Closed-form Estimation Method for Multidimensional Parameters of MIMO Radar Based on the Principle of Orthogonality of Polarization Subspaces

Abstract

The present invention discloses a closed-form estimation method for multi-dimensional parameters of a MIMO radar based on the principle of orthogonality of polariton spaces, mainly solving the problem of multi-dimensional parameter estimation under the model error of the transceiver array in the existing method based on a polarimetric MIMO radar. The implementation steps are as follows: based on array measurements, obtain the horizontal and vertical polarization sampling matrices of the transceiver array; based on the measured sampling matrices of the transceiver array, establish a signal model for the polarimetric MIMO radar; perform matched filtering processing on the received signal; separate and decouple the multi-dimensional parameters of the signal after the matched filtering processing; reduce the multi-dimensional parameter estimation problem to a direction-of-arrival estimation problem, and solve the direction-of-arrival based on the covariance matrix of the received array output after the matched filtering processing; solve the estimation of the polarization parameters. Based on the principle of orthogonality of polariton spaces, the present invention provides an effective solution for closed-form estimation of multi-dimensional parameters of a polarimetric MIMO radar with arbitrary multi-polarization configurations considering the model error of the transceiver array, and can be used for radar target detection and positioning.

Description

Technical Field

[0001] The invention belongs to the technical field of signal processing, and in particular relates to a multidimensional parameter estimation method for multi-polarization MIMO radar, specifically a closed-form estimation method for multidimensional parameters of MIMO radar based on the orthogonality principle of polarization subspace, which can be used for radar target detection and positioning. Background Art

[0002] MIMO radar can achieve the effect of increasing the array aperture by generating a virtual array and using fewer array elements, and has attracted widespread attention. At present, most research is devoted to the application of scalar MIMO radar in the estimation of the wave departure direction and wave arrival direction, which has higher angle measurement accuracy than traditional phased array radar in direction finding. However, none of the above existing methods considers the polarization diversity of the signal, and the polarization mismatch problem that is prevalent in practical applications will lead to a decrease in the accuracy of parameter estimation.

[0003] Taking into account the polarization diversity of signals, the present invention proposes a closed-form estimation method for multi-dimensional parameters of polarization MIMO radars whose transmitting array and receiving array are both arbitrary multi-polarization arrays. Specifically, the present invention adopts the transmitting array and receiving array of a single-base multi-polarization MIMO radar to transmit and receive signals respectively, and adopts a polynomial root-finding method based on the orthogonal principle of polarization subspace to perform closed-form joint estimation of the direction of arrival and polarization parameters. Existing single-base MIMO radars used for angle estimation mostly use scalar antennas, which do not consider the signal polarization diversity that is closer to actual application scenarios, resulting in polarization mismatch, thereby reducing the accuracy of angle estimation. Therefore, the present invention adopts a transmitting and receiving multi-polarization antenna array to improve this problem and effectively avoid the loss of angle estimation accuracy caused by polarization mismatch.

[0004] However, most of the existing multi-dimensional parameter estimation methods for multi-polarization MIMO radars do not take into account model errors, such as gain, phase and position errors that may exist in the transceiver antenna array. The above model errors limit their application in practice, and there is an urgent need to explore the multi-dimensional parameter estimation of multi-polarization MIMO radars considering model errors. The present invention refers to the polarization manifold separation technology, adopts a joint estimation method of the direction of arrival and polarization parameters based on polynomial root finding, and proposes a multi-dimensional parameter closed-form estimation method based on the orthogonality principle of polarization subspace considering the array model errors of the transceiver antennas. The method proposed in the present invention utilizes the construction of a virtual domain to reconstruct the Fourier basis corresponding to the reconstructed sampling matrix of the transceiver array, and realizes the joint closed-form estimation of the direction of arrival and polarization parameters by the method of polynomial root finding. It has the advantages of automatic pairing of multi-dimensional parameters and avoiding spectral peak search, and does not cause estimation ambiguity. Summary of the invention

[0005] The object of the present invention is to propose a closed-form multi-dimensional parameter estimation method based on the principle of polarization subspace orthogonality considering the transceiver array model error in view of the fact that most of the existing multi-polarized MIMO radar multi-dimensional parameter estimation methods do not consider the accuracy loss caused by model errors. This method is applicable to MIMO radar arrays with any multi-polarized configuration, has the advantages of automatic pairing of multi-dimensional parameters and avoiding spectral peak search, and does not cause estimation ambiguity. It provides a feasible idea and an effective solution for the closed-form estimation of multi-dimensional parameters of any multi-polarized MIMO radar considering model errors.

[0006] The object of the present invention is achieved by the following technical solutions: A closed-form multi-dimensional parameter estimation method for MIMO radar based on the principle of polarization subspace orthogonality, comprising the following steps:

[0007] (1) Based on array measurement, obtain the horizontal polarization and vertical polarization sampling matrices of the transmitting array and the receiving array; Consider a multi-polarized monostatic MIMO radar with a multi-polarized transmitting antenna array and a multi-polarized receiving antenna array having an arbitrary structure, where the number of transmitting antenna elements is M (orthogonal signals are transmitted by antennas with different polarizations), and the number of receiving antenna elements is L; The transmitting antennas and receiving antennas are placed arbitrarily, but no estimation ambiguity is caused; After array response measurement, Γ t,h and Γ t,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarized transmitting antenna array; Γ r,h and Γ r,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarized receiving antenna array, and respectively represent the polarization vectors of the transmitted signal and the received signal corresponding to the p-th target, where γ t,p and η t,p respectively represent the polarization auxiliary angle and polarization phase difference of the transmitted signal of the p-th target, γ r,p and η r,p respectively represent the polarization auxiliary angle and polarization phase difference of the received signal of the p-th target, [·] T represents the transpose operation;

[0008] (2) Based on the measured transmitting and receiving array sampling matrices, perform signal model modeling for the multi-polarized MIMO radar; For the considered multi-polarized MIMO radar, assume that the transmitting antennas transmit orthogonal fully polarized signals, and the targets to be estimated are assumed to be in the same range cell; Assume that the direction of arrival of the p-th target is θ p , according to the reciprocity of the transmitting and receiving antennas, the steering vectors of the transmitting antennas with different polarizations and the receiving antennas with different polarizations are respectively expressed as

[0009]

[0010] and

[0011]

[0012] where \(I_2\) represents the second - order identity matrix, represents the \((2N + 1)\times1\) truncated one - dimensional Fourier basis, \(N\) represents the mode number corresponding to the transceiver array, \(w\) t and \(w\) r respectively represent the modeling errors of the transmit steering vector and the receive steering vector caused by truncation and calibration noise; it should be noted that when the mode number is correctly selected, \(w\) t and \(w\) r will be small enough;

[0013] then the received signal of the receive multi - polarization antenna array is

[0014]

[0015] where the time range is \(T_0\) is the starting time, \(T\) k is the \(k\) - th pulse duration, \(K\) is the number of pulses, \(P\) is the number of targets, \(\mu\) p,k represents the scattering coefficient of the \(p\) - th target during the \(k\) - th pulse; \(u(t)=[u_1(t),u_2(t),\cdots,u\) M (t)] T represents the vector of orthogonal transmission signals; \(e\) k (t) represents the additive Gaussian white noise vector;

[0016] (3) Perform matched - filtering processing on the received signal; after the matched - filtering processing corresponding to the \(m\) - th transmit signal, the matched - filtering output of the receive multi - polarization antenna array is expressed as

[0017]

[0018] where ( (a complex number) is a vector with the \(m\) - th element being 1 and the remaining elements being 0, \(n\) k,m represents the Gaussian white noise corresponding to the \(m\) - th signal under the \(k\) - th pulse; after using a series of matched - filtering corresponding to \(M\) transmit signals for the output signal of each receive element, the received signal vector is re - expressed as

[0019]

[0020] where represents the steering matrix composed of the steering vectors corresponding to \(P\) targets, \(s\) k =[\(\mu\) 1,k ,\(\mu\) 2,k…, μ P,k T represents the transmitted signal vector, represents the noise vector, represents the steering vector corresponding to the p-th target;

[0021] (4) Perform multi-dimensional parameter separation and decoupling on the signal after matched filtering; First, the steering vectors of the transmitting and receiving antenna arrays are respectively represented as follows by using the polarization manifold separation technique through appropriate truncation and omission of the modeling error

[0022]

[0023]

[0024] Perform multi-dimensional parameter separation and decoupling on the signal after matched filtering; The received signal vector x k (t) is expressed as

[0025]

[0026] where

[0027]

[0028] where represents the reconstructed sampling matrix, ( (where represents a rational number) represents the selection matrix, which can be expressed as J = blkdiag(J1, J1), and blkdiag(J1, J1) represents the block diagonal matrix composed of J1, and J1 is expressed as

[0029] J1[1 + 2(l - 1)(2N + 1):2(l - 1)(2N + 1) + 2N + 1, l:l + 2N] = I 2N+1 ,

[0030] J1[1 + 2(l - 1)(2N + 1) + 2N + 1:2(l - 1)(2N + 1) + 2(2N + 1), l + 4N + 1:l + 6N + 1] = I 2N+1 ;

[0031] where l = 1, 2,..., 2N + 1; represents the reconstructed one-dimensional Fourier basis, represents the reconstructed polarization vector;

[0032] (5) Reduce the multi-dimensional parameter estimation problem to a direction-of-arrival estimation problem, and solve the direction-of-arrival based on the covariance matrix of the received array output after matched filtering;

[0033] ​(6) According to the direction of arrival obtained in step (5) and the Rayleigh-Ritz theorem, solve for the estimation of the polarization parameters.

[0034] Furthermore, in step (4), reconstruct the one-dimensional Fourier basis and the polarization vectors to achieve the separation of multi-dimensional parameters. Specifically: the reconstructed one-dimensional Fourier basis and the polarization vectors are respectively expressed as

[0035]

[0036]

[0037] Since the steering vector corresponding to the received signal vector x k (t) can be expressed as and the reconstructed sampling matrix the reconstructed Fourier basis and the reconstructed polarization vector are only related to the amplitude-phase response characteristics of the array, the direction of arrival corresponding to the p-th target, and the polarization parameters of the transmitted signal and the received signal corresponding to the p-th target respectively. Therefore, the steering vector of the multi-polarization MIMO radar realizes the separation of multi-dimensional parameters.

[0038] Furthermore, in step (5), the direction of arrival estimation is performed using the following methods: the polynomial root-finding method based on the polarization subspace orthogonality principle, and the one-dimensional spectral search method based on the polarization subspace orthogonality principle.

[0039] Furthermore, in step (5), the polynomial root-finding method based on the polarization subspace orthogonality principle is used to solve for the direction of arrival estimation of P targets. Specifically, according to the polarization subspace orthogonality principle, the following formula holds

[0040]

[0041] where the matrix U n is the noise subspace, and its column vectors are composed of the eigenvectors corresponding to the ML - P smallest eigenvalues of the covariance matrix In practice, R xx can be obtained by solving with K pulses, that is (·) H represents the conjugate transpose; according to the rank deficiency principle, it can be known that because so holds for the direction of arrival of all P targets; express as the matrix Q, and evenly partition Q into 16 sub-matrices Q ij , where i, j = 1, …, 4 and all have the same dimension (4N + 1) × (4N + 1);

[0042]

[0043] Thus can be expressed as:

[0044]

[0045] The polynomial coefficient vector e is obtained by applying Laplace's theorem, and its sub-vectors are [e ij u = sum[diag(Q ij , u - 2N - 1)], where u = 1, 2, …, 4N + 1, sum[·] represents the sum of the elements of the vector within the brackets, and diag(A, n) represents the vector composed of the nth diagonal elements of matrix A; thus the formula can be rewritten as e T z, where

[0046]

[0047] z = [z -32N , z -32N+1 , …, 1, …, z 32N-1 , z 32N T ; According to the previous discussion, the roots of the P polynomials of e T z = 0 can be used to estimate the direction of arrival of the target by the following formula:

[0048]

[0049] where arg{·} represents the phase of the complex number within the brackets.

[0050] Furthermore, in step (6), according to the Rayleigh - Ritz theorem, the polarization parameter estimates of the P targets are obtained. Specifically, from it can be seen that:

[0051]

[0052] It should be noted that Therefore, according to the Rayleigh - Ritz theorem, is collinear with the eigenvector q corresponding to the minimum eigenvalue of p , that is where q p is a non - zero variable; therefore, the polarization parameters of the transmitted signal corresponding to the pth target are obtained by the following formula:

[0053] γ​​t,p = arctan[|q p (3) / q p (1)|],

[0054] η t,p = arg[q p (4) / q p (2)],

[0055] where |·| represents taking the modulus, and q p (n) represents the nth element of q p ; the polarization parameter of the received signal corresponding to the pth target is obtained by the following formula:

[0056] γ r,p = arctan[|q p (2) / q p (1)|],

[0057] η r,p = arg[q p (2) / q p (1)].

[0058] The present invention has the following advantages compared with the existing methods:

[0059] (1) Different from traditional scalar MIMO radar, the present invention proposes a closed-form estimation method for multi-dimensional parameters applicable to multi-polarization MIMO radar, which can realize the joint estimation of direction of arrival and polarization parameters, and can effectively avoid the accuracy loss of multi-dimensional parameter estimation caused by polarization mismatch;

[0060] (2) The method proposed by the present invention can be used for any multi-polarization array, has no strict requirements on the configuration and position of multi-polarization array elements, and is applicable to the case where there are model errors in the transmitting and receiving arrays, can avoid the accuracy loss of multi-dimensional parameter estimation caused by non-ideal factors such as gain, phase and position errors, and does not cause estimation ambiguity;

[0061] (3) The method proposed by the present invention is a closed-form self-pairing implementation method, which can avoid the multi-dimensional spectrum peak search and multi-dimensional parameter pairing processes, and has high computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 is the overall flowchart of the present invention;

[0063] Figure 2 、 Figure 3 and Figure 4 are the relationships between the multi-dimensional parameter estimation performance and the number of modes N, where Figure 2 is the relationship between the root mean square error of multi-dimensional parameter estimation of the method proposed by the present invention and the number of modes N; Figure 3 is the relationship between the root mean square error of direction of arrival estimation of the proposed method and the number of modes N;Figure 4 is the relationship between the root mean square error (RMSE) of direction of arrival (DOA) estimation based on existing methods and the number of modes N;

[0064] Figure 5 is the DOA estimation result of the proposed method;

[0065] Figure 6 is the 2D scatter plot of polarization parameter estimation of the proposed method;

[0066] Figure 7 is the relationship between the RMSE of the proposed method and existing multi-dimensional parameter estimation accuracy and the signal-to-noise ratio;

[0067] Figure 8 is the relationship between the RMSE of the proposed method and existing multi-dimensional parameter estimation accuracy and the number of pulses. Detailed implementation manners

[0068] The technical solution of the present invention will be further described in detail with reference to the accompanying drawings below.

[0069] To solve the problem that the traditional single-polarization MIMO radar cannot avoid the loss of angle estimation accuracy caused by polarization mismatch, and most of the existing multi-dimensional parameter estimations based on multi-polarization MIMO radar do not consider model errors, the present invention provides a closed-form estimation method for multi-dimensional parameters of a polarization MIMO radar with an arbitrary multi-polarization array based on the polarization subspace orthogonality principle considering model errors, so as to avoid polarization mismatch and achieve closed-form estimation of multi-dimensional parameters of a polarization MIMO radar with an arbitrary multi-polarization array when considering model errors. Refer to Figure 1 The implementation steps of the present invention are as follows:

[0070] Step 1: Obtain the horizontal polarization and vertical polarization sampling matrices of the transmitting array and the receiving array based on array measurements. Consider a multi-polarization monostatic MIMO radar with a multi-polarization transmitting antenna array and a multi-polarization receiving antenna array having an arbitrary structure, where the number of transmitting antenna elements is M (transmitting orthogonal signals with different polarization antennas), and the number of receiving antenna elements is L. The transmitting antennas and receiving antennas are placed arbitrarily, but no estimation ambiguity is caused. After array response measurement, Γ t,h and Γ t,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarization transmitting antenna array; Γ r,h and Γ r,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarization receiving antenna array, and respectively represent the polarization vectors of the transmitted signal and the received signal corresponding to the p-th target, where γ t,p and η t,p respectively represent the polarization auxiliary angle and polarization phase difference of the transmitted signal of the p-th target, γr,p and η r,p respectively represent the received signal polarization auxiliary angle and polarization phase difference of the p-th target, "[·] T " represents the transpose operation.

[0071] Step 2: Based on the measured transmit and receive array sampling matrices, model the signal model of the multi-polarization MIMO radar. For the considered multi-polarization MIMO radar, assume that the transmit antennas transmit orthogonal fully polarized signals, and the target to be estimated is assumed to be in the same range cell. Assume that the direction of arrival of the p-th target is θ p , according to the reciprocity of the transmit and receive antennas, the steering vectors of the transmit different polarization antennas and receive different polarization antennas can be respectively expressed as

[0072]

[0073] and

[0074]

[0075] where I2 represents the second-order identity matrix, represents the (2N + 1)×1 truncated one-dimensional Fourier basis, N represents the number of modes corresponding to the transceiver array, w t and w r respectively represent the modeling errors of the transmit steering vector and the receive steering vector caused by truncation and calibration noise. It should be noted that when the number of modes is correctly selected, w t and w r will be small enough.

[0076] Then the received signal of the receive multi-polarization antenna array is

[0077]

[0078] where the time range is T0 is the starting time, T k is the k-th pulse duration, K is the number of pulses, P is the number of targets, μ p,k represents the scattering coefficient of the p-th target during the k-th pulse (which is related to the radar cross section). u(t) = [u1(t), u2(t), …, u M (t)] T represents the vector of orthogonal transmission signals; e k (t) represents the additive Gaussian white noise vector.

[0079] Step 3: Perform matched filtering on the received signal. After the matched filtering corresponding to the m-th transmit signal, the matched filtering output of the receive multi-polarization antenna array can be expressed as

[0080]

[0081] wherein is a vector with the m-th element being 1 and the remaining elements being 0, and n k,m represents the Gaussian white noise corresponding to the m-th signal under the k-th pulse. After using a series of matched filters corresponding to M transmitted signals for the output signals of each receiving array element, the received signal vector can be re-expressed as

[0082]

[0083] wherein represents the steering matrix composed of the steering vectors corresponding to P targets, and s k = [μ 1,k , μ 2,k …, μ P,k T represents the transmitted signal vector, represents the noise vector, represents the steering vector corresponding to the p-th target.

[0084] Step 4: Perform multi-dimensional parameter separation and decoupling on the signals after the matched filtering process. First, the steering vectors of the transmitting and receiving antenna arrays are respectively processed by using the polarization manifold separation technique through appropriate truncation and omission

[0085] The modeling error can be expressed as follows

[0086]

[0087]

[0088] Perform multi-dimensional parameter separation and decoupling on the signals after the matched filtering process. The received signal vector x k (t) can be expressed as

[0089]

[0090] wherein

[0091]

[0092] wherein represents the reconstructed sampling matrix, represents the selection matrix, which can be expressed as J = blkdiag(J1, J1), and blkdiag(J1, J1) represents the block diagonal matrix composed of J1, and J1 is expressed as

[0093] J1[1 + 2(l - 1)(2N + 1):2(l - 1)(2N + 1) + 2N + 1, l:l + 2N] = I 2N+1 ​,

[0094] J1[1 + 2(l - 1)(2N + 1)+2N + 1:2(l - 1)(2N + 1)+2(2N + 1),l + 4N + 1:l + 6N + 1]=I 2N+1 ,

[0095] where l = 1, 2, ..., 2N + 1;

[0096] The reconstructed one - dimensional Fourier basis and polarization vector can be expressed as

[0097]

[0098]

[0099] Since the steering vector corresponding to the received signal vector x k (t) can be expressed as and the reconstructed sampling matrix the reconstructed Fourier basis and the reconstructed polarization vector are only related to the amplitude - phase response characteristics of the array, the direction of arrival of the p - th target, and the polarization parameters of the transmitted and received signals corresponding to the p - th target respectively. Therefore, the steering vector of the multi - polarization MIMO radar realizes the separation of multi - dimensional parameters.

[0100] Step 5: First, reduce the multi - dimensional parameter estimation problem to a direction - of - arrival estimation problem, and solve the direction - of - arrival estimation based on the covariance matrix of the received array output after matched filtering. According to the polarization subspace orthogonality principle, the following formula holds

[0101]

[0102] where the matrix U n is the noise subspace, and its column vectors are composed of the eigenvectors corresponding to the ML - P smallest eigenvalues of the covariance matrix In practice, the covariance matrix R xx can be obtained by solving with K pulses, that is According to the rank - deficiency principle, because so holds for the directions of arrival of all P targets. Represent as the matrix Q, and divide Q evenly into 16 sub - matrices Q ij , where i, j = 1, …, 4 and all have the same dimension (4N + 1)×(4N + 1).

[0103]

[0104] Therefore can be expressed as:

[0105]

[0106] The direction of arrival of P targets can be estimated by finding the roots of the polynomial equation e T z = 0. The polynomial coefficient vector e can be obtained by applying Laplace's theorem, and its sub-vectors [e ij u = sum[diag(Q ij , u - 2N - 1)], where u = 1, 2,..., 4N + 1, sum[·] represents the sum of the vector elements within the brackets, and diag(A, n) represents the vector composed of the nth diagonal elements of matrix A. Thus, the formula can be rewritten as e T z, where

[0107]

[0108] z = [z -32N , z -32N+1 , …, 1, …, z 32N-1 , z 32N T ,

[0109] According to the previous discussion, the P roots of the polynomial e T z = 0 can be used to estimate the direction of arrival of the target by the following formula:

[0110]

[0111] where arg{·} represents the phase of the complex number within the brackets.

[0112] Step 6: Solve for the estimated values of the polarization parameters. From it can be seen that:

[0113]

[0114] It should be noted that Therefore, according to the Rayleigh - Ritz theorem, we can obtain collinear with the eigenvector q p corresponding to the minimum eigenvalue of that is p where q

[0115] γ t,p ​​= arctan[|q p (3) / q p (1)|],

[0116] η t,p = arg[q p (4) / q p (2)],

[0117] where |·| represents taking the modulus, and q p (n) represents the nth element of q p . The polarization parameters of the received signal corresponding to the pth target can be obtained by the following formula:

[0118] γ r,p = arctan[|q p (2) / q p (1)|],

[0119] η r,p = arg[q p (2) / q p (1)].

[0120] The effects of the present invention will be further described below in conjunction with simulation examples.

[0121] Simulation examples: Three simulation examples are given below to illustrate the performance of the proposed scheme and compare it with the Manifold Separation Technique - Polarimetric Element Space (MST - PES) method. In the three simulation examples, three targets to be estimated with directions of arrival of (20°, 60°, 80°) are considered. The simulation uses a non - uniform monostatic multi - polarization MIMO radar. The transmitting array consists of two standard tri - dipole antennas located at positions (1,0,0)d and (1.9,0,0)d in the x - y - z Cartesian coordinate system respectively, and the receiving array consists of three standard tri - dipole antennas located at positions (0.1,3,0)d, (1.05,3,0)d and (2.1,3,0)d in the x - y - z Cartesian coordinate system respectively, where d represents the half - wavelength. For the sake of comparison, we use six single - polarization antennas (assuming without loss of generality that the axial direction of the dipole antenna is parallel to the z - axis) to form the transmitting array, which is used as the transmitting and receiving array for the comparative MST - PES method. The receiving arrays of the proposed method and the comparative MST - PES method are the same. The gain error of each antenna in the transmitting and receiving arrays follows a normal distribution, and the phase error of each antenna in the transmitting and receiving arrays follows a normal distribution. The radar cross - section coefficient follows the Swerling II target model.

[0122] First, select an appropriate number of modes \(N\), and design different estimation parameters to simulate the relationship between the root mean square error performance and the number of modes. The polarization parameters \((\gamma\) t,p , \(\eta\) t,p ) of the transmitted signals corresponding to the 3 targets, \(p = 1, 2, 3\) are all set to \((45^{\circ}, 25^{\circ})\), and the polarization parameters \((\gamma\) r,p , \(\eta\) r,p ) of the received signals corresponding to the 3 targets, \(p = 1, 2, 3\) are \((10^{\circ}, 40^{\circ})\), \((50^{\circ}, 50^{\circ})\) and \((70^{\circ}, 10^{\circ})\) respectively. The simulation results are as Figure 2 shown, where the number of pulses is 200 and the signal-to-noise ratio (SNR) is 15 dB. It can be seen from the simulation results that when \(N\) increases, the estimation performance gets better. And for a certain \(N\), there is a fixed threshold for the root mean square error. The fixed threshold in this simulation example is about 17. In addition, we simulated the relationship between the root mean square error performance and the number of modes for different signal-to-noise ratios. We can observe in Figure 3 that when the signal-to-noise ratio is set to 30 dB, the root mean square error tends to a stable value when \(N\) is greater than 19; when the signal-to-noise ratio is set to 10 dB and 20 dB, the root mean square error tends to a stable value when \(N\) is greater than 17. This result shows that the fixed threshold increases with the increase of the signal-to-noise ratio. This phenomenon can be explained as a trade-off between the sampling accuracy (higher \(N\) means higher sampling matrix accuracy) and the noise level. When the signal-to-noise ratio is large enough, the sampling accuracy will be the main factor affecting the parameter estimation performance. For comparison, the simulation results of the root mean square error varying with the number of modes of the MST-PES method are also given as Figure 4 shown. It can be seen that compared with the proposed method, the MST-PES method corresponds to a higher fixed threshold. This is because the array aperture of a single-polarization antenna array is usually larger than that of a multi-polarization antenna array composed of multi-polarization antennas with a co-point configuration, and this fixed threshold is always positively correlated with the array aperture.

[0123] Next, study and compare the performance of the closed-form multi-dimensional parameter estimation under the appropriate number of modes. The polarization parameters \((\gamma\) t,p , \(\eta\) t,p ) of the transmitted signals corresponding to the 3 targets, \(p = 1, 2, 3\) are set to \((45^{\circ}, 25^{\circ})\), \((80^{\circ}, 60^{\circ})\) and \((15^{\circ}, 70^{\circ})\) respectively, and the polarization parameters \((\gamma\) r,p , \(\eta\) r,p ) of the received signals corresponding to the 3 targets, \(p = 1, 2, 3\) are set to \((10^{\circ}, 40^{\circ})\), \((50^{\circ}, 50^{\circ})\) and \((70^{\circ}, 10^{\circ})\) respectively. In this example, \(N = 20\) which is higher than the fixed threshold is used to verify the closed-form and automatic pairing characteristics of the proposed method, and 100 Monte Carlo experiments are carried out in total. Figure 5 andFigure 6 The estimation results of direction of arrival (DOA) estimation and polarization parameter estimation are shown, where the number of pulses is 200 and the signal-to-noise ratio is 15 dB. It can be seen from the simulation results that the proposed method has excellent estimation performance and does not require a pairing process.

[0124] Finally, the statistical estimation performances of the proposed method and the existing MST-PES method are studied and compared. Referring to the results of the first simulation example, N is selected as 26. The polarization parameters (γ t,p , η t,p ) of the transmitted signals corresponding to the 3 targets, p = 1, 2, 3 are all set to (45°, 25°), and the polarization parameters (γ r,p , η r,p ) of the received signals corresponding to the 3 targets are set to (10°, 40°), (50°, 50°) and (70°, 10°) respectively. The simulation results are as shown in Figure 7 and Figure 8 , where Figure 7 has 200 pulses, and Figure 8 has 20 dB of pulses. The simulation results show that the proposed method has excellent estimation performance and is superior to MST-PES, while the MST-PES method may be affected by polarization mismatch. Therefore, the method proposed in the present invention has great advantages in terms of estimation performance and anti-polarization mismatch, and its transmitting and receiving arrays can be non-ideal and arbitrarily configured in practical applications.

[0125] In summary, based on the principle of orthogonality of the polarization subspace, the present invention proposes a polynomial root-finding method for joint estimation of direction of arrival and polarization parameters applied to a polarization MIMO radar with an arbitrary configuration, and derives the polynomial coefficients, realizing the automatic pairing of the direction of arrival and polarization parameter estimations of each signal source, and achieving accurate estimations of the direction of arrival and polarization parameters while improving the calculation efficiency. The proposed closed-form method takes into account the model errors of the transmitting and receiving arrays and is applicable to polarization MIMO radars with any multi-polarization configuration.

Claims

1. A closed-form estimation method for multi-dimensional parameters of MIMO radar based on the principle of orthogonality of polaron subspaces, characterized in that, The steps include: (1) Obtain the horizontal polarization and vertical polarization sampling matrices of the transmitting array and the receiving array based on array measurements; consider a multi-polarization monostatic MIMO radar with a multi-polarization transmitting antenna array and a multi-polarization receiving antenna array having an arbitrary structure, where the number of transmitting antenna elements is M and the number of receiving antenna elements is L; the transmitting antennas and the receiving antennas are placed arbitrarily without causing estimation ambiguity; after array response measurement, Γ t,h and Γ t,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarization transmitting antenna array; Γ r,h and Γ r,v respectively represent the array sampling matrices corresponding to the horizontal polarization and vertical polarization of the multi-polarization receiving antenna array, and respectively represent the transmitting signal polarization vector and the receiving signal polarization vector corresponding to the p-th target, where γ t,p and η t,p respectively represent the transmitting signal polarization auxiliary angle and the polarization phase difference of the p-th target, γ r,p and η r,p respectively represent the receiving signal polarization auxiliary angle and the polarization phase difference of the p-th target, [·] T represents the transpose operation; (2) Model the signal model of a multi-polarization MIMO radar based on the measured transmit and receive array sampling matrix; for the considered multi-polarization MIMO radar, assume that the transmit antennas emit orthogonal fully polarized signals, and the target to be estimated is assumed to be in the same range cell; assume that the direction of arrival of the p-th target is θ p , according to the reciprocity of the transmit and receive antennas, the steering vectors of the transmit different polarization antennas and the receive different polarization antennas are respectively expressed as and where \(I_2\) represents the second-order identity matrix, represents the \((2N + 1)\times1\) truncated one-dimensional Fourier basis, \(N\) represents the mode number corresponding to the transceiver array, \(w\) t and \(w\) r respectively represent the modeling errors of the transmit steering vector and the receive steering vector caused by truncation and calibration noise; it should be noted that when the mode number is correctly selected, \(w\) t and \(w\) r will be small enough; The received signal of the multi-polarized antenna array is where the time range is k = 1, 2, …, K - 1, T0 is the starting time, T k is the k-th pulse duration, K is the number of pulses, P is the number of targets, μ p,k represents the scattering coefficient of the p-th target during the k-th pulse; u(t) = [u1(t), u2(t), …, u M (t)] T represents the vector of orthogonal transmission signals; e k (t) represents the additive Gaussian white noise vector; (3) Perform matched filtering on the received signal; after the matched filtering corresponding to the m-th transmitted signal, the matched filtering output of the received multi-polarized antenna array is expressed as wherein is a vector with the m-th element being 1 and the remaining elements being 0, and n k,m represents Gaussian white noise corresponding to the m-th signal at the k-th pulse; after using a series of matched filters corresponding to M transmitted signals for the output signals of each receiving array element, the received signal vector is re-expressed as where denotes the steering matrix composed of the steering vectors corresponding to P targets, s k = [μ 1,k , μ 2,k …, μ P,k T denotes the transmitted signal vector, denotes the noise vector, denotes the steering vector corresponding to the p-th target;​ (4) Perform multi-dimensional parameter separation and decoupling on the signal after the matched filtering; first, the steering vectors of the transmitting and receiving antenna arrays are respectively expressed as follows after appropriate truncation and omission of the modeling error using the polarization manifold separation technique Perform multi-dimensional parameter separation and decoupling on the signal after matched filtering processing; the received signal vector x k (t) is expressed as where wherein represents the reconstructed sampling matrix, represents the selection matrix, and blkdiag(J1, J1) represents the block diagonal matrix composed of J1, and J1 is expressed as Among them represents the reconstructed one-dimensional Fourier basis represents the reconstructed polarization vector; (5) Reduce the multi-dimensional parameter estimation problem to a direction-of-arrival estimation problem, and solve the direction-of-arrival based on the covariance matrix of the received array output after the matched filtering; (6) According to the direction-of-arrival obtained in step (5) and the Rayleigh-Ritz theorem, solve the estimation of the polarization parameters.

2. The closed - form estimation method for multi - dimensional parameters of MIMO radar based on the orthogonality principle of polaron subspace according to claim 1, characterized in that, In step (4), reconstruct the one-dimensional Fourier basis and the polarization vector to achieve the separation of multi-dimensional parameters. Specifically: the reconstructed one-dimensional Fourier basis and the polarization vector are respectively expressed as Since the steering vector corresponding to the received signal vector x k (t) can be expressed as and the reconstructed sampling matrix the reconstructed Fourier basis and the reconstructed polarization vector are only related to the amplitude-phase response characteristics of the array, the direction of arrival of the p-th target, and the polarization parameters of the transmitted and received signals corresponding to the p-th target, respectively. Therefore, the steering vector of the multi-polarization MIMO radar realizes the separation of multi-dimensional parameters.

3. The closed-form estimation method for multi-dimensional parameters of a MIMO radar based on the principle of orthogonality of polaron subspaces according to claim 1, characterized in that, In step (5), the direction-of-arrival estimation is performed using the following methods: the polynomial root-finding method based on the polarization subspace orthogonality principle, the one-dimensional spectrum search method based on the polarization subspace orthogonality principle.

4. The closed-form estimation method for multi-dimensional parameters of a MIMO radar based on the principle of orthogonality of polariton subspaces according to claim 1, characterized in that In step (5), the polynomial root-finding method based on the polarization subspace orthogonality principle is used to solve the direction-of-arrival estimation of P targets. Specifically, according to the polarization subspace orthogonality principle, the following formula holds where the matrix U n is the noise subspace, and its column vectors are composed of the eigenvectors corresponding to the ML - P smallest eigenvalues of the covariance matrix . In practice, R xx is obtained by solving with K pulses, that is denotes the conjugate transpose; According to the rank deficiency principle, since Therefore holds for the directions of arrival of all P targets; Represent as matrix Q, and divide Q equally into 16 submatrices Q ij , where i, j = 1, …, 4 and all have the same dimension (4N + 1) × (4N + 1); Thus It can be expressed as: The polynomial coefficient vector e is obtained by applying Laplace's theorem, and its sub-vectors are [e ij u = sum[diag(Q ij , u - 2N - 1)], where u = 1, 2, …, 4N + 1, sum[·] represents the sum of the vector elements within the parentheses, and diag(A, n) represents the vector composed of the nth diagonal element of matrix A; thus the formula is rewritten as e T z, where​ e = (e 11 * e 22 - e 21 * e 12 ) * (e 33 * e 44 - e 43 * e 34 ) -(e 11 *e 23 -e 21 *e 13 )*(e 32 *e 44 -e 42 *e 34 ) +(e 11 *e 24 -e 21 *e 14 )*(e 32 *e 43 -e 42 *e 33 ) +(e 12 *e 23 -e 22 *e 13 )*(e 31 *e 44 -e 41 *e 34 ), -(e 12 *e 24 -e 22 *e 14 )*(e 31 *e 43 -e 41 *e 33 ) +(e 13 *e 24 -e 23 *e 14 )*(e 31 *e 42 -e 41 *e 32 ) z = [z -32N , z -32N+1 , …, 1, …, z 32N-1 , z 32N T ; According to the previous discussion, e T The roots of P polynomials with z = 0 Estimate the direction of arrival of the target by the following formula:​ where arg{·} represents the phase of the complex number in the brackets.

5. The closed-form estimation method for multi-dimensional parameters of a MIMO radar based on the principle of orthogonality of polaron subspaces according to claim 1, characterized in that In step (6), according to the Rayleigh-Ritz theorem, the polarization parameter estimates of P targets are obtained. Specifically, from it can be seen that: It should be noted that Therefore, according to the Rayleigh-Ritz theorem, is collinear with the eigenvector q corresponding to the minimum eigenvalue of p , that is where q p is a non-zero variable; therefore, the polarization parameters of the transmitted signal corresponding to the p-th target are obtained by the following formula: γ t,p = arctan[|q p (3) / q p (1)|], η t,p = arg[q p (4) / q p (2)], where |·| represents taking the modulus, and q p (n) represents the n-th element of q p ; The polarization parameters of the received signal corresponding to the p-th target are obtained through the following formula: γ r,p = arctan[|q p (2) / q p (1)|], η r,p = arg[q p (2) / q p (1)].

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