for The vector norm of , , , It is the iterative intermediate variable generated according to the augmented Lagrange multiplier and constraints;

[0112]

[0113] Obviously, the matrix C depends on the number of receivers in (5). When the noise covariance When the covariance of the noise is known, Generally, it can be obtained through measurement or modeling. The update expression of formula (22) can be simplified as:

[0114]

[0115] in,

[0116]

[0117] Obviously, for a certain M, and , G and is a constant.

[0118] According to formula (12), by Extracted from 、 and and insert it into formulas (21a)-(21c). Therefore, u, and The value of is also updated in the following way:

[0119]

[0120] From formula (9a) to formula (9c), we can know that u, 、 、 The relationship between and p can be determined and the values ​​of p, that is, the position of the transmitter and the position of the target.

[0121] The above is the theoretical basis for decomposing the original positioning problem into the subproblems corresponding to Equations (8a) and (8b). In practical applications, when jointly estimating the target position and the transmitter position, iterative calculations are required. During each iteration, the two decomposed subproblems are solved.

[0122] The stopping criterion of the iterative calculation is determined based on the original residual. is defined as:

[0123] ;

[0124] Where B is the constraint matrix and c is the constant vector.

[0125] In the present invention, the stopping criterion of ADMM iteration is that the original residual is small enough, that is:

[0126] ;

[0127] The parameters is the convergence tolerance, set to a value close enough to zero, such as In addition, the penalty parameter Need to change to adjust the convergence speed of the two blocks. In order to simplify the algorithm, ADMM-ST algorithm will Fixed, for example, it can be set to 0.5. The algorithm iteration stops and the target position estimate p and the single transmitter position estimate are obtained. .

[0128] For the single transmitter problem, the algorithm is as follows:

[0129] Input parameters include: d, s, ∑, , ;

[0130] Output parameters include: p, t;

[0131] Please refer to Figure 2 , the corresponding steps of the ADMM-ST algorithm are:

[0132] S11: Initialization 、 、 is a non-zero random value;

[0133] S12: Import A, B, C and c according to formula (5) and formula (14);

[0134] S13: Generate according to formula (18) and formula (25) , G, ;

[0135] S14: Enter the ADMM update loop, and the initial number of iterations is k=1;

[0136] S15: Update according to formula (17) ,in Calculated by formula (15);

[0137] S16: Update according to formulas (26a)-(26c) in Obtained by formula (24);

[0138] S17: Update according to formula (8c) ;

[0139] S18: Update k=k+1;

[0140] S19: If the accuracy satisfies formula (28), the iterative calculation is terminated. If the accuracy does not satisfy formula (28), steps S15 to S19 are repeated. That is, the iteration stopping condition is that the original residual is less than the convergence tolerance.

[0141] After the above steps, according to formula (26a) to formula (26c) and formula (9a) to formula (9c), we can calculate That is, the joint estimation of the target position and the transmitter position is completed.

[0142] The two-block ADMM algorithm problem (11) is equivalent to the MLE minimization problem (4), and the two sub-problems have linear constraints; the correction of the iterative solution by the augmented Lagrange multiplier further suppresses the influence of noise and ensures that the expected value of the final estimator converges to the actual position; the mean square error (MSE) of the final estimator can reach the Cramer-Rao lower bound value, and the bias approaches zero.

[0143] Example 2

[0144] This embodiment is an improvement made on the basis of the first embodiment, that is, the application of this method is extended from the single-transmitter problem to the multi-transmitter problem.

[0145] In many cases, multiple transmitters are used to improve the performance of target positioning. The frequencies of the signals transmitted by multiple transmitters are set to different frequencies, and the receiver can distinguish the transmitters by the signal frequency.

[0146] For the ADMM-MT (multi-transmitter) problem, assume that N transmitters are deployed in K-dimensional space. The transmitter sends a signal, which is reflected by the target and finally received by M receivers. Similarly, the position of the target is represented by p and the position of the receiver is represented by Indicates that, unlike the case of a single transmitter (ST), the position information of multiple transmitters (MT) is used Denote, where n = 1, 2, ..., N. Therefore, the DTD measurement is modeled as:

[0147] ;

[0148] Therefore, the expression of its noise is:

[0149] ;

[0150] The vector form is:

[0151] ;

[0152] Similarly, additional noise Assuming a Gaussian distribution with zero mean, the covariance is In the case of multiple transmitters, the first variable x is defined as:

[0153] ;

[0154] Therefore, the maximum likelihood (ML) estimator is also expressed as (4), where:

[0155] ;

[0156] Due to the different dimensions of the first variable, the definition of the intermediate variable is slightly different from that in the single-transmitter scenario. The specific definitions are as follows:

[0157] ;

[0158] in =1,2,...,M, 1,...,N. Therefore, the second variable z is defined as:

[0159]

[0160] The minimization problem (11) becomes the following form:

[0161]

[0162] in, =1,2,...,M, 1,...,N. Function Defined as:

[0163]

[0164] Similarly, formula (36) can be rewritten as formula (6), where and The definition of is the same as that of formula (13a) and formula (13b). In addition, A, B, and c are defined as:

[0165]

[0166] When the positioning problem is expressed in the same form as formula (6), the augmented Lagrangian function of formula (7) can be easily obtained similar to the derivation steps in embodiment 1. The iterative update process of ADMM is the same as the expressions of formula (8a) to formula (8c). In the multi-transmitter (MT) scenario, the size of the first variable x increases to , the size of the second variable z increases to Although the sizes of the first variable x and the second variable z are different from those of the single transmitter (ST), the expressions of the analytical solutions are almost the same. Due to the same expression (15), the analytical solution of the subproblem corresponding to formula (8a) in the multi-transmitter problem can also be expressed by formula (16), where A is given by formula (38). Similarly, The update expression is also simplified to the form of formula (17). The solution of the subproblem corresponding to formula (8b) is also similar to the case of a single transmitter. The minimization problem is expressed in the form of formula (17). By applying a similar method to ADMM-ST, it can be obtained:

[0167]

[0168] in , Therefore, update the value Also obtained through formula (22), represents the norm of h, where:

[0169]

[0170] Similarly, The update of can be simplified to formula (24), where G and Defined in (25). For a given M, N, and , G and are constants with fixed values. Therefore, they can be determined in advance before the ADMM step begins.

[0171] Similarly, the vector norm is obtained by , and insert it into equations (39a) to (39c), thereby updating , and We obtain formulas (41a) to (41c),

[0172]

[0173] in , .

[0174] In the above process, the ADMM-MT algorithm is almost the same as the ADMM-ST algorithm. The difference is that the dimensions of the two update variables are increased. Using the original residual The same stopping criteria as for applies, and the penalty parameter is fixed.

[0175] In this embodiment, for the multi-transmitter problem, the input parameters are: d, s, ∑, , ; Output: .

[0176] Please refer to Figure 3 , the steps of ADMM-MT algorithm are as follows:

[0177] S21: Initialization 、 、 is a non-zero random value;

[0178] S22: import A, B, C and c according to formula (33) and formula (38);

[0179] S23: Generated according to formula (18) and formula (25) , G, ;

[0180] S24: Enter the ADMM update loop, and the initial number of iterations is k=1;

[0181] S25: Update according to formula (17) , where A is defined by formula (38);

[0182] S26: Update according to formula (41a) to formula (41c) in Obtained by formula (24);

[0183] S27: Update according to formula (8c) ;

[0184] S28: Update k=k+1;

[0185] S29: If the accuracy satisfies the condition (28), the iterative calculation is terminated; if the accuracy does not satisfy the formula (28), repeat steps S25 to S29.

[0186] In the above process, when the iterative calculation is completed, the position of the target object and the position of the transmitter can be calculated according to formulas (41a) to (41c) and formulas (34a) to (34c).

[0187] Through the solutions disclosed in Examples 1 and 2 above, the present invention proposes an innovative solution based on the standard two-block alternating direction multiplier method (ADMM) to the problem of asynchronous estimation based on differential time delay (DTD) in passive target positioning. By introducing intermediate position variables, the non-convex positioning problem is decomposed into two convex optimization sub-problems with analytical solutions, breaking through the estimation bias and computational complexity bottlenecks caused by over-reliance on auxiliary variables in traditional weighted least squares (WLS) and semi-definite programming (SDP) methods. A separable objective function is established through a variable partitioning strategy, transforming the original nonlinear maximum likelihood estimation (MLE) problem into a standard ADMM form. Secondly, a projection operator and proximal mapping with closed-form solutions are developed to achieve efficient solution of the sub-problems. Finally, geometric constraints between parameters are used for iterative correction, significantly improving positioning accuracy while ensuring global convergence.

[0188] Compared with the prior art, the first and second embodiments proposed in the present invention have at least the following beneficial effects: Reducing the number of required receivers: Since a solution equivalent to the maximum likelihood estimator (MLE) is provided, the number of receivers required to uniquely determine the position of the target object is reduced. High efficiency: The ADMM estimator can provide performance close to the Clamer-Rao lower bound (CRLB) accuracy and has high computational efficiency. The proposed algorithm exhibits stronger robustness when measurement data is insufficient, and its convergence speed is improved compared to traditional methods while having higher accuracy, providing a new theoretical paradigm for real-time target tracking in distributed radar networks. It can also be extended to other range-based positioning, such as using direct range positioning, range difference positioning, and range sum positioning.

[0189] Example 3

[0190] This embodiment proposes a system applicable to the methods described in the first and second embodiments, wherein please refer to Figure 4 , comprising: at least one transmitter for transmitting a measurement signal; a plurality of receivers for receiving the measurement signal; a processor, communicatively connected to the receivers, the processor being configured to obtain the location information of each receiver and the measurement result of each receiver, and estimate the location of the target object; the processor using the method described in Example 1 or Example 2 to solve the positioning problem of the target object by converting it into two sub-problems with analytical solutions. The measurement result of each receiver includes the differential time delay corresponding to each receiver. The number of the transmitters is 1-20, and they are deployed separately; the number of the receivers is 2 to 20, and they are deployed separately.

[0191] This embodiment is used in conjunction with the first or second embodiment, that is, the method described in the first or second embodiment is implemented based on the hardware provided by this embodiment.

[0192] Through the above examples through Example 3, in Example 1, DTD-based single-transmitter (ST) target positioning is constructed as a standard ADMM form, consisting of two convex subproblems. The ADMM-ST solution is achieved by obtaining analytical solutions to these two subproblems. In Example 2, the ADMM-ST solution is extended to the scenario of multiple-transmitter (MT) target positioning, and an ADMM-MT solution method is designed for MT target positioning. Because the decomposed subproblems are convex, ADMM has a natural global convergence property. This invention adopts a standard two-stage ADMM method based on DTD to achieve efficient and accurate estimation of target and transmitter positions. In ADMM, the complex optimization problem is decomposed into several subproblems, effectively simplifying the computational process. As a result, the ADMM solution method demonstrates fast processing speed and good convergence. This method has also been extended to the multi-transmitter scenario, demonstrating good performance.

[0193] The above describes in detail the structure, features and effects of the present invention based on the embodiments shown in the drawings. The above is only a preferred embodiment of the present invention, but the scope of implementation of the present invention is not limited to what is shown in the drawings. Any changes made in accordance with the concept of the present invention, or modifications to equivalent embodiments with equivalent changes, which do not exceed the spirit covered by the description and drawings, should be within the scope of protection of the present invention.

Claims

1. A positioning method based on differential time delay and alternating direction multiplier method, characterized in that: include: Obtain receiver position information and differential time delay; The position of the target object and the position of the transmitter are jointly estimated based on the alternating direction multiplier method. During the joint estimation, the first variable and the second variable are introduced, and during the iterative calculation in the joint estimation process, the original optimization problem is decomposed into two sub-problems related to the first variable, the second variable and the Lagrange multiplier coefficient. In each iteration, the analytical solutions of the two sub-problems are solved respectively, and the first variable, the second variable and the Lagrange multiplier coefficient are updated.

2. The positioning method based on differential time delay and alternating direction multiplier method according to claim 1, characterized in that: In the standard two-block alternating multiplier method, the objective function of the original optimization problem is expressed in the form of a two-part sum when the first and second variables have linear constraints, and the augmented Lagrangian function of the objective function of the original optimization problem is determined.

3. The positioning method based on differential time delay and alternating direction multiplier method according to claim 2, characterized in that: In the process of jointly estimating the target position and the transmitter position based on the alternating direction multiplier method, the iteration stopping condition is that the original residual is less than the convergence tolerance.

4. The positioning method based on differential time delay and alternating direction multiplier method according to claim 2, characterized in that: The original optimization problem is, ； Among them, x is the first variable and z is the second variable; ； ； is the location coordinate of the receiver, M is the number of receivers; K is the spatial dimension.

5. The positioning method based on differential time delay and alternating direction multiplier method according to claim 4, characterized in that: The augmented Lagrangian function of the objective function of the original optimization problem is: ； in, are the Lagrange multiplier coefficients, is the penalty parameter.

6. The positioning method based on differential time delay and alternating direction multiplier method according to claim 5, characterized in that: During the iterative update of step k, the first of the two sub-problems after decomposition is: ； The second of the two sub-problems after decomposition is: ； The update formula of the Lagrange multiplier coefficient is: 。 7. The positioning method based on differential time delay and alternating direction multiplier method according to claim 6, characterized in that: The analytical solution to the first subproblem is: ;in, ； The analytical solution to the second subproblem is: ； Among them, G and is a constant, ； In the above formula, , and For The three vectors extracted from ; in, .

8. A positioning system based on differential time delay and alternating direction multiplier method, comprising: at least one transmitter for transmitting a measurement signal; a plurality of receivers for receiving measurement signals; A processor is communicatively connected to the receiver, and is used to obtain the position information of each receiver and the measurement result of each receiver, and estimate the position of the target object; characterized in that, The processor uses the method according to any one of claims 1 to 7 to solve the problem of locating the target object by converting the problem into two sub-problems with analytical solutions.

9. The positioning system based on differential time delay and alternating direction multiplier method according to claim 8, characterized in that: The measurement results of each receiver include the differential time delay corresponding to each receiver.

10. The positioning system based on differential time delay and alternating direction multiplier method according to claim 8, characterized in that: The number of the transmitters is 1-20 and they are deployed separately; the number of the receivers is 2 to 20 and they are deployed separately.

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