A multi-target robust positioning method based on two-order differential received signal strength

Abstract

A robust multi-target localization method based on second-order differential received signal strength is proposed. This method includes: selecting a reference receiver and a reference target; constructing a dual-difference RSS (D²RSS) observation model using receiver differential and target differential signals, while simultaneously eliminating unknown transmit power and receiver gain difference (RGD); constructing a pseudo-linear equation system by performing variable substitution and first-order approximation on the observation equations based on modified polar coordinate representation (MPR); and solving the system using a two-stage weighted least squares (WLS) algorithm, first obtaining a coarse estimate including inverse distance and direction vectors, and then using auxiliary variable constraints to solve for the final position. This invention requires no prior transmit power or additional calibration, achieves efficient computation through closed-form solutions, and is free from initial value sensitivity issues. Simulations show that this method maintains centimeter-level positioning accuracy even with RGD as high as 1 dB, and its performance is close to the Cramer-Rao lower bound, making it suitable for multi-target passive localization scenarios such as UAV swarms and ground base stations.

Description

Technical Field

[0001] This invention belongs to the field of wireless passive positioning technology based on radio electromagnetic signals, and specifically relates to a robust multi-target positioning method based on the intensity of second-order differential received signals. Background Technology

[0002] Existing passive positioning technologies typically rely on various observational information related to the target location, such as Time Difference of Arrival (TDOA), Frequency Difference of Arrival (FDOA), Direction of Arrival (DOA), and Received Signal Strength (RSS). Among these, the TDOA method achieves high positioning accuracy but requires strict time synchronization between receiving nodes, thus it is mostly used in fixed infrastructure network scenarios. The DOA method relies on antenna arrays to estimate signal direction, requiring high consistency in array structure and synchronized signal acquisition, resulting in relatively high system costs. The FDOA method usually requires relative motion between the observation platform and the target, limiting its applicability. In contrast, the RSS positioning method has lower hardware requirements, a simpler system structure, less stringent requirements for node synchronization, good scalability, and lower deployment costs, thus it has been widely researched and applied in applications such as WiFi indoor positioning, LoRa IoT positioning, and 5G network positioning.

[0003] However, RSS-based passive localization still faces several key challenges. First, in practical applications, the transmit power of the target signal is usually unknown, which introduces uncertainty into RSS-based distance estimation. Second, due to manufacturing differences in the RF hardware and signal acquisition links of different receiving nodes, there is often a receiver gain discrepancy (RGD) between nodes, which introduces system bias into RSS measurements and significantly reduces positioning accuracy. Therefore, how to construct an RSS localization method that requires no additional calibration, has low computational complexity, and is robust, when both unknown transmit power and receiver gain inconsistency exist, remains an important problem that urgently needs to be solved in current passive localization research.

[0004] To address the critical issue of unknown target transmit power in RSS positioning, the academic community has proposed Differential RSS (DRSS) technology. By performing differential calculations on the RSS measurements of two receivers, it can effectively cancel out the unknown target transmit power, becoming the mainstream improvement direction for RSS positioning. Regarding receiver gain noise (RGD), another core issue affecting positioning accuracy, existing technologies mainly fall into two categories: one is calibration technology relying on dedicated anchor points. This involves deploying anchor nodes with known locations and precise transmit power to uniformly calibrate the receiver gain, thereby eliminating the impact of RGD. However, this approach is difficult to deploy in dynamic scenarios or environments with limited infrastructure, limiting its practicality. The other category is joint estimation technology. Research has proposed the concept of "united-RSS," which simultaneously estimates the target location and RGDs by constructing a specific model. This approach has been extended to cooperative positioning scenarios, showing better performance than traditional methods that ignore RGD. However, this type of method is only applicable to scenarios where the target transmit power is known and cannot handle the actual situation where unknown transmit power coexists with RGD.

[0005] In attempts to simultaneously handle unknown transmit power and RGD, some studies have proposed alternating iterative optimization algorithms to approximate the optimal solution by iteratively updating the target position and RGD estimate. However, these algorithms have significant drawbacks: high computational cost, making it difficult to meet the real-time requirements of large-scale multi-target localization; and extreme sensitivity to initial values, easily leading to local convergence or divergence problems, resulting in unstable positioning accuracy. Furthermore, existing RSS and DRSS positioning technologies do not fully exploit the measurement redundancy information in multi-target scenarios and lack measurement models specifically designed for multi-target characteristics. Most rely on optimization approaches similar to single-target localization or employ iterative solutions, resulting in low computational efficiency and difficulty adapting to the growing multi-target localization needs in scenarios such as 6G, IoT, and integrated air-space-ground networks.

[0006] The paper "Three-dimensional location estimation using biased RSS measurements" proposes a 3D positioning method to address the systematic bias in RSS measurements. This method constructs a biased RSS measurement model and models and estimates the bias parameters to jointly solve for the target position and measurement bias. While this method can improve 3D positioning accuracy to some extent, its high model complexity due to the need to simultaneously estimate both position and bias parameters, its reliance on iterative optimization requiring initial values, its tendency to get trapped in local optima, and its significant computational cost limit its application in multi-target or real-time scenarios.

[0007] The paper "Semidefinite programming for wireless cooperative localization using biased RSS measurements" proposes an optimization method based on semidefinite programming (SDP) for the RSS cooperative localization problem with measurement bias. By employing convex relaxation techniques, the original non-convex problem is transformed into a solvable convex optimization problem, thereby improving positioning accuracy and stability. Theoretically, this method possesses good global optimality, but its computational complexity is high, especially as the node size increases significantly, making it difficult to meet the needs of large-scale networks or real-time positioning.

[0008] The paper "Optimal sensor placement for RSS-based localization with biased measurement model" addresses the issue of degraded positioning performance due to RSS measurement bias. From the perspective of sensor placement, it optimizes sensor spatial distribution by analyzing the Cramer-Rao lower bound (CRLB) to improve positioning accuracy. While this method improves positioning performance at the system design level, it relies on a pre-known error model and network structure, lacking the ability to adaptively handle unknown biases in real-world dynamic environments.

[0009] The paper "Robust differential received signal strength-based localization" addresses the insufficient robustness of traditional DRSS methods under noise and model error conditions by proposing a robust DRSS localization method. This method reduces the impact of anomalous measurements on the localization results by improving the cost function or introducing a weighting mechanism. While this method enhances noise immunity to some extent, it still relies on the DRSS model and fails to fundamentally eliminate systematic errors such as receiver gain bias. Therefore, its localization performance remains limited in heterogeneous network environments. Summary of the Invention

[0010] To overcome the problems of the existing technology, the present invention discloses a robust multi-target localization method based on second-order differential received signal strength. Utilizing the measurement redundancy of multiple targets, a second-order differential of the DRSS is performed to simultaneously cancel out the unknown target transmit power and the receiver RGD. Combined with an improved Modified Polar Representation (MPR) method, the target positions are remodeled, decoupling the multi-target position variables. A two-stage WLS closed-form solution algorithm is proposed to achieve efficient estimation of multi-target positions. This invention requires no anchor point calibration and no known target transmit power, enabling robust passive localization of multiple targets in heterogeneous networks. The algorithm is a closed-form solution, exhibiting high computational efficiency and no sensitivity to initial conditions.

[0011] To achieve the above objectives, the present invention adopts the following technical solution: A robust multi-target localization method based on second-order differential received signal strength includes the following steps: (1) In a two-dimensional positioning scenario containing N known position sensor nodes and M unknown position target signal sources, the first sensor node is selected as the reference receiver and the first target signal source is selected as the reference target. The target transmission power is eliminated by receiver differential, and the receiver gain difference RGD is eliminated by target differential. The D²RSS observation value is only related to the distance ratio, without the need for prior transmission power and additional calibration. (2) Based on the D²RSS observations, the modified polar coordinate representation MPR is used to convert the target position parameters into a combination of inverse distance variables and direction vectors. The D²RSS observation equations are subjected to exponential transformation and first-order Taylor approximation to map the D²RSS observations into a pseudo-linear system of equations about the unknown position parameters. (3) Solve the pseudo-linear equation system using the WLS algorithm to obtain a coarse estimation vector containing the product of the squared inverse distance and the inverse distance direction of each target; (4) Based on the coarse estimation vector, construct the auxiliary variable constraint equation, and use the WLS algorithm again to solve it to obtain the final position estimate of each target.

[0012] Step (1) specifically includes: (1.1) Obtaining the raw RSS measurement: The signal power received by sensor node j (j=1,...,N) from target i (i=1,...,M) Following the free space path loss model: in The goal Unknown transmission power For free space path loss, It is the true position of target i. to the actual location of sensor node j distance, Let i be the carrier frequency of the target i, in MHz; For the unknown front-end RF receiver gain of the sensor node; (1.2) Receiver Differential: Calculate the differential RSS between target i at sensor node j and the reference receiver: Through this difference operation, the target Unknown transmit power Eliminated, introducing receiver gain difference , indicating that node j is relative to the reference node Systematic bias; (1.3) Target difference: Calculate the D²RSS double difference between target i and reference target at node pair (j,1): This second differential operation eliminates receiver gain differences. We obtain observations that are only related to the distance ratio; considering measurement noise. The actual observed value is: ,in, That is, target i and reference target 1 at node and The D²RSS observations between.

[0013] In step (1.3), the equivalent measurement noise It follows a zero-mean Gaussian distribution, and its covariance matrix is... From the DRSS noise covariance matrix It is derived through the difference transformation matrix Φ.

[0014] Step (2) specifically includes: (2.1) Define the inverse distance of target i With direction vector The target location is represented as And set the reference receiver to the origin. ; (2.2) Express the square of the distance ratio as: in Let be the coordinate vector of node j; (2.3) Perform an exponential transformation and first-order Taylor approximation on the D²RSS observations, substitute them into the range ratio expression, and stack the D²RSS observations of all targets and receivers to construct a pseudo-linear system of equations in a unified matrix form: Among them, the variables to be estimated , , This is the noise figure matrix for the first stage. It is the equivalent measurement noise vector. It is the pseudo-linear coefficient matrix of the first stage. This is the first-stage observation vector, and T represents the transpose operation.

[0015] In step (3), the weight matrix of the WLS solution in the first stage The covariance matrix of noise is measured using D²RSS. Sure: The WLS solution is: The covariance matrix of the estimation error is .

[0016] Step (4) specifically includes: (4.1) Define auxiliary variables and establish constraint relationships. , where ⊙ represents the Schur product, that is, the product of corresponding elements; (4.2) Based on the coarse estimate of the first stage Constructing about auxiliary variables linear equations ,in This represents the estimation error in the first stage. It is the coefficient matrix of the estimation error. This is the second-stage observation vector. It is the pseudo-linear coefficient matrix of the second stage; Solving for auxiliary variables and according to Calculate the final position estimate of target i: Among them, sgn( The sign function () is used to determine the sign of a number. Indicates taking The two elements in the middle correspond to the target i.

[0017] 7. The method according to claim 6, characterized in that, in step (4.2), the weight matrix Wy of the second-stage WLS solution is based on the covariance matrix of the first-stage estimation error. Sure: The weighted least squares estimate of the auxiliary variable is: .

[0018] The sensor nodes are either drone clusters or ground base stations, the target signal sources are non-cooperative radiation sources, and the frequency bands of the target signals do not overlap.

[0019] The method further includes step (5): taking the final position estimate obtained in step (4) as the initial value and inputting it into the maximum likelihood estimator (MLE) for iterative optimization to approximate the Cramerlow lower bound (CRLB).

[0020] A multi-target passive positioning device, comprising: The data acquisition module is used to receive the RSS measurement values ​​of M targets collected by N sensor nodes; The model building module is used to perform the operations of steps (1) and (2) in the method of any one of claims 1 to 8 to build a D²RSS pseudo-linear model; The calculation module is used to perform the operations of steps (3) and (4) in the method of any one of claims 1 to 8, and calculate the final position estimates of the M targets; The output module is used to output the positioning results.

[0021] Compared with the prior art, the present invention has the following beneficial effects: 1. The dual differential received signal strength (D²RSS) measurement model proposed in this invention can effectively suppress the impact of receiver hardware inconsistencies on positioning performance without requiring known target transmit power or additional deployment of calibration nodes or receiver gain calibration, and significantly improve positioning robustness in complex heterogeneous network environments.

[0022] 2. To address the problems of traditional RSS or DRSS positioning methods, which generally rely on iterative optimization, have high computational complexity, and are easily affected by initial values, this invention introduces an MPR correction method that converts the target position parameters into a combination of inverse distance and direction vectors, and constructs a D²RSS pseudolinear observation model. Based on this, a two-stage weighted least squares (WLS) closed-form solution algorithm is proposed, which can directly obtain the target position estimation result without iterative search, effectively avoiding local optima and convergence failure problems. It has advantages such as high computational efficiency, good real-time performance, and simple implementation.

[0023] 3. This invention fully utilizes the observation redundancy information inherent in multi-target scenarios, organically combining the D²RSS measurement model, MPR parameterization method, and two-stage WLS localization algorithm to form a complete multi-target robust passive localization framework. Simulation results show that when the receiver gain difference reaches 1 dB, the localization error of the traditional DRSS method deteriorates significantly with the increase of RGD, while the method proposed in this invention can maintain a basically stable localization accuracy, unaffected by RGD. Under different measurement noise levels, different numbers of targets, and different numbers of receivers, this invention can achieve localization performance close to the Cramero lower bound (CRLB), demonstrating excellent localization accuracy, robustness, and scalability, and is suitable for multi-target passive localization scenarios in UAV swarm cooperative localization, wireless sensor networks, and complex electromagnetic environments. Attached Figure Description

[0024] Figure 1 This is a multi-target localization scene diagram; Figure 2 The comparison shows the positioning errors of DRSS and the proposed method under different RGD conditions. Figure 3(a) shows the estimated performance of target 1; Figure 3(b) shows the estimated performance of target 2; Figure 3(c) shows the estimated performance of target 3; Figure 3(d) shows the estimated performance of target 4; Figure 4(a) shows the impact of the number of targets on target localization performance; Figure 4(b) shows the effect of the number of receivers on target positioning performance. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0026] This invention proposes a Double-Differential RSS (D²RSS) measurement model. Utilizing the measurement redundancy of multiple targets, it performs a second difference on the DRSS to simultaneously cancel out the unknown target transmit power and receiver RGD. Combined with an improved Modified Polar Representation (MPR) method, it remodels the target position, decoupling the multi-target position variables. A two-stage WLS closed-form solution algorithm is proposed to achieve efficient estimation of multi-target positions. This invention requires no anchor point calibration and no known target transmit power, enabling robust passive localization of multiple targets in heterogeneous networks. The algorithm is a closed-form solution, offering high computational efficiency and avoiding initial value sensitivity issues.

[0027] Scene building: Unmanned cluster collaborative positioning scenario Consider as Figure 1 The two-dimensional positioning scene shown includes sensor nodes and A non-cooperative target signal source, the location of the sensor node is known, is The target's actual location is To simplify subsequent derivations, let's assume... Furthermore, each target operates with unknown and different power. Transmitted signal, carrier frequency is Furthermore, it is assumed that the frequency bands of the target signals do not overlap. All sensor nodes are equipped with broadband receivers, capable of simultaneously listening to and separating all target signals. Under line-of-sight propagation conditions, the target... The signal arrived at the The received power of each receiver follows the free space path loss model: (1) in, (dB) is the target The unknown transmission power. (dB) is from target i to The path loss, its theoretical value is: (2) here This is the actual distance between the target and the drone, a constant of 27.55, derived from the propagation formula. It is derived from this. (dB) represents the unknown front-end RF receiver gain of the cluster node.

[0028] To simplify the expression, the dB unit is omitted in the subsequent derivation, and all variables are values ​​on the logarithmic scale (dB).

[0029] Technical path: Step 1: Construct the D²RSS measurement model.

[0030] 1.1) Select If the reference node is used, then the target At the node and DRSS on (3) in, It is a receiver The receiver gain difference RGD relative to reference receiver 1 is the system bias parameter to be estimated.

[0031] 1.2) Select target 1 as the reference target and define the target pair. In node pairs D2 RSS on (4) 1.3) In practical systems, RSS measurements are inevitably affected by factors such as thermal noise, hardware non-ideals, and environmental disturbances. Assume the RSS measurement noise... In the logarithmic field, it follows a mean of 0 and a variance of . If the target is an additive Gaussian distribution, then... At the receiving node With reference node The DRSS measurements between can be expressed as (5) in For DRSS, measure the noise term, and satisfy... , .

[0032] It should be noted that "eliminating target transmission power" mentioned in step (1.2) of this invention refers to eliminating the target's transmission power when targeting the same target. When performing differential calculations on RSS measurements from different receivers, the target's transmit power... All measurement equations contain constant unknown parameters, which can theoretically be completely eliminated through subtraction, as shown in equation (5). This elimination process only targets the additive constant term in the model and does not involve correction of the statistical characteristics of the measurement noise. The absolute values ​​of the transmit power of different targets may differ, but since they have been eliminated in the receiver differential step, they will not introduce systematic bias into the subsequent D²RSS observation model. The change in transmit power mainly affects the signal-to-noise ratio of the received signal, and thus affects the variance of the equivalent measurement noise. In the two-stage WLS estimation, this invention introduces a noise covariance matrix. As a weight matrix, it has undergone optimal statistical processing.

[0033] Furthermore, the target can be obtained. and At the node and The D²RSS measurement value between (6) in Let be the equivalent measurement noise of D²RSS, and . .

[0034] 1.4) For One goal and With one receiver, selecting target 1 and node 1 as references, the total number of DRSS observations that can be constructed is... The total number of D²RSS observations is .

[0035] Arranging all parameter values ​​into a matrix form yields: (7) in, (8) The covariance matrix of the noise measured by DRSS; The covariance matrix of the noise measured by D²RSS.

[0036] Step 2: Rewrite the D²RSS measurement equation based on the modified MPR and construct a D²RSS pseudo-linear model.

[0037] 2.1) By transforming both sides of formula (6), we can obtain (9) make and noise item By performing a first-order Taylor approximation, we can obtain (10) in, .

[0038] 2.2) Regarding the target Position is represented by its inverse distance and direction vector, i.e. (11) Therefore, the target location can be represented as (12) Because the reference receiver is set at the origin. ,have And the goal To the The distance between the receivers can be written as (13) Therefore, the distance ratio can be expressed as: (14) Squaring both sides, we get (15) (16) 2.3) Substituting formulas (15) and (16) into formula (10), we get (17) Will , , and Treating them as unknown variables, rearranging and simplifying them yields pseudolinear equations. (18) 2.4) Regarding the target For all By stacking the observations, we can obtain... (19) in, (20) With the estimated variable as The specific form is (twenty one) 2.5) For formula (19), all By stacking the observations, we can obtain... (twenty two) in, (twenty three) Step 3: Use the WLS algorithm to refine the target parameters. A rough estimate is then performed to obtain a solution.

[0039] The WLS solution to formula (22) is obtained based on the WLS algorithm principle. (twenty four) in Let be the weight matrix. The covariance matrix of the error between the WLS solution and the true solution is: (25) Step 4, consider unknown variables middle and The correlation between them is then further estimated using the WLS algorithm.

[0040] 4.1) Define a new auxiliary variable vector ,in Represents the Schur product. and The formula between them is satisfied Therefore, we can conclude that... (26) in It is a 2×1 vector of all 1s.

[0041] 4.2) The first stage The estimation error is denoted as Then there is ,neglect The higher-order error term can be transformed into (26) as follows: (27) For (27), all targets Stacking the observations yields the following linear equation in matrix form. (28) in (29) here This represents the Kronecker product. Therefore, the auxiliary variable... The weighted least squares estimate is (30) The weight matrix for the second stage is as follows: .

[0042] when When the error caused by noise is small, the covariance matrix of the estimator can be approximated as: (31) Therefore, the final estimated location of the target is (32) This invention provides a comprehensive performance evaluation of the proposed D²RSS-based multi-target localization algebraic algorithm. First, by comparing it with DRSS-based methods, the fundamental robustness of the proposed D²RSS framework is verified, demonstrating its independence from RGD. Second, the estimation accuracy of the proposed algorithm is evaluated by benchmarking its performance against the root CRLB (RCRLB) of the localization error theoretical bound and a Taylor-iterated maximum likelihood estimator (MLE). Furthermore, the scalability and robustness of the algorithm are investigated by examining its performance with increasing target and receiver numbers and under random network configurations. Here, CRLB is defined as the inverse of the Fisher Information Matrix (FIM), which gives the lower bound of the variance of any unbiased estimator.

[0043] The feasibility, effectiveness, and positioning performance of this invention are verified through simulation experiments.

[0044] This section provides the specific configuration of the simulation scenario, including the spatial distribution of the receiver and target, the measurement model, and related parameter settings. In the experiment used to verify the accuracy of the proposed algorithm in estimating the target position, consider a scenario containing... A network of receivers, with fixed locations as follows: m、 m、 m、 m、 m, at the same time, set There are 1 target, and their locations are respectively m、 m、 m and In the scalability analysis, when studying the impact of the number of receivers on target positioning performance, the target positions remain fixed as described above. The positions of the first five receivers remain unchanged, while the positions of newly added receivers are randomly obtained from a two-dimensional Gaussian distribution with a mean of zero and a standard deviation of 1000m. Conversely, when studying the impact of the number of targets on target positioning performance, the receiver configuration remains unchanged, the positions of the first four targets are fixed, and the positions of newly added targets are also randomly generated from a Gaussian distribution with a mean of zero and a standard deviation of 1000m.

[0045] The measurement model uses the free-space path loss model, and the carrier frequency is set to... f = 900 MHz. Assume that the RSS errors at different receivers are independent and identically distributed, with the same standard deviation. In target location estimation analysis, from Change to In scalability analysis, Fixed as The corresponding DRSS error covariance matrix is ​​in off-diagonal form and can be expressed as follows: ,in .here, The size is Correspondingly, the covariance matrix of the D²RSS measurement error... It is also a non-diagonal matrix, for ,in The difference transformation matrix has the following form: (33) Simulation Experiment To conduct benchmark performance comparisons, the initial values ​​for MLE (Mean Optimization Error) in the simulations were set to the true values, the estimated values ​​from the proposed algorithm, or random values. The random initial values ​​were generated by adding zero-mean Gaussian noise to the average coordinates of all receiver positions, with a standard deviation of 40m. The target position estimation performance was evaluated using the root mean square error (RMSE). For a single target, the RMSE for position estimation is defined as... in This represents the number of Monte Carlo trials, set to 2000 in the simulation. This represents the estimated position of the i-th target in the l-th trial. This corresponds to the true location of the target. Overall localization performance is measured by the average RMSE of all targets, i.e. in Let be the target number. To further evaluate the algorithm's performance under different geometric configurations, the positioning errors obtained from different random geometric configurations are averaged, and the expression is: in This is the number of geometric configurations, which is set to 200 in the scalability analysis.

[0046] Figure 2 The changes in localization RMSE of the proposed algorithm and the traditional DRSS localization method without considering the influence of RGD are presented under different RGD amplitude conditions. It can be observed that as the RGD amplitude... As the RGD gradually increases, the positioning error of traditional methods based on the DRSS measurement model increases significantly, exhibiting an approximately linear growth trend. When the RGD reaches 1 dB, its positioning error exceeds 30 m. In contrast, the proposed algorithm based on the D²RSS measurement model, shown by the red curve in the figure, maintains an almost constant positioning error throughout the entire RGD range, and its RMSE remains basically stable, without deteriorating with the increase of the RGD amplitude, indicating that the proposed method has strong robustness to RGD.

[0047] Figure 3 shows the variation of RMSE for single target location estimation under different measurement noise levels, where Figures 3(a)–3(d) correspond to target 1, target 2, target 3, and target 4, respectively. By gradually changing the measurement noise intensity, the localization performance of the proposed algorithm and the MLE method under three different initialization strategies was evaluated and compared with the theoretical RCRLB. It can be observed that… Figures 3(a)-3(d) The proposed algorithm exhibits a similar overall trend. Within the low to moderate noise level range, it achieves high-precision target localization, with its RMSE curve closely approximating the RCRLB curve, indicating near-optimal statistical performance within this noise range. Furthermore, compared to the MLE method using random initial values, the proposed algorithm demonstrates significantly better localization accuracy, and its performance is essentially consistent with the MLE method using true value initialization. This demonstrates that the proposed algorithm can achieve near-optimal localization results without requiring precise prior initialization.

[0048] When the measurement noise level further increases and exceeds approximately (corresponding to linear scale) When the RMSE curve of the proposed algorithm gradually deviates from the theoretical RCRLB, it can be observed that the threshold effect is common in closed positioning algorithms. That is, when the measurement noise exceeds a certain level, the linearization or approximation conditions of the algorithm are no longer fully satisfied, which leads to a gradual decrease in positioning accuracy.

[0049] Furthermore, if the estimation results obtained by the proposed algorithm are used as initial values ​​for further MLE iterative optimization, the estimation accuracy can be further improved. This shows that the proposed algebraic solution can not only independently achieve stable positioning estimation, but also serve as an efficient initialization strategy to provide initial values ​​for subsequent iterative optimization algorithms.

[0050] Therefore, a practical implementation strategy is to first use the proposed algebraic solution to quickly obtain a stable initial position estimate, and then use this result as the initial value to further refine the estimation result through iterative optimization methods such as MLE, thereby achieving a positioning accuracy close to the theoretical limit while ensuring computational efficiency.

[0051] Figure 4 shows the number of receivers. and target quantity Impact on target estimation performance. For Figure 4(a), increasing the number of receivers provides more measurement information, enhances spatial observability, and reduces the impact of random noise. Therefore, the proposed algorithm and the RMSE of initializing MLE using true or estimated values, as well as the corresponding RCRLB, improve with... The increase shows a clear downward trend. When MLE uses random initial values, the optimization process is prone to getting trapped in local optima, and the algorithm often fails to converge effectively. The proposed algorithm can consistently closely approximate RCRLB under different random receiver deployments, demonstrating good robustness and network scalability.

[0052] For Figure 4(b), as the number of targets... As the number of receivers increases, the overall positioning error shows an upward trend. This is because, with the number of receivers remaining constant, more targets need to share the spatial geometric information provided by receivers with known locations, thus diluting the prior spatial constraints that each target can utilize to some extent. Therefore, the RMSE of the proposed algorithm and the MLE method initialized with true or estimated values ​​both increase with... The value gradually increases with the increase in the value. At the same time, the corresponding location estimate RCRLB also shows the same growth trend.

[0053] Furthermore, under all random geometric configurations, the estimation performance of the proposed algorithm consistently approximates the corresponding RCRLB and maintains a consistent trend with the theoretical lower bound, further verifying the stability and robustness of the proposed method under different target numbers. It also demonstrates that the algorithm can achieve near-optimal estimation performance in complex network scenarios.

Claims

1. A robust multi-target localization method based on second-order differential received signal strength, characterized in that, Includes the following steps: (1) In a two-dimensional positioning scenario containing N known position sensor nodes and M unknown position target signal sources, the first sensor node is selected as the reference receiver and the first target signal source is selected as the reference target. The target transmission power is eliminated by receiver differential, and the receiver gain difference RGD is eliminated by target differential. The D²RSS observation value is only related to the distance ratio, without the need for prior transmission power and additional calibration. (2) Based on the D²RSS observations, the modified polar coordinate representation MPR is used to convert the target position parameters into a combination of inverse distance variables and direction vectors. The D²RSS observation equations are subjected to exponential transformation and first-order Taylor approximation to map the D²RSS observations into a pseudo-linear system of equations about the unknown position parameters. (3) Solve the pseudo-linear equation system using the WLS algorithm to obtain a coarse estimation vector containing the product of the squared inverse distance and the inverse distance direction of each target; (4) Based on the coarse estimation vector, construct the auxiliary variable constraint equation, and use the WLS algorithm again to solve it to obtain the final position estimate of each target.

2. The method according to claim 1, characterized in that, Step (1) specifically includes: (1.1) Obtaining the raw RSS measurement: The signal power received by sensor node j (j=1,...,N) from target i (i=1,...,M) Following the free space path loss model: in The goal Unknown transmission power For free space path loss, It is the true position of target i. to the actual location of sensor node j distance, Let i be the carrier frequency of the target i, in MHz; For the unknown front-end RF receiver gain of the sensor node; (1.2) Receiver Differential: Calculate the differential RSS between target i at sensor node j and the reference receiver: Through this difference operation, the target Unknown transmit power Eliminated, introducing receiver gain difference , indicating that node j is relative to the reference node Systematic bias; (1.3) Target difference: Calculate the D²RSS double difference between target i and reference target at node pair (j,1): This second differential operation eliminates receiver gain differences. We obtain observations that are only related to the distance ratio; considering measurement noise. The actual observed value is: ,in, That is, target i and reference target 1 at node and The D²RSS observations between.

3. The method according to claim 2, characterized in that, In step (1.3), the equivalent measurement noise It follows a zero-mean Gaussian distribution, and its covariance matrix is... From the DRSS noise covariance matrix It is derived through the difference transformation matrix Φ.

4. The method according to claim 1, characterized in that, Step (2) specifically includes: (2.1) Define the inverse distance of target i With direction vector The target location is represented as And set the reference receiver to the origin. ; (2.2) Express the square of the distance ratio as: in Let be the coordinate vector of node j; (2.3) Perform an exponential transformation and first-order Taylor approximation on the D²RSS observations, substitute them into the range ratio expression, and stack the D²RSS observations of all targets and receivers to construct a pseudo-linear system of equations in a unified matrix form: Among them, the variables to be estimated , , This is the noise figure matrix for the first stage. It is the equivalent measurement noise vector. It is the pseudo-linear coefficient matrix of the first stage. This is the first-stage observation vector, and T represents the transpose operation.

5. The method according to claim 1, characterized in that, In step (3), the weight matrix of the WLS solution in the first stage The covariance matrix of noise is measured using D²RSS. Sure: The WLS solution is: The covariance matrix of the estimation error is .

6. The method according to claim 1, characterized in that, Step (4) specifically includes: (4.1) Define auxiliary variables and establish constraint relationships. , where ⊙ represents the Schur product, that is, the product of corresponding elements; (4.2) Based on the coarse estimate of the first stage Constructing about auxiliary variables linear equations ,in This represents the estimation error in the first stage. It is the coefficient matrix of the estimation error. This is the second-stage observation vector. It is the pseudo-linear coefficient matrix of the second stage; Solving for auxiliary variables and according to Calculate the final position estimate of target i: Among them, sgn( The sign function () is used to determine the sign of a number. Indicates taking The two elements in the middle correspond to the target i.

7. The method according to claim 6, characterized in that, In step (4.2), the weight matrix Wy of the second-stage WLS solution is based on the covariance matrix of the estimation error in the first stage. Sure: The weighted least squares estimate of the auxiliary variable is: 。 8. The method according to claim 1, characterized in that, The sensor nodes are either drone clusters or ground base stations, the target signal sources are non-cooperative radiation sources, and the frequency bands of the target signals do not overlap.

9. The method according to any one of claims 1 to 8, characterized in that, It also includes step (5): taking the final position estimate obtained in step (4) as the initial value and inputting it into the maximum likelihood estimator (MLE) for iterative optimization to approximate the Cramerlow lower bound (CRLB).

10. A multi-target passive positioning device, characterized in that, include: The data acquisition module is used to receive the RSS measurement values ​​of M targets collected by N sensor nodes; The model building module is used to perform the operations of steps (1) and (2) in the method of any one of claims 1 to 8 to build a D²RSS pseudo-linear model; The calculation module is used to perform the operations of steps (3) and (4) in the method of any one of claims 1 to 8, and calculate the final position estimates of the M targets; The output module is used to output the positioning results.

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