A 3D positioning device and method for MIMO radar targets

Abstract

A three-dimensional positioning device and method for MIMO radar targets. The device includes a transceiver antenna array composed of a sparse uniform rectangular array (URA), and each element is composed of a complete electromagnetic vector sensor (EMVS). By measuring the two-dimensional direction of departure (2D-DOD) of the target relative to the transmitting array and the two-dimensional direction of arrival (2D-DOA) of the target relative to the receiving array, and then combining the specific coordinates of the transmitting and receiving arrays, the three-dimensional coordinates of the target can be obtained. This method first represents the array signal after matched filtering of the bistatic MIMO radar as a third-order parallel factor tensor model, then uses parallel factor decomposition to obtain the estimation of the factor matrices, and uses the rotational invariance characteristics of the spatial domain and polarization domain of the array to obtain high-resolution 2D-DOD and 2D-DOA estimations with non-ambiguity characteristics. The algorithm proposed in the present invention can obtain parameter estimations with low computational complexity, high precision, and automatic pairing.

Description

Technical Field

[0001] The present invention relates to the technical field of MIMO radar target positioning, and particularly relates to a three-dimensional positioning device and a positioning method for MIMO radar targets. Background Art

[0002] Multiple-input Multiple-output (MIMO) technology is the mainstream direction for the future development of radar. MIMO radar uses multiple transmitting antennas to transmit orthogonal waveforms, and at the receiving end, a matched filter is used to separate the signals received by multiple receiving antennas, thereby forming multiple virtual channels between the transmitting and receiving ends. Using the idea of diversity, MIMO radar can greatly improve the target detection performance. It has potential advantages in terms of resolution, anti-fading, identifiability, and noise suppression, thus attracting extensive attention from the academic and engineering circles at home and abroad.

[0003] Joint Direction-of-Departure (DOD) and Direction-of-Arrival (DOA) estimation is one of the important tasks for bistatic MIMO radar target positioning and also one of the hot issues in current MIMO radar research. So far, a large number of excellent estimation algorithms have emerged, such as the Estimation Method of Signal Parameters via Rotational (ESPRIT) algorithm, the Parallel Factor (PARAFAC) algorithm, etc. However, the existing methods can only effectively obtain the one-dimensional Direction-of-Departure (DOD) and the one-dimensional Direction-of-Arrival (DOA). Only a few literatures have studied the 2D-DOD and 2D-DOA estimation problems. For example:

[0004] Literature [1]: C. Chen and X. Zhang, “A low-complexity joint 2D-DOD and 2D-DOA estimation algorithm for MIMO radar with arbitrary arrays,” Int J. Electron., vol. 100, no. 10, pp. 1455–1469, Jan. 2013.;

[0005] Reference [2]: J. Li and X. Zhang, “Closed-form blind 2D-DOD and 2D-DOA estimation for MIMO radar with arbitrary arrays,” Wireless Per. Commun., vol. 69, no. 1, pp. 175–186, Mar. 2013.;

[0006] Reference [3]: T. Xia, “Joint diagonalization based 2D-DOD and 2D-DOA estimation for bistatic MIMO radar,” Signal Process., vol. 116, pp. 7–12, Nov. 2015.;

[0007] A common feature of the above estimation methods is the use of traditional two-dimensional non-linear scalar sensor arrays, such as: L-shaped arrays, rectangular arrays, and arbitrary arrays. However, scalar array sensors have low degrees of freedom and low accuracy in parameter estimation.

[0008] Different from scalar sensors, an Electromagnetic Vector Sensor (EMVS) can measure two-dimensional (2D) azimuth and elevation angles. In addition, it has several inherent advantages, such as: having better recognition ability, being able to achieve automatic pairing of two-dimensional angles, and providing additional polarization information of the signal source.

[0009] As recorded in Reference [4]: K. T. Wong and X. Yuan, “vector cross-product direction-finding with an electromagnetic vector-sensor of six orthogonally oriented but spatially noncollocating dipoles / loops,” IEEE Trans. Signal Process., vol. 59, no. 1, pp. 160–171, Jan. 2011.

[0010] Reference [5]: S. Chintagunta and P. Palanisamy, “2D-DOD and 2D-DOA estimation using the electromagnetic vector sensors,” Signal Process., vol. 147, pp. 163–172, Jun. 2018. proposed a bistatic electromagnetic vector sensor-multiple input multiple output (EMVS-MIMO) radar system, which uses electromagnetic vector sensor arrays at both the transmitter and receiver. At the same time, this reference proposed an improved ESPRIT algorithm. However, this algorithm has a series of defects. First, this algorithm involves eigenvalue decomposition and has a very high computational complexity. Second, the tensor structure between multi-dimensional samples is ignored, and the accuracy of the algorithm needs to be improved. In addition, this method requires additional pairing of the estimated 2D-DOD and 2D-DOA.

[0011] Reference [6]: F. Wen, J. Shi. “Fast direction finding for bistatic EMVS-MIMO radar without pairing,” Signal Process., 2020, 173, 107512. proposed an improved ESPRIT algorithm;

[0012] Reference [7]: F. Wen, J. Shi, Z. Zhang. “Joint 2D-DOD, 2D-DOA and polarization angles estimation for bistatic EMVS-MIMO radar via PARAFAC analysis,” IEEE Trans. Veh. Technol., 2020, 29(2): 1626-1638. proposed a PARAFAC algorithm;

[0013] The above algorithms are all only applicable to uniform linear array EMVS-MIMO radars, and the PARAFAC algorithm has better performance than existing matrix decomposition algorithms.

[0014] In addition, Reference [8]: F. Wen, J. Shi, Z. Zhang. “Closed-form estimation algorithm for EMVS-MIMO radar with arbitrary sensor geometry,” Signal Process., 2021, 186, 108117. proposed an ESPRIT algorithm applicable to arbitrary arrays;

[0015] Reference [9]: F. Wen, J. Shi, G. Gui, et al. “3D positioning method for anonymous UAV based on bistatic polarized MIMO radar,” IEEE Internet Things J., 2022, doi: 10.1109 / JIOT.2022.3204267. proposed a PARAFAC algorithm;

[0016] Although the above algorithm is insensitive to the array manifold, the accuracy of the algorithm is limited by the array aperture, and new methods are urgently needed to improve the positioning accuracy. Summary of the Invention

[0017] To solve the above technical problems, the present invention provides a MIMO radar target three-dimensional positioning device and positioning method. The method first represents the array signal after matched filtering of the bistatic MIMO radar as a third-order parallel factor tensor model; then, uses parallel factor decomposition to obtain an estimate of the factor matrix, and uses the spatial rotation invariance property of the array and the rotation invariance property of the polarization domain to obtain high-resolution 2D-DOD and 2D-DOA estimates with non-ambiguous characteristics; in particular, parameter estimates with low computational complexity, high accuracy, and automatic pairing can be obtained.

[0018] The technical solution adopted by the present invention is as follows:

[0019] A MIMO radar target three-dimensional positioning method includes the following steps:

[0020] Step 1: Establish a parallel factor tensor model and perform parallel factor decomposition on the matched filtering output signal of the MIMO radar;

[0021] Step 2: Utilize the uniform property of the sparse uniform rectangular array URA to obtain the two-dimensional wave departure angle parameter of the target relative to the transmitting array and the two-dimensional wave arrival angle parameter of the target relative to the receiving array. The two-dimensional wave departure angle parameter and two-dimensional wave arrival angle parameter obtained in this step have high resolution and ambiguity characteristics;

[0022] Step 3: Based on the vector cross product technology of the electromagnetic vector sensor EMVS array, obtain two-dimensional wave departure angle parameters and two-dimensional wave arrival angle parameters with low resolution but no ambiguity;

[0023] Step 4: Combine the results of Step 2 and Step 3 to obtain two-dimensional wave departure angle parameters and two-dimensional wave arrival angle parameters with high resolution and non-ambiguous characteristics;

[0024] Step 5: Based on the results of Step 4 and combined with the three-dimensional coordinates of the transmitting and receiving arrays, obtain the three-dimensional coordinates of the target.

[0025] In step 1, a third-order parallel factor tensor model is constructed for the matched filtering output signal of the MIMO radar, as follows:

[0026] Let M1 and M2 represent the number of rows and columns of the MIMO transmit array EMVS, respectively, where M1 and M2 are integers;

[0027] Let N1 and N2 represent the number of rows and columns of the MIMO receive array EMVS, respectively, where N1 and N2 are integers;

[0028] Let λ represent the operating wavelength of the MIMO radar;

[0029] Let d t = I t ·λ / 2 and d r = I r ·λ / 2 represent the element spacing of the transmit EMVS and the element spacing of the receive EMVS, respectively, where I t and I r are both integers much larger than 2; the positions of the transmit and receive elements are assumed as shown in Figure 1 ;

[0030] Let K represent the number of far-field targets of the MIMO radar, where K is an integer;

[0031] Let θ t,k , φ t,k , γ k , η k be the elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the transmit array, respectively; k = 1, 2,..., K; (θ t,k , φ t,k ) constitutes the two-dimensional wave departure angle of the target;

[0032] Let θ r,k , φ r,k , γ r,k , η r,k be the elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the receive array, respectively; (θ r,k , φ r,k ) constitutes the two-dimensional wave arrival angle of the target;

[0033] Assume that each EMVS channel of the transmit array transmits mutually orthogonal waveforms, and the output of the receive array signal after matched filtering can be expressed as:

[0034]

[0035] In formula (1), ⊙ represents the KhatriRao product; M = M1 × M2; represents definition; Denote the complex number field, and the superscript represents its dimension;

[0036] b tx,k and b ty,k are respectively the spatial response vectors of the k-th target of the transmitting array on the x-axis and y-axis; Denote the Kronecker product;

[0037] is the corresponding transmitting polarization response vector;

[0038] N = N1 × N2;

[0039] b rx,k and b ry,k are respectively the spatial response vectors of the k-th target of the receiving array on the x-axis and y-axis;

[0040] is the corresponding receiving polarization response vector;

[0041] f l,k denotes the echo coefficient of the k-th target in the l-th pulse;

[0042] is the noise sample, and L represents the number of samples;

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049] [·] T Denote the transpose.

[0050] Definition: The specific forms of the above response vectors are as follows:

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] Definition: The matrix data in Equation (1) can also be stacked into a third-order parallel factor tensor Its specific tensor form is as follows:

[0058]

[0059] In Equation (3), is the corresponding noise tensor, and the subscript ×n represents the modulo-n product of the tensor. In fact, the relationship between Z3 and is denotes the modulo-n expansion of

[0060] In step 1, the third-order parallel factor tensor model established is subjected to parallel factor decomposition, specifically as follows:

[0061] According to the definition of tensor expansion, can also be expanded into the form of other matrices:

[0062]

[0063]

[0064] According to the connotation of parallel factor decomposition, the decomposition of the tensor in Equation (3) can be solved by the trilinear alternating least squares method, and its optimal solutions are respectively

[0065]

[0066]

[0067]

[0068] The least squares in Equations (6a)-(6c) are iterated alternately, which is also called the trilinear decomposition algorithm. The COMFAC algorithm is used to accelerate the iterative convergence. The estimated values of the factor matrices and can be expressed as:

[0069]

[0070]

[0071]

[0072] Where: denote a permutation matrix; Δ1, Δ2, Δ3 are K×K real diagonal matrices, whose diagonal elements are scaling factors, and Δ1Δ2Δ3 = I K ; N1, N2, and N3 denote fitting errors.

[0073] In step 2, by utilizing the uniform property of the URA, the high-resolution 2D-DOD and 2D-DOA estimates with ambiguity characteristics of the target are obtained from the estimated value of the factor matrix obtained in step 1 . The specific calculation process includes the following steps:

[0074] S2.1: Construct a spatial domain selectivity matrix and estimate the permutation matrix:

[0075] Define:

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084] In the formula: I M denotes the identity matrix of dimension M×M; 0 M denotes the zero vector of dimension M×1, and all values are 0;

[0085] Then let

[0086] Calculate and perform eigenvalue decomposition on it, and K eigenvalues λ tu,1 , λ tu,2 , …, λ tu,K and the corresponding eigenvectors v t,1 , v t,2 , …, v t,K ;

[0087] Let the diagonal matrix composed of eigenvalues matrix diag{·} represents the diagonalization calculation.

[0088] S2.2: Estimate the array spatial rotation invariance matrix and obtain u with ambiguity characteristics t,k , v t,k , u r,k and v r,k estimation values:

[0089] Next, calculate the left side of the following formula:

[0090]

[0091]

[0092]

[0093] Three diagonals can be obtained in sequence and

[0094] Then, obtain the u with ambiguity characteristics by calculating the following formula t,k , v t,k , u r,k and v r,k estimation values:

[0095]

[0096]

[0097]

[0098]

[0099] In the above formula, angle(·) represents taking the phase angle.

[0100] In step 3, using the cross-product property of the EMVS elements, obtain the low-resolution 2D-DOD and 2D-DOA parameters of the target without ambiguity characteristics from the estimated value of the factor matrix obtained in step 1 . The specific calculation process includes the following steps:

[0101] Construct the polarization selectivity matrix and estimate the normalized polarization factor:

[0102] Define: i 6,q represents the q-th row of I6,

[0103] Calculate the left side of the following formula:

[0104]

[0105]

[0106] where \(q = 2, 3, \cdots, 6\); let the obtained and the \(k\)th diagonal elements be denoted as and

[0107] Then, construct:

[0108]

[0109]

[0110]

[0111]

[0112] Next, calculate the right - hand side of the following formula:

[0113]

[0114]

[0115] In the above formula, ▲ represents the vector cross - product, and the superscript * represents the conjugate; the and obtained through the above formula are the unambiguous estimates of \(u\) t,k , \(v\) t,k , \(u\) r,k and \(v\) r,k . Since the spatial rotation invariance of the array is not utilized, and have relatively low resolution.

[0116] In step 4, using the results of step 2 and step 3, high - resolution 2D - DOD and 2D - DOA parameters with target unambiguous characteristics are obtained. The specific calculation process is as follows:

[0117] For the estimated \(u\) t,k and \(v\) t,k , they each have \(I\) t possible solutions, and for the estimated \(u\) r,k and \(v\) r,k , they each have \(I\) r possible solutions. List all possible solutions of the above parameters:

[0118]

[0119]

[0120]

[0121] High - resolution with unambiguous characteristics and are respectively as follows:

[0122]

[0123]

[0124]

[0125]

[0126] Finally, the parameters of the high-resolution 2D-DOD and 2D-DOA without ambiguity characteristics are:

[0127]

[0128]

[0129] In the said step 5, by using the high-resolution 2D-DOD and 2D-DOA parameters of the target without ambiguity characteristics obtained in step 5, the target three-dimensional position is obtained, and its specific calculation process is:

[0130] The target three-dimensional positioning principle is as Figure 2 shown. Taking the target three-dimensional position as the coordinate origin to establish a three-dimensional coordinate system, assuming the coordinate position of the target is (X0, Y0, Z0), and the positions of the transceiver arrays are known, with coordinates being (P x1 , P y1 , P z1 ) and (P x2 , P y2 , P z2 ) respectively. Through the array antennas, the elevation angles between the target and the transceiver are measured as θ1 and θ2 respectively, and the azimuth angles are measured as φ1 and φ2 respectively. The position coordinates of the target are calculated as follows:

[0131]

[0132]

[0133]

[0134] A MIMO radar target three-dimensional positioning device, which device includes:

[0135] A set of transmitting and receiving antenna elements composed of a sparse uniform rectangular array URA, and each antenna element includes an electromagnetic vector sensor EMVS;

[0136] By measuring the two-dimensional direction of departure 2D-DOD of the target relative to the transmitting array and the two-dimensional direction of arrival 2D-DOA of the target relative to the receiving array, and then combining the specific coordinates of the transmitting and receiving arrays, the target three-dimensional coordinates can be obtained.

[0137] The transmitting array consists of M1×M2 EMVSs to form a URA array geometric structure, and the spacing between EMVSs is d t = I t ·λ / 2, where: λ is 1 / 2 of the radar operating wavelength, and I t is an integer much larger than 2;

[0138] The receiving array consists of N1×N2 EMVSs to form a URA array geometric structure, and the spacing between EMVSs is d r = I r ·λ / 2, where: I r is an integer much larger than 2.

[0139] A three-dimensional target positioning device and a positioning method for a MIMO radar according to the present invention have the following technical effects:

[0140] 1) The present invention proposes a bistatic MIMO radar target positioning device based on a sparse uniform rectangular array, and each transceiver element thereof is composed of a complete electromagnetic vector sensor. Compared with the existing uniform linear array transceiver array scheme, the proposed sparse transceiver array scheme has a larger physical aperture, so the positioning device can obtain a two-dimensional transceiver angle estimate with higher resolution for the target. Combining the spatial coordinates of the transceiver arrays, the three-dimensional spatial coordinates of the target can be easily obtained.

[0141] 2) Based on the sparse bistatic MIMO radar, the present invention proposes a high-resolution positioning algorithm based on tensor decomposition. Among them, the introduction of tensor decomposition is beneficial for the algorithm to utilize the multi-dimensional structure information of the array signal. Compared with the existing matrix decomposition algorithms, the proposed tensor decomposition algorithm can improve the accuracy of angle estimation.

[0142] 3) The present invention proposes a two-step angle estimation algorithm based on the idea of rough estimation and fine estimation. Among them, in the rough estimation process, a single electromagnetic vector sensor has the ability to provide a low-resolution and unambiguous two-dimensional angle estimate. Through the rotational invariance technique and the normalized vector cross product technique, a rough but unambiguous two-dimensional transceiver angle estimate is obtained. The fine estimation process mainly utilizes the uniform characteristic of the array in the spatial domain. Through the rotational invariance technique or a high-resolution but ambiguous two-dimensional transceiver angle estimate, combined with the unambiguous two-dimensional transceiver angle estimate obtained in the previous step, a high-resolution and unambiguous two-dimensional transceiver angle estimate of the target is finally obtained. Compared with the existing algorithms, the algorithm proposed by the present invention has higher estimation accuracy.

[0143] 4) The present invention is based on a target three-dimensional coordinate positioning method. This method calculates the three-dimensional spatial coordinates of the target according to the estimated two-dimensional transceiver angles of the target and the spatial coordinates of the transceiver arrays. This method only requires two base stations (receiving and transmitting arrays) to obtain the three-dimensional coordinates of the target, greatly saving the cost of building stations. Description of the Drawings

[0144] Figure 1 It is a schematic diagram of the MIMO radar device of the present invention.

[0145] Figure 2 It is the 2D-DOD and 2D-DOA scatter plots of the estimator proposed by the present invention.

[0146] Figure 3 It is a schematic diagram of the comparison of the average RMSE of the 2D-DOA estimation parameters varying with SNR.

[0147] Figure 4 It is a schematic diagram of the comparison of the average RMSE of the 2D-DOA estimation parameters varying with M.

[0148] Figure 5 It is a schematic diagram of the comparison of the average running time of different schemes varying with M. Detailed implementation manners

[0149] A three-dimensional positioning device and method for MIMO radar targets, which uses a uniform rectangular array (URA) composed of sparse co-located EMVS as the transmitting and receiving antenna array of the MIMO system, and uses PARAFAC decomposition to improve the accuracy of data decomposition. It utilizes the uniform characteristics of the URA to obtain the 2D-DOD and 2D-DOA with high resolution and ambiguity characteristics of the target, uses the vector cross product technology of the EMVS array to obtain the 2D-DOD and 2D-DOA estimations with low resolution but no ambiguity, combines the above results to obtain the 2D-DOD and 2D-DOA estimations with high resolution and no ambiguity characteristics, and finally combines the positions of the transmitting and receiving arrays to obtain the three-dimensional space coordinates of the target. This framework has low complexity and high accuracy.

[0150] (1): Use M1 and M2 to represent the number of rows and columns of the EMVS of the MIMO transmitting array, where M1 and M2 are integers; use N1 and N2 to represent the number of rows and columns of the EMVS of the MIMO receiving array, where N1 and N2 are integers; use λ to represent the operating wavelength of the MIMO radar, and use d t = I t ·λ / 2 and d r = I r ·λ / 2 to represent the element spacing of the transmitting EMVS array and the element spacing of the receiving EMVS array respectively, where I t and I r are both integers much larger than 2; the positions of the transmitting and receiving elements are as Figure 1 shown.

[0151] Use K to represent the number of far-field targets of the MIMO radar, where K is an integer; let θ t,k , φ t,k , γ k , ηk The elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the transmitting array are respectively, where k = 1, 2, …, K; let θ r,k , φ r,k , γ r,k , η r,k The elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the receiving array are respectively, where k = 1, 2, …, K; (θ t,k , φ t,k ) is also called the 2D-DOD of the target, and (θ r,k , φ r,k ) is also called the target 2D-DOA;

[0152] Suppose each EMVS channel of the transmitting array transmits mutually orthogonal waveforms, and the output of the received array signal after matched filtering can be expressed as:

[0153]

[0154] where, ⊙ represents the Khatri-Rao product, b tx,k and b ty,k are respectively the spatial response vectors of the k-th target of the transmitting array on the x-axis and y-axis, represents the Kronecker product; is the corresponding transmitting polarization response vector; b rx,k and b ry,k are respectively the spatial response vectors of the k-th target of the receiving array on the x-axis and y-axis; is the corresponding receiving polarization response vector; f l,k represents the echo coefficient of the k-th target in the l-th pulse, is the noise sample; [·] T represents the transpose.

[0155] Define The specific forms of the above response vectors are as follows

[0156]

[0157]

[0158]

[0159]

[0160]

[0161]

[0162] Definition The matrix data in Equation (1) can also be stacked into a third-order parallel factor tensor Its specific tensor form is as follows

[0163]

[0164] Among them, is the corresponding noise tensor, and the subscript ×n represents the modulo n product of the tensor. In fact, the relationship between Z3 and is denotes The modulo n expansion of

[0165] According to the definition of tensor expansion, the noise-free can also be expanded into the form of other matrices:

[0166]

[0167]

[0168] Combining Equation (1), Equation (4a) and Equation (4b), the estimation of the factor matrices D t , D r , F can be obtained by simultaneously solving the following optimization problem:

[0169]

[0170]

[0171]

[0172] In the formula, ||·|| F represents the Frobenius norm of the matrix. The above problem is transformed into a least squares problem and can be solved by trilinear alternating least squares method. Its optimal solutions are respectively

[0173]

[0174]

[0175]

[0176] In the above formula, conforming to represents the pseudoinverse. The least squares alternating iteration in Equation (6a) - Equation (6c) is also called the trilinear decomposition algorithm. Here, we use the COMFAC algorithm to accelerate the iterative convergence.

[0177] As is well known, matrix factorization is not unique unless the matrix satisfies some strict conditions, such as being orthogonal. However, the decomposition of tensor rank is often unique. The following theorem in reference

[10] : T.G. Kolda and B.W. Bader, “Tensor decompositions and applications,” SIAM Rev., vol. 51, no. 3, pp. 455–500, Aug. 2009. provides a sufficient condition for the uniqueness of PARAFAC decomposition. According to this theorem, if

[0178]

[0179] then the estimated values of the factor matrices D t , D r , F are unique up to column permutation and scaling. Permutation ambiguity means that the columns of the coefficient matrix are reordered, and scaling refers to the columns being scaled by multiplying by a constant. Specifically, the estimated values of the factor matrices and can be expressed as:

[0180]

[0181]

[0182]

[0183] where denotes a permutation matrix, and Δ1, Δ2, Δ3 are K×K real diagonal matrices with diagonal elements being scaling factors, and Δ1Δ2Δ3 = I K . N1, N2, and N3 represent the fitting errors.

[0184] (II): High-resolution 2D-DOD and 2D-DOA parameter estimation with ambiguity characteristics:

[0185] Define

[0186] where I M denotes the identity matrix of dimension M×M, and 0 M denotes the zero vector of dimension M×1 with all values being 0. It is easy to obtain:

[0187] J M,2 B t = J M,1 B t ψ t,u (9a);

[0188] J M,4 Bt = J M,3 B t ψ t,v (9b);

[0189] J N,2 B r = J N,1 B r ψ r,u (9c);

[0190] J N,4 B r = J N,3 B r ψ r,v (9d);

[0191] wherein,

[0192] Let again It can be obtained that:

[0193] J t,2 D t = J t,1 D t ψ t,u (10a);

[0194] J t,4 D t = J t,3 D t ψ t,v (10b);

[0195] J r,2 D r = J r,1 D r ψ r,u (10c);

[0196] J r,4 D r = J r,3 D r ψ r,v (10d);

[0197] Substitute the estimated values in Equation (8a) and Equation (8b) for the true values in Equation (10a) - Equation (10d), and ignore the noise terms, it can be obtained that:

[0198]

[0199]

[0200]

[0201]

[0202] For any permutation matrix ∏ and any scaling matrix Δ, it can be easily proven that: ∏Δ = Δ∏ -1 , (∏Δ) -1 = ∏Δ -1 , for any diagonal matrix there is Δ -1 ψΔ = ψ, then there is (∏Δ) -1 ψ∏Δ = ∏ -1 ψ∏. Substituting it into equations (11a) - (11d) gives:

[0203]

[0204]

[0205]

[0206]

[0207] According to the relationship in equation (11a), for using eigenvalue decomposition, K eigenvalues λ tu,1 , λ tu,2 , …, λ tu,K and the corresponding eigenvectors v t,1 , v t,2 , …, v t,K can be obtained. Obviously, the diagonal matrix constituted by the eigenvalues is the estimated value of ψ t,u , and the matrix is the estimated value of ∏. Then, substituting for ∏ and calculating the left - hand sides of equations (11b) - (11d) in turn, three diagonals

[0208] and can be obtained respectively. They are the estimated values of ψ t,v , ψ t,v and ψ t,v . Then u t,k , v t,k , u r,k and v r,k can be estimated by the following formula

[0209]

[0210]

[0211]

[0212]

[0213] In the above formula, angle(·) represents taking the phase angle. Since -I t π ≤ I t πu t,k ≤ I t π, -I t π ≤ I t πv t,k ≤ I t π, -I r π ≤ I r πu r,k ≤ I r π and -I r π ≤ I r πv r,k ≤ I r π, and -π ≤ angle(I t πu t,k ) ≤ π, -I t ≤ angle(I t πv t,k ) ≤ I t , -π ≤ angle(I r πu r,k ) ≤ π and -π ≤ angle(I r πv r,k ) ≤ π;

[0214] Therefore, the estimated values of u t,k , v t,k , u r,k and v r,k obtained through equations (12a) - (12d) are ambiguous.

[0215] (III): Low - resolution 2D - DOD and 2D - DOA parameter estimation with non - ambiguity characteristics

[0216] For a complete EMVS, its polarization response vector satisfies:

[0217]

[0218]

[0219] In the above formula, ▲ represents the vector cross - product, and the superscript * represents the conjugate. Since

[0220]

[0221]

[0222] where represents the q - th element in a t,k (q = 1, 2,..., 6). Since:

[0223]

[0224]

[0225] Considering that and are constants, let Then:

[0226]

[0227]

[0228] Therefore, the key to obtaining 2D-DOD and 2D-DOA estimates is to estimate and Since:

[0229]

[0230]

[0231] Where:

[0232]

[0233]

[0234]

[0235]

[0236] Define i 6,q denoted as the q-th row of I6, we can obtain:

[0237]

[0238]

[0239] That is, there exists So, Eqs. (17a), (17b) are rewritten as:

[0240]

[0241]

[0242] Substituting the estimated values in Eqs. (8a) and (8b) for the true values in Eqs. (19a)-(19b) and ignoring the noise terms, we can get:

[0243]

[0244]

[0245] Then, use to replace ∏, and calculate the left - hand sides in equations (20a) - (20b) in sequence. When q takes 2, 3, …, 6 in sequence, and can be obtained. The k - th diagonal element of each corresponds to and respectively. The estimated values are denoted as and Then, can be constructed.

[0246]

[0247]

[0248] The and obtained through the above formula are the unambiguous u t,k , v t,k , u r,k and v r,k estimated values. Since the spatial rotation invariance of the array is not utilized, and have low resolution.

[0249] (IV): High - resolution 2D - DOD and 2D - DOA parameter estimation with ambiguity - free property:

[0250] For the estimated u t,k and v t,k in equations (12a) - (12b), they should have I t possible solutions respectively. The i t th possible solution and are related to and respectively as

[0251]

[0252]

[0253] Theoretically, and the true values in should coincide with the estimated values and in equation (21a) respectively. Therefore, the high - resolution and with ambiguity - free property should be

[0254]

[0255]

[0256] Finally, the estimation of the high-resolution 2D-DOD without ambiguity is:

[0257]

[0258] Similarly, for the estimated u r,k and v r,k in equations (12c) - (12d), there should be I r possible solutions respectively. The i r -th possible solution and are related to and respectively as follows:

[0259]

[0260]

[0261] Similarly, the high-resolution and without ambiguity should be respectively:

[0262]

[0263]

[0264] Finally, the estimation of the high-resolution 2D-DOA without ambiguity is:

[0265]

[0266] (V): Calculation of the target three-dimensional coordinate position:

[0267] The principle of target three-dimensional positioning is as Figure 2 shown. A three-dimensional coordinate system is established with the target three-dimensional position as the coordinate origin. Assume the coordinate position of the target is (X0, Y0, Z0), and the positions of the transceiver arrays are known, with coordinates (P x1 , P y1 , P z1 ) and (P x2 , P y2 , P z2 ) respectively. Through the array antennas, the elevation angles between the target and the transceivers are measured as θ1 and θ2 respectively, and the azimuth angles are φ1 and φ2 respectively. The relationships of the above parameters are specifically:

[0268]

[0269]

[0270]

[0271]

[0272] By simultaneously solving the above four equations, the solutions are:

[0273]

[0274]

[0275]

[0276] Example:

[0277] To verify the effectiveness of the proposed framework of the present invention, the Monte Carlo method is adopted to evaluate the estimation performance. Here, it is assumed that the MIMO radar consists of a transmitting array of URA with M1×M2 and a receiving array of URA with N1×N2. Each transceiver element is a complete co-located EMVS. The spacing between the transmitting elements and the receiving elements is d t = 5λ and d r = 7.5λ, where λ is the reciprocal of the operating wavelength and frequency of the system.

[0278] Suppose there are K = 3 far target signals, and their parameters are θ t = (10°, 36°, 78°), φ t = (15°, 50°, -53°), γ t = (10°, 52°, 75°), η t = (18°, -31°, 67°), θ r = (5°, 45°, 78°), φ t = (15°, -23°, 30°), γ t = (32°, 10°, 65°), η t = (16°, 56°, 38°). In addition, it is assumed that L = 200 samples have been collected. The results of each figure in the simulation depend on 500 independent trials. In the simulation, the signal-to-noise ratio (SNR) is defined as SNR = 10lg(||Z3 - N|| 2 / ||N|| 2 ), where both Z3 and N are data matrices in expression (1). The performance evaluation is carried out using the root mean square error (RMSE).

[0279] First, the scatter plot results of the 2D-DOD and 2D-DOA estimations of the proposed estimator are shown by Figure 3Given that: M1 = 2, M2 = 2, N1 = 2, N2 = 3 and SNR = 0 dB. It can be clearly seen that all parameters are correctly estimated and automatically paired. The results show that the proposed framework can provide closed-form solutions for target 2D-DOD and 2D-DOA estimations.

[0280] Secondly, Figure 4 The average RMSE performance of target 2D-DOD and 2D-DOA estimations at different SNRs is given, where M1 = 2, M2 = 3, N1 = 4, N2 = 2. To highlight the reliability of the proposed solution of the present invention, the proposed solution of the present aspect is compared with the PARAFAC(ULA) algorithm in reference [9], the ESPRIT-Like algorithm in reference [8], the PARAFAC-Like algorithm in reference

[10] , and the Cramer-Rao bounds of the uniform linear array and URA. The Cramer-Rao bounds of URA are labeled as CRB(ULA) and CRB(URA), respectively.

[0281] It is worth noting that when the SNR increases, all algorithms will provide better RMSE performance. However, the RMSE of the method proposed in the present invention is higher than that of all the compared algorithms, especially when the SNR is relatively high, such as SNR > 0 dB, which indicates that the proposed solution of the present invention can provide more accurate estimation performance.

[0282] In addition, the performance comparisons of different algorithms under different numbers of samples L are compared, where the SNR is set to 0 dB, and the results are as Figure 5 shown. It can be seen that the method proposed in the present invention far outperforms the existing solutions under the condition of a large number of samples. Under the condition of a small number of samples, the performance of the algorithm of the present invention will decline, but the estimation performance is still better than that of the existing solutions.

Claims

1. A three-dimensional positioning method for MIMO radar targets, characterized in that It includes the following steps: Step 1: Establish a parallel factor tensor model and perform parallel factor decomposition on the matched filtering output signal of the MIMO radar; Step 2: Utilize the uniform characteristics of the sparse uniform rectangular array URA to obtain the two-dimensional wave departure angle parameters of the target relative to the transmitting array and the two-dimensional angle of arrival parameters of the target relative to the receiving array. The two-dimensional wave departure angle parameters and two-dimensional angle of arrival parameters obtained in this step have high resolution and ambiguity characteristics; Step 3: Based on the vector cross product technology of the electromagnetic vector sensor EMVS array, obtain two-dimensional wave departure angle parameters and two-dimensional angle of arrival parameters with low resolution but no ambiguity; Step 4: Combine the results of Step 2 and Step 3 to obtain two-dimensional wave departure angle parameters and two-dimensional angle of arrival parameters with high resolution and no ambiguity characteristics; Step 5: Based on the results of Step 4, combine the three-dimensional coordinates of the receiving and transmitting arrays to obtain the three-dimensional coordinates of the target; In the said Step 5, use the high-resolution 2D-DOD and 2D-DOA parameters without ambiguity characteristics obtained in Step 4 to obtain the three-dimensional position of the target. The specific calculation process is as follows: Establish a three-dimensional coordinate system with the target three-dimensional position as the coordinate origin. Let the coordinate position of the target be (X0, Y0, Z0). The positions of the transceiver arrays are known, with coordinates (P x1 , P y1 , P z1 ) and (P x2 , P y2 , P z2 ). Through the array antennas, the elevation angles between the target and the transceiver are measured as θ1 and θ2 respectively, and the azimuth angles are φ1 and φ2 respectively. The position coordinates of the target are calculated as follows:

2. The three-dimensional target positioning method of a MIMO radar according to claim 1, wherein: In the said Step 1, construct a third-order parallel factor tensor model for the matched filtering output signal of the MIMO radar, specifically as follows: Let M1 and M2 represent the number of rows and columns of the MIMO transmitting array EMVS, and M1, M2 are integers; Let N1 and N2 represent the number of rows and columns of the MIMO receiving array EMVS, and N1, N2 are integers; Let λ represent the operating wavelength of the MIMO radar; Let d t = I t ·λ / 2, d r = I r ·λ / 2 represent the transmitting EMVS element spacing and the receiving EMVS element spacing respectively, and I t , I r are both integers far greater than 2; Let K represent the number of far-field targets of the MIMO radar, and K is an integer; Let θ t,k , φ t,k , γ k , η k be the elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the transmitting array, respectively; k = 1, 2, …, K; (θ t,k , φ t,k ) form the two-dimensional wave departure angle of the target. Let θ r,k , φ r,k , γ r,k , η r,k be the elevation angle, azimuth angle, auxiliary polarization angle, and polarization phase difference of the k-th target relative to the receiving array, respectively; (θ r,k , φ r,k ) form the two-dimensional angle of arrival of the target; Let each EMVS channel of the transmitting array transmit mutually orthogonal waveforms, and the output of the receiving array signal after matched filtering can be expressed as: In formula (1), ⊙ represents the Khatri-Rao product; M = M1 × M2; denotes definition; denotes the complex number field, and the superscript represents its dimension; b tx,k and b ty,k are the spatial response vectors of the k-th target of the transmitting arrays on the x-axis and y-axis respectively; denotes the Kronecker product; is the corresponding transmitting polarization response vector; N = N1 × N2; b rx,k and b ry,k are respectively the spatial domain response vectors of the k-th target of the receiving array on the x-axis and y-axis; is the corresponding received polarization response vector; f l,k represents the echo coefficient of the k-th target within the l-th pulse; is a noise sample, and L represents the number of samples; [·] T Denotes transpose; Definition: The specific forms of the above response vectors are as follows: Definition: The matrix data in Equation (1) can also be stacked into a third-order parallel factor tensor Its specific tensor form is as follows: In formula (3), is the corresponding noise tensor, and the subscript ×n represents the modulo-n product of the tensor; actually, the relationship between Z3 and is denotes the modulo-n expansion of.

3. The method for three-dimensional positioning of MIMO radar targets according to claim 2, characterized in that: In the said Step 1, perform parallel factor decomposition on the established third-order parallel factor tensor model, specifically as follows: According to the definition of tensor expansion, it can also be expanded into the form of other matrices: According to the connotation of parallel factor decomposition, the tensor decomposition in Equation (3) can be solved by the trilinear alternating least squares method, and its optimal solutions are respectively The least-squares alternating iteration in Equations (6a)-(6c), also known as the trilinear decomposition algorithm, is carried out; the COMFAC algorithm is used to accelerate the iterative convergence; the estimated values of the factor matrices obtained and can be expressed as: Wherein: represents a permutation matrix; Δ1, Δ2, and Δ3 are K×K real diagonal matrices, the diagonal elements of which are scaling factors, and Δ1Δ2Δ3 = I K ; N1, N2, and N3 represent fitting errors.

4. The MIMO radar target three-dimensional positioning method according to claim 3, characterized in that: In the said step 2, by using the uniform characteristics of the URA, the high-resolution 2D-DOD and 2D-DOA parameters with fuzzy characteristics of the target are obtained from the estimated value of the factor matrix obtained in step 1 The specific calculation process includes the following steps: S2.1: Construct a spatial domain selectivity matrix and estimate the permutation matrix: Define: Where: I M represents an identity matrix of dimension M×M; 0 M represents a zero vector of dimension M×1, all values of which are 0; Let again Calculate and perform eigenvalue decomposition on it to obtain K eigenvalues λ tu,1 , λ tu,2 , …, λ tu,K and the corresponding eigenvectors v t,1 , v t,2 , …, v t,K ; Let the diagonal matrix composed of eigenvalues matrix diag{·} represents the diagonalization calculation; S2.2: Estimate the array spatial rotation invariance matrix and obtain the u with ambiguity characteristics t,k , v t,k , u r,k and v r,k estimated values of: Then, calculate the left side of the following formula: Can obtain three diagonals in sequence and Then, the estimated values of u t,k , v t,k , u r,k and v r,k with fuzzy characteristics are obtained by calculating the following formula: In the above formula, angle(·) represents taking the phase angle.

5. A three-dimensional positioning method for MIMO radar targets according to claim 3, characterized in that: In step 3, by using the cross-product property of the EMVS array elements, the low-resolution 2D-DOD and 2D-DOA parameters with target ambiguity-free characteristics are obtained from the estimated value of the factor matrix obtained in step 1 The specific calculation process is as follows: Construct a polarization selectivity matrix and estimate the normalized polarization factor: Definition: i 6,q represents the q-th row of I6, Calculate the left side of the following formula: where q = 2, 3, …, 6; let the obtained and the k-th diagonal element of be denoted as and Then, construct: Then calculate the right side part of the following formula: In the above formula, ▲ represents the vector cross product, and the superscript * represents the conjugate; the and are the unambiguous u t,k , v t,k , u r,k and v r,k estimation values; since the spatial rotation invariance of the array is not utilized, and have relatively low resolution.

6. The MIMO radar target three-dimensional positioning method according to claim 5, characterized in that: In the said Step 4, use the results in Step 2 and Step 3 to obtain high-resolution 2D-DOD and 2D-DOA parameters without ambiguity characteristics of the target. The specific calculation process is as follows: For the estimated u t,k and v t,k which each have I t possible solutions, and the estimated u r,k and v r,k which each have I r possible solutions, list all possible solutions of the above parameters respectively: High-resolution without blurring characteristics and are respectively: Finally, the parameters of high-resolution 2D-DOD and 2D-DOA without ambiguity characteristics are:

7. A three-dimensional positioning device for MIMO radar targets, which is used to execute the three-dimensional positioning method for MIMO radar targets according to any one of claims 1 to 6, and is characterized in that: The device includes: receiving and transmitting antenna elements composed of a sparse uniform rectangular array URA, and each antenna element includes an electromagnetic vector sensor EMVS; By measuring the two-dimensional wave departure angle 2D-DOD of the target relative to the transmitting array and the two-dimensional angle of arrival 2D-DOA of the target relative to the receiving array, and then combining the specific coordinates of the receiving and transmitting arrays, the three-dimensional coordinates of the target can be obtained.

8. The MIMO radar target three-dimensional positioning device according to claim 7, wherein: The transmitting array consists of an URA array geometry composed of M1×M2 EMVSs, and the spacing between EMVSs is d t = I t ·λ / 2, where: λ is 1 / 2 of the radar operating wavelength, and I t is an integer much greater than 2; The receiving array consists of an URA array geometry composed of N1×N2 EMVSs, and the spacing between EMVSs is d r = I r ·λ / 2, where: I r is an integer much larger than 2

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