Joint multidimensional parameter estimation method for sparse multipolarization arrays based on orthogonal magnetic rings and dipoles

Abstract

The present invention discloses a method for joint estimation of multidimensional parameters of a sparse multipolarization array based on orthogonal magnetic rings and dipoles. The method mainly solves the problem that existing technologies cannot realize joint processing of spatial and polarization domain information. The implementation steps are: constructing a sparse multipolarization array using orthogonal magnetic rings and dipole array elements; modeling the received signals of the proposed array and decoupling the horizontal polarization and vertical polarization received signals; performing multidimensional parameter separation on the decoupled signals; obtaining a virtual signal corresponding to a virtual domain equivalent uniform multipolarization array; reconstructing the virtual domain received signal covariance matrix; and performing joint estimation of two-dimensional direction of arrival and polarization parameters based on the reconstructed virtual domain received signal covariance matrix. The present invention effectively utilizes joint smoothing technology in the spatial and polarization domains to provide an effective solution for joint estimation of multidimensional parameters of a sparse multipolarization array, which can be used for passive detection and target positioning.

Description

Technical Field

[0001] The present invention belongs to the field of signal processing technology, and in particular relates to a parameter estimation method for a sparse multipolarization array. Specifically, the present invention is a multidimensional parameter joint estimation method for a sparse multipolarization array based on orthogonal magnetic rings and dipoles, which can be used for passive detection and target positioning. Background Art

[0002] Sparse arrays can achieve the effect of increasing the array aperture using fewer elements and have attracted widespread attention. Currently, most research focuses on the application of sparse linear arrays for one-dimensional direction-of-arrival estimation, such as minimum redundancy arrays, coprime arrays, and nested arrays. Other representative studies focus on the application of sparse planar arrays for two-dimensional direction-of-arrival estimation, such as parallel coprime arrays, parallel nested arrays, L-shaped nested arrays, and concentric rectangular arrays. However, none of these existing methods consider the polarization diversity of the signal, and polarization mismatch, a common problem in practical applications, can lead to reduced parameter estimation accuracy.

[0003] In order to avoid polarization mismatch and achieve a larger array aperture and better parameter estimation effect with fewer array elements, the present invention proposes a new type of sparse multi-polarization array: a sparse multi-polarization array using orthogonal magnetic rings and dipoles. Specifically, the present invention uses magnetic rings and dipole array elements to receive the horizontal polarization component and the vertical polarization component of the signal respectively. However, existing research on multi-polarization arrays based on orthogonal magnetic rings and dipoles only focuses on the estimation of one-dimensional direction of arrival and polarization parameters, and does not consider the two-dimensional direction of arrival estimation problem that is closer to actual application scenarios, nor does it deeply explore the application of sparse multi-polarization arrays in the joint estimation problem of two-dimensional direction of arrival and polarization parameters. In addition, the existing spatial smoothing technology cannot be directly applied to the multi-dimensional parameter estimation method of sparse multi-polarization arrays of orthogonal magnetic rings and dipoles. It is urgent to explore the joint smoothing technology of spatial domain and polarization domain to reconstruct its corresponding virtual domain uniform multi-polarization array covariance matrix, so as to achieve the joint estimation of two-dimensional direction of arrival and polarization parameters. Summary of the Invention

[0004] The present invention aims to address the problem that existing spatial smoothing techniques cannot be directly applied to multidimensional parameter estimation for sparse multipolarization arrays of orthogonal magnetic rings and dipoles. By proposing a joint multidimensional parameter estimation method for sparse multipolarization arrays based on orthogonal magnetic rings and dipoles, this method, based on the proposed joint smoothing techniques in the spatial and polarization domains, provides a feasible approach and effective solution for the optimized design of sparse multipolarization arrays and the corresponding joint multidimensional parameter estimation.

[0005] The object of the present invention is achieved through the following technical solution: a method for joint estimation of multidimensional parameters of a sparse multipolarization array based on orthogonal magnetic rings and dipoles, comprising the following steps:

[0006] (1) A sparse multi-polarization array is constructed at the receiving end using orthogonal magnetic rings and dipole array elements. The sparse multi-polarization array consists of four parallel sub-arrays, where sub-arrays 1H and 2H are composed of magnetic rings, and sub-arrays 1V and 2V are composed of dipoles. Each sub-array is a fully scalable sparse array containing L0 array elements. Here, a fully scalable sparse array refers to a sparse array without holes in the derived virtual differential array. Sub-arrays 1V and 2V are located directly above sub-arrays 1H and 2H, respectively, with a distance of 0 < d. z ≤λ / 2, where λ represents the wavelength of the incident narrowband signal, i.e. the spacing between subarrays 1V and 1H and the spacing between subarrays 2V and 2H are both d z Subarrays 2H and 2V are located at a distance 0<d from subarrays 1H and 1V in the positive direction of the x-axis. x ≤λ / 2, that is, the distance between sub-array 2H and sub-array 1H and the distance between sub-array 2V and sub-array 1V are both d x ; The normal direction of the magnetic ring array element and the axial direction of the dipole array element in the array are both parallel to the z-axis;

[0007] (2) The received signal of the constructed orthogonal magnetic ring and dipole sparse multi-polarization array is modeled. The received signal x(t) of the array at time t is expressed as:

[0008]

[0009] where x 1H (t) and x 2H (t) represents the horizontal polarization component of the signal received by subarray 1H and subarray 2H at time t, respectively, and x 1V (t) and x 2V (t) represents the vertical polarization component of the signal received by subarray 1V and subarray 2V at time t, respectively, [·] T represents the transposition operation, M is the number of non-correlated far-field narrowband signal sources, represents the 4L0×2-dimensional angle-domain steering matrix corresponding to the m-th signal source of the designed sparse multi-polarization array:

[0010]

[0011] Among them, θ m 、φ m Respectively represent the azimuth and elevation angles of the mth signal source, m = 1, 2, ..., M, represents the spatial steering vector of subarray 1H, y l , l=1,2,…,L0, represents the distance between the lth array element and the first array element, y1=0, represents the L0×1 dimensional zero vector, The polarization vector corresponding to the mth signal source (including the parameter cosγ corresponding to the horizontal polarization component m and the parameters corresponding to the vertical polarization component ), γ m ,η m They represent the polarization auxiliary angle and polarization phase difference of the mth signal source, s m (t) represents the waveform corresponding to the mth signal source, n(t) is the Gaussian white noise component with zero mean and independent of each signal source;

[0012] To facilitate the guidance vector Characterize the sub-array 1H angle domain steering vector in and other parameter factors related to the angle parameter sinφ m 、 and the polarization vector After decoupling, the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array can be expressed as the following decoupling form of horizontal polarization and vertical polarization:

[0013]

[0014] Among them I n represents the n×n dimensional identity matrix, represents the Kronecker product, h m Indicates the delay factor and the horizontal polarization component parameter cosγ m and vertical polarization component parameters A 4×1 dimensional vector:

[0015]

[0016] The delay factor includes and

[0017] (3) In order to perform multi-dimensional parameter separation on the decoupled signal, define α m , β m are the angles between the arrival direction of the mth signal source and the y-axis and x-axis, so θ m 、φ m , α m and β m The following relationship is satisfied:

[0018] sinφ m sinθ m =cosα m , sinφ m cosθ m =cosβ m ;

[0019] Then the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array can be expressed as:

[0020]

[0021] The 4L0×4 dimensional matrix Defined as:

[0022]

[0023] is the spatial steering vector corresponding to subarray 1H The equivalent representation of is:

[0024]

[0025] Correspondingly, h m It can be equivalently expressed as:

[0026]

[0027] After the above operation, the orthogonal magnetic ring and the dipole sparse multi-polarization array receive the signal x(t) with the angle α m The associated block diagonal matrix It is separated out to facilitate the separation and solution of multi-dimensional parameters;

[0028] (4) To obtain the virtual signal corresponding to the virtual domain equivalent uniform multi-polarization array, first calculate the covariance matrix R of the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array xx :

[0029]

[0030] where E{·} represents the mathematical expectation, [·] H represents the conjugate transpose operation, represents the power of the mth signal source, σ 2 Indicates the noise power; in actual situations, R xx It can be approximately calculated based on K sampling snapshots, that is:

[0031]

[0032] based on This property, R xx The average block is 16 block matrices R with dimensions L0×L0 related to spatial and polarization domain parameters. pq :

[0033]

[0034] in 1≤p,q≤4, represents the matrix The element in the pth row and qth column of ; When p=1, 2, 3, 4, and p=1, R pp represents the autocorrelation of the received signal of subarray 1H. When p=2, R pp Represents the autocorrelation of the received signal of subarray 1V. When p=3, R pp represents the autocorrelation of the received signal of subarray 2H. When p=4, R pp represents the autocorrelation of the received signal of sub-array 2V, p,q=1,2,3,4 and p≠q, total R pq , represents the cross-correlation between the received signals of subarray 1H, subarray 1V, subarray 2H and subarray 2V; pq , p, q = 1, 2, 3, 4, perform vectorization operation to obtain vector r pq :

[0035]

[0036] Among them, vec(R pq ) represents the matrix R pq Vectorized operation, that is, the matrix R pq The columns in are stacked in sequence to form a new vector, a L0×M dimensional matrix represents the part of the steering matrix of each sub-array of the designed array that is only related to the parameter α, (·) * represents the conjugation operation, represents the Khatri-Rao product, Dimensional Matrix represents the virtual array steering matrix related only to the parameter α, an M×1 dimensional vector represents the parameter corresponding to the decoupling of α in the proposed array steering matrix, Since the position of each sub-array element corresponds to a fully scalable sparse matrix, the vector r pq The corresponding virtual array can be represented as a uniform array containing 2L-1 virtual continuous array elements 0<d≤λ / 2 represents the unit interval, The vector r pq The elements in the virtual uniform array The equivalent received signals corresponding to each array element are sorted into vector signals υ pq :

[0037]

[0038] where e 2L-1 represents a 2L-1 dimensional column vector, whose Lth element is 1 and the rest are zero. Expressed as:

[0039]

[0040] It corresponds to the spatial steering vector in the virtual uniform differential array of each sub-array, with the polarization component removed and only related to the angle parameter α;

[0041] (5) Reconstruct the virtual domain received signal covariance matrix; first, the virtual domain The equivalent received signal υ corresponding to each array element pq Decomposed into L L×1-dimensional virtual equivalent received signal sub-vectors:

[0042] υ pq，v =Ω v υ pq ,

[0043] where Ω v =[O L×(v-1) , I L , O L×(L-v) ], v = 1, 2, ..., L, Represents an m1×m2 dimensional all-zero matrix; for L υ pq，v By performing the column vector merging operation, the autocorrelation / cross-correlation matrix of the received signals of each sub-array of the virtual domain equivalent uniform multi-polarization array can be obtained:

[0044]

[0045] in Represents the one-dimensional angle parameter α m The spatial domain steering vector of subarray 1H in the virtual domain equivalent uniform multi-polarization array; the covariance matrix R of the received signal of the virtual orthogonal magnetic ring and the dipole uniform multi-polarization array υυ Can be refactored as:

[0046]

[0047] in Represents The corresponding 4L×4 dimensional block diagonal matrix;

[0048] (6) According to the reconstructed virtual domain equivalent uniform multi-polarization array received signal covariance matrix R υυ Solve the polynomial coefficients that fuse spatial and polarization domain information;

[0049] (7) Joint estimation of two-dimensional direction of arrival and polarization parameters is performed based on the polynomial root-finding principle and subspace orthogonality principle.

[0050] Furthermore, in step (1), a sparse multipolarization array is constructed using orthogonal magnetic rings and dipole array elements placed at non-co-points, specifically: the normal direction of the magnetic ring sparse linear sub-array 1H is parallel to the axial direction of the dipole sparse linear sub-array 1V; the normal direction of the magnetic ring of sub-array 2H is parallel to the axial direction of the dipole in sub-array 2V; the normal directions of sub-arrays 1H and 2H are both parallel to the z-axis, and the axial directions of sub-arrays 1V and 2V are both parallel to the z-axis; the array element positions of the four sub-arrays correspond to the same fully scalable sparse array, and the above four fully scalable sparse sub-arrays select the minimum redundancy array, nested array, super nested array, and augmented nested array.

[0051] Furthermore, in step (4), vector r pq The elements in are rearranged into the virtual signal υ corresponding to the virtual domain equivalent uniform multi-polarization array pq This can be achieved by selecting the matrix, specifically: first, R xx The average block is 16 block matrices R with dimensions L0×L0 related to spatial and polarization domain parameters. pq , then R pq Perform vectorization operation to obtain vector r pq =vec(R pq ); then, define Dimension selection matrix J:

[0052]

[0053] Where l′=1, 2, ..., 2L-1, 2L-1 represents the number of virtual continuous array elements in the differential array of each sub-array of the proposed array, , L0-1≥i≥0, L0≥j≥1, ω(l′-L, i, j) represents a function of l′-L, i and j, which represents the combination of i and j values ​​that satisfies y i -y j =(,′-L)d logarithm; For the constructed virtual domain equivalent uniform multi-polarization array, after the following spatial domain and polarization domain joint smoothing process related to the selection matrix J, the virtual signal vector υ corresponding to the autocorrelation / cross-correlation of the received signal of each subarray pq It can be expressed as:

[0054] υ pq =Jr pq .

[0055] Furthermore, in step (6), the covariance matrix R of the received signal of the reconstructed virtual domain equivalent uniform multi-polarization array is υυ Solve the polynomial coefficients that fuse spatial and polarization domain information. Specifically, for R υυPerform eigendecomposition and arrange the corresponding eigenvalues ​​from large to small. The subspace formed by the eigenvectors corresponding to the last 4L-M smaller eigenvalues ​​is recorded as the noise subspace U n According to R υυ The expression of virtual array steering vector and noise subspace U n The orthogonal relationship between Where det(·) represents the determinant operation; the matrix Divided into 16 4×4 blocks of sub-matrices P with dimensions of L×L pq =P[(p-1)L+1:pL,(q-1)L+1:qL], 1≤p,q≤4, and according to We can get:

[0056]

[0057] To solve the polynomial coefficients corresponding to the above determinant, each element in the 4×4 dimensional matrix in the above formula 1≤p,q≤4, re-expressed as:

[0058]

[0059] where r pq is a (2L-1)×1 dimensional vector, whose uth element can be expressed as r pq (u)=∑diag(P pq , uL), that is, for the matrix P pq Sum the elements in the uLth diagonal of , where uL=0 represents the elements in the main diagonal, uL>0 represents the elements in the diagonal above the main diagonal, and uL<0 represents the elements in the diagonal below the main diagonal. According to Laplace's theorem for calculating the determinant of a 4×4 matrix, the entire determinant can be re-expressed as:

[0060]

[0061] Where r is a (8L-7)×1 dimensional vector, corresponding to the polynomial coefficients of the above 4×4 dimensional matrix determinant, which can be specifically expressed as:

[0062]

[0063] Furthermore, in step (7), the two-dimensional direction of arrival and polarization parameters are jointly estimated based on the obtained polynomial coefficients and the polynomial root-finding principle and the subspace orthogonality principle. Specifically, according to the polynomial root-finding principle, z = e j2πd cosα / λ , for polynomials

[0064] r T [z-(4L-4) , z -(4L-5) ,…,1,…,z 4L-5 , z 4L-4 ] T =0,

[0065] Find the M roots whose modulus is within the unit circle and closest to the unit circle You can get α m The closed-form solution of :

[0066]

[0067] Where ∠(·) represents the phase of the complex number; according to the subspace orthogonality principle, the estimated α m Substitute in the block diagonal matrix and noise subspace U n Correlation matrix And solve the minimum eigenvalue corresponding to h m Parallel eigenvectors where h m is a non-zero constant; then according to step (3) h m The expression of β m The closed-form solution of :

[0068]

[0069] in Represents a vector The nth element of m 、φ m , α m and β m The relationship sinφ m sinθ m =cosα m , sinφ m cosθ m =cosβ m It can be seen that and The closed-form solution of can be obtained by the following formula: If cosα m ≠0, then

[0070] θ m =arccot(cosβ m / cosα m ), cosβ m / cosα m ≥0, cosβ m ≥0

[0071] θ m =arctan(cosα m / cosβm ) + π, cosβ m / cosα m ≥0, cosβ m <0

[0072] θ m =arctan(cosα m / cosβ m ) + π, cosβ m / cosα m <0, cosβ m ≥0

[0073] θ m =arctan(cosα m / cosβ m ),cosβ m / cosα m <0, cosβ m <0

[0074]

[0075] If cosβ m ≠0, then

[0076] θ m =arctan(cosα m / cosβ m ),cosα m / cosβ m ≥0, cosα m ≥0

[0077] θ m =arctan(cosα​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​<0, cosα m <0

[0080]

[0081] If cosα m =0 and cosβ m =0, then θ m =90° and φ m =0°;

[0082] Further according to step (3) h m The expression of can be used to obtain the closed-form solution of the polarization parameter:

[0083]

[0084]

[0085] in is the estimated value of the pitch angle of the mth signal source.

[0086] Compared with the prior art, the present invention has the following advantages:

[0087] (1) The present invention utilizes the orthogonality of magnetic ring and dipole array elements to construct a new orthogonal magnetic ring and dipole sparse multi-polarization array. Polarization decoupling is achieved by taking advantage of the characteristics of the magnetic ring and dipole receiving the horizontal polarization component and the vertical polarization component of the signal respectively. The parallel symmetry of the four sub-arrays of the designed array is used to achieve the separation of the multi-dimensional angle of the signal and the polarization parameters.

[0088] (2) The present invention achieves joint estimation of two-dimensional direction of arrival and polarization parameters based on joint smoothing processing in the spatial and polarization domains. The resulting closed-form solution effectively avoids the impact of the predefined spatial grid points introduced in the spectrum peak search step on the parameter estimation performance, and the closed-form solutions of the two-dimensional direction of arrival and polarization parameters corresponding to each signal source are automatically matched.

[0089] (3) The present invention utilizes the non-co-point configuration characteristics of the sparse multi-polarization array of orthogonal magnetic rings and dipoles. While increasing the array's degrees of freedom, it can also reduce the mutual coupling between array elements and alleviate the mutual coupling effect caused by the non-absolute orthogonality of the magnetic rings and dipole array elements. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] Figure 1 It is an overall flow chart of the present invention.

[0091] Figure 2 It is a schematic diagram of the sparse multipolarization array structure of orthogonal magnetic rings and dipoles proposed in the present invention.

[0092] Figure 3It is a schematic diagram of the virtual domain equivalent uniform multi-polarization array structure derived by the present invention.

[0093] Figure 4 It is a scatter plot of two-dimensional direction of arrival estimation under underdetermined conditions using the method proposed in this invention.

[0094] Figure 5 It is a scatter plot of two-dimensional direction of arrival estimation under certain conditions using the method proposed in this invention.

[0095] Figure 6 It is a scatter plot of polarization parameter estimation under over-determined conditions using the method proposed in the present invention. DETAILED DESCRIPTION

[0096] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0097] In order to solve the problem that traditional sparse array parameter estimation methods do not take signal polarization domain information into account, resulting in polarization mismatch and inability to estimate signal polarization parameters, the present invention provides a sparse multi-polarization array multi-dimensional parameter joint estimation method based on orthogonal magnetic rings and dipoles to avoid polarization mismatch and achieve joint estimation of wave direction of arrival and polarization parameters under underdetermined (i.e., the number of signal sources is greater than the number of physical array elements) and overdetermined (i.e., the number of signal sources is less than the number of physical array elements) conditions. Figure 1 , the implementation steps of the present invention are as follows:

[0098] Step 1: Construct a sparse multi-polarization array using orthogonal magnetic rings and dipole array elements. In order to expand the array aperture and reduce the mutual coupling between array elements while keeping the number of array elements constant, a sparse multi-polarization array consisting of four parallel sparse linear subarrays (each sparse linear subarray contains L0 array elements) is constructed: the magnetic ring sparse linear subarray on the y-axis is named 1H, and the dipole sparse linear subarray directly above the y-axis is named 1V. The normal direction of the magnetic ring in the two subarrays is parallel to the axial direction of the dipole, and the distance between the two subarrays is d. z Greater than zero and less than or equal to half of the wavelength λ of the incident narrowband signal, that is, 0<d z ≤λ / 2; located in the positive x-axis direction of subarray 1H, and the distance between it and subarray 1H is 0<d x The sparse linear subarray of magnetic rings ≤λ / 2 is named 2H. The sparse linear subarray of dipoles located directly above subarray 2H is named 2V. The normal direction of the magnetic ring of subarray 2H is parallel to the axial direction of the dipole in subarray 2V. The distance between subarrays 2H and 2V is d. z , the distance between subarray 1V and subarray 2V is d x. The designed sparse multi-polarization array is not only sparsely placed in the airspace, but also has magnetic rings and dipole elements with different polarization sensing capabilities and non-co-point placement, which effectively reduces the mutual coupling effect of the elements caused by the small spacing between the elements, as well as the mutual coupling effect of the elements caused by the non-absolute orthogonality of the magnetic rings and dipole elements in practical applications. The normal directions of subarrays 1H and 2H are both parallel to the z-axis, and the axial directions of subarrays 1V and 2V are both parallel to the z-axis. The element positions of the above four subarrays are all the same fully augmentable sparse array. Here, the fully augmentable sparse array refers to a sparse array without holes in the derived virtual difference co-array. You can choose a minimum redundant array, a nested array, a super nested array, an augmented nested array, etc. When L0=4 and the y-axis coordinates of each sub-array element are (0, d, 2d, 5d), the sparse multi-polarization array structure of the orthogonal magnetic ring and dipole is constructed as follows: Figure 2 shown.

[0099] Step 2: Model the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array, and decouple the horizontal polarization and vertical polarization received signals. Assume that there are M non-correlated far-field narrowband signal sources incident on the orthogonal magnetic ring and dipole sparse multi-polarization array designed in step 1, θ m 、φ m Represent the azimuth and elevation angles of the mth signal source, γ m ,η m Respectively represent the polarization auxiliary angle and polarization phase difference of the mth signal source, where m = 1, 2, ..., M. Use x 1H (t) and x 2H (t) represents the horizontal polarization component of the signal received by subarray 1H and subarray 2H at time t; x 1V (t) and x 2V (t) represents the vertical polarization component of the signal received by subarray 1V and subarray 2V at time t. Since the axis of the dipole and the normal of the magnetic ring in the array are parallel to the z-axis, subarray 1H and subarray 2H only receive the horizontal polarization component of the incident signal, and subarray 1V and subarray 2V only receive the vertical polarization component of the incident signal. Since subarray 1V, subarray 2H, subarray 2V and subarray 1H are parallel and symmetrical in space, there are only different time delays between the spatial domain steering vectors of subarray 1V, subarray 2H, subarray 2V and subarray 1H. Therefore, the angle domain steering matrix corresponding to the mth signal source of the designed sparse multi-polarization array can be expressed as a 4L0×2-dimensional matrix

[0100]

[0101] in represents the spatial steering vector corresponding to subarray 1H, [·] T represents the transpose operation, y l , l=1,2,…,L0, represents the y-axis coordinate of the l-th array element, y1=0, represents the spatial steering vector corresponding to the sub-array 1V ( is the delay factor of subarray 1V relative to subarray 1H), represents the spatial steering vector corresponding to subarray 2V ( is the delay factor of subarray 2H relative to subarray 1H), represents the spatial steering vector corresponding to subarray 2V ( is the delay factor of subarray 2V relative to subarray 1H), Represents the L0×1-dimensional zero vector, then the received signal x(t) of the designed orthogonal magnetic ring and dipole sparse multi-polarization array at time t can be modeled as:

[0102]

[0103] in The polarization vector corresponding to the mth signal source (including the parameter cosγ corresponding to the horizontal polarization component m and the parameters corresponding to the vertical polarization component ), s m (t) represents the waveform corresponding to the mth signal source, and n(t) is a Gaussian white noise component with zero mean and independent of each signal source.

[0104] To facilitate the guidance vector Characterize the sub-array 1H angle domain steering vector in and other factors related to the angle parameter sinφ m 、 and the polarization vector Decoupled, the definition includes the delay factor ( and ), horizontal polarization component parameter cosγ m and vertical polarization component parameters A 4×1 dimensional vector:

[0105]

[0106] Then the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array can be further expressed as:

[0107]

[0108] Among them I n represents the n×n dimensional identity matrix, represents the Kronecker product. Therefore It is decoupled from other factors related to angle parameters and polarization parameters.

[0109] Step 3: Perform multi-dimensional parameter separation on the decoupled received signals of the orthogonal magnetic ring and the dipole sparse multi-polarization array. m , β m Respectively represent the angles between the direction of arrival of the mth signal source and the y-axis and the x-axis (such as Figure 1 shown), θ m 、φ m , α m and β m The following relationship is satisfied:

[0110] sinφ m sinθ m =cosα m , sinφ m cosθ m =cosβ m ,

[0111] Then the spatial steering vector corresponding to sub-array 1H is can be re-expressed as:

[0112]

[0113] The coupling of multi-dimensional parameters in the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array makes it difficult to estimate each parameter. In order to achieve joint smoothing in the spatial and polarization domains and thus complete the multi-dimensional parameter estimation, the decoupled received signal x(t) of the orthogonal magnetic ring and the dipole sparse multi-polarization array is decomposed into an expression related to only a single parameter and an expression related to the separated multi-dimensional parameters. Specifically, the vector and vector Equivalently, if we define a 4L0×4 dimensional matrix

[0114]

[0115] Then the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array in step 2 can be further expressed as m Related expressions And the expressions related to the separated multidimensional parameters:

[0116]

[0117] From the above formula, we can see that the signal x(t) received by the orthogonal magnetic ring and dipole sparse multi-polarization array can be separated into the signal that is only related to a single parameter α m The associated block diagonal matrix Variable sinφ m and vector h m According to sinφ m cosθ m =cosβ m It can be seen that h m It can be further expressed as:

[0118]

[0119] After the above operation, the orthogonal magnetic ring and the dipole sparse multi-polarization array receiving signal with the angle α can be m The associated block diagonal matrix Separating them separately facilitates the subsequent joint smoothing operation in the spatial and polarization domains.

[0120] Step 4: Quantize each sub-matrix of the proposed array covariance matrix to obtain the virtual signal corresponding to the virtual domain equivalent uniform multi-polarization array. The covariance matrix R of the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array xx It can be expressed as:

[0121]

[0122] where E{·} represents the mathematical expectation, [·] H represents the conjugate transpose operation, represents the power of the mth signal source, σ 2 In actual situations, R xx It can be approximately calculated based on K sampling snapshots, that is:

[0123]

[0124] because Then R xx It can be re-expressed as a 4×4 block matrix with 16 parameters related to the spatial and polarization domains:

[0125]

[0126] in 1≤p,q≤4, represents the matrix The p-th row and q-th column elements of each block matrix R pq The dimensions are all L0×L0. p=1, 2, 3, 4 represent the autocorrelation of the received signal of subarray 1H (p=1), the autocorrelation of the received signal of subarray 1V (p=2), the autocorrelation of the received signal of subarray 2H (p=3), and the autocorrelation of the received signal of subarray 2V (p=4), respectively; p,q=1,2,3,4 and p≠q(total Rpq ) represents the cross-correlation between the received signals of subarrays 1H, 1V, 2H, and 2V. Because each submatrix has a data structure similar to the covariance matrix of a traditional linear sparse array, it facilitates subsequent joint smoothing operations in the spatial and polarization domains.

[0127] Next, based on the covariance matrix R xx Perform joint smoothing operation in spatial and polarization domains. First, R pq , p, q = 1, 2, 3, 4, perform vectorization operation to obtain vector r pq :

[0128]

[0129] Among them, vec(R pq ) represents the matrix R pq Vectorized operation, that is, the matrix R pq The columns in are stacked in sequence to form a new vector, a L0×M dimensional matrix represents the part of the steering matrix of each sub-array of the designed array that is only related to the parameter α, (·) * represents the conjugation operation, represents the Khatri-Rao product, Dimensional Matrix represents the virtual array steering matrix related only to the parameter α, an M×1 dimensional vector represents the parameter corresponding to the decoupling of α in the proposed array steering matrix (which can facilitate the joint smoothing operation in spatial and polarization domains), Since the position of each sub-array element corresponds to a fully scalable sparse matrix, the vector r pq The corresponding virtual array can be represented as a uniform array containing 2L-1 virtual continuous array elements 0<d≤λ / 2 represents the unit interval, In order to transform the vector r pq The elements in are rearranged into the virtual signal corresponding to the virtual domain equivalent uniform multi-polarization array, and the definition is Dimension selection matrix J:

[0130]

[0131] Where l′=1, 2, ..., 2L-1, 2L-1 represents the number of virtual continuous array elements in the differential array of each sub-array of the proposed array, L0-1≥i≥0, L0≥j≥1, ω(l′-L, i, j) represents the function of l′-L, i and j (represents the combination of i and j values ​​that satisfies y i -y j=(l′-L)d logarithm). For the constructed virtual domain equivalent uniform multi-polarization array, after the following spatial domain and polarization domain joint smoothing process related to the selection matrix J, the virtual signal vector υ corresponding to the autocorrelation / cross-correlation of the received signal of each sub-array pq It can be expressed as:

[0132]

[0133] in The spatial steering vector corresponding to the virtual uniform differential array of each subarray, with the polarization component removed and only related to the angle parameter α, can be expressed as:

[0134]

[0135] Step 5: Based on the virtual signal corresponding to the virtual domain equivalent uniform multi-polarization array, reconstruct the covariance matrix of the received signal of the virtual domain equivalent uniform multi-polarization array. First, the virtual vector υ pq Decomposed into L virtual sub-vectors of L×1 dimension in sequence:

[0136] υ pq，v =Ω v υ pq ,

[0137] where Ω v =[O L×(v-1) , I L , O L×(L-v) ], v = 1, 2, ..., L, represents an m1×m2 dimensional all-zero matrix, then υ pq，1 Indicates υ pq The vector consisting of the 1st to Lth elements in ,υ pq，2 Indicates υ pq The vector consisting of the 2nd to L+1th elements in ,υ pq，L Indicates υ pq A vector consisting of the Lth to 2L-1th elements in . pq，v By performing the column vector merging operation, the autocorrelation / cross-correlation of the received signals of each sub-array corresponding to the virtual domain equivalent uniform multi-polarization array can be obtained:

[0138]

[0139] in Represents the one-dimensional angle parameter α m The spatial domain steering vector of the subarray 1H in the virtual domain equivalent uniform multi-polarization array. Then, the covariance matrix R of the received signal of the virtual orthogonal magnetic ring and the dipole uniform multi-polarization array can be obtained. υυ :

[0140]

[0141] in Represents The corresponding 4L×4 dimensional block diagonal matrix. Therefore, the covariance matrix R of the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array is xx It can be reconfigured into its corresponding virtual domain equivalent uniform multi-polarization array (such as Figure 3 The covariance matrix R of the received signal is shown as υυ . The reconstructed covariance matrix R υυ Contains the decoupling of horizontal polarization parameters and vertical polarization parameters and parameter α m The data information is separated from other angle and polarization parameters, thus facilitating the subsequent joint estimation of multi-dimensional parameters.

[0142] Step 6: According to the reconstructed virtual domain equivalent uniform multi-polarization array received signal covariance matrix R υυ Solve the polynomial coefficients. First, R υυ Perform eigendecomposition and arrange the corresponding eigenvalues ​​from large to small. The subspace formed by the eigenvectors corresponding to the last 4L-M smaller eigenvalues ​​is recorded as the noise subspace U n According to R υυ The expression of virtual array steering vector and noise subspace U n The orthogonal relationship between Where det(·) represents the determinant operation. Divided into 16 4×4 blocks of sub-matrices P with dimensions of L×L pq =P[(p-1)L+1:pL,(q-1)L+1:qL], 1≤p,q≤4, and according to We can get:

[0143]

[0144] To solve the polynomial coefficients corresponding to the above determinant, each element in the 4×4 dimensional matrix in the above formula 1≤p,q≤4, re-expressed as:

[0145]

[0146] where r pq is a (2L-1)×1 dimensional vector, whose uth element can be expressed as r pq (u)=∑diag(P pq , uL), that is, the matrix.P pqSum the elements in the uLth diagonal of . uL = 0 represents the elements in the main diagonal, uL > 0 represents the elements in the diagonal above the main diagonal, and uL < 0 represents the elements in the diagonal below the main diagonal. Furthermore, according to Laplace's Theorem for calculating the determinant of a 4×4 matrix, the entire determinant can be re-expressed as:

[0147]

[0148] Where r is a (8L-7)×1 dimensional vector, corresponding to the polynomial coefficients of the above 4×4 dimensional matrix determinant, which can be specifically expressed as:

[0149]

[0150] Step 7: Solve the closed-form solution for the joint estimation of the two-dimensional direction of arrival and polarization parameters based on polynomial root finding and subspace orthogonality principle. Define z = e j2πd cosα / λ According to the polynomial root-finding principle, we know that for polynomial

[0151] r T [z -(4L-4) , z -(4L-5) ,…,1,…,z 4L-5 , z 4L-4 ] T =0,

[0152] Find the M roots whose modulus is within the unit circle and closest to the unit circle You can get α m The closed-form solution of :

[0153]

[0154] Where ∠(·) represents the phase of the complex number. According to the subspace orthogonality principle, the estimated α m Substitute into step 6 and the block diagonal matrix and noise subspace U n Correlation matrix And solve the minimum eigenvalue corresponding to h m Parallel eigenvectors where h m is a non-zero constant. Then according to h in step 3 m The expression of β m The closed-form solution of :

[0155]

[0156] in Represents a vector The nth element of . According to θm , φ m , α m and β m The relationship of sinφ m sinθ m = cosα m , sinφ m cosθ m = cosβ m It can be obtained that and The closed - form solution: If cosα m ≠0, then

[0157] θ m = arccot(cosβ m / cosα m ), cosβ m / cosα m ≥0, cosβ m ≥0

[0158] θ m = arctan(cosα m / cosβ m )+π, cosβ m / cosα m ≥0, cosβ m <0

[0159] θ m = arctan(cosα m / cosβ m )+π, cosβ m / cosα m <0, cosβ m ≥0

[0160] θ m = arctan(cosα m / cosβ[[ID=​​​​​​​​​​​​​​​​​​​​​​​​​​​≥0, cosα m ≥0

[0164] θ m =arctan(cosα m / cosβ m )+π,cosα m / cosβ m ≥0, cosα m <0

[0165] θ m =arctan(cosα m / cosβ m )+π,cosα m / cosβ m <0, cosα m ≥0

[0166] θ m =arctan(cosα m / cosβ m )+2π,cosα m / cosβ m <0, cosα m <0

[0167]

[0168] If cosα m =0 and cosβ m =0, then θ m =90° and φ m =0°.

[0169] Further according to step 3 h m The expression of can be used to obtain the closed-form solution of the polarization parameter:

[0170]

[0171]

[0172] in is the estimated value of the pitch angle of the mth signal source.

[0173] The effects of the present invention are further described below with reference to simulation examples.

[0174] Simulation example: The proposed sparse multi-polarization array is used to receive the incident signal. The arrangement of the elements of each subarray is a nested array. The Cartesian coordinates of the elements in subarray 1H on the y-axis are {(0, 0, 0), (0, d, 0), (0, 2d, 0), (0, 5d, 0), (0, 8d, 0)}, where d = d x =dz =λ / 2, and the remaining subarrays 1V, 2H, and 2V are parallel to subarray 1H. Considering the cases where the number of signals is underdetermined and overdetermined, 100 Monte Carlo experiments are performed for each case.

[0175] First, consider the case where the number of incident signals is underdetermined. Assume that the azimuth and elevation angles (θ, φ) of the 17 quaternary phase-shift keying narrowband incident signal sources are (10°+15°r1, 35°+10°r1), r1=0, 1, …, 5, (132°-15°r2, 32°+10°r2), r2=0, 1, 2, 3, (68°, 50°) and (175°-15°r3, 30°+10 °r3), r3=0, 1, ..., 5, the polarization auxiliary angle and polarization phase difference (γ, η) of the first 5 signal sources are all (45°, 0°), the polarization auxiliary angle and polarization phase difference (γ, η) of the middle 4 signal sources are all (45°, -90°), and the polarization auxiliary angle and polarization phase difference (γ, η) of the last 8 signal sources are all (45°, 90°); the signal-to-noise ratio is set to 30dB, and the number of sampling snapshots is set to 20000. Under the above conditions, the two-dimensional wave direction of arrival estimation results of the method proposed in the present invention are as follows Figure 4 As shown in the figure, the proposed method can effectively resolve these 17 signal sources. For a conventional uniform multi-polarization array composed of orthogonal magnetic rings and dipoles with the same number of physical elements (each subarray contains 5 physical elements, for a total of 20 physical elements), the corresponding DOA estimation method can resolve up to 16 signal sources. These results demonstrate that the proposed method can overcome the bottleneck problem of conventional uniform arrays where the degrees of freedom are limited by the number of physical elements, thereby increasing the degrees of freedom.

[0176] Then, considering the case where the number of incident signals is overdetermined, it is assumed that the azimuth and elevation angles (θ, φ) of the three Gaussian random narrowband incident signal sources are (15°, 35°), (41°, 50°), and (60°, 65°), respectively; the polarization assistance and polarization phase difference (γ, η) are (35°, 60°), (50°, 45°), and (45°, 30°), respectively; the signal-to-noise ratio is set to 20dB, and the number of sampling snapshots is set to 500. Under the above conditions, the two-dimensional direction of arrival estimation and polarization parameter estimation results of the method proposed in the present invention are as follows: Figure 5 and Figure 6 As shown, the above simulation results demonstrate that the method proposed in the present invention can accurately estimate the two-dimensional direction of arrival and polarization parameters.

[0177] In summary, the present invention proposes a method for joint estimation of multidimensional parameters of a sparse multi-polarization array based on orthogonal magnetic rings and dipoles, which improves the array's degrees of freedom, constructs a virtual domain uniform multi-polarization array covariance matrix based on joint smoothing of spatial and polarization domain data, and provides a closed-form solution method based on polynomial root finding, thereby realizing automatic pairing of two-dimensional direction of arrival and polarization parameter estimation of each signal source, and achieving accurate estimation of two-dimensional direction of arrival and polarization parameters while ensuring the advantageous performance of a sparse array.

Claims

1. A method for joint multidimensional parameter estimation of a sparse multipolarization array based on orthogonal magnetic rings and dipoles, characterized in that: The following steps are involved: (1) A sparse multi-polarization array is constructed at the receiving end using orthogonal magnetic rings and dipole array elements. The sparse multi-polarization array consists of four parallel sub-arrays, where sub-arrays 1H and 2H are composed of magnetic rings, and sub-arrays 1V and 2V are composed of dipoles. Each sub-array is a fully scalable sparse array containing L0 array elements. Here, a fully scalable sparse array refers to a sparse array without holes in the derived virtual differential array. Sub-arrays 1V and 2V are located directly above sub-arrays 1H and 2H, respectively, with a distance of 0 < d. z ≤λ / 2, where λ represents the wavelength of the incident narrowband signal, i.e. the spacing between subarrays 1V and 1H and the spacing between subarrays 2V and 2H are both d z Subarrays 2H and 2V are located at a distance 0<d from subarrays 1H and 1V in the positive direction of the x-axis. x ≤λ / 2, that is, the distance between sub-array 2H and sub-array 1H and the distance between sub-array 2V and sub-array 1V are both d x ; The normal direction of the magnetic ring array element and the axial direction of the dipole array element in the array are both parallel to the z-axis; (2) The received signal of the constructed orthogonal magnetic ring and dipole sparse multi-polarization array is modeled. The received signal x(t) of the array at time t is expressed as: where x 1H (t) and x 2H (t) represents the horizontal polarization component of the signal received by subarray 1H and subarray 2H at time t, respectively, and x 1V (t) and x 2V (t) represents the vertical polarization component of the signal received by subarray 1V and subarray 2V at time t, respectively, [·] T represents the transposition operation, M is the number of non-correlated far-field narrowband signal sources, represents the 4L0×2-dimensional angle-domain steering matrix corresponding to the m-th signal source of the designed sparse multi-polarization array: Among them, θ m 、φ m Respectively represent the azimuth and elevation angles of the mth signal source, m=1,2,…,M, represents the spatial steering vector of subarray 1H, y l , l=1,2,…,L0, represents the distance between the lth array element and the first array element, y1=0, represents the L0×1 dimensional zero vector, The polarization vector corresponding to the mth signal source includes the parameter cosγ corresponding to the horizontal polarization component m and the parameters corresponding to the vertical polarization component γ m ,η m They represent the polarization auxiliary angle and polarization phase difference of the mth signal source, s m (t) represents the waveform corresponding to the mth signal source, n(t) is the Gaussian white noise component with zero mean and independent of each signal source; To facilitate the guidance vector Characterize the sub-array 1H angle domain steering vector in and other parameter factors related to the angle parameter sinφ m 、 and the polarization vector After decoupling, the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array is expressed as the following decoupling form of horizontal polarization and vertical polarization: Among them I n represents the n×n dimensional identity matrix, represents the Kronecker product, h m Indicates the delay factor and the horizontal polarization component parameter cosγ m and vertical polarization component parameters A 4×1 dimensional vector: The delay factor includes and (3) In order to perform multi-dimensional parameter separation on the decoupled signal, define α m , β m are the angles between the arrival direction of the mth signal source and the y-axis and x-axis, so θ m 、φ m , α m and β m The following relationship is satisfied: sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m ; Then the received signal of the orthogonal magnetic ring and dipole sparse multi-polarization array is expressed as: The 4L0×4 dimensional matrix Defined as: is the spatial steering vector corresponding to subarray 1H The equivalent representation of is: Correspondingly, h m The equivalent expression is: After the above operation, the orthogonal magnetic ring and the dipole sparse multi-polarization array receive the signal x(t) with the angle α m The associated block diagonal matrix It is separated out to facilitate the separation and solution of multi-dimensional parameters; (4) To obtain the virtual signal corresponding to the virtual domain equivalent uniform multi-polarization array, first calculate the covariance matrix R of the received signal of the orthogonal magnetic ring and the dipole sparse multi-polarization array xx : where E{·} represents the mathematical expectation, [·] H represents the conjugate transpose operation, represents the power of the mth signal source, σ 2 Indicates the noise power; in actual situations, R xx It is calculated based on K sampling snapshots, namely: based on This property, R xx The average block is 16 block matrices R with dimensions L0×L0 related to spatial and polarization domain parameters. pq : in 1≤p,q≤4, represents the matrix The element in the pth row and qth column of ; When p=1,2,3,4, and p=1, R pp represents the autocorrelation of the received signal of subarray 1H. When p=2, R pp Represents the autocorrelation of the received signal of subarray 1V. When p=3, R pp represents the autocorrelation of the received signal of subarray 2H. When p=4, R pp represents the autocorrelation of the received signal of sub-array 2V, p,q=1,2,3,4 and p≠q, total R pq , represents the cross-correlation between the received signals of subarray 1H, subarray 1V, subarray 2H and subarray 2V; pq ,p,q=1,2,3,4, perform vectorization operation to obtain vector r pq : Among them, vec(R pq ) represents the matrix R pq Vectorized operation, that is, the matrix R pq The columns in are stacked in sequence to form a new vector, a L0×M dimensional matrix represents the part of the steering matrix of each sub-array of the designed array that is only related to the parameter α, (·) * represents the conjugation operation, represents the Khatri-Rao product, Dimensional Matrix represents the virtual array steering matrix related only to the parameter α, an M×1 dimensional vector represents the parameter corresponding to the decoupling of α in the proposed array steering matrix, Since the position of each sub-array element corresponds to a fully scalable sparse matrix, the vector r pq The corresponding virtual array is represented as a uniform array containing 2L-1 virtual continuous array elements 0<d≤λ / 2 represents the unit interval, The vector r pq The elements in the virtual uniform array The equivalent received signals corresponding to each array element are sorted into vector signals υ pq : where e 2L-1 represents a 2L-1 dimensional column vector, whose Lth element is 1 and the rest are zero. Expressed as: It corresponds to the spatial steering vector in the virtual uniform differential array of each sub-array, with the polarization component removed and only related to the angle parameter α; (5) Reconstruct the virtual domain received signal covariance matrix; first, the virtual domain The equivalent received signal υ corresponding to each array element pq Decomposed into L L×1-dimensional virtual equivalent received signal sub-vectors: u pq,v =Oh v u pq , where Ω v =[O L×(v-1) ,I L ,O L×(L-v) ],v=1,2,…,L, Represents an m1×m2 dimensional all-zero matrix; for L υ pq,v Perform column vector merging operations to obtain the autocorrelation / cross-correlation matrix of the received signals of each sub-array of the virtual domain equivalent uniform multi-polarization array: in Represents the one-dimensional angle parameter α m The spatial domain steering vector of subarray 1H in the virtual domain equivalent uniform multi-polarization array; the covariance matrix R of the received signal of the virtual orthogonal magnetic ring and the dipole uniform multi-polarization array υυ Refactored to: in Represents The corresponding 4L×4 dimensional block diagonal matrix; (6) According to the reconstructed virtual domain equivalent uniform multi-polarization array received signal covariance matrix R υυ Solve the polynomial coefficients that fuse spatial and polarization domain information; (7) Joint estimation of two-dimensional direction of arrival and polarization parameters is performed based on the polynomial root-finding principle and subspace orthogonality principle.

2. The method for joint multidimensional parameter estimation of a sparse multipolarization array based on orthogonal magnetic rings and dipoles according to claim 1 is characterized in that: In step (1), a sparse multipolarization array is constructed using orthogonal magnetic rings and dipole array elements placed at non-co-points, specifically: the normal direction of the magnetic ring sparse linear sub-array 1H is parallel to the axial direction of the dipole sparse linear sub-array 1V; the normal direction of the magnetic ring of sub-array 2H is parallel to the axial direction of the dipole in sub-array 2V; the normal directions of sub-arrays 1H and 2H are both parallel to the z-axis, and the axial directions of sub-arrays 1V and 2V are both parallel to the z-axis; the array element positions of the four sub-arrays correspond to the same fully scalable sparse array, and the above four fully scalable sparse arrays are selected from minimum redundancy arrays, nested arrays, super nested arrays or augmented nested arrays.

3. The method for joint multidimensional parameter estimation of a sparse multipolarization array based on orthogonal magnetic rings and dipoles according to claim 1 is characterized in that: In step (4), vector r pq The elements in are rearranged into the virtual signal υ corresponding to the virtual domain equivalent uniform multi-polarization array pq , specifically by selecting the matrix: First, R xx The average block is 16 block matrices R with dimensions L0×L0 related to spatial and polarization domain parameters. pq , then R pq Perform vectorization operation to obtain vector r pq =vec(R pq ); then, define Dimension selection matrix J: Where l′=1,2,…,2L-1, 2L-1 represents the number of virtual continuous array elements in the differential array of each sub-array of the proposed array, L0-1≥i≥0, L0≥j≥1, ω(l′-L,i,j) represents a function of l′-L, i and j, which represents the combination of i and j values ​​that satisfies y i -y j =(l′-L)d logarithm; for the constructed virtual domain equivalent uniform multi-polarization array, after the following spatial domain and polarization domain joint smoothing process related to the selection matrix J, the virtual signal vector υ corresponding to the autocorrelation / cross-correlation of the received signal of each subarray pq Expressed as: υ pq =Jr pq 。 4. The method for joint multidimensional parameter estimation of a sparse multipolarization array based on orthogonal magnetic rings and dipoles according to claim 1 is characterized in that: In step (6), the covariance matrix R of the received signal of the reconstructed virtual domain equivalent uniform multi-polarization array is υυ Solve the polynomial coefficients that fuse spatial and polarization domain information. Specifically, for R υυ Perform eigendecomposition and arrange the corresponding eigenvalues ​​from large to small. The subspace formed by the eigenvectors corresponding to the last 4L-M smaller eigenvalues ​​is recorded as the noise subspace U n According to R υυ The expression of virtual array steering vector and noise subspace U n The orthogonal relationship between Where det(·) represents the determinant operation; the matrix Divided into 16 4×4 blocks of sub-matrices P with dimensions of L×L pq =P[(p-1)L+1:pL,(q-1)L+1:qL],1≤p,q≤4, and according to get: To solve the polynomial coefficients corresponding to the above determinant, each element in the 4×4 dimensional matrix in the above formula 1≤p,q≤4, re-expressed as: where r pq is a (2L-1)×1 dimensional vector, whose uth element is represented by r pq (u)=Σdiag(P pq ,uL), that is, the matrix P pq Sum the elements in the uLth diagonal of , where uL=0 represents the elements in the main diagonal, uL>0 represents the elements in the diagonal above the main diagonal, and uL<0 represents the elements in the diagonal below the main diagonal. Then, according to Laplace's theorem for calculating the determinant of a 4×4 matrix, the entire determinant is re-expressed as: Where r is a (8L-7)×1 dimensional vector, corresponding to the polynomial coefficients of the above 4×4 dimensional matrix determinant, specifically expressed as:

5. The method for joint multidimensional parameter estimation of a sparse multipolarization array based on orthogonal magnetic rings and dipoles according to claim 4 is characterized in that: In step (7), the two-dimensional direction of arrival and polarization parameters are jointly estimated based on the obtained polynomial coefficients and the polynomial root-finding principle and the subspace orthogonality principle. Specifically, according to the polynomial root-finding principle, z = e j2πdcosα / λ , for polynomials r T [With -(4L-4) ,With -(4L-5) ,…,1,…,z 4L-5 ,With 4L-4 ] T =0, Find the M roots whose modulus is within the unit circle and closest to the unit circle Get α m The closed-form solution of : Where ∠(·) represents the phase of the complex number; according to the subspace orthogonality principle, the estimated α m Substitute in the block diagonal matrix and noise subspace U n Correlation matrix And solve the minimum eigenvalue corresponding to h m Parallel eigenvectors where h m is a non-zero constant; then according to step (3) h m The expression of β m The closed-form solution of : in Represents a vector The nth element of m 、φ m , α m and β m The relationship sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m It can be seen that and The closed-form solution of is obtained by the following formula: If cosα m ≠0, then θ m =arccot(cosβ m / cosα m ),cosβ m / cosα m ≥0,cosβ m ≥0 i m =arctan(cosα m / cosβ m )+π,cosβ m / cosα m ≥0,cosβ m <0 i m =arctan(cosα m / cosβ m )+π,cosβ m / cosα m <0,cosβ m ≥0 θ m =arctan(cosα m / cosβ m ),cosβ m / cosα m <0,cosβ m <0 If cosβ m ≠0, then θ m =arctan(cosα m / cosβ m ),cosα m / cosβ m ≥0,cosα m ≥0 θ m =arctan(cosα m / cosβ m )+π,cosα m / cosβ m ≥0,cosα m <0 θ m =arctan(cosα m / cosβ m )+π,cosα m / cosβ m <0,cosα m ≥0 θ m =arctan(cosα m / cosβ m )+2π,cosα m / cosβ m <0,cosα m <0 If cosα m =0 and cosβ m =0, then θ m =90° and φ m =0°; Further according to step (3) h m The closed-form solution of the polarization parameter is obtained from the expression of in is the estimated value of the pitch angle of the mth signal source.

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