A target source positioning method based on maximum entropy criterion in non-Gaussian noise environment

Abstract

A target source positioning method based on maximum entropy criterion in non-Gaussian noise environment, comprising the following steps: constructing a non-cooperative target source positioning scene; collecting distance difference measurement values in the positioning scene; constructing a distance difference measurement model according to the obtained distance difference measurement values, and then obtaining a measurement model in the form of measurement value mixed noise and non-line-of-sight error according to the distance difference measurement model; processing the distance difference measurement values; constructing a pseudo-linear equation set to form a weighted least square problem; solving the target position and weight coefficient vector of the least square problem; and solving the method by alternating iteration to obtain the final estimation result; in the case that the target source is non-cooperative, the arrival time difference measurement values obtained by processing the sensor system are used to effectively estimate the coordinate position of the target source, and the influence of non-Gaussian noise on the estimation result is reduced; and the method has the advantages of wide application range and good robustness.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of wireless communication signal positioning, and particularly relates to a target source positioning method based on a maximum entropy criterion in a non-Gaussian noise environment. BACKGROUND

[0002] As a positioning method based on time information, the Time Difference of Arrival (TDOA) has attracted extensive attention of many researchers due to its relatively reliable positioning accuracy and relatively low equipment cost. The basic principle of TDOA is to measure the time difference of arrival of a signal from a target to multiple receivers, model the time difference information between different receivers as multiple hyperbolas, and then calculate the intersection points of the hyperbolas to calculate the specific position of the target. Due to its simple principle and low equipment cost, TDOA has been widely applied in many fields such as wireless communication, radar, sonar and navigation system.

[0003] Unlike the Time of Arrival (TOA) based positioning method, TDOA relies on the time difference information between receivers rather than the absolute arrival time of a signal. Therefore, the TDOA system only needs to ensure the clock synchronization between the receivers, and does not need to synchronize the clock with the target, which reduces the complexity of the system and also reduces the requirement for clock accuracy. However, the TDOA positioning system also faces great challenges in complex signal propagation environments. Especially in urban, indoor or other complex scenarios, the signal may be blocked or reflected by obstacles during transmission, thereby introducing a Non-Line-of-Sight (NLOS) error. This error is usually much larger than the general measurement noise, and does not conform to the distribution characteristics of Gaussian noise, which seriously affects the positioning accuracy and performance of the positioning system.

[0004] For the target positioning problem in such a non-Gaussian noise environment, although many literatures have conducted in-depth research, most of the methods rely on accurate measurement error prior statistical information (Ouyang R W, Wong A K S. An enhanced TOA-based wireless location estimation algorithm for dense NLOS environments [C]. 2009 IEEE Wireless Communications and Networking Conference. IEEE, 2009: 1-6. and CHONG C C, WATANABE F, et al. NLOS identification and weighted least-squares localization for UWB systems using multipath channel statistics [J]. EURASIP Journal on Advances in Signal Processing, 2007, 2008: 1-14), which uses the statistical distribution, variance and mean of the error. However, these prior information is often difficult to obtain in actual scenarios, especially in dynamic environments, and the measurement error may also change over time. To solve this problem, researchers have proposed some improved methods, such as the method based on variable center maximum entropy criterion (WANG W and et al. Robust TDOA localization based on maximum correntropy criterion with variable center [J]. Signal Processing, 2023, 205: 108860), which can better handle non-Gaussian noise and improve positioning accuracy in the absence of prior information. Although this method solves the problem of non-Gaussian noise to some extent, its computational complexity is high, and it needs to constantly iterate between the target position, kernel function center and kernel bandwidth three sub-problems, with high computational complexity. In addition, in the sub-problem of solving the target position, the conventional processing of this method results in additional unknown variables in the final target function, increasing the computational complexity and causing multiple local solutions when using the Laplace kernel as the kernel function. SUMMARY

[0005] In order to overcome the shortcomings of the prior art, the purpose of the present application is to provide a target source positioning method in a non-Gaussian noise environment based on the maximum entropy criterion. In the case of non-cooperative target source, the arrival time difference measurement value obtained by the sensor system is processed to effectively estimate the coordinate position of the target source, while reducing the influence of non-Gaussian noise on the estimation result. It has the advantages of wide application range and good robustness.

[0006] In order to achieve the above purpose, the technical scheme adopted by the present application is as follows:

[0007] A target source positioning method in a non-Gaussian noise environment based on the maximum entropy criterion, comprising the following steps:

[0008] Step 1. Construct a non-cooperative target source positioning scene;

[0009] Step 2. Collect distance difference measurements using the non-cooperative target source localization scenario constructed in Step 1;

[0010] Step 3. Based on the distance difference measurement values ​​obtained in Step 2, construct a distance difference measurement model, and then based on the distance difference measurement model, obtain a measurement model in the form of mixed noise and non-line-of-sight error in the measurement values;

[0011] Step 4. For the measurement model with mixed noise and non-line-of-sight error in the measured values, perform shifting, squaring, and subtraction processing on the distance difference measured values;

[0012] Step 5. Based on the results of transposing and subtracting the distance difference measurements in Step 4, construct a pseudo-linear system of equations to form a weighted least squares problem;

[0013] Step 6. Use the maximum entropy criterion framework to solve the target position and weight coefficient vector of the least squares problem described in Step 5;

[0014] Step 7. Based on the area size of the non-cooperative target source location scene, select a scaling factor β to scale the location scene; then iterate alternately according to the solution method in Step 6 to obtain the final estimation result.

[0015] The specific method of step 1 is as follows: In a non-cooperative target source localization scenario, establish... p Using a 3D coordinate system as the reference coordinate system, we assume there is one target source and N+1 radar sensors to receive the target source signal and obtain the signal arrival time. The true coordinate positions of the radar sensors are known, while the true coordinate positions of the target source are unknown. We assume the radar sensors are clock-synchronized, and the radar sensors and the target source are in a non-cooperative relationship. The start time of the target source's signal transmission is unknown. Let s denote the true coordinate position of the i-th radar sensor in the reference coordinate system. i The (N+1)th radar sensor is designated as the reference radar sensor, and the true position of the target source in the reference coordinate system, i.e., the target position, is denoted as u. o .

[0016] The specific method of step 2 is as follows: Collect the specific time when the signal emitted by the target source arrives at each radar sensor; subtract the specific time when the signal emitted by the target source arrives at each radar sensor from the specific time when the signal emitted by the target source arrives at the (N+1)th radar sensor, respectively, to obtain the time difference; define the time difference as the time difference measurement value; then, based on the time difference measurement value and the known signal propagation speed, obtain the distance difference measurement value between the first N radar sensors and the (N+1)th radar sensor; and denoise the un-noiseed distance difference measurement value between the i-th radar sensor and the (N+1)-th radar sensor as... where i = 1,...,N.

[0017] The specific method of the step 3 is: according to the set of un-noised distance difference measurement values of the i th radar sensor about the N+1 th radar sensor in the step 2 The distance difference measurement model is constructed and described as: Then, according to the distance difference measurement model, the measurement value mixed noise and the measurement model in the form of non-line-of-sight error are obtained: Substitute into The measurement model in the form of measurement value mixed noise and non-line-of-sight error is described as: where "|| ||" represents the two norm, ||u o -s i || represents the real distance of the target to the i th radar sensor, ||u o -s N+1 || represents the real distance of the target to the N+1 th radar sensor, s N+1 represents the real coordinate position of the N+1 th radar sensor in the reference coordinate system, d i represents the noisy distance difference measurement value information of the source emitted signal to the i th radar sensor and the N+1 th radar sensor, represents the non-line-of-sight error of d i . represents the measurement noise of d i .

[0018] The specific process of the step 4 is:

[0019] According to the measurement model in the form of measurement value mixed noise and non-line-of-sight error Substitute into the left side of the equation, square the two sides of the equation and ignore the second order noise and error terms on the right side of the equation The flat way corresponding to the i th radar sensor is obtained

[0020] The specific method of the step 5 is: eliminating the norm term 2d i ||u o -s N+1 || in the flat way corresponding to the i th radar sensor, and multiplying the flat way corresponding to the i+1 th radar sensor by The result is

[0021] The flat way corresponding to the i th radar sensor The difference between the two equations is obtained: Further arrangement obtains where, called ξi For the error model; at this time, Constructed as a weighted least squares problem, described as follows: Where min() is the minimization function, (Ay) T Let WAy be the objective function, y be the optimization variable, and y = [u T ,1] T u represents the target coordinate position variable, and A is the introduced coefficient matrix. Using the optimization variable y and the introduced coefficient matrix A, the error model ξ i Represented as ξ i =A i y; where A i Let represent the vector formed by the elements of the i-th row of the introduced coefficient matrix A, where s1 represents the true coordinate position of the first radar sensor in the reference coordinate system, s2 represents the true coordinate position of the second radar sensor in the reference coordinate system, and s... N-1 s represents the true coordinate position of the (N-1)th radar sensor in the reference coordinate system. N Let dn represent the true coordinate position of the Nth radar sensor in the reference coordinate system, d1 represent the distance difference measurement value obtained by the 1st radar sensor and the reference radar sensor, and d2 represent the distance difference measurement value obtained by the 2nd radar sensor and the reference radar sensor. N-1 d represents the range difference measurement value obtained by the (N-1)th radar sensor and the reference radar sensor. N Let represent the distance difference measurement obtained by the Nth radar sensor and the reference radar sensor; W is the introduced weight matrix. w1 is the weight coefficient corresponding to d1, w N For d N The corresponding weighting coefficients; the aforementioned weighted least squares problem In this context, the optimization variable y includes the target position u. o and weighting coefficients w1,...,w N .

[0022] The specific method for step 6 is as follows: using the maximum entropy criterion framework to solve the target position u of the least squares problem described in step 5. o and weighting coefficients w1,...,w N :

[0023] Entropy is used to represent the similarity between two random variables X and Y, expressed by the formula: in, κ represents the expected value. σ () denotes a kernel function with a kernel bandwidth of σ, where the Silverman criterion defines the kernel bandwidth as σ = 1.06 × min{σ}. EL / 1.34} x M -0.2 ; where σ E is the standard deviation between random variables X and Y, L is the interquartile range between two variables, and M is the sample size; when random variable Y is zero, random variable X can be expressed as random error, and assuming that a represents an unknown variable related to random variable X, the estimation of the unknown variable is achieved by maximizing the entropy, which is expressed as

[0024] Combining the weighted least squares problem described in step 5 The unknown variable a is an optimization variable y in the weighted least squares problem; the mathematical expectation is approximately calculated by a limited number of samples: denoted as the maximum entropy problem, where ξ i (y), i = 1,..., N represents a limited number of error samples;

[0025] Using the Laplace kernel function Further transformation of the maximum entropy problem: joint and The maximum entropy problem is arranged as: where the constant coefficient is irrelevant to the optimization variable y and is ignored in the following; according to the error function expression ξ i = A i y in step 5, the objective function is further written as Based on the properties of convex conjugate functions, the weight matrix W is written as a weight coefficient vector w = [w1,..., w N ] T , and the objective function is further transformed into a new augmented problem for solving the weight coefficient vector w, denoted as where w i is the d i th weight coefficient corresponding to is the convex conjugate function of ; the augmented problem is substituted into the objective function to obtain At this time the optimization variable y and the weight coefficient vector w are jointly solved, denoted as the joint optimization problem where represents the function form with respect to the optimization variable y and the weight coefficient vector w; since the joint optimization problem is difficult to solve, the formula is transformed into two sub-problems to be solved iteratively:

[0026] Sub-problem 1:

[0027] When the weight coefficient vector w is known, the joint optimization problem It can be simplified to in, Indicates w i The estimated value is obtained by introducing an intermediate variable τ, since the joint optimization problem also involves an absolute value term and is therefore non-convex. i As |A i The upper bound of y| transforms the joint optimization problem into a convex problem for solution; the final optimization problem of subproblem 1 is expressed as: Where st represents the constraint; after solving the final optimization problem of subproblem 1, the solution result is denoted as This represents the estimated value of the optimization variable y;

[0028] Subproblem 2:

[0029] When the optimization variable y is known, the joint optimization problem Simplified to At this point, the weight coefficient vector w has a corresponding closed-form solution. Let w be the solution result, representing the estimated value of the weight coefficient vector w.

[0030] The specific method of step 7 is as follows: based on the size of the location scene area, a scaling factor β is selected to scale the location scene.

[0031] Then set Solve subproblem 1 of step 6. The result is directly substituted into Solve subproblem 2 in step 6; by iteratively solving subproblem 1 and subproblem 2, when the L2 norm of the difference between the two iterations is less than 10... -10 If the number of iterations reaches 10, the final estimation result is considered to be the optimal estimate of the target source coordinate position, denoted as u. * ,in, y * Let be the optimal estimate of y. Indicates by y * The subvector formed by the first to pth elements; This represents the estimated value of the weight coefficient vector w1. This represents the estimated value of the weight coefficient vector w2. Represents the weight coefficient vector w N The estimated value is w2, which is the weight coefficient vector corresponding to d2.

[0032] Compared with the prior art, the present invention has the following advantages:

[0033] 1) The existing target source positioning problem mostly considers that the measurement noise conforms to Gaussian distribution, thereby causing the application scene of the proposed positioning method to be limited; and the present application considers that the system works in a non-Gaussian environment, and a corresponding positioning analysis method is proposed for the environment, through using a maximum entropy criterion framework, the target position and the weight coefficient are both treated as unknown quantities for the positioning scene of non-Gaussian noise, two sub-problems are solved alternately, and the estimation about the target position is obtained after the sub-problems converge. The method is suitable for the positioning scene of Gaussian noise, the application range of the method is wide, and the robustness is good.

[0034] 2) The present application considers the numerical relationship between the measurement values from different radar sensors, through step 5, the non-convex norm item is eliminated by difference based on the characteristic that the measurement values contain a common norm item, step 6 sub-problem 1 is simplified, and the problem that the algorithm converges to a local optimum in the solving process of sub-problem 1 due to the non-convex norm item is avoided, so that the positioning problem is suitable for solving by using a Laplace kernel, and compared with solving by using a Gaussian kernel, the method has a faster decay speed.

[0035] 3) The simulation experiment result of the method of the present application shows that the estimation precision of the target source position of the proposed method has great advantages compared with the positioning method for ordinary Gaussian noise, and the method has good performance in mean square error (MSE).

[0036] In conclusion, in the case that the target source is non-cooperative, the coordinate position of the target source is effectively estimated by processing the time difference of arrival measurement values obtained by the radar sensor system, and the influence of non-Gaussian noise on the estimation result is reduced; the method has the advantages of wide application range and good robustness. BRIEF DESCRIPTION OF DRAWINGS

[0037] Figure 1 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) of the present application and the two-step weighted least square (TSWLS) when N=6 is given.

[0038] Figure 2 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) of the present application and the two-step weighted least square (TSWLS) when SNR=1 is given.

[0039] Figure 3 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) of the present application and the two-step weighted least square (TSWLS) when SNR=1 is given.

[0040] Figure 4 It is a flow chart of the method of the present application. Detailed Implementation

[0041] The present invention will now be described in further detail with reference to the accompanying drawings.

[0042] See Figure 4 A target source localization method based on the maximum entropy criterion in a non-Gaussian noise environment includes the following steps:

[0043] Step 1. Construct a non-cooperative target source localization scenario; the specific method is as follows:

[0044] In a non-cooperative target source localization scenario, establish p A 3D coordinate system is used as the reference coordinate system, and a target source is assumed to exist. N+1 There are 10 radar sensors used to receive signals from a target source and obtain the signal arrival time. The true coordinates of the radar sensors are known, while the true coordinates of the target source are unknown. The radar sensors are clock-synchronized, and the relationship between the radar sensors and the target source is non-cooperative. The start time of the target source's signal transmission is unknown. Let s be the true coordinates of the i-th radar sensor in the reference coordinate system. i The (N+1)th radar sensor is designated as the reference radar sensor, and the true position of the target source in the reference coordinate system, i.e., the target position, is denoted as u. o .

[0045] Step 2. Collect distance difference measurements using the non-cooperative target source localization scenario constructed in Step 1; the specific method is as follows:

[0046] The system collects the specific times when the signals emitted by the target source arrive at each radar sensor. It then subtracts the arrival times of the signals emitted by the target source corresponding to the first N radar sensors from the arrival times of the signals emitted by the target source corresponding to the (N+1)th radar sensor, obtaining the time difference. This time difference is defined as the time difference measurement. Based on the time difference measurement and the known signal propagation speed, the system obtains the distance difference measurement between the first N radar sensors and the (N+1)th radar sensor. The un-noiseed distance difference measurement between the i-th radar sensor and the (N+1)-th radar sensor is denoted as... Where i = 1, ..., N.

[0047] Step 3. Based on the distance difference measurement values ​​obtained in Step 2, construct a distance difference measurement model. Then, based on the distance difference measurement model, obtain a measurement model in the form of mixed noise and non-line-of-sight error in the measurement values. The specific method is as follows:

[0048] Based on the set of un-noiseed range difference measurements of the i-th radar sensor with respect to the N+1-th radar sensor in step 2. Construct a distance difference measurement model, described as follows: Then, based on the distance difference measurement model, a measurement model in the form of mixed noise and non-line-of-sight error is obtained: Will Substitution The measurement model, which incorporates mixed noise and non-line-of-sight errors, is described as follows: Where "|| ||" represents taking the 2-norm, ||u o -s i || represents the true distance from the target to the i-th radar sensor, ||u o -s N+1 || represents the true distance from the target to the (N+1)th radar sensor, s N+1 d represents the true coordinate position of the (N+1)th radar sensor in the reference coordinate system. i This represents the noisy distance difference measurement between the source-transmitted signal arriving at the i-th radar sensor and the N+1-th radar sensor. d i Non-line-of-sight error, d i Measurement noise.

[0049] Step 4. For the measurement model involving mixed noise and non-line-of-sight errors in the measured values, the distance difference measurements are shifted, squared, and subtracted; the specific process is as follows:

[0050] Measurement models for mixed noise and non-line-of-sight errors in measurements Will Move to the left side of the equation, square both sides, and ignore the second-order noise and error terms on the right side of the equation. Obtain the square root corresponding to the i-th radar sensor

[0051] Step 5. Based on the results of rearranging, squaring, and subtracting the distance difference measurements in Step 4, construct a pseudo-linear system of equations to form a weighted least squares problem; the specific method is as follows:

[0052] Eliminate the norm term 2d in the square root of the i-th radar sensor. i ||u o -s N+1 ||, multiply both sides of the square corresponding to the (i+1)th radar sensor. get

[0053] The square corresponding to the i-th radar sensor The difference between the two equations is obtained Further sorting yielded in, ξ iis the error model; at this time, is constructed as a weighted least squares problem, described as: where min() is a minimization function, (Ay) T is an objective function, y is an optimization variable, and y = [u T ,1] T , u represents a target coordinate position variable, A is an introduced coefficient matrix Using the optimization variable y, the introduced coefficient matrix A, and the error model ξ i is expressed as ξ i = A i y; where A i represents a vector composed of the i-th row elements of the introduced coefficient matrix A, s1 represents the true coordinate position of the first radar sensor in the reference coordinate system, s2 represents the true coordinate position of the second radar sensor in the reference coordinate system, s N-1 represents the true coordinate position of the N-1th radar sensor in the reference coordinate system, and s N represents the true coordinate position of the Nth radar sensor in the reference coordinate system, d1 represents the distance difference measurement value obtained by the first radar sensor and the reference radar sensor, d2 represents the distance difference measurement value obtained by the second radar sensor and the reference radar sensor, d N-1 represents the distance difference measurement value obtained by the N-1th radar sensor and the reference radar sensor, and d N represents the distance difference measurement value obtained by the Nth radar sensor and the reference radar sensor; W is an introduced weight matrix, w1 is the weight coefficient corresponding to d1, w N is the weight coefficient corresponding to d N ; in the above weighted least squares problem , the optimization variable y includes the target position u o and the weight coefficients w1,..., w N .

[0054] Step 6. Solve the target position and weight coefficient vector of the least squares problem described in step 5 using the maximum entropy criterion framework; the specific method is:

[0055] Solve the target position u o and the weight coefficients w1,..., w N of the least squares problem described in step 5 using the maximum entropy criterion framework:

[0056] Use entropy to represent the similarity between two random variables X and Y, which is expressed as where represents the mathematical expectation, and κ σ() denotes a kernel function with bandwidth σ, and the Silverman's rule defines the kernel bandwidth as σ = 1.06 x min{σ E , L / 1.34} x M -0.2 ; where σ E is the standard deviation between random variables X and Y, L is the interquartile range between two variables, and M is the sample size; when the random variable Y is zero, the random variable X can be expressed as a random error, and assuming that α represents an unknown variable related to the random variable X, the estimation of the unknown variable is realized by maximizing the entropy, which is expressed as

[0057] Combining the weighted least squares problem described in step 5 The unknown variable α is an optimization variable y in the weighted least squares problem; the mathematical expectation is approximately calculated by a finite number of samples: denoted as the maximum entropy problem, where ξ i (y), i = 1,..., N represents a finite number of error samples;

[0058] The Laplace kernel function is used Further transformation is made to the maximum entropy problem: joint The maximum entropy problem is arranged as: where the constant coefficient is irrelevant to the optimization variable y and is ignored in the following; according to the error function expression ξ i = A i y in step 5, the objective function is further written as Based on the properties of convex conjugate functions, the weight matrix W is written as a weight coefficient vector w = [w1,..., w N ] T , and the objective function is further transformed into a new augmented problem for solving the weight coefficient vector w, which is expressed as where w i is the d i th weight coefficient corresponding to is the convex conjugate function of ; the augmented problem is substituted into the objective function to obtain At this time the optimization variable y and the weight coefficient vector w are jointly solved, which is expressed as a joint optimization problem where represents the function form about the optimization variable y and the weight coefficient vector w; since the joint optimization problem is difficult to solve, the formula is transformed into two sub-problems to be solved iteratively:

[0059] Sub-problem 1:

[0060] When the weight coefficient vector w is known, the joint optimization problem can be simplified as where, denotes the estimation value of w i , at this time, since the joint optimization problem is also a non-convex problem with an absolute value term, by introducing an intermediate variable τ i as an upper bound of |A i y|, the joint optimization problem is changed into a convex problem for solving; the final optimization problem of subproblem 1 is expressed as: where, s.t. denotes being constrained to; after solving the final optimization problem of subproblem 1, the solution result is denoted as denotes the estimation value of the optimization variable y;

[0061] Subproblem 2:

[0062] When the optimization variable y is known, the joint optimization problem is simplified as At this time, the weight coefficient vector w has a corresponding closed-form solution The solution result is denoted as denotes the estimation value of the weight coefficient vector w.

[0063] Step 7. According to the size of the non-cooperative target source positioning scene, a scaling factor β is self-defined and selected to scale the positioning scene; then, the final estimation result is obtained by alternately and iteratively solving according to the solving method of step 6; the specific method is:

[0064] According to the size of the positioning scene (in the simulation, the sensor position is in the range area of [-2.5, 2.5]km×[-2.5, 2.5]km, and the non-cooperative source target position is in the range area of [-0.5, 0.5]km×[-0.5, 0.5]km), a scaling factor β is self-defined and selected to scale the positioning scene;

[0065] Then, the result of solving subproblem 1 of step 6 is directly brought into step 6 subproblem 2 for solving; Through the alternately and iteratively solving of subproblem 1 and subproblem 2, when the two iteration changes difference norms are less than 10 -10 or the iteration number reaches 10, the final estimation result, that is, the optimal estimation value of the target source coordinate position, is considered to be obtained, which is denoted as u * , where, y * is the optimal estimation value of y, denotes a subvector composed of the first to p-th elements of y * ; denotes an estimated value of a weight coefficient vector w1, denotes an estimated value of a weight coefficient vector w2, denotes an estimated value of a weight coefficient vector w N , and w2 is a weight coefficient vector corresponding to d2.

[0066] Simulation experiment:

[0067] To verify the feasibility and effectiveness of the method, a simulation experiment is performed.

[0068] It is assumed that there are N+1 radar sensors and one non-cooperative source target in a two-dimensional space, wherein the sensor positions are randomly generated in a range area of [-2.5, 2.5] km x [-2.5, 2.5] km, and the non-cooperative source target position is randomly generated in a range area of [-0.5, 0.5] km x [-0.5, 0.5] km; the measurement noise of the distance measurement value is assumed to have a mean of 0 and a covariance matrix of Q = σ 2 I N , wherein σ 2 is the noise power of the distance measurement value measurement noise, and The non-line-of-sight error is randomly generated in the range of [-0.5, 0.5] km.

[0069] Based on the above parameter settings, the performance of the method in three cases of different signal-to-noise ratios, different non-line-of-sight error upper limits, and different numbers of radar sensors is tested, including the mean square error of the method. To verify the robustness of the method, ten random scenes are generated, and 100 Monte Carlo experiments are performed for each scene, and the final mean square error is obtained by integrating all results and taking the average. To verify the performance of the method, a two-step weighted least squares method for Gaussian noise is introduced for comparison.

[0070] Figure 1 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) and the two-step weighted least squares (TSWLS) method when N = 6 is given.

[0071] Figure 2 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) and the two-step weighted least squares (TSWLS) method when SNR = 1 is given.

[0072] Figure 3 A comparison chart of the log mean square error (MSE) of the coordinate position estimation of the non-cooperative target source by the method (MCC) and the two-step weighted least squares (TSWLS) method when SNR = 1 is given.

[0073] From Figure 1 it can be seen that the performance of the method of the present application is better than that of the two-step weighted least squares method in estimating the target position at the given SNR level as the measurement SNR changes; as the SNR changes, the performance of the two methods does not change significantly, the two-step weighted least squares method maintains the estimation of the target position at about 45.5dB, and the method of the present application maintains the estimation of the target position at about 44dB, the method of the present application has an advantage of about 1.5dB over the two-step weighted least squares method; from Figure 2 it can be seen that the performance of the method of the present application is better than that of the two-step weighted least squares method in estimating the target position at the given NLOS error level as the NLOS error upper limit changes; as the NLOS error level increases, the performance of the two-step weighted least squares method and the method of the present application in estimating the target position both deteriorate, but the method of the present application has an advantage of 1.5dB-2.5dB over the two-step weighted least squares method in estimating the target position; from Figure 3 it can be seen that the performance of the method of the present application is better than that of the two-step weighted least squares method in estimating the target position at the given number of radar sensors as the number of radar sensors changes; as the number of sensors increases, although more NLOS errors are introduced, the performance of the two methods in estimating the target position overall improves, and the method of the present application maintains an advantage of nearly 2dB. In summary, the method of the present application can effectively suppress the influence of NLOS errors on the performance of target positioning in the tested range, and is more robust than the existing method.

Claims

1. A target source positioning method in a non-Gaussian noise environment based on a maximum entropy criterion, characterized in that, The method comprises the following steps: Step 1. Constructing a non-cooperative target source positioning scene; Step 2. Collecting distance difference measurement values of the first N radar sensors with respect to the N+1th radar sensor by using the non-cooperative target source positioning scene constructed in step 1; Step 3. Constructing a distance difference measurement model according to the distance difference measurement values obtained in step 2, and then obtaining a measurement model in the form of measurement mixed noise and non-line-of-sight error according to the distance difference measurement model; Step 4. Moving and squaring the distance difference measurement values for the measurement model in the form of measurement mixed noise and non-line-of-sight error, and performing difference processing; Step 5. Constructing a pseudo-linear equation set according to the results of moving and squaring the distance difference measurement values and performing difference processing in step 4, and forming a weighted least square problem; Step 6. Solving the target position and weight coefficient vector of the least square problem in step 5 by using a maximum entropy criterion framework; Step 7. Defining and selecting a scaling factor β according to the size of the non-cooperative target source positioning scene, and scaling the positioning scene; Then, the final estimation result is obtained by alternately iterating the solving method in step 6.

2. The target source localization method based on the maximum entropy criterion in a non-Gaussian noise environment according to claim 1, characterized in that, The specific method of step 1 is that in a non-cooperative target source positioning scene, a p-dimensional coordinate system is established as a reference coordinate system, and it is assumed that there is one target source and N+1 radar sensors for receiving target source signals to obtain signal arrival times, the real coordinate positions of the radar sensors are known, the real coordinate position of the target source is unknown, the radar sensors are clock-synchronized, the radar sensors and the target source are in a non-cooperative relationship, and the starting time of the target source signal transmission is unknown. Let the real coordinate position of the i-th radar sensor in the reference coordinate system be denoted as s i where the N+1-th radar sensor is denoted as the reference radar sensor, and the real position of the target source in the reference coordinate system, i.e., the target position, is denoted as u o .

3. The method according to claim 2, wherein, The specific method of step 2 is: collecting the specific time when the signal emitted by the target source reaches each radar sensor, subtracting the specific time when the signal emitted by the target source corresponding to the first N radar sensors reaches each radar sensor from the specific time when the signal emitted by the target source corresponding to the N+1 radar sensor reaches the N+1 radar sensor, to obtain the time difference; define the time difference as the time difference measurement value; then according to the time difference measurement value and the known signal propagation speed, obtain the distance difference measurement value of the first N radar sensors relative to the N+1 radar sensor, and the un-noised distance difference measurement value of the i-th radar sensor relative to the N+1 radar sensor is denoted as Where i = 1,...,N.

4. The method according to claim 3, wherein, The specific method of step 3 is: according to the set of un-noised range difference measurement values of the i-th radar sensor with respect to the N+1-th radar sensor in step 2 The distance difference measurement model is constructed and described as: Then, according to the distance difference measurement model, the measurement model in the form of mixed noise and non-line-of-sight error is obtained: Substituting into The measurement model in the form of mixed noise and non-line-of-sight error is described as: Wherein, "||" represents the two-norm, and ||u o -s i || represents the true distance of the target to the i-th radar sensor, and ||u o -s N+1 || represents the true distance of the target to the N+1-th radar sensor, and s N+1 represents the true coordinate position of the N+1-th radar sensor in the reference coordinate system, d i represents the noisy range difference measurement value information of the source transmitted signal reaching the i-th radar sensor and the N+1-th radar sensor, represents the non-line-of-sight error of d i , represents the measurement noise of d i .

5. The method according to claim 4, wherein, The specific process of step 4 is as follows: Measurement model for mixed noise and non-line-of-sight error forms of measurement values Moving to the left side of the equation, square both sides of the equation and ignore the second order noise, error terms on the right side of the equation Moving to the left side of the equation, square both sides of the equation and ignore the second order noise, error terms on the right side of the equation Get the corresponding flat way of the i-th radar sensor 6. The method according to claim 5, wherein, The specific method of step 5 is to eliminate the norm 2d in the plane mode corresponding to the i-th radar sensor i ||u o -s N+1 ||,and the same multiplication on both sides of the plane mode corresponding to the i+1-th radar sensor Obtained The plane mode corresponding to the i-th radar sensor The difference between the two equations is obtained: Further arrangement obtains Wherein, Call ξ i Error model; at this time, construct As a weighted least squares problem, described as: Wherein, min() is the minimization function, (Ay) T WAy is the objective function, y is the optimization variable, and y = [u T ,1] T , u represents the target coordinate position variable, A is the introduced coefficient matrix Using the optimization variable y, the introduced coefficient matrix A, and the error model ξ i Expressed as ξ i =A i y; wherein, A i represents a vector composed of the i-th row elements of the introduced coefficient matrix A, s1 represents the true coordinate position of the first radar sensor in the reference coordinate system, s2 represents the true coordinate position of the second radar sensor in the reference coordinate system, s N-1 represents the true coordinate position of the N-1-th radar sensor in the reference coordinate system, s N represents the true coordinate position of the N-th radar sensor in the reference coordinate system, d1 represents the distance difference measurement value obtained by the first radar sensor and the reference radar sensor, d2 represents the distance difference measurement value obtained by the second radar sensor and the reference radar sensor, d N-1 represents the distance difference measurement value obtained by the N-1-th radar sensor and the reference radar sensor, d N represents the distance difference measurement value obtained by the N-th radar sensor and the reference radar sensor; W is the introduced weight matrix, w1 is the weight coefficient corresponding to d1, w N is the weight coefficient corresponding to d N In the above weighted least squares problem , the optimization variable y includes the target position u o and the weight coefficients w1,..., w N .

7. The method according to claim 6, wherein, The specific method of step 6 is to solve the target position u of the least square problem in step 5 using the maximum entropy criterion framework o and the weight coefficients w1,..., wn N : The similarity between two random variables X and Y is expressed by entropy, which is expressed by the formula E[κ σ (X-Y)]; wherein E[·] represents the mathematical expectation, κ σ () represents a kernel function with a kernel bandwidth σ, and the kernel bandwidth σ is defined by the Silverman rule as σ = 1.06 x min{σ E , L / 1.34} x M -0.2 ; wherein σ E is the standard deviation between the random variables X and Y, L is the interquartile range between the two variables, and M is the sample size; when the random variable Y is zero, the random variable X represents a random error, and assuming that α represents an unknown variable related to the random variable X, the estimation of the unknown variable is achieved by maximizing the entropy, which is expressed by the formula The weighted least squares problem as described in step 5 The unknown variable a is an optimization variable y in the weighted least squares problem; the mathematical expectation is approximated by a finite number of samples: The maximization of entropy problem is denoted as max H(y | p(x)), where i (y), i = 1,..., N represent a finite number of error samples; Using Laplacian kernel function Further transformation on the maximum entropy problem: joint With The maximum entropy problem is arranged as: Where constant coefficient Is irrelevant to optimization variable y, which is ignored in the following; according to the error function expression ξ i = A i y in step 5, the objective function is further written as Based on the properties of convex conjugate function, the weight matrix W is written in the form of weight coefficient vector w = [w1,..., wd] N ] T , and the objective function is further transformed into a new augmented problem for solving the weight coefficient vector w, denoted as Where w i is the d i th weight coefficient, is the convex conjugate function of ; the augmented problem is substituted into the objective function to obtain At this time The optimization variable y and the weight coefficient vector w are solved jointly, denoted as the joint optimization problem Where represents the function form about optimization variable y and weight coefficient vector w; since the joint optimization problem is difficult to solve, the formula is transformed into two sub-problems to be solved iteratively: Sub-problem 1: When the weight coefficient vector w is known, the joint optimization problem is simplified as where, represents the estimation value of w i , at this time, since the joint optimization problem also has an absolute value item, it is also a non-convex problem, by introducing an intermediate variable τ i as an upper bound of |A i y|, the joint optimization problem is changed into a convex problem for solving; the final optimization problem of the sub-problem 1 is represented as: where, s.t. represents being constrained by; after solving the final optimization problem of the sub-problem 1, the solution result is recorded as represents the estimation value of the optimization variable y; Sub-problem 2: When the optimization variable y is known, the joint optimization problem is simplified to At this time, the weight coefficient vector w has a corresponding closed-form solution Let the solution be denotes the estimated value of the weight coefficient vector w.

8. The method according to claim 7, wherein, The specific method of step 7 is that a scaling factor β is defined and selected according to the size of the positioning scene, and the positioning scene is scaled. Then set Solve step 6 sub-problem 1 The results are directly brought into Solve step 6 sub-problem 2; through the iterative solution of sub-problems 1 and 2, when the two iterations change difference norm is less than 10 -10 or the number of iterations reaches 10, the final estimation result is considered to be obtained, that is, the optimal estimation value of the target source coordinate position, which is recorded as u * , wherein y * is the optimal estimation value of y, represents the sub-vector composed of the first to p-th elements of y * ; represents the estimation value of the weight coefficient vector w1, represents the estimation value of the weight coefficient vector w2, represents the estimation value of the weight coefficient vector w N , and w2 is the weight coefficient vector corresponding to d2.

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