A moving target elliptical positioning method with unknown transmitter position and velocity

Abstract

The present invention relates to an elliptical positioning method for a moving target with unknown transmitter position and velocity. The method uses time delay and Doppler shift measurements of two paths to jointly locate the position and velocity of the target and transmitter. Based on a model of the two paths, the problem of jointly locating the position and velocity of the target and transmitter, as well as the time and frequency offset, is mathematically described as a non-convex constrained optimization problem. This problem is then relaxed into a semi-definite programming problem using a convex relaxation technique, ultimately obtaining accurate estimates of the position and velocity of the target and transmitter. This method requires neither prior knowledge of the transmitter's position and velocity nor time synchronization between the transmitter and receiver to accurately estimate the target's position and velocity.

Description

Technical Field

[0001] The present invention relates to the technical field of target positioning, in particular to an elliptical positioning method for a moving target with unknown transmitter position and speed. Background Art

[0002] Target positioning technology has a wide range of applications in both military and civilian fields. In the military, radar is used to detect enemy targets, while in civilian applications, it can be used in autonomous driving, underground search and rescue, and shared bicycle positioning. Therefore, target positioning is closely related to national defense security and daily life.

[0003] Target localization has been a hot research topic in recent years. In practice, target localization can be achieved by collecting measurements using radar, sonar, and wireless sensor networks. Typically, distance information is collected for the purpose of target localization, including time of arrival (TOA), time difference of arrival (TDOA), and signal power, such as received signal strength or acoustic signal level. Recent advances, such as wideband ranging and representative observation selection, can obtain accurate distance information even in non-line-of-sight (NLOS) and multipath environments. In addition, frequency information, including frequency of arrival (FOA) and frequency difference of arrival (FDOA), can be combined with TOA or TDOA for target localization with mobility.

[0004] Ellipse positioning estimates the target's position by finding the intersection of several ellipses. Specifically, a transmitter sends a signal, which is reflected by the target and received by the receiver. The propagation time delay (TD) of the signal from the transmitter to the receiver defines an ellipse (ellipsoid in 3D), where the positions of the transmitter and receiver serve as the focus of the target. The ellipse positioning model has been applied to traditional bistatic / multistatic radar and sonar systems and the recently developed MIMO radar systems. This model is also used in passive coherent location (PCL) systems, which are passive radar systems without a dedicated transmitter. In these systems, when there is relative motion between the transmitter and the target, not only the distance but also the Doppler frequency shift (DFS) information can be observed from the signal reflected by the target. The DFS measurement can be used to improve positioning accuracy and estimate the target's velocity.

[0005] If the transmitter's position and velocity are completely known, the direct path contains no information about the target, so only the reflected path information is needed for positioning. However, in real-world situations, the transmitter's position and velocity may be unknown or highly inaccurate. For example, when the transmitter's position and velocity change over time, their position and velocity may be unreliable or non-real-time; or, the transmitter may be located in a location where GPS cannot communicate, resulting in GPS being unable to provide its position and velocity estimates. In fact, in many system designs, the transmitter's position and velocity are intentionally set so that the transmitter serves only as an illumination source, significantly reducing hardware requirements. However, in this case, the elliptical positioning methods found in existing literature are rarely applicable. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide an elliptical positioning method for a mobile target with unknown transmitter position and speed, which has high positioning accuracy in a small noise situation and also high positioning accuracy in a large noise situation.

[0007] The technical solution adopted by the present invention is a method for elliptical positioning of a moving target with unknown transmitter position and velocity, comprising the following steps:

[0008] (1) Establish a coordinate system: Establish a plane coordinate system as a reference coordinate system in the MIMO radar system; assume that there are M transmitters for transmitting signals in the MIMO radar system and their true positions and velocities are unknown, where the position and velocity of the i-th transmitter in the reference coordinate system are respectively recorded as and Where 1≤i≤M, assume that there are N receivers for receiving signals in the MIMO radar system, whose real positions are known, and the position of the jth receiver in the reference coordinate system is recorded as Where 1≤j≤N, and the transmitter and receiver clocks are assumed to be asynchronous, where the time and frequency offsets from the i-th transmitter are respectively and In addition, assume that there is a target whose true position and velocity are unknown in the MIMO radar system, and its position and velocity in the reference coordinate system are denoted as p o and

[0009] (2) Obtain the distance measurement values and Doppler frequency shift measurement values of the indirect path and the direct path: The time it takes for the signal transmitted by the i-th transmitter to be reflected by the target and then received by the j-th receiver is recorded as τ r，ij , calculate the measurement value r of the sum of the distances from the i-th transmitter to the target and the distance from the target to the j-th receiver ij , r ij =τ r，ij×c, where c is the signal propagation speed, and the time partial derivative of the distance of the indirect path is used to obtain the corresponding Doppler shift measurement value The indirect path refers to M transmitters with unknown positions and velocities transmitting signals through a moving target p o After reflection, it is received by N receivers whose positions are known. This path is an indirect path. The time it takes for the signal transmitted by the i-th transmitter to be directly received by the j-th receiver is recorded as τ d，ij , calculate the measured value of the distance between the i-th transmitter and the j-th receiver, that is, d ij , d ij =τ d，ij ×c, and taking the time partial derivative of the distance of the direct path to obtain the corresponding Doppler shift measurement value The direct path means that the signals transmitted by M transmitters whose positions and velocities are unknown can also be directly received by N receivers whose positions are known. This path is a direct path.

[0010] (3) The measured value and the corresponding noise obtained in step (2) are expressed as phasors: The distance measurement value r of the indirect path obtained in step (2) is ij The model form is expressed as: The vector form of the measurement value of all indirect path distances is: r = [r 11 ,…,r 1N ,…,r M1 ,…,r MN ] T , the corresponding noise vector form is:

[0011] e r =[e r,11 ,...,e r,1N ,…,e r,M1 ,…,e r,MN ] T , the Doppler shift measurement value of the indirect path obtained in step (2) The model form is expressed as: The Doppler shift measurement Also called distance rate of change Then the vector form of the distance change rate of all indirect paths is:

[0012] The corresponding noise vector form is:

[0013] The distance measurement d of the direct path obtained in step (2) ij The model form is expressed as: The vector form of the measured values of all direct path distances is:

[0014] d=[d 11 ,…,d 1N ,…,d M1 ,…,d MN ] T , the corresponding noise vector form is: e d =[e d，11 ,...,e d，1N ,…,e d，M1 ,…,e d，MN ] T , the Doppler shift measurement value of the direct path obtained in step (2) The model form is expressed as: The Doppler shift measurement Also called distance rate of change Then the vector form of the distance change rate of all direct paths is: The corresponding noise vector form is: Write the noise corresponding to the distance measurement value of the indirect path and the direct path and the noise corresponding to the Doppler frequency shift measurement value in the form of a vector Assume that e follows a Gaussian distribution with mean 0 and covariance Q;

[0015] (4) Perform mathematical processing on the model in step (3): First, let The indirect path distance measurement value r obtained in step (3) ij The model form is transformed into: Then, squaring both sides and ignoring the second-order noise term, we can get the processed indirect path distance value r ij The model form is:

[0016] Where j = 1, ..., N; the processed indirect path distance value r ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler frequency shift measurement value model form: Where i=1,...,M,j=1,...,N, The direct path distance measurement value d obtained in step (3) ij The processed direct path distance value d can be obtained by squaring both sides of the model form and ignoring the second-order noise term. ij The model form is: The processed direct path distance measurement value d ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler frequency shift measurement value model form:

[0017] (5) Constructing a least squares problem with constraints: Constructing a constrained optimization problem based on the distance measurement value model form and Doppler measurement value model form of the processed direct path and indirect path obtained in step (4);

[0018] (6) Forming a semi-definite programming problem: Using the semi-definite relaxation technique, the constrained optimization problem obtained in step (5) is relaxed into a convex semi-definite programming problem;

[0019] (7) Obtaining the semi-positive definite programming problem: The interior point method is used to solve the semi-positive definite programming problem, and finally the optimal solution of the target position and velocity as well as the optimal solution of the transmitter position and velocity are obtained.

[0020] The beneficial effects of the present invention are as follows: the above method not only takes into account the unknown position and velocity of the transmitter, but also the time asynchrony between the transmitter and the receiver. By utilizing the distance and distance change rate of the indirect path and the direct path to jointly estimate the position and velocity of the target and the transmitter, as well as the unknown time and frequency offset, the above method does not require prior knowledge of the transmitter's position and velocity, nor does it require time synchronization between the transmitter and the receiver, and can accurately estimate the position and velocity of the target. The positioning accuracy of this method in low-noise conditions can reach the Cramer-Rao lower bound, and its performance is superior to that of the two-step weighted least squares method in the prior art. Moreover, compared with the split-step semi-definite relaxation method in the prior art, this method exhibits higher positioning accuracy in high-noise conditions.

[0021] Preferably, in step (5), the constrained optimization problem is constructed as follows:

[0022]

[0023] x (4k+4) =b τ,i b f,i ,

[0024]

[0025] x (4k+7) =||p|| 2 ,

[0026] x (4k+8) =||t i || 2 ,

[0027]

[0028] x (4k+13) =ξ i b τ,i ,

[0029] x (4k+14) =ξ i b f,i ,

[0030]

[0031] Among them, min means "minimize", st means "subject to constraints", R=DVD T ,

[0032]

[0033] Preferably, in step (6), by setting X=xx T , use the semi-positive relaxation technique to relax the constrained optimization problem obtained in step (5) into a convex semi-positive programming problem, which is:

[0034]

[0035] x (L+2M+i) =X (L+i，L+i) ,

[0036] x (L+3M+i) =X (L+i，L+M+i) ,

[0037] x (L+4M+1) =tr{X (1：k，(k+1：2k))},

[0038] x (L+4M+1+i) =tr{X ((i+1)k+1：(i+2)k，(i+1+M)k+1：(i+2+M)k)},

[0039] x (L+5M+2) =tr{X (1：k，1：k)},

[0040] x (L+5M+2+i) =tr{X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k)},

[0041] X (L+6M+2+i，L+6M+2+i) =tr{X (1：k，1：k)}-2tr{X (1：k，(i+1)k+1：(i+2)k)}+tr{X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k)},

[0042] X (L+7M+2+i，L+7M+2+i) ≤tr{X ((k+1)：2k，(k+1)：2k)}-2tr{X ((k+1)：2k，(i+M+1)k+1：(i+M+2)k)}+tr{X ((i+M+1)k+1：(i+M+2)k，(i+M+1)k+1：(i+M+2)k)},

[0043] x (L+8M+2+i) =X (L+6M+2+i，L+7M+2+i) ,

[0044] x (L+8M+2+i) =tr{X (1：k，(k+1)：2k)}-tr{X (1：k，(i+M+1)k+1：(i+M+2)k)}-tr{X ((k+1)：2k，(i+1)k+1：(i+2)k)}+tr{X ((i+1)k+1：(i+2)k，(i+1+M)k+1：(i+2+M)k)},

[0045] x (L+9M+2+i) =X (L+6M+2+i，L+6M+2+i) ,

[0046] x (L+10M+2+i) =X (L+6M+2+i，L+i) ,

[0047] x (L+11M+2+i) =X (L+6M+2+i，L+M+i) ,

[0048] x (L+12M+2+i) =X (L+7M+2+i，L+i) ,

[0049] Where i = 1, ..., M, L = (2M + 2) k,

[0050] As a preference, in step (7), let the optimal solutions of target position and velocity be p and The optimal solutions for the transmitter's position and velocity are t and According to the solved semi-positive programming problem, we can get: p = x (1：k) , t=x (2k+1：3k) , BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 A flowchart of a method for elliptical positioning of a moving target with unknown transmitter position and velocity according to the present invention;

[0052] Figure 2 A model diagram corresponding to a moving target elliptical positioning method with unknown transmitter position and velocity according to the present invention;

[0053] Figure 3 A comparison chart of the logarithmic mean square error of the target position of the method of the present invention and the existing two-step weighted least squares method, the continuous positive semi-definite relaxation method and the Cramer-Rao lower bound method;

[0054] Figure 4 The figure is a comparison chart of the logarithmic mean square error of the target velocity of the method of the present invention and the existing two-step weighted least squares method, the continuous semi-definite relaxation method and the Cramer-Rao lower bound method. DETAILED DESCRIPTION

[0055] The invention will be further described below with reference to the accompanying drawings and in combination with specific implementations, so that those skilled in the art can implement the invention with reference to the description. The protection scope of the invention is not limited to the specific implementations.

[0056] The present invention relates to a method for elliptical positioning of a moving target with unknown transmitter position and speed, such as Figure 1 As shown, the following steps are included:

[0057] (1) Establish a coordinate system: Establish a plane coordinate system as a reference coordinate system in the MIMO radar system. Assume that there are M transmitters in the MIMO radar system for transmitting signals and their true positions and velocities are unknown. The position and velocity of the i-th transmitter in the reference coordinate system are respectively recorded as and Where 1≤i≤M, assume that there are N receivers for receiving signals in the MIMO radar system, whose real positions are known, and the position of the jth receiver in the reference coordinate system is recorded as Where 1≤j≤N, and the transmitter and receiver clocks are assumed to be asynchronous, where the time and frequency offsets from the i-th transmitter are respectively and In addition, assume that there is a target whose true position and velocity are unknown in the MIMO radar system, and its position and velocity in the reference coordinate system are denoted as p o and

[0058] (2) Obtain the distance measurement values and Doppler frequency shift measurement values of the indirect path and the direct path: The time it takes for the signal transmitted by the i-th transmitter to be reflected by the target and then received by the j-th receiver is recorded as τ r，ij , calculate the measurement value r of the sum of the distances from the i-th transmitter to the target and the distance from the target to the j-th receiver ij , r ij =τ r，ij ×c, where c is the signal propagation speed, and the time partial derivative of the distance of the indirect path is used to obtain the corresponding Doppler shift measurement value The indirect path refers to M transmitters with unknown positions and velocities transmitting signals through a moving target p o After reflection, it is received by N receivers whose positions are known. This path is an indirect path. The time it takes for the signal transmitted by the i-th transmitter to be directly received by the j-th receiver is recorded as τ d，ij , calculate the measured value of the distance between the i-th transmitter and the j-th receiver, that is, d ij , d ij =τ d，ij ×c, and taking the time partial derivative of the distance of the direct path to obtain the corresponding Doppler shift measurement value The direct path means that the signals transmitted by M transmitters whose positions and velocities are unknown can also be directly received by N receivers whose positions are known. This path is a direct path.

[0059] In step (2), the time of flight of the signal can be calculated based on the timestamp of the received signal recorded by the receiver, and the measured distance of the direct path and the indirect path can be calculated. In addition, due to the relative motion between the transmitter, receiver and target, the Doppler measurement of the indirect path and the direct path can be calculated.

[0060] (3) The measured value and the corresponding noise obtained in step (2) are expressed as phasors: The distance measurement value and Doppler shift measurement value of the indirect path obtained in step (2) are described in the form of a model: The distance measurement value r of the indirect path ij The model form is expressed as: The vector form of the measurement value of all indirect path distances is: r = [r 11 ,…,r 1N ,…,r M1 ,…,r MN ] T , the corresponding noise vector form is:

[0061] e r =[e r，11 ,...,e r，1N ,…,e r，M1 ,…,e r，MN ] T ; Taking the time partial derivative of the distance measurement value of the indirect path can obtain the Doppler frequency shift measurement value of the indirect path Distance change rate The Doppler shift measurement of the indirect path obtained in step (2) The model form is: Then the vector form of the distance change rate of all indirect paths is: The corresponding noise vector form is: The distance measurement value and Doppler frequency shift measurement value of the direct path obtained in step (2) are described in the form of a model: the distance measurement value d of the direct path ij The model form is:

[0062] The vector form of the measurement value of all direct path distances is: d = [d 11 ,…,d 1N ,…,d M1 ,…,d MN ] T , the corresponding noise vector form is: e d =[e d，11,...,e d，1N ,…,e d，M1 ,…,e d，MN ] T , the measurement value d of the direct path distance ij The time partial derivative can be used to obtain the Doppler shift measurement of the direct path Distance change rate Doppler shift measurements for the direct path The model form is: Then the vector form of the distance change rate of all direct paths is:

[0063] The corresponding noise vector form is: Write the noise corresponding to the distance measurement value of the indirect path and the direct path and the noise corresponding to the Doppler frequency shift measurement value in the form of vectors Assume that e follows a Gaussian distribution with mean 0 and covariance Q;

[0064] In step (3), the vector r represents the distance measurement value of all indirect paths, and the corresponding measurement noise vector is e r ;vector Represents the distance change rate measurement value of all indirect paths, and the corresponding measurement noise vector is The vector d represents the measurement value of all direct paths, and its corresponding measurement noise vector is denoted by e d ;vector Represents the distance change rate measurement value of all direct paths, and its corresponding measurement noise vector is expressed as

[0065] In step (3), the symbol “||||” is used to find the Euclidean norm. and represent the coordinate position and velocity of the i-th transmitter in the reference coordinate system respectively; represents the coordinate position of the jth receiver in the reference coordinate system; p o and Represent the coordinate position and velocity of the target in the reference coordinate system, and denote the time and frequency offsets from the ith transmitter, respectively, and denote the distance and distance change rate between the i-th transmitter and the target, e r,ij 、 e d,ij and denote the distance and Doppler measurement errors of the indirect and direct paths for the i-th transmitter and j-th receiver pair, respectively, [g]T represents the transpose of g, symbol ρ a Represents ρ a =a / ||a||;

[0066] (4) Perform mathematical processing on the model in step (3): First, let The indirect path distance measurement value r ij The model form is transformed into: Then, squaring both sides and ignoring the second-order noise term, we can get the processed indirect path distance value r ij The model form is: Where j = 1, ..., N; the processed indirect path distance value r ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler measurement value model form:

[0067] where i = 1, ..., M, j = 1, ..., N; for the direct path distance measurement d ij Model Squaring both sides and ignoring the second-order noise term yields the processed direct path distance measurement d ij The model form is:

[0068] The processed direct path distance measurement value d ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler frequency shift measurement model form:

[0069]

[0070] In step (4), It is an auxiliary variable introduced to facilitate the processing of the measurement model and to help add constraints later to form a tighter semi-definite programming problem;

[0071] In step (4), the symbol “||||” is used to find the Euclidean norm. and represent the coordinate position and velocity of the i-th transmitter in the reference coordinate system respectively; represents the coordinate position of the jth receiver in the reference coordinate system; p o and Represent the coordinate position and velocity of the target in the reference coordinate system, and denote the time and frequency offsets from the i-th transmitter, respectively. r,ij 、 e d,ij and denote the distance and Doppler measurement errors of the indirect and direct paths for the i-th transmitter and j-th receiver pair, respectively, [g] T represents the transpose of g, symbol ρ a represents ρ a =a / ||a||

[0072] (5) Constructing a constrained least squares problem: Constructing a constrained optimization problem based on the distance measurement value model and Doppler frequency shift measurement value model of the processed direct path and indirect path obtained in step (4). The constrained optimization problem is:

[0073]

[0074] x (4k+4) =b τ,i b f,i ,

[0075]

[0076]

[0077] x (4k+7) =||p|| 2 ,

[0078] x (4k+8) =||t i || 2 ,

[0079]

[0080] x (4k+13) =ξ i b τ,i ,

[0081] x (4k+14) =ξ i b f,i ,

[0082]

[0083] Where min means "minimize", st means "subject to constraints",

[0084]

[0085] V represents a zero-mean matrix,

[0086]

[0087] Dd =blkdiag(D d,1 ,D d,2 ,...,D d,M ),

[0088]

[0089] In step (5), the mathematical transformation of the model forms a constrained weighted least squares problem, which lays the foundation for the next step of using the semi-definite relaxation technique to form a semi-definite programming problem.

[0090] In step (5), and denote the coordinate position and velocity of the i-th transmitter in the reference coordinate system, p o and Represent the coordinate position and velocity of the target in the reference coordinate system, and denote the time and frequency offsets from the ith transmitter, respectively, and Represent the distance and distance change rate between the i-th transmitter and the target, x (4k+3) Represents the 4k+3th element of vector x, x (4k+4) Represents the 4k+4th element of vector x, x (4k+5) Represents the 4k+5th element of vector x, x (4k+6) Represents the 4k+6th element of vector x, x (4k+7) Represents the 4k+7th element of vector x, x (4k+8) Represents the 4k+8th element of vector x, x (4k+9) Represents the 4k+9th element of vector x, x (4k+10) Represents the 4k+10th element of vector x, x (4k+11) Represents the 4k+11th element of vector x, x (4k+12) Represents the 4k+12th element of vector x, x (4k+13) Represents the 4k+13th element of vector x, x (4k+14) Represents the 4k+14th element of vector x, x (4k+15) Represents the 4k+15th element of vector x;

[0091] (6) Form a semi-positive programming problem: Let X = xx T , use the semi-positive relaxation technique to relax the constrained optimization problem obtained in step (5) into a convex semi-positive programming problem, which is:

[0092]

[0093] x (L+2M+i) =X(L+i，L+i) ,

[0094] x (L+3M+i) = X (L+i，L+M+i) ,

[0095] x (L+4M+1) = tr{X (1：k，(k+1:2k))},

[0096] x (L+4M+1+i) = tr{X ((i+1)k+1：(i+2)k，(i+1+M)k+1：(i+2+M)k)},

[0097] x (L+5M+2) = tr{X (1：k，1:k)},

[0098] x (L+5M+2+i) = tr{X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k)},

[0099] X (L+6M+2+i，L+6M+2+i) = tr{X (1：k，1：k)}-2tr{X (1：k，(i+1)k+1：(i+2)k)}+tr{X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k)},

[0100] X (L+7M+2+i，L+7M+2+i) ≤ tr{X ((k+1)：2k，(k+1)：2k)}-2tr{X ((k+1)：2k，(i+M+1)k+1：(i+M+2)k)}+tr{X ((i+M+1)k+1：(i+M+2)k，(i+M+1)k+1：(i+M+2)k)},

[0101] x (L+8M+2+i) = X (L+6M+2+i，L+7M+2+i) ,

[0102] x (L+8M+2+i) = tr{X (1：k，(k+1)：2k)}-tr{X (1：k，(i+M+1)k+1：(i+M+2)k)}-tr{X ((k+1)：2k，(i+1)k+1：(i+2)k)}+tr{X ((i+1)k+1：(i+2)k，(i+1+M)k+1：(i+2+M)k)}, <o000599>x (L+9M+2+i) = X[[ID=o000210> ,

[0104] x (L+10M+2+i) = X (L+6M+2+i，L+i) [[ID=9"2]],

[0105] x (L+11M+2+i) = X (L+6M+2+i，L+M+i) ,

[0106] x (L+12M+2+i) = X (L+7M+2+i，L+i) ,

[0107] where i = 1,..., M, L = (2M + 2)k

[0108] It should be noted that there seems to be an "o000599" and an "o000210" in the original text which might be incorrect. I've translated them as they are but they may need to be corrected in the source for a more accurate translation overall.In step (6), by introducing auxiliary variable X=xx T The highly non-convex nonlinear weighted least squares problem with constraints is transformed, and then the non-convex constraint rank(X)=1 is dropped using the semi-definite relaxation technique, thus forming a semi-definite programming problem.

[0109] In step (6), the symbol tr{} is the trace of the matrix, x (L+2M+i) Represents the L+2M+ith element of vector x, X (L+i,L+i) represents the L+i-th row and L+i-th column element of matrix X, x (L+3M+i) represents the L+3M+ith element of vector x, X (L+i，L+M+i) represents the L+ith row and L+M+ith column element of matrix X, x (L+4M+1) Represents the L+4M+1th element of vector x, X (1：k，(k+1：2k)) represents the 1st to kth rows and k+1th to 2kth columns of the matrix X, x (L+4M+1+i) represents the L+4M+1+ith element of vector x, X ((i+1)k+1：(i+2)k，(i+1+M)k+1：(i+2+M)k) represents the matrix X with rows (i+1)k+1 to (i+2)k and columns (i+M+1)k+1 to (i+M+2)k, x (L+5M+2) Represents the L+5M+2th element of vector x, X (1：k，1：k) represents the 1st to kth columns of matrix X, x (L+5M+2+i) Represents the L+5M+2+ith element of vector x, X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k) represents the elements of the matrix X from (i+1)k+1 to (i+2)k rows and (i+1)k+1 to (i+2)k columns, X (L+6M+2+i,L+6M+2+i) represents the L+6M+2+ith row and L+6M+2+ith column element of matrix X, X (1：k，(i+1)k+1：(i+2)k) represents the elements in the 1st to kth rows and the (i+1)k+1th to (i+2)kth columns of the matrix X, X ((i+1)k+1：(i+2)k，(i+1)k+1：(i+2)k) represents the elements of the matrix X from (i+1)k+1 to (i+2)k rows and (i+1)k+1 to (i+2)k columns, X (L+7M+2+i，L+7M+2+i) represents the L+7m+2+ith row and L+7m+2+ith column element of matrix X, X ((k+1)：2k，(i+M+1)k+1：(i+M+2)k) represents the matrix X with rows (i+1)k+1 to (i+2)k and rows (i+M+1)k+1 to (i+M+2)k, X ((i+M+1)k+1：(i+M+2)k，(i+M+1)k+1：(i+M+2)k) represents the matrix X with rows (i+M+1)k+1 to (i+M+2)k and columns (i+M+1)k+1 to (i+M+2)k, x (L+8M+2+i) Represents the L+8M+2+ith element of vector x, x (L+8M+2+i) Represents the L+8M+2+ith element of vector x, x (L+9M+2+i) Represents the L+9M+2+ith element of vector x, X (L+6M+2+i,L+6M+2+i)represents the L+6M+2+ith row and L+6M+2+ith column element of matrix X, x (L+10M+2+i) Represents the L+10M+2+ith element of vector x, X (L+6M+2+i,L+i) represents the L+6M+2+ith row and L+ith column element of matrix X, x (L+11M+2+i) Represents the L+11M+2+ith element of vector x, X (L+6M+2+i,L+M+i) represents the L+6M+2+ith row and L+M+ith column element of matrix X, x (L+12M+2+i) Represents the L+12M+2+ith element of vector x, X (L+7M+2+i,L+i) Represents the element in the L+7M+2+i row and L+i column of the matrix X;

[0110] (7) Obtaining the semi-positive definite programming problem: The interior point method is used to solve the semi-positive definite programming problem. The interior point method is the method used by the underlying function of matlab. It only needs to call the underlying function to solve the semi-positive definite programming problem, and finally obtain the optimal solution of the target position and speed and the optimal solution of the transmitter position and speed; let the optimal solutions of the target position and speed be p and The optimal solutions for the transmitter's position and velocity are t and Therefore, p = x (1:k) , t=x (2k+1:3k) ,

[0111] In step (7), p and represent the final estimated values of target position and velocity, t i and denote the final estimated values of the position and velocity of the i-th transmitter, respectively, x (1:k) Represents the vector x from the 1st row to the kth element, x (k+1:2k) Represents the vector x consisting of the elements from row k+1 to row 2k, x ((i+1)k+1:(i+2)k) represents the vector x consisting of the elements from row (i+1)k+1 to row (i+2)k, x ((i+1+M)k+1:(i+2+M)k) Represents the vector consisting of the elements from the (i+1+M)k+1th row to the (i+2+M)kth row of vector x.

[0112] The method of the present invention jointly locates the position and velocity of a target and a transmitter by using two path delay and Doppler shift measurements: an indirect path delay and Doppler shift measurement model, i.e., the sum of the algebraic distances between the transmitter and the target and the receiver, and the corresponding indirect path Doppler shift measurements; and a direct path delay and Doppler shift measurement model, i.e., the algebraic distance between the transmitter and the receiver, and the corresponding direct path Doppler shift measurements. Based on these two path models, the problem of jointly locating the position and velocity of the target and the transmitter, as well as the time and frequency offset, is mathematically described as a non-convex constrained optimization problem, which is then relaxed into a semi-definite programming problem using a convex relaxation technique. Finally, the problem is solved using the CVX toolbox in MATLAB, thereby obtaining relatively accurate estimates of the position and velocity of the target and the transmitter. The method has the advantage that the position and velocity of the target can be accurately estimated without requiring prior knowledge of the transmitter's position and velocity, or time synchronization between the transmitter and the receiver.

[0113] In order to verify the feasibility and effectiveness of the method of the present invention, a simulation test was conducted on the method of the present invention, and the results are as follows:

[0114] Assume that there are six receivers and three transmitters in three-dimensional space, and their real position coordinates [x, y, z] T Randomly generated, x~(-4000,4000)m, y~(-4000,4000)m and z~(1000,3000)m, true speed Also randomly generated, x ~ (-30, 30) m, y ~ (-30, 30) m and z ~ (10, 30) m, the position and speed of the target are set to [-1000, 500, 1500] T m and [15,15,30] T m / s, the time and frequency offsets are set to 200m, 150m, 180, 5m / s, 8m / s and 10m / s respectively, and the covariance matrix Q of the measurement noise is uncorrelated noise;

[0115] The performance of the method is tested with the measurement error power σ 2 Change, the range is from 10 -1 m 2 to 10 3 m 2 ,like Figure 3 and Figure 4A comparison chart of the logarithmic mean square error of the proposed method with the existing two-step weighted least squares method (Two Step Weighted Least Squares, TSWLS), continuous semidefinite relaxation method (Successive-SDP), and Cramer-Rao lower bound method (Cramer-Rao lower bound, CRLB) is given. Figure 3 and Figure 4 In the figure, Multi-Tx-Joint-SDP represents the method of the present invention, Multi-Tx-TSWLS represents the two-step weighted least squares method, Multi-Tx-Successive-SDP represents the continuous semidefinite relaxation method, and Multi-Tx-CRLB represents the Cramer-Rao lower bound method. The advantages of the method of the present invention over other methods are as follows:

[0116] ① The positioning accuracy of the method proposed in the present invention can reach the Cramer-Rao lower bound under low noise conditions, and its performance is better than the two-step weighted least squares method in the prior art;

[0117] ② The method proposed in this invention also has good accuracy in the case of large noise, while the two-step weighted least squares method in the prior art has very poor positioning accuracy in large noise conditions, especially in poor scene conditions.

[0118] ③ Compared with the prior art split-step semi-definite relaxation method, the positioning accuracy of the method proposed in the present invention has higher accuracy in the case of large noise.

Claims

1. A method for elliptical positioning of a moving target with unknown transmitter position and velocity, characterized by: The following steps are involved: (1) Establishing a coordinate system: Establish a plane coordinate system as a reference coordinate system in the MIMO radar system; Assume that there are M transmitters for transmitting signals in a multiple-input multiple-output radar system and their true positions and velocities are unknown, where the position and velocity of the i-th transmitter in the reference coordinate system are respectively recorded as and Where 1≤i≤M, assume that there are N receivers for receiving signals in the MIMO radar system, whose real positions are known, and the position of the jth receiver in the reference coordinate system is recorded as Where 1≤j≤N, and the transmitter and receiver clocks are assumed to be asynchronous, where the time and frequency offsets from the i-th transmitter are respectively and In addition, assume that there is a target whose true position and velocity are unknown in the MIMO radar system, and its position and velocity in the reference coordinate system are denoted as p o and (2) Obtain the distance measurement values and Doppler frequency shift measurement values of the indirect path and the direct path: The time it takes for the signal transmitted by the i-th transmitter to be reflected by the target and then received by the j-th receiver is recorded as τ r，ij , calculate the measurement value r of the sum of the distances from the i-th transmitter to the target and the distance from the target to the j-th receiver ij , r ij =τ r，ij ×c, where c is the signal propagation speed, and the time partial derivative of the distance of the indirect path is used to obtain the corresponding Doppler shift measurement value The indirect path refers to M transmitters with unknown positions and velocities transmitting signals through a moving target p o After reflection, it is received by N receivers whose positions are known. This path is an indirect path. The time it takes for the signal transmitted by the i-th transmitter to be directly received by the j-th receiver is recorded as τ d,ij , calculate the measured value of the distance between the i-th transmitter and the j-th receiver, that is, d ij , d ij =τ d，ij ×c, and taking the time partial derivative of the distance of the direct path to obtain the corresponding Doppler shift measurement value The direct path means that the signals transmitted by M transmitters whose positions and velocities are unknown can also be directly received by N receivers whose positions are known. This path is a direct path. (3) The measured value and the corresponding noise obtained in step (2) are expressed as phasors: The distance measurement value r of the indirect path obtained in step (2) is ij The model form is expressed as: The vector form of the measurement value of all indirect path distances is: r = [r 11 ,…,r 1N ,…,r M1 ,…,r MN ] T , the corresponding noise vector form is: e r =[e r,11 ,...,e r,1N ,…,e r,M1 ,…,e r,MN ] T ; The Doppler shift measurement value of the indirect path obtained in step (2) The model form is expressed as: The Doppler shift measurement Also called distance rate of change Then the vector form of the distance change rate of all indirect paths is: The corresponding noise vector form is: The distance measurement d of the direct path obtained in step (2) ij The model form is expressed as: The vector form of the measurement value of all direct path distances is: d = [d 11 ,…,d 1N ,…,d M1 ,…,d MN ] T , the corresponding noise vector form is: e d =[e d，11 ,...,e d，1N ,…,e d，M1 ,…,e d，MN ] T , the Doppler shift measurement value of the direct path obtained in step (2) The model form is expressed as: The Doppler shift measurement Also called distance rate of change Then the vector form of the distance change rate of all direct paths is: The corresponding noise vector form is: Write the noise corresponding to the distance measurement value of the indirect path and the direct path and the noise corresponding to the Doppler frequency shift measurement value in the form of a vector Assume that e follows a Gaussian distribution with mean 0 and covariance Q; (4) Perform mathematical processing on the model in step (3): First, let The indirect path distance measurement value r obtained in step (3) ij The model form is transformed into: Then, squaring both sides and ignoring the second-order noise term, we can get the processed indirect path distance value r ij The model form is: i=1,…,M, where j=1,…,N; for the processed indirect path distance value r ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler frequency shift measurement value model form: Where i = 1, ..., M, j = 1, ..., N; the direct path distance measurement value d obtained in step (3) ij The processed direct path distance value d can be obtained by squaring both sides of the model form and ignoring the second-order noise term. ij The model form is: The processed direct path distance measurement value d ij The model form of the time partial derivative can be obtained to obtain the corresponding Doppler frequency shift measurement value model form: i=1,...,M,j=1,...,N; (5) Constructing a least squares problem with constraints: Constructing a constrained optimization problem based on the distance measurement value model form and Doppler measurement value model form of the processed direct path and indirect path obtained in step (4); (6) Forming a semi-definite programming problem: Using the semi-definite relaxation technique, the constrained optimization problem obtained in step (5) is relaxed into a convex semi-definite programming problem; (7) Obtaining the semi-positive definite programming problem: The interior point method is used to solve the semi-positive definite programming problem, and finally the optimal solution of the target position and velocity as well as the optimal solution of the transmitter position and velocity are obtained.

2. The method for elliptical positioning of a moving target with unknown transmitter position and velocity according to claim 1, characterized in that: In step (5), the constrained optimization problem is constructed as follows: x (4k+4) =b τ,i b f,i , x (4k+7) =||p|| 2 , x (4k+8) =||t i || 2 , x (4k+13) =ξ i b τ，i , x (4k+14) =ξ i b f,i , i=1,...,M, Among them, min means "minimize", st means "subject to constraints", R=DVD T , 3. The method for elliptical positioning of a moving target with unknown transmitter position and velocity according to claim 2, characterized in that: In step (6), by setting X = xx T , use the semi-positive relaxation technique to relax the constrained optimization problem obtained in step (5) into a convex semi-positive programming problem, which is: x (L+2M+i) =X (L+i,L+i) , x (L+3M+i) =X (L+i,L+M+i) , x (L+4M+1) =tr{X (1:k,(k+1:2k)) }, x (L+4M+1+i) =tr{X ((i+1)k+1:(i+2)k,(i+1+M)k+1:(i+2+M)k) }, x (L+5M+2) =tr{X (1:k,1:k) }, x (L+5M+2+i) =tr{X ((i+1)k+1:(i+2)k,(i+1)k+1:(i+2)k) }, X (L+6M+2+i,L+6M+2+i) =tr{X (1:k,1:k) }-2tr{X (1:k,(i+1)k+1:(i+2)k) } +tr{X ((i+1)k+1:(i+2)k,(i+1)k+1:(i+2)k) }, X (L+7M+2+i,L+7M+2+i) ≤tr{X ((k+1):2k,(k+1):2k) }-2tr{X ((k+1):2k,(i+M+1)k+1:(i+M+2)k) } +tr{X ((i+M+1)k+1:(i+M+2)k,(i+M+1)k+1:(i+M+2)k) }, x (L+8M+2+i) =X (L+6M+2+i,L+7M+2+i) , x (L+8M+2+i) =tr{X (1:k,(k+1):2k) }-tr{X (1:k,(i+M+1)k+1:(i+M+2)k) } -tr{X ((k+1):2k,(i+1)k+1:(i+2)k) }+tr{X ((i+1)k+1:(i+2)k,(i+1+M)k+1:(i+2+M)k) }, x (L+9M+2+i) =X (L+6M+2+i,L+6M+2+i) , x (L+10M+2+i) =X (L+6M+2+i,L+i) , x (L+11M+2+i) =X (L+6M+2+i,L+M+i) , x (L+12M+2+i) =X (L+7M+2+i,L+i) , Where i = 1, ..., M, L = (2M + 2) k, 4. The method for elliptical positioning of a moving target with unknown transmitter position and velocity according to claim 3, characterized in that: In step (7), let the optimal solutions of target position and velocity be p and The optimal solutions for the transmitter's position and velocity are t and According to the solved semi-positive programming problem, we can get:

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