Closed-form Joint Estimation Method for Two-dimensional Direction of Arrival and Polarization Parameters Based on Cascade Sparse Multipolarization Arrays

Abstract

The present invention discloses a closed-form joint estimation method for two-dimensional direction of arrival (DOA) and polarization parameters based on a cascaded sparse multi-polarization array, which mainly solves the problems of high computational complexity and limited degrees of freedom in existing methods. The implementation steps are as follows: constructing a cascaded sparse multi-polarization array; modeling the received signal model of the cascaded sparse multi-polarization array and decoupling multi-dimensional parameters; performing block smoothing processing on the received signal covariance matrix; constructing the received signal covariance matrix of a virtual cascaded uniform multi-polarization array; performing eigenvalue decomposition on the virtual covariance matrix to obtain the estimation of the angle between the direction of arrival of each signal source and a single coordinate axis; obtaining the estimation of the angle between the direction of arrival of each signal source and another coordinate axis, and obtaining the closed-form estimation of azimuth, elevation angle, and polarization parameters. Based on the multi-domain sub-array smoothing technology, the present invention provides a feasible and effective solution for realizing the fast joint estimation of two-dimensional DOA and polarization parameters, and can be used for sensing and positioning in complex scenarios such as industrial Internet of Things.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and particularly relates to parameter estimation for a cascaded sparse multi-polarization planar array. Specifically, it is a closed-form joint estimation method for two-dimensional direction of arrival (DOA) and polarization parameters based on a cascaded sparse multi-polarization planar array, which can be used for sensing and positioning in complex scenarios such as industrial Internet of Things. Background Art

[0002] Multi-polarization arrays have the ability to stably and reliably jointly sense the two-dimensional DOA and polarization parameters of signals. These stable multi-dimensional sensing parameters can be used for sensing and positioning in complex scenarios in fields such as industrial Internet of Things, communication, and radar. Existing uniform multi-polarization arrays have mutual coupling effects due to small element spacings, and the degrees of freedom of the array are limited by the number of physical elements, resulting in a limited array aperture size, which in turn affects the estimation effects of two-dimensional DOA and polarization parameters.

[0003] To reduce the mutual coupling effect, increase the degrees of freedom of the array, and achieve a larger array aperture with fewer elements, the present invention proposes a brand-new cascaded sparse multi-polarization planar array structure, and conducts a closed-form joint estimation of two-dimensional DOA and polarization parameters based on this new multi-polarization sparse planar array. Specifically, the present invention sparsely cascades magnetic ring and dipole elements with different polarization types in a hierarchical and sub-array manner to receive different polarization components of signals. On the other hand, existing related methods for joint estimation of two-dimensional DOA and polarization parameters require multi-dimensional spectral peak search and pairing processes, with relatively high computational complexity. Therefore, there is an urgent need to design methods that avoid multi-dimensional spectral peak search and pairing, reduce the computational complexity of joint estimation of two-dimensional DOA and polarization parameters, and achieve joint estimation of two-dimensional DOA and polarization parameters applicable to cascaded sparse multi-polarization planar arrays. Summary of the Invention

[0004] The purpose of the present invention is to address the problems of relatively high computational complexity and limited degrees of freedom of the array in the above-mentioned joint estimation method for two-dimensional DOA and polarization parameters, and propose a closed-form joint estimation method for two-dimensional DOA and polarization parameters based on a cascaded sparse multi-polarization planar array. For the designed cascaded sparse multi-polarization planar array, the method proposes corresponding multi-domain sub-array smoothing processing techniques, providing feasible ideas and effective solutions for realizing joint estimation of two-dimensional DOA and polarization parameters and improving its computational efficiency.

[0005] The purpose of the present invention is achieved through the following technical solutions: A closed-form joint estimation method for two-dimensional DOA and polarization parameters based on a cascaded sparse multi-polarization planar array, comprising the following steps:

[0006] (1) Construct a cascaded sparse multi-polarization planar array: Construct an array composed of N pA cascaded sparse multi-polarization planar array composed of ≥2 sparse sub-arrays in cascade, with the distance between adjacent sub-arrays being d, and each sub-array being composed of L0N m magnetic rings or dipoles with the same polarization type. The normal direction of the magnetic ring is parallel to a certain coordinate axis, and the axial direction of the dipole is parallel to a certain coordinate axis, and L0N m array elements are each composed of N m ≥2 mutually parallel layers, with the distance between adjacent layers being d x , and the number of array elements in each layer being L0. Then, each layer of the cascaded sparse multi-polarization planar array has a total of L = L0N p array elements; the sparse arrangement patterns of the array elements in each layer are all the same fully expandable sparse arrays; the number of virtual array elements corresponding to each layer of each sub-array is L V , and the distance between adjacent virtual array elements is d;

[0007] (2) Suppose there are M uncorrelated far-field narrowband signal sources incident on the designed cascaded sparse multi-polarization planar array, and θ m , φ m represent the azimuth angle and elevation angle of the m-th signal source respectively, and γ m , η m represent the polarization auxiliary angle and polarization phase difference of the m-th signal source respectively, where m = 1, 2,..., M. Then, the received signal x(t) of the designed cascaded sparse multi-polarization planar array at time t is modeled as:

[0008]

[0009] where s m (t) represents the waveform corresponding to the m-th signal source, and n(t) is a Gaussian white noise component with a mean of zero and independent of each signal source. The spatial and polarization domain joint domain steering vector of the designed cascaded sparse multi-polarization planar array is expressed as

[0010]

[0011] where a h (θ m , φ m ), a v (θ m , φ m ) represent the horizontal polarization steering vector and vertical polarization steering vector respectively, corresponding to a signal with a wave direction of (θ m , φ m ), represents the polarization vector corresponding to the m-th signal, blkdiag[B s×t , C p×q represents a diagonal block matrix constructed from the matrices in the parentheses, that is:

[0012]

[0013] where Β s×t、 C p×q represent an s×t dimensional matrix and a p×q dimensional matrix respectively, and O a×b represents an a×b dimensional zero matrix; the spatial and polarization domain joint domain steering vector of the designed cascaded sparse multi-polarization array is further expressed as

[0014]

[0015] where a s,m is the pure spatial domain steering vector of the designed cascaded sparse multi-polarization array that is only related to the spatial domain parameters, and is expressed as

[0016]

[0017] where [·] T represents the transpose operation, represents the pure spatial domain steering vectors of the 1st, 2nd, …, N m layers of the designed cascaded sparse multi-polarization array, λ represents the signal wavelength, d l , l = 1, 2, …, L represents the distance from the l-th array element in each layer to the origin of coordinates along the y-axis direction, d1 = 0, diag(a) represents the diagonal matrix generated based on the elements in the vector a, represents the polarization domain steering vector of the designed cascaded sparse multi-polarization array, D is an N m L×N p dimensional matrix, each row has and only has one element as 1, and the rest of the elements are all 0, which characterizes the polarization type selection method of all the array elements of the array after stacking the array elements layer by layer to form N m L array elements. The designed cascaded sparse multi-polarization array has N p sub-arrays, [D] a,b = 1 indicates that the a-th array element belongs to the polarization type of the b-th sub-array, a = 1, 2, …, N m L, b = 1, 2, …, N p , [D] a,b represents the element in the a-th row and b-th column of the matrix D. D is block-diagonalized by layer as

[0018]

[0019] where is an L×N p dimensional matrix. D is block-diagonalized by array element as

[0020]

[0021] where d (l) is an N p ×1 dimensional vector, [d (l) b = 1 indicates that the l-th array element belongs to the polarization type of the b-th sub-array, l = 1, 2, …, N m L, b = 1, 2, …, N p , [d (l) b represents the b-th element of the vector d (l) ; J is an N p ×6 dimensional selection matrix, with exactly one element equal to 1 in each row and the remaining elements equal to 0, which represents the array elements of N p sub-arrays selected from 3 dipoles with axes parallel to the x, y, and z axes and 3 magnetic loop antennas with normal directions parallel to the x, y, and z axes, is a 6×2 dimensional matrix representing the polarization types of 6 multi-polarization array elements:

[0022]

[0023] (3) Denote α m , β m as the angles between the direction of arrival of the m-th signal source and the y-axis and the x-axis respectively, then θ m , φ m , α m and β m satisfy the following relationships:

[0024] sinφ m sinθ m = cosα m , sinφ m cosθ m = cosβ m ,

[0025] Then the spatial domain steering vector a s,m is rewritten as where In addition, according to the mathematical structure of the polarization domain steering vector and its relationship with a s,m , the spatial and polarization domain joint domain steering vector is rewritten as

[0026]

[0027] where I n represents the n×n dimensional identity matrix, represents the Kronecker product, The parameter is only related to the spatial domain parameter θ​​m , φ m is related to only related to the antenna polarization type of the multi-polarized planar array, only related to the polarization parameters. Therefore, through the above operations, the spatial domain parameters, the remaining angular parameters related to the antenna polarization characteristics, and the polarization parameters are decoupled;

[0028] (4) The spatial domain steering vectors of each layer of the designed cascaded sparse multi-polarized planar array are further block-represented as:

[0029]

[0030]

[0031]

[0032] Among them, the spatial domain steering vectors belonging to different layers and different sub-arrays are represented as

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042] Then the received signal covariance matrix R of the cascaded sparse multi-polarized planar array xx is represented as:

[0043]

[0044] Among them, represents the power of the m-th signal source, σ 2 represents the noise power, (·) H represents the conjugate transpose operation, is an L×L dimensional matrix, p,q = 1,2,…,N m , R xxis partitioned into sub - arrays. Next, the transformation of R xx can also be divided into steps and carried out according to the following method:

[0045]

[0046] where is expressed as

[0047]

[0048] where D (1) (1:L0:end,:) represents the sub - matrix composed of the elements in the 1, 1 + L0, …, [1+(N (1) -1)L0] - th rows of D p , and (·) * represents the conjugate operation; is further simplified to

[0049]

[0050] where ρ 11,m =δ 11,m , ρ 22,m =δ 22,m , …,

[0051] R xx is block - represented as A(m1:m2,n1:n2) represents the sub - matrix composed of the elements at the intersection of the m1 - th to m2 - th rows and the n1 - th to n2 - th columns of matrix A; in practice, R xx is approximately calculated based on K sampling snapshots, that is:

[0052]

[0053] where t k represents the time corresponding to the k - th sampling snapshot; Next, the spatial and polarization joint - domain smoothing processing operation is performed on each block sub - matrix of the proposed array covariance matrix. First, is vectorized to obtain the vector

[0054]

[0055] where, represents the vectorization operation on the matrix , that is, vectorizing the matrix The columns in it are stacked in sequence to form a new vector, denotes a pure spatial domain steering matrix, denotes the Khatri-Rao product, vector The corresponding virtual array is represented as a uniform array containing 2L V -1 virtual continuous array elements Define dimensional selection matrix Δ:

[0056]

[0057] where represents the element in the l'-th row and the -th column of matrix Δ, l' = 1, …, 2L V -1, L0 - 1 ≥ i1 ≥ 0, L0 ≥ i2 ≥ 1, ω(l' - L V , i1, i2) represents a function of l' - L V , i1 and i2, representing the number of index combinations where the value combinations of i1 and i2 satisfy ; Based on the selection matrix Δ, the elements in vector are reordered into an equivalent virtual signal corresponding to

[0058]

[0059] where is the pure spatial domain steering vector obtained by removing the polarization component from the virtual uniform array corresponding to each subarray represents a 2L V -1 dimensional column vector with only the L V -th element being 1 and the rest being 0;

[0060] (5) The virtual vector is sequentially decomposed into L V V ×1 dimensional virtual subvectors:

[0061]

[0062] where then represents the vector composed of the first to the L V elements in represents the vector composed of the second to the L V +1 elements in represents the vector starting from the L​​​V to the 2L V -1 element vector; for L V a Performing a column vector merging operation on L, the autocorrelation / cross-correlation matrix corresponding to the received signals of each sub-array in each layer of the virtual equivalent cascaded uniform multi-polarization array can be obtained:

[0063]

[0064] where represents the pure spatial domain steering vector of the first sub-array in the first layer of the virtual equivalent cascaded uniform multi-polarization array; and then the corresponding covariance matrix of the received signals of the virtual equivalent cascaded uniform multi-polarization array is obtained

[0065]

[0066] where is expressed as

[0067]

[0068] equivalently expressed as

[0069]

[0070] where

[0071]

[0072]

[0073]

[0074] Then the covariance matrix R of the received signals of the cascaded sparse multi-polarization array xx is reconstructed into the covariance matrix of the received signals of its corresponding virtual equivalent uniform multi-polarization array

[0075]

[0076] The spatial and polarization joint domain steering vector of the virtual equivalent uniform multi-polarization array is expressed as:

[0077]

[0078] where N p L V ×N p dimensional matrix There is exactly one element equal to 1 in each row, and the remaining elements are all 0, which represents the polarization type selection method of each element in the first layer of the virtual equivalent uniform multi-polarization planar array. The virtual equivalent uniform multi-polarization planar array has N p sub-arrays, represents the polarization type of the a-th element belonging to the b-th sub-array, where a = 1, 2, …, N p L V , and b = 1, 2, …, N p ;

[0079] (6) Perform eigenvalue decomposition on the virtual received signal covariance matrix, and obtain the estimation of the angle between the direction of arrival of each signal source after separation and decoupling and the y-axis in the reference coordinate system;

[0080] (7) Obtain the closed-form estimations of the azimuth angle, elevation angle, and polarization parameters.

[0081] The present invention has the following advantages compared with the prior art:

[0082] (1) The present invention constructs a cascaded sparse multi-polarization planar array by using magnetic rings and dipole elements. The array aperture is relatively large, and each element has a systematic array structure;

[0083] (2) By cascading magnetic rings and dipole elements with different polarization types, the present invention effectively reduces the mutual coupling effect caused by the co-located configuration of multi-polarization elements;

[0084] (3) Based on the multi-domain sub-array smoothing processing, the present invention obtains the joint estimation of the two-dimensional direction of arrival and polarization parameters. The proposed method is closed-form and automatically paired, avoiding the computationally complex multi-dimensional spectral peak search and pairing process, and effectively improving the computational efficiency. Description of the Drawings

[0085] Figure 1 is the overall flow block diagram of the present invention.

[0086] Figure 2 is the structural schematic diagram of the cascaded sparse multi-polarization planar array in the present invention.

[0087] Figure 3 is the structural schematic diagram of the virtual equivalent cascaded uniform multi-polarization planar array corresponding to the cascaded sparse multi-polarization planar array in the present invention.

[0088] Figure 4 is the relationship curve between the cosine value of the angle between the steering vector of the array proposed in the present invention and the actual steering vector of the co-located multi-polarization array and the two-dimensional direction of arrival angle.

[0089] Figure 5 is the scatter plot of the two-dimensional direction of arrival estimation under underdetermined conditions by the method proposed in the present invention.

[0090] Figure 6 It is the scatter plot of the two-dimensional direction of arrival and polarization parameter estimation of the method proposed by the present invention under overdetermined conditions.

[0091] Figure 7 It is the relationship curve between the mean square error of each estimated parameter and the signal-to-noise ratio of the method proposed by the present invention under overdetermined conditions. Detailed implementation manners

[0092] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings.

[0093] In order to solve the problems such as high computational complexity and limited array degrees of freedom in the existing joint estimation methods for two-dimensional direction of arrival and polarization parameters, the present invention provides a closed-form joint estimation method for two-dimensional direction of arrival and polarization parameters based on a cascaded sparse multi-polarization planar array, so as to achieve a larger array aperture with fewer array elements, improve the array degrees of freedom, and improve the computational efficiency of the joint estimation method for two-dimensional direction of arrival and polarization parameters.

[0094] Refer to Figure 1 , the implementation steps of the present invention are as follows:

[0095] Step 1: Construct a cascaded sparse multi-polarization planar array. In order to ensure the polarization information processing ability of the array, expand the array aperture and reduce the mutual coupling effect between array elements when the number of array elements is certain, a cascaded sparse multi-polarization planar array composed of N p ≥2 sparse sub-arrays in cascade is constructed, the distance between adjacent sub-arrays is d, and each sub-array is composed of L0N m magnetic rings (whose normal directions are parallel to a certain coordinate axis) or dipoles (whose axial directions are parallel to a certain coordinate axis) with the same polarization type, and L0N m array elements are all composed of N m ≥2 mutually parallel layers, the distance between adjacent layers is d x , the number of array elements in each layer is L0, and the sparse arrangement patterns of the array elements in each layer are the same. A specific virtual co-array without holes and fully augmentable sparse array can be selected, and these sparse arrays can be selected as the minimum redundancy array, nested array, super nested array. Then, each layer of the cascaded sparse multi-polarization planar array has L = L0N p array elements. In the virtual uniform array deduced by each sub-array, the distance between adjacent array elements in each layer is d, and the number of array elements is L V . Thus, the position of each array element in the n p th sub-array and the n m th layer can be expressed as a position coordinate matrix of L0×2 dimensions:

[0096]

[0097] where \(n\) p = 1, 2, …, \(N\) p , \(n\) m = 1, 2, …, \(N\) m , \(v\) 1,1 is an \(L_0\times1\) vector representing the \(y\)-axis coordinates of the 1st to \(L_0\)th array elements in the 1st layer of the 1st subarray, is an \(L_0\times1\) all-ones vector. When taking \(L_0 = 3\) and \(v\) 1,1 = [0d3d] T , the constructed cascaded sparse multi-polarization planar array structure is as Figure 2 shown, where [·] T represents the transpose operation.

[0098] Step 2: Modeling the received signals of the cascaded sparse multi-polarization planar array. Suppose there are \(M\) uncorrelated far-field narrowband signal sources incident on the cascaded sparse multi-polarization planar array designed in Step 1. \(\theta\) m , \(\varphi\) m respectively represent the azimuth angle and elevation angle of the \(m\)th signal source, \(\gamma\) m , \(\eta\) m respectively represent the polarization auxiliary angle and polarization phase difference of the \(m\)th signal source, where \(m = 1, 2, …, M\). Then the received signal \(x(t)\) of the designed cascaded sparse multi-polarization linear array at time \(t\) can be modeled as:

[0099]

[0100] where \(s\) m (t) represents the waveform corresponding to the \(m\)th signal source, \(n(t)\) is a Gaussian white noise component with zero mean and independent of each signal source, and the spatial and polarization domain joint domain steering vector of the designed sparse multi-polarization array can be expressed as

[0101]

[0102] where \(a\) h (\(\theta\) m , \(\varphi\) m ), \(a\) v (\(\theta\) m , \(\varphi\) m ) respectively represent the horizontal polarization steering vector and the vertical polarization steering vector corresponding to the signal with the incoming wave direction of (\(\theta\) m , \(\varphi\) m ), represents the polarization vector corresponding to the \(m\)th signal (including the parameters \(\cos\gamma\) corresponding to the horizontal polarization component and vertical polarization separation m and blkdiag[B s×t , C p×qdenotes a diagonal block matrix constructed from the matrix within the brackets, i.e.,

[0103]

[0104] where Β s×t and C p×q represent an s×t dimensional matrix and a p×q dimensional matrix respectively, and O a×b represents an a×b dimensional zero matrix.

[0105] The spatial and polarization domain joint domain steering vector of the designed cascaded sparse multi-polarization planar array can be further expressed as

[0106]

[0107] where a s,m is the pure spatial domain steering vector of the designed cascaded sparse multi-polarization planar array that is only related to the spatial domain parameters and can be expressed as

[0108]

[0109] where represents the pure spatial domain steering vectors of the 1st, 2nd, …, N m layers of the designed cascaded sparse multi-polarization planar array, λ represents the signal wavelength, d l , l = 1, 2, …, L represents the distance from the l-th array element in each layer to the origin along the y-axis direction, d1 = 0, diag(a) represents the diagonal matrix generated based on the elements in vector a, represents the polarization domain steering vector of the designed cascaded sparse multi-polarization planar array, D is an N m L×N p dimensional matrix, and each row has exactly one element equal to 1 and the rest of the elements are 0. It characterizes the polarization type selection method of all array elements of the planar array after stacking the array elements layer by layer to form N m L array elements. The designed cascaded sparse multi-polarization planar array has N p sub-arrays, [D] a,b = 1 indicates that the a-th array element belongs to the polarization type of the b-th sub-array, a = 1, 2, …, N m L, b = 1, 2, …, N p , [D] a,b represents the element in the a-th row and b-th column of matrix D. D can be block-represented by layer as

[0110]

[0111] where is an L×N p dimensional matrix, and D can also be block-represented by array element as

[0112]

[0113] where d (l) is an N p ×1 dimensional vector, [d (l) b = 1 indicates that the l-th array element belongs to the polarization type of the b-th sub-array, l = 1, 2, …, N m L, b = 1, 2, …, N p , [d (l) b represents the b-th element of the vector d (l) ; J is an N p ×6 dimensional selection matrix, with exactly one element equal to 1 in each row and the remaining elements equal to 0, which represents the array elements of N p sub-arrays selected from 3 dipoles with axes parallel to the x, y, and z axes and 3 magnetic loop antennas with normals parallel to the x, y, and z axes. Specifically, represents that the n p -th sub-array element is a dipole with its axial direction parallel to the x-axis; represents that the n p -th sub-array element is a dipole with its axial direction parallel to the y-axis; represents that the n p -th sub-array element is a dipole with its axial direction parallel to the z-axis; represents that the n p -th sub-array element is a magnetic loop with its normal direction parallel to the x-axis; represents that the n p -th sub-array element is a magnetic loop with its normal direction parallel to the y-axis; represents that the n p -th sub-array element is a magnetic loop with its normal direction parallel to the z-axis. is a 6×2 dimensional matrix characterizing the polarization types of 6 multi-polarization array elements:

[0114]

[0115] Step 3: Perform multi-dimensional parameter separation and decoupling on the received signals of the cascaded sparse multi-polarization planar array. Denote α m , β m as the angles between the direction of arrival of the m-th signal source and the y-axis and the x-axis respectively. Then θ m , φ m , α m and β m satisfy the following relationship:

[0116] sinφ m sinθ m = cosα m ​​, sinφ m cosθ m = cosβ m ,

[0117] Then the spatial domain steering vector a s,m can be rewritten as where In addition, according to the mathematical structure of the polarization domain steering vector and its relationship with a s,m the joint domain steering vector of the spatial domain and the polarization domain can be rewritten as

[0118]

[0119] where I n represents the n×n dimensional identity matrix, represents the Kronecker product, The parameter is only related to the spatial domain parameters θ m , φ m and is only related to the antenna polarization type of the multi-polarization planar array, is only related to the polarization parameters. Therefore, through the above operations, the spatial domain parameters, the remaining angle parameters related to the antenna polarization characteristics, and the polarization parameters are decoupled.

[0120] Step 4: Block smoothing processing of the covariance matrix of the received signals of the cascaded sparse multi-polarization planar array. For the convenience of the joint domain smoothing processing operation in the spatial and polarization domains, and further to achieve the closed-form joint estimation of the two-dimensional direction of arrival and the polarization parameters, it is necessary to first represent the covariance matrix of the received signals of the cascaded sparse multi-polarization planar array in a block form according to the multi-subarray structure of the cascaded sparse multi-polarization planar array. The spatial domain steering vectors of each layer of the designed cascaded sparse multi-polarization planar array can be further represented in a block form as:

[0121]

[0122]

[0123]

[0124] where the spatial domain steering vectors belonging to different layers and different subarrays can be expressed as

[0125]

[0126]

[0127]

[0128]

[0129]

[0130]

[0131]

[0132]

[0133]

[0134] Then the received signal covariance matrix \(R\) of the cascaded sparse multi-polarization array xx can be expressed as:

[0135]

[0136] where represents the power of the \(m\)-th signal source, \(\sigma\) 2 represents the noise power, \((\cdot)^{H}\) H represents the conjugate transpose operation, is an \(L\times L\) matrix, \(p,q = 1,2,\cdots,N\) m , \(R\) xx is partitioned into sub-arrays. Next, the transformation of \(R\) xx can also be divided into steps and carried out according to the following method:

[0137]

[0138] where can be expressed as

[0139]

[0140] where \(D\) (1) \((1:L0:end,:)\) represents the sub-matrix composed of the elements in the 1st, \(1 + L0,\cdots,[1+(N\) (1) - 1)\(L0]\) rows of \(D\), \((\cdot)^{*}\) p represents the conjugate operation. *

[0141] can be further simplified to

[0142]

[0143] where \(\rho\) 11,m =\(\delta\) 11,m, ρ 22,m = δ 22,m , …,

[0144] Therefore, R xx can be partitioned into (N m N p ) 2 sub - matrices, which can be expressed as

[0145] A(m1:m2,n1:n2) represents the sub - matrix composed of the elements at the intersection of the m1 - to - m2 rows and the n1 - to - n2 columns of matrix A. Since each sub - matrix has a data structure similar to the received signal covariance matrix of each hierarchical sub - array that composes the cascaded sparse multi - polarized array, subsequent joint spatial and polarization domain smoothing processing in a partitioned form is very easy to operate. In practical situations, R xx can be approximately calculated based on K sampling snapshots, that is:

[0146]

[0147] where t k represents the time corresponding to the k - th sampling snapshot.

[0148] Next, perform spatial and polarization joint domain smoothing processing operations on each partitioned sub - matrix of the proposed array covariance matrix. First, perform a vectorization operation on to obtain the vector

[0149]

[0150] where, represents the vectorization operation on matrix , that is, stacking the columns of matrix in sequence to form a new vector, represents the pure spatial domain steering matrix, represents the Khatri - Rao product, Since the positions of the sub - array elements correspond to a fully expandable sparse array, the virtual array corresponding to the vector can be represented as a uniform array containing 2L V - 1 virtual continuous elements To re - order the elements in the vector into the equivalent virtual signals corresponding to , define the - dimensional selection matrix Δ:

[0151]

[0152] where \(l' = 1,\cdots,2L\) V -1, \(L_0 - 1\geq i_1\geq0\), \(L_0\geq i_2\geq1\), \(\omega(l' - L\) V , \(i_1\), \(i_2)\) represents a function of \(l' - L\) V , \(i_1\) and \(i_2\) (representing the number of index combinations where the value combinations of \(i_1\) and \(i_2\) satisfy . Next, based on the selection matrix \(\Delta\), the elements in the vector are reordered to the equivalent virtual signal corresponding to .

[0153]

[0154] where is the pure spatial domain steering vector obtained by removing the polarization components from the virtual uniform array corresponding to each subarray , represents a \(2L\) V -1-dimensional column vector with only the \(L\) V -th element being 1 and the remaining elements being 0.

[0155] Step 5: Construct the covariance matrix for fitting the received signal space and polarization joint domain of the virtual cascaded uniform multi-polarization planar array. First, the virtual vector is successively decomposed into \(L\) V \(L\) V ×1-dimensional virtual sub-vectors:

[0156]

[0157] where Then represents the vector composed of the first to the \(L\) V elements in represents the vector composed of the second to the \(L\) V +1 elements in represents the vector composed of the \(L\) V -th to the \(2L\) V -1 elements in V For the \(L\) a column vector merging operation is performed to obtain the auto-correlation / cross-correlation matrix corresponding to the received signals of each layer and each subarray of the virtual equivalent cascaded uniform multi-polarization planar array:

[0158]

[0159] where Represents the pure spatial domain steering vector of the first sub-array in the first layer of the virtual equivalent cascaded uniform multi-polarization planar array. It should be noted that, for the convenience of joint estimation of two-dimensional direction of arrival and polarization parameters, the is the autocorrelation / cross-correlation matrix of the received signals of different sub-arrays in different layers of the virtual equivalent cascaded uniform multi-polarization planar array. Then, the corresponding covariance matrix of the received signals of the virtual equivalent cascaded uniform multi-polarization planar array

[0160]

[0161] where can be expressed as

[0162]

[0163] can also be expressed as

[0164]

[0165] where

[0166]

[0167]

[0168]

[0169] Then, the covariance matrix of the received signals of the cascaded sparse multi-polarization planar array can be reconstructed as the covariance matrix of the received signals of its corresponding virtual equivalent uniform multi-polarization planar array

[0170]

[0171] The spatial and polarization joint domain steering vector of the virtual equivalent uniform multi-polarization planar array can be expressed as:

[0172]

[0173] where N p L V ×N p dimensional matrix Each row has and only has one element as 1, and the rest of the elements are 0, which characterizes the polarization type selection method of each element in the first layer of the virtual equivalent uniform multi-polarization planar array. The virtual equivalent uniform multi-polarization planar array has N p sub-arrays, represents the polarization type of the a-th element belonging to the b-th sub-array, a = 1, 2,..., N p LV , b = 1, 2, …, N p . The covariance matrix corresponds to a virtual equivalent cascaded uniform multi-polarization planar array with the structure shown in Figure 3 . The reconstructed covariance matrix contains the decoupled data information of horizontal polarization parameters and vertical polarization parameters, as well as the separated data information of spatial domain parameters and polarization parameters, thus facilitating the subsequent closed-form joint estimation of two-dimensional direction of arrival and polarization parameters.

[0174] Step 6: Perform eigen-decomposition on the covariance matrix fitted in the virtual space and polarization joint domain to obtain the estimation of the angle between the direction of arrival of each signal source and a single coordinate axis. By introducing a multi-dimensional joint sparse representation method, a deep learning method, a polarization multiple signal classification method, a polarization subspace rotation invariant method, or a polarization multiple signal classification root finding method, the estimation of α m after separation and decoupling can be obtained. Taking the polarization multiple signal classification root finding method as an example, first perform eigen-decomposition on and arrange its corresponding eigenvalues from largest to smallest. Take the subspace spanned by the eigenvectors corresponding to the last L V N m N p -M smaller eigenvalues and denote it as the noise subspace According to 's expression and the orthogonality relationship between the virtual array steering vector and the noise subspace , it can be known that:

[0175]

[0176] Furthermore, according to the rank deficiency principle, it can be deduced that

[0177]

[0178] where det(·) represents the determinant operation, thus transforming the four-dimensional parameter estimation into the estimation of α m first. Then there is the following equation:

[0179]

[0180] where z is a variable, and the solution of this equation is Therefore, the estimated value of the angle α m between each signal source and the y-axis can be obtained by a method based on polynomial root finding, and the polynomial coefficients can be obtained based on the matrix determinant calculation principle: can be written as

[0181]

[0182] Among them

[0183]

[0184]

[0185]

[0186] Then the following equation holds:

[0187]

[0188] Among them, P is an L V N m N p ×L V N m N p dimensional matrix,

[0189]

[0190] Then P is partitioned into (N m N p ) 2 submatrices, and then we can obtain

[0191]

[0192] Among them

[0193] can be respectively expressed as

[0194] Among them

[0195]

[0196] And the 2L V -1 dimensional vector c pq , p, q = 1, 2,..., N p N m can be obtained by the following formula:

[0197] [c pq u = Σdiag(P pq , u - L V ),

[0198] where u = 1, 2,... 2L V -1, Σdiag(X, n) represents the sum of the elements on the nth diagonal of matrix X, then the above determinant equation can be rewritten as​

[0199]

[0200] wherein, Based on the obtained c pq , p, q = 1, 2, …, N p N m and obtained by the Laplace theorem for solving the matrix determinant.

[0201] The angle α between each signal source and the y-axis m The closed-form solution is given by the following formula:

[0202] α m = arccos[λ(∠z m ) / 2πd],

[0203] where ∠(·) represents the phase of the complex number.

[0204] Step Seven: Obtain the estimated angle between the direction of arrival of each signal source and another coordinate axis, and obtain the closed-form estimates of the azimuth angle, elevation angle, and polarization parameters. Next, perform eigenvalue decomposition on R xx and arrange its corresponding eigenvalues from largest to smallest. Denote the subspace spanned by the eigenvectors corresponding to the last N m L - M smaller eigenvalues as U. According to the structure and rank deficiency principle of the joint spatial and polarization domain steering vector , the matrix The eigenvector h corresponding to the minimum eigenvalue m can be expressed as

[0205]

[0206] where h m is a non-zero constant. Using all the information contained in h m , the closed-form solution of β m can be obtained:

[0207]

[0208] It can be seen that the above steps transform the estimation of the azimuth angle θ m and elevation angle φ m into the estimation of α m and β m . Furthermore, according to the relationship between θ m , φ m and α m , β m :

[0209] sinφ m sinθ m = cosα m , sinφm cosθ m = cosβ m ,

[0210] Then the closed - form solutions of θ m and φ m can be obtained without the pairing process.

[0211] Matrix

[0212]

[0213]

[0214] The eigenvector v corresponding to the minimum eigenvalue m can be expressed as

[0215]

[0216] where v m is a non - zero constant. Therefore, the polarization parameters γ m , η m can be obtained with the closed - form solutions:

[0217]

[0218]

[0219] Next, the effects of the present invention will be further described in combination with simulation examples.

[0220] Simulation example: The proposed cascaded sparse multi - polarization array is used to receive the incident signals. Each incident signal is a complex Gaussian white signal with a mean of zero, and d = d x = λ / 2.

[0221] First, consider the analysis of the mutual coupling effect. Take N m = 2, N p = 3, v 1,1 = [0 d 2d 5d 8d] T , J = blkdiag(I2, [0 1 0 0]), d (1) = … = d (5) = [1, 0, 0] T , d (6) = … = d (10) = [0, 1, 0] T , d (11) = … = d (15) = [0, 0, 1] T , assuming there are 3 incident signals. For simplicity and without loss of generality, assume that the polarization auxiliary angles and polarization phase differences (γ, η) of all incident signals are (45°, 0°). Let a0 be the assumed steering vector, is the actual steering vector obtained through full-wave electromagnetic simulation. Here, the cosine value of the angle between the two vectors is calculated, where ||·|| represents the Euclidean norm. As can be seen from Figure 4 , the cosine value of the angle of the cascaded sparse multi-polarization planar array proposed in the present invention is always close to 1, while the cosine value of the angle of the multi-polarization antenna array placed at the same point is always less than 1, indicating that compared with the multi-polarization array placed at the same point, the array proposed in the present invention effectively reduces the mutual coupling effect between array elements.

[0222] Next, consider the case where the number of incident signals is underdetermined. Let N m = N p = 2, v 1,1 = [0d 2d 5d 8d] T , J = [I2, O 2×4 , d (1) =... = d (5) = [1, 0] T , d (6) =... = d (10) = [0, 1] T . Assume that there are 11 signal sources incident, and the azimuth and elevation angles (θ, φ) are (12° + 15°r1, 30° + 10°r1), r1 = 0, 1, 2, (166° - 15°r2, 13° + 10°r2), r2 = 0, 1, 2, 3, 4, (58°, 55°), (73°, 65°), and (98°, 70°). The polarization auxiliary angle and polarization phase difference (γ, η) of the first 7 signals are both (45°, 90°), and the polarization auxiliary angle and polarization phase difference (γ, η) of the last 4 signals are both (45°, -90°); the signal-to-noise ratio is set to 30 dB, and the number of sampling snapshots is 500000. The relatively large number of snapshots set here is used to fit the true covariance matrix under underdetermined conditions. The results of the two-dimensional direction of arrival automatic pairing closed-form estimation are as Figure 5 shown, where 200 Monte Carlo experiments are conducted in total. For traditional methods such as root multi-signal classification in the polarization element space, an array composed of 12 antennas with different polarization types can estimate at most 10 signal sources, while the method proposed in the present invention can estimate up to 14 signal sources, achieving an increase in the degrees of freedom of the array.

[0223] Finally, consider the case where the number of incident signals is overdetermined. The array structure is the same as in the overdetermined case. Assume that there are 3 signal sources incident, and the azimuth and elevation angles (θ, φ) are (10°, 25°), (30°, 40°), (80°, 60°) respectively; the polarization auxiliary and polarization phase difference (γ, η) are (85°, -80°), (25°, 30°), (65°, -5°) respectively; the signal-to-noise ratio is set to 20 dB, and the number of sampling snapshots is set to 500. The results of the closed-form joint estimation of the two-dimensional direction of arrival and polarization parameters are as Figure 6 shown, where 200 Monte Carlo experiments were conducted in total. The curve of the mean square error versus the signal-to-noise ratio is as Figure 7 shown, where the number of sampling snapshots is set to 500, and 5000 Monte Carlo experiments were conducted in total. From Figure 6 the results of the two-dimensional direction of arrival and polarization parameter estimation in and Figure 7 the small gap between the mean square error of each parameter estimation and the Cramer-Rao bound in, it can be seen the closed-form automatic pairing property and effectiveness of the method proposed in the present invention. In addition, the average calculation time of each experiment of the method proposed in the present invention is 5.21 milliseconds, where the simulation platform used is MATLAB 2020a, and the central processing unit is Intel(R) Core(TM) i7-10875H 2.30 GHz; while when the search ranges of the azimuth angle θ and the elevation angle φ are both from 0° to 90°, and the search intervals are both 0.2°, the traditional two-dimensional spectral peak search method takes 1.89 seconds on average for each experiment, showing the high calculation efficiency of the method proposed in the present invention.

[0224] In summary, the present invention constructs a brand-new cascaded sparse multi-polarization planar array based on the polarization diversity of antennas and using the idea of multi-subarray cascading, reduces the mutual coupling effect between array elements, increases the degrees of freedom of the array, and enables the array to have polarization diversity. Moreover, based on the multi-domain subarray smoothing processing technology, it realizes the conversion of the output covariance matrix of the cascaded sparse multi-polarization planar array to the output covariance matrix of the virtual cascaded uniform multi-polarization planar array. At the same time, a polynomial root-finding method for the joint estimation of the two-dimensional direction of arrival and polarization parameters is proposed, a closed-form solution method based on polynomial root-finding is given, and the polynomial coefficients are derived, realizing the automatic pairing of the two-dimensional direction of arrival and polarization parameter estimation of each signal source, and achieving the accurate estimation of the two-dimensional direction of arrival and polarization parameters while improving the calculation efficiency.

Claims

1. A closed-form joint estimation method for two-dimensional direction of arrival and polarization parameters based on a cascaded sparse multi-polarization planar array, characterized in that It includes the following steps: (1) Construct a cascaded sparse multi-polarization planar array; (2) Model the received signals of the cascaded sparse multi-polarization planar array; (3) Perform multi-dimensional parameter separation and decoupling on the received signals of the cascaded sparse multi-polarization planar array; (4) Perform block smoothing processing on the covariance matrix of the received signals of the cascaded sparse multi-polarization planar array; (5) Construct a covariance matrix for fitting the spatial and polarization joint domain of the received signals of a virtual cascaded uniform multi-polarization planar array; (6) Perform eigenvalue decomposition on the covariance matrix for fitting the virtual spatial and polarization joint domain, and obtain the estimation of the angle between the direction of arrival of each signal source and the single coordinate axis; (7) Obtain the closed-form estimation of the azimuth angle, elevation angle, and polarization parameters. Specifically, for the covariance matrix R of the received signals of the cascaded sparse multi-polarization planar array xx perform eigenvalue decomposition, and arrange the corresponding eigenvalues in descending order. Take the subspace spanned by the eigenvectors corresponding to the last N m L - M smaller eigenvalues and denote it as U. The matrix The eigenvector h corresponding to the minimum eigenvalue m Denoted as where M is the number of non-signal sources, and θ m , φ m represent the azimuth angle and elevation angle of the m-th signal source respectively, and γ m , η m represent the polarization auxiliary angle and polarization phase difference of the m-th signal source respectively; The cascaded sparse multi-polarization planar array is composed of N p ≥2 cascaded sparse sub-arrays. Each sub-array is composed of L0N m magnetic rings or dipoles with the same polarization type, and L0N m array elements are all composed of N m ≥2 parallel layers. The distance between adjacent layers is d x , and the number of array elements in each layer is L0. Each layer of the cascaded sparse multi-polarization planar array has L = L0N p array elements; represents the N m ×N m dimensional identity matrix, D (1) is an L×N p dimensional matrix, represents the pure spatial domain steering vector of the first layer of the cascaded sparse multi-polarization planar array, h' m is a non-zero constant, J is an N p ×6 dimensional selection matrix, is a 6×2 dimensional matrix characterizing the polarization types of 6 multi-polarization array elements, represents the polarization vector corresponding to the m-th signal, λ represents the signal wavelength; Obtain the closed-form solution of β m : Furthermore, based on θ m , φ m and α m , β m relationship sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m , Namely, the closed-form solutions of θ m and φ m can be obtained without a pairing process; the matrix The eigenvector v corresponding to the minimum eigenvalue m Denoted as where v' m is a non-zero constant; obtaining the closed-form solutions of the polarization parameters γ m , η m :

2. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 1, wherein, In step (1), a cascaded sparse multi-polarization array is constructed as follows: construct a cascaded sparse multi-polarization array composed of N p ≥ 2 cascaded sparse sub-arrays, the distance between adjacent sub-arrays is d, and each sub-array is composed of L0N m magnetic rings or dipoles with the same polarization type. The normal direction of the magnetic ring is parallel to a certain coordinate axis, and the axial direction of the dipole is parallel to a certain coordinate axis. And L0N m array elements are all composed of N m ≥ 2 parallel layers, the distance between adjacent layers is d x , the number of array elements in each layer is L0, then each layer of the cascaded sparse multi-polarization array has a total of L = L0N p array elements; the sparse arrangement of the array elements in each layer is the same fully expandable sparse array; The number of virtual array elements corresponding to each layer of each sub-array is L V , and the spacing between adjacent virtual array elements is d.

3. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 2, characterized in that, In step (1), the sparse arrangement modes of each layer of array elements are selected as the minimum redundant array, nested array, and super nested array without holes in the virtual domain difference array; the number of virtual array elements corresponding to each layer of each sub-array is L V , the n p th sub-array, the positions of array elements in the n m th layer are represented as a position coordinate matrix of L0×2 dimensions: where n p = 1, 2, …, N p , n m = 1, 2, …, N m , v 1,1 is an L0×1 vector representing the y-axis coordinates of the first to the L0th array elements in the first layer of the first subarray, is an L0×1 all-ones vector.

4. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 2, characterized in that, In step (2), the modeling of the received signals of the cascaded sparse multi-polarization planar array is specifically as follows: Suppose there are M uncorrelated far-field narrowband signal sources incident on the designed cascaded sparse multi-polarization planar array, and θ m , φ m represent the azimuth angle and elevation angle of the m-th signal source respectively, and γ m , η m represent the polarization auxiliary angle and polarization phase difference of the m-th signal source respectively, where m = 1, 2, …, M. Then, the received signal x(t) of the designed cascaded sparse multi-polarization planar array at time t is modeled as: where s m (t) represents the waveform corresponding to the m-th signal source, n(t) is a Gaussian white noise component with zero mean and independent of each signal source, and the spatial and polarization domain joint domain steering vector of the designed cascaded sparse multi-polarization array is expressed as where a h (θ m , φ m ), a v (θ m , φ m ) represent the horizontally polarized steering vector and the vertically polarized steering vector respectively, corresponding to the signal with the incoming wave direction of (θ m , φ m ), represents the polarization vector corresponding to the m-th signal, blkdiag[B s×t , C p×q represents the block diagonal matrix constructed by the matrices in the brackets, that is: where Β s×t , C p×q represent an s×t dimensional matrix and a p×q dimensional matrix respectively, and O a×b represents an a×b dimensional zero matrix; Spatial and Polarization Domain Joint Domain Steering Vector of the Designed Cascade Sparse Multipolarization Array It is further expressed as where a s,m is the pure spatial domain steering vector related only to the spatial domain parameters of the designed cascaded sparse multi-polarization planar array, expressed as where [·] T represents the transpose operation, represents the pure spatial domain steering vectors of the 1st, 2nd, …, N m layers of the designed cascaded sparse multi-polarization planar array, λ represents the signal wavelength, d l , l = 1, 2, …, L represents the distance of the l-th element in each layer from the coordinate origin along the y-axis direction, d1 = 0, diag(a) represents the diagonal matrix generated based on the elements in vector a, represents the polarization domain steering vector of the designed cascaded sparse multi-polarization planar array, D is an N m L×N p dimensional matrix, with exactly one element equal to 1 in each row and the remaining elements equal to 0, which characterizes the polarization type selection method of all elements of the planar array after stacking the elements layer by layer to form N m L elements. The designed cascaded sparse multi-polarization planar array has N p sub-arrays, [D] a,b = 1 indicates that the a-th element belongs to the polarization type of the b-th sub-array, a = 1, 2, …, N m L, b = 1, 2, …, N p , [D] a,b represents the element in the a-th row and b-th column of matrix D. D is block-diagonalized by layer as wherein is an L×N p dimensional matrix, and D is represented in terms of element blocks as where d (l) is an N p ×1 dimensional vector, [d (l) b = 1 indicates that the l-th array element belongs to the polarization type of the b-th sub-array, l = 1, 2, …, N m L, b = 1, 2, …, N p , [d (l) b represents the b-th element of the vector d (l) ; J is an N p ×6 dimensional selection matrix, with exactly one element equal to 1 in each row and the remaining elements equal to 0, which represents the array elements of N p sub-arrays selected from 3 dipoles with axes parallel to the x, y, and z axes and 3 magnetic loop antennas with normal directions parallel to the x, y, and z axes, is a 6×2 dimensional matrix characterizing the polarization types of 6 multi-polarization array elements:​​ 5. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 4, characterized in that In step (2), N p The 6 - dimensional selection matrix J is constructed as follows: Different column elements of J being 1 represent dipoles and magnetic rings with different polarization characteristics. Specifically, n p = 1, 2, …, N p indicates that the n p -th sub - array element is a dipole with its axial direction parallel to the x - axis; n p = 1, 2, …, N p indicates that the n p -th sub - array element is a dipole with its axial direction parallel to the y - axis; n p = 1, 2, …, N p indicates that the n p -th sub - array element is a dipole with its axial direction parallel to the z - axis; n p = 1, 2, …, N p indicates that the n p -th sub - array element is a magnetic ring with its normal direction parallel to the x - axis; n p = 1, 2, …, N p indicates that the n p -th sub - array element is a magnetic ring with its normal direction parallel to the y - axis; n p = 1, 2, …, N p indicates that the n p -th sub - array element is a magnetic ring with its normal direction parallel to the z - axis.

6. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 4, characterized in that, In step (3), the multi-dimensional parameter separation and decoupling of the received signals of the cascaded sparse multi-polarization planar array is specifically as follows: Denote α m and β m as the angles between the direction of arrival of the m-th signal source and the y-axis and the x-axis respectively. Then θ m and φ m , α m and β m satisfy the following relationships: sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m , Then the spatial domain steering vector a s,m is rewritten as where In addition, according to the mathematical structure of the polarization domain steering vector and its relationship with a s,m the joint domain steering vector of the spatial domain and the polarization domain is rewritten as where I n represents an n×n dimensional identity matrix, represents the Kronecker product, parameter is only related to the spatial domain parameter θ m , φ m , and is only related to the antenna polarization type of the multi-polarization planar array, is only related to the polarization parameter. Therefore, through the above operations, the spatial domain parameter, the remaining angular parameters related to the antenna polarization characteristics, and the polarization parameter are decoupled.

7. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 6, characterized in that In step (4), the block smoothing processing of the covariance matrix of the received signals of the cascaded sparse multi-polarization planar array is specifically as follows: Spatial steering vectors of each layer of the designed cascaded sparse multi-polarization planar array It is further block-represented as: The spatial steering vectors belonging to different layers and different sub-arrays are represented as Then the received signal covariance matrix \(R\) of the cascaded sparse multi-polarization planar array xx is expressed as: Among them, represents the power of the m-th signal source, and σ 2 represents the noise power, and (·) H represents the conjugate transpose operation. is an L×L dimensional matrix, where p, q = 1, 2, …, N m , R xx is partitioned into submatrices. Next, the transformation of R xx is divided into steps and carried out according to the following method: wherein is represented as Among them D (1) (1:L0:end,:) represents the submatrix composed of the elements in the 1st, 1+L0, …, [1+(N (1) -1)L0] rows of D, and (·) p represents the conjugate operation; * denotes the conjugate operation; is further simplified to where ρ 11,m = δ 11,m , R xx is block - represented as p,q = 1,2,…,N m , u,v = 1,2,…,N p , and A(m1:m2,n1:n2) represents the sub - matrix composed of the elements at the intersection of the m1 - to - m2 rows and the n1 - to - n2 columns of matrix A; in practice, R xx is approximately calculated according to K sampling snapshots, that is: where \(t\) k represents the moment corresponding to the \(k\)-th sampling snapshot; Next, perform spatial and polarization joint-domain smoothing processing operations on each sub-block matrix of the proposed array covariance matrix. First, perform a vectorization operation on to obtain a vector Among them, represents the vectorization operation on the matrix That is, stacking the columns of the matrix in sequence to form a new vector, represents the pure spatial domain guiding matrix, represents the Khatri-Rao product, The vector The corresponding virtual array is represented as a uniform array containing 2L V -1 virtual continuous array elements Define dimensional selection matrix Δ: where [Δ] l′,l represents the element in the \(l'\)-th row and \(l\)-th column of the matrix Δ, \(l' = 1,\ldots,2L\) V -1, \(l = i_1L_0 + i_2\), \(L_0 - 1\geq i_1\geq0\), \(L_0\geq i_2\geq1\), \(\omega(l' - L V ,i_1,i_2)\) represents \(l' - L V 、a function of \(i_1\) and \(i_2\), representing the number of index combinations where the value combinations of \(i_1\) and \(i_2\) satisfy ; based on the selection matrix Δ, the elements in the vector are reordered to the equivalent virtual signal corresponding to ​ wherein is the virtual uniform array corresponding to each subarray is the pure spatial domain steering vector obtained by removing the polarization component from denotes a 2L V -dimensional column vector with only the L V -th element being 1 and the remaining elements being 0 8. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization planar array according to claim 7, characterized in that In step (5), the construction of the covariance matrix for fitting the spatial and polarization joint domain of the received signals of a virtual cascaded uniform multi-polarization planar array is specifically as follows: Decompose the virtual vector into L V L V ×1-dimensional virtual sub-vectors in sequence: where l″ = 1, 2, …, L V , then denotes the vector composed of the 1st to the Lth V elements in denotes the vector composed of the 2nd to the (L + 1)th V elements in denotes the vector composed of the Lth to the (2L - 1)th V elements in V ; for L V pieces of perform column vector merging operations to obtain the auto - correlation / cross - correlation matrix corresponding to the received signals of each sub - array in each layer of the virtual equivalent cascaded uniform multi - polarization planar array: wherein denotes the pure spatial domain steering vector of the first sub-array in the first layer of the virtual equivalent cascaded uniform multi-polarization planar array; and then obtain the corresponding received signal covariance matrix of the virtual equivalent cascaded uniform multi-polarization planar array wherein is represented as Equivalently expressed as where Then the covariance matrix \(R\) of the received signals of the cascaded sparse multi-polarization planar array xx is reconstructed into the covariance matrix of the received signals of its corresponding virtual equivalent uniform multi-polarization planar array Spatial and Polarization Joint Domain Steering Vector of Virtual Equivalent Uniform Multipolarization Planar Array It is expressed as: Among them N p L V ×N p dimensional matrix Each row has exactly one element equal to 1 and the remaining elements are all 0, which represents the polarization type selection method of each element in the first layer of the virtual equivalent uniform multi-polarization array. The virtual equivalent uniform multi-polarization array has a total of N p sub-arrays, represents the polarization type of the a-th element belonging to the b-th sub-array, where a = 1, 2, …, N p L V , b = 1, 2, …, N p .

9. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization planar array according to claim 8, wherein, In step (6), for the estimation of the angle between the direction of arrival of each signal source after separation and decoupling and the y-axis in the reference coordinate system, the following methods are adopted: multi-dimensional joint sparse representation method, deep learning method, polarization multiple signal classification method, polarization subspace rotation invariant method or polarization multiple signal classification root finding method.

10. The two-dimensional direction of arrival and polarization parameter closed-form joint estimation method based on a cascaded sparse multi-polarization array according to claim 8, characterized in that In step (6), the root MUSIC method is used to obtain the estimation of the angle between the direction of arrival of each signal source after separation and decoupling and the y-axis in the reference coordinate system. Specifically: First, perform eigen-decomposition on and arrange the corresponding eigenvalues in descending order. Take the subspace spanned by the eigenvectors corresponding to the last L V N m N p -M smaller eigenvalues and denote it as the noise subspace According to the expression of and the orthogonality relationship between the virtual array steering vector and the noise subspace , we get: Furthermore, according to the rank deficiency principle, it is deduced that where det(·) represents the determinant operation, and there is the following equation: Among them z is a variable, and the solution of this equation is Therefore, α m The estimated value of is obtained by a method based on polynomial root finding, and the polynomial coefficients are obtained based on the principle of matrix determinant calculation: Written as where Then the following equation holds: Where P is L V N m N p ×L V N m N p dimensional matrix Then P is partitioned into (N m N p ) 2 submatrices, and thus obtain Among them Respectively represented as wherein while 2L V -dimensional vector c pq , p, q = 1, 2, …, N p N m is obtained by the following formula: [c pq u = Σdiag(P pq , u - L V ),​ where \(u = 1, 2, \ldots, 2L\) V -1, and \(\sum \text{diag}(X, n)\) represents the sum of the elements on the \(n\)-th diagonal of matrix \(X\), then we get In the formula, According to the obtained c pq , p, q = 1, 2, …, N p N m Obtained by the Laplace theorem for solving the matrix determinant; α m The closed-form solution of is given by the following formula: α m = arccos[λ(∠z m ) / 2πd], where ∠(·) represents the phase of a complex number.

Images