Multi-cumulant matrix rotation-invariant near-field cold precise model direction finding method

Abstract

The application belongs to the technical field of array signal processing of near-field sources, and more particularly relates to a multi-cumulant matrix rotation-invariant near-field COLD accurate model direction finding method. A uniform COLD array is used to construct a fourth-order cumulant matrix and cascade to obtain a joint cumulant long matrix. The covariance matrix of the joint matrix is calculated and eigenvalue decomposition is performed. According to the rotation invariance between the signal subspaces, the spatial domain information and the polarization information are successfully separated from the spatial domain polarization domain. The least square method and the polarization vector relationship are used to step by step solve the azimuth angle and distance parameters and the polarization auxiliary angle and polarization phase difference parameters. The method considers the parameter estimation problem under the accurate model, considers the amplitude attenuation of the signal on different array elements, can effectively use the information of all array elements in the array, and has higher estimation accuracy.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of array signal processing of near-field sources, and more particularly relates to a multi-cumulant matrix rotation-invariant near-field COLD accurate model direction finding method. BACKGROUND

[0002] Direction of Arrival (DOA) estimation is a major problem in the field of array signal processing, which plays an important role in the research of wireless communication and radar. According to the distance of the signal source from the reference array element, the signal source can be divided into far-field sources and near-field sources. The far-field source is located in the far-field region, and the near-field source is located in the Fresnel region. The wave front of the far-field source corresponds to the plane wave front, and the wave front of the near-field source corresponds to the spherical wave front. Unlike far-field sources, the phase factor of near-field sources is a bivariate function of angle and distance, and due to the path loss of electromagnetic waves in space, the signals received by different array elements from the same signal source have amplitude attenuation.

[0003] In order to simplify the calculation of near-field source positioning, a large number of existing near-field DOA estimation algorithms use approximate models, but the accuracy of these algorithms will be affected by the approximation error. A large number of existing algorithms based on near-field algorithms have been proposed, but most of these algorithms are based on uniform linear arrays and cannot estimate the motion characteristics of electromagnetic wave vectors. Compared with ordinary arrays, the Co-centered Orthogonal Loop and Dipole (COLD) array not only can obtain the complex amplitude information of the signal, but also can obtain the polarization information of the electromagnetic signal. When the signal cannot be well distinguished in the spatial domain, joint parameter estimation in the spatial and polarization domains can be performed. At the same time, these algorithms only consider providing a closed-form solution for parameter estimation, and cannot effectively utilize all the information provided by the array elements. SUMMARY

[0004] In view of the above problems existing in the prior art, the application provides a multi-cumulant matrix rotation-invariant near-field COLD accurate model direction finding method. The method utilizes the rotation invariance between the fourth-order cumulant matrices of the array elements to estimate the spatial steering vector and the polarization steering vector, estimates the position parameters by constructing a linear equation system, and estimates the polarization parameters by utilizing the polarization vector information.

[0005] To achieve the above object, the application adopts the following technical scheme:

[0006] The multi-cumulant matrix rotation-invariant near-field COLD accurate model direction finding method comprises the following steps:

[0007] Step 1: a near-field one-dimensional uniform COLD array model of an accurate propagation model is established, array element position information is constructed, multiple narrow-band near-field sources in space are incident to the array, according to the geometric propagation relationship of the accurate spherical wave front, the spatial amplitude phase factor in the free space is obtained, the spatial amplitude phase factor constitutes the space domain steering vector, the polarization vector is obtained, the steering vector is obtained through the space domain steering vector and the polarization vector, the spatial amplitude phase factor includes two-dimensional parameter information of the azimuth angle and the distance, the polarization vector includes two-dimensional parameter information of the polarization azimuth angle and the polarization phase difference;

[0008] Step 2: using the array element output data and the array receiving data vector of the COLD array, first, the fourth-order cumulative quantity matrix of each array element is constructed, and then the joint cumulative quantity long matrix is obtained by cascading;

[0009] Step 3: the covariance matrix of the joint cumulative quantity long matrix is calculated and is subjected to eigenvalue decomposition, the signal subspace corresponding to the fourth-order cumulative quantity of each array element is extracted in the index order, the spatial amplitude phase factor can be separated by using the rotational invariance between the subspaces, and the array manifold estimation value of the space domain is obtained;

[0010] Step 4: the signal subspace corresponding to the fourth-order cumulative quantity of each array element is further decomposed, and the polarization information can be separated from the signal subspace containing the space domain polarization domain. The first half of the subspace corresponding to the fourth-order cumulative quantity matrix of each array element is further cascaded to obtain the signal subspace E h , the second half of the subspace corresponding to the fourth-order cumulative quantity matrix of each array element is further cascaded to obtain the signal subspace E v , according to the rotational invariance between the subspaces, the matrix is constructed, the polarization auxiliary angle estimation value and the polarization phase difference estimation value are obtained by performing eigenvalue decomposition on the matrix and extracting the eigenvalues;

[0011] Step 5: according to the array manifold matrix estimation value of the space domain, the spatial amplitude phase factor estimation value is extracted and the corresponding amplitude attenuation term is obtained, the matrix equations about the azimuth angle and the distance are solved, the coarse estimation value of the azimuth angle and the distance is obtained by using the least square method;

[0012] Step 6: according to the coarse estimation value of the azimuth angle and the distance in step 5, the accurate delay phase unambiguous coarse estimation value is calculated. According to the array manifold matrix estimation value of the space domain, the spatial amplitude phase factor estimation value is obtained, and the corresponding delay phase term is extracted, the accurate delay phase estimation value is obtained by using the accurate delay phase unambiguous coarse estimation value to solve the ambiguity, and the accurate estimation value of the azimuth angle and the distance is obtained by solving the matrix equations about the azimuth angle and the distance by using the least square method.

[0013] Further optimization of the technical scheme is as follows:

[0014] A near-field one-dimensional uniform COLD array model is established to build an accurate propagation model, the array uniformly places COLD elements on the axis x , and takes the central element as the reference element and the coordinate origin, the total number of elements is 2L+1, the element spacing is d, d=λ / 2, λ is the signal wavelength; it is assumed that there are K narrowband near-field sources incident in the space to the array, the position parameters of the kth signal are (θ k , r k ), the polarization parameters are (γ k , η k ), r k is the distance from the kth signal source to the reference element of the array, θ k is the azimuth angle, that is, the included angle between the incident direction of the kth signal source and the y-axis θ k ∈[-π / 2, π / 2], γ k is the polarization auxiliary angle of the kth signal source γ k ∈[0, π / 2], η k is the polarization phase difference of the kth signal source η k ∈[-π, π], the distance from the kth signal source to the lth element is:

[0015]

[0016] Where l=-L,...,0,...,L;

[0017] According to the geometric propagation relationship of the accurate spherical wave front, the spatial amplitude and phase factor of the lth element receiving from the kth signal source is:

[0018]

[0019] b(θ k , r k )=[b -L，k ,..., b l，k ,..., b L，k ] T represents the spatial steering vector of the kth signal source, the dimension is (2L+1)×1, the accurate delay phase τ l，k , the amplitude attenuation term ρ k，k ;

[0020] The polarization vector of the kth signal source is represented as:

[0021]

[0022] The steering vector of the kth signal source is:

[0023]

[0024] Where Denotes the Kronecker product, a(θ) k r k γ k η k The dimension of ) is 2(2L+1)×1;

[0025] The received data vector of the array at time t is x(t).

[0026]

[0027] The dimension of x(t) is 2(2L+1)×1, ⊙ represents the Khatri-Rao product, and A=[a(θ1,r1,γ1,η1),...,a(θ k r k γ k η k ), ..., a(θ) K r K γ K η K [b(θ1, r1), ..., b(θ)] is an array manifold matrix with dimension 2(2L+1)×K, where B = [b(θ1, r1), ..., b(θ)]. k r k ), ..., b(θ) K r K Let G = [g(γ1, η1), ..., g(γ)] be the spatial array manifold matrix. k η k ), ..., g(γ) K η K ] is the polarization domain array manifold matrix with dimension 2×K, s(t)=[s1(t),...,s k (t), ..., s K (t)] T It is a signal vector of dimension K×1, and n(t) is a Gaussian white noise vector of dimension 2(2L+1)×1;

[0028] because The received data vector can be further represented as:

[0029]

[0030] Where, x1(t)=[x 1，-L (t), ..., x 1，0 (t), ..., x 1，L (t)] T It is a vector consisting of the first 2L+1 elements of x(t), x2(t) = [x 2，-L (t), ..., x 2，0 (t), ..., x 2，L(t)] T is a vector consisting of the last 2L+1 elements of x(t), and P2=diag[cosγ1,..., cosγ is a diagonal matrix. k ,..., cosγ K ]; thus, the steering vector can be rewritten as a(θ k , r k , γ k , η k ) = [a 1，-L，k ,..., a 1,L,k， a 2，-L，k ,..., a 2,L,k ] T , and a k will be used to replace a(θ k , r k , γ k , η k ) in the following.

[0031] 8. The multiple-cumulant matrix rotation-invariant near-field COLD precise direction finding method according to claim 2, wherein the step 2 is specifically as follows:

[0032] Using the output data of the m-th element and the array receiving data vector, a fourth-order cumulant matrix C l ,

[0033]

[0034] where (·) * denotes conjugation, and (·) H denotes conjugate transpose. The fourth-order cumulant matrix C l can be further expressed as C l = AΦ l C 4s A H , and the dimension of the matrix is 2(2L+1)×2(2L+1), wherein P2=diag[cosγ1,..., cosγ The diagonal elements of the matrix represent the cumulants of the k-th signal, the symbol cum(·) represents calculation of the cumulants, and P2=diag[cosγ1,..., cosγ According to the construction method of the above fourth-order cumulant matrix, the fourth-order cumulant matrices corresponding to each element are obtained, and there are 2L+1 fourth-order cumulant matrices in total, and a joint cumulant matrix Q of the dimension of 2(2L+1) 2 ×2(2L+1) is obtained.

[0035]

[0036] where Φ0 is a unit matrix.

[0037] The further optimization of the technical solution is that the step 3 is specifically as follows:

[0038] The covariance matrix R of the joint cumulant long matrix Q is calculated, R = E{QQ H} with the dimension of 2(2L+1) 2 ×2(2L+1) 2 The matrix R is subjected to eigenvalue decomposition, that is, Wherein, U s is a signal subspace composed of eigenvectors corresponding to K large eigenvalues, U N is a noise subspace composed of eigenvectors corresponding to 2(2L+1) 2 -K small eigenvalues, ∑s is a diagonal matrix composed of K large eigenvalues, and ∑ N is a diagonal matrix composed of 2(2L+1) 2 -K small eigenvalues.

[0039] The signal subspace U s can be decomposed into 2L+1 parts, the number of which corresponds to the number of array elements, that is,

[0040]

[0041] Wherein is composed of the 2(2L+1)L+1th row to the 2(2L+1)L+2(2L+1)th row of the signal subspace U s , and is denoted as a signal subspace corresponding to a reference array element cumulant matrix.

[0042] Since the space spanned by the signal subspace is the same as the space spanned by the spatial array manifold matrix, and the signal subspace corresponding to the reference array element cumulant matrix and the signal subspace corresponding to each array element cumulant matrix satisfy the rotational invariance, there is a non-singular matrix such that the following relationship is established:

[0043]

[0044] Therefore, the signal subspace and the signal subspace corresponding to the reference array element satisfy the relationship, Wherein ψ l =T -1 Φ l T, the matrix ψ l is similar to the diagonal matrix Φ l , and the diagonal elements of the matrix Φ l are eigenvalues of the matrix ψ l ; according to the least square criterion, the matrix ψ l can be solved,

[0045]

[0046] in(·) + This represents the pseudo-inverse of the matrix. The solved matrix ψ... l By performing eigenvalue decomposition, we can obtain the diagonal matrix Φ. l ;

[0047] Then, the diagonal matrix Φ is obtained from the eigenvalue decomposition. l The diagonal elements are spread into row vectors and concatenated according to the position index order of the corresponding array elements to obtain the estimated value of the spatial array manifold matrix.

[0048]

[0049] Among them, diag -1 (·) indicates that the diagonal elements of the matrix are spanned into row vectors, and the spatial guided vector estimate is obtained.

[0050] This technical solution is further optimized, and step 4 is as follows:

[0051] Since matrix A = G⊙B, the fourth-order cumulant matrix C l It can be decomposed into:

[0052]

[0053] Therefore, the signal subspace can be further decomposed into:

[0054]

[0055] in It is composed of signal subspace It consists of the first 2L+1 row vectors, It is composed of signal subspace It consists of the last 2L+1 row vectors;

[0056] Extract from signal subspace Constructing the signal subspace E h ,

[0057]

[0058] Extract from signal subspace Constructing the signal subspace E v ,

[0059]

[0060] Based on the rotational invariance between the signal subspaces corresponding to the fourth-order cumulant matrices of different array elements and the signal subspace corresponding to the fourth-order cumulant matrix of the reference array element, it can be known that...

[0061]

[0062] According to the matrix sum matrix Computable matrix E h and E v Substituting into the above equation and simplifying further, we can obtain... in Substituting the diagonal matrices P1 and P2 into the above equation, we can calculate the matrix:

[0063]

[0064] Because P = TDT -1 Matrix D is similar to diagonal matrix P, and the diagonal elements of matrix P are the eigenvalues ​​of matrix D. Therefore, eigenvalue decomposition of matrix D yields an estimate of diagonal matrix P. Therefore, the estimated polarization auxiliary angle of the k-th signal can be obtained. Polarization phase difference estimate of the k-th signal in Represents a diagonal matrix The kth diagonal element.

[0065] This technical solution is further optimized, and step 5 is as follows:

[0066] Due to the amplitude attenuation term ρ l，k for,

[0067]

[0068] Expanding and simplifying the above equation, we get...

[0069]

[0070] Where d1 = ld represents the distance between the l-th array element and the reference array element; the spatial amplitude phase factor estimate obtained in step 3 is used. By performing a modulo operation, an estimate of the amplitude attenuation term can be obtained. Where abs(·) represents the modulo operation; the estimated value Substituting into the above equation, we can obtain the equation relationship of the unknowns.

[0071]

[0072] Apart from the reference array element, the above equations can be listed as 2L equations, resulting in the following system of equations:

[0073]

[0074] where two unknowns of the equation set are and r k sinθ k ; the coefficient matrix of the equation set is

[0075]

[0076] and the unknown vector is

[0077] The solution of the unknown vector by using the least square method is

[0078] The expression of the distance estimation value and the azimuth estimation value is:

[0079]

[0080] where

[0081] represents the first element of the unknown vector, and represents the second element of the unknown vector. The further optimization of the technical solution is that the step 6 is specifically as follows:

[0082] The angle operation is performed on the space amplitude phase factor estimation value obtained in the step 3, and the delay phase estimation value

[0083] where angle(·) represents the angle operation; it is worth noting that the delay phase obtained by the operation is the principal value of the angle, and is not the real delay phase, that is, the phase is ambiguous; since the delay phase estimation value has an integer multiple relationship of 2πn with the accurate delay phase τ l，k in the step 1, a set S is constructed for ambiguity resolution, and the accurate delay phase estimation value is an element of the set S l . The delay phase non-ambiguous rough estimation value

[0084] is constructed by using the distance estimation value and the angle estimation value obtained in the step 5.

[0085]

[0086] where the accurate delay phase estimation value​ is the set S l with the delay phase without ambiguity coarse estimate value the value with the minimum deviation, i.e.

[0087]

[0088] For simplifying the operation, the obtained accurate delay phase estimate value from the above formula is normalized by 2π / λ, and the result is recorded as

[0089]

[0090] the delay term and the angle and distance parameters of the signal source satisfy the relationship:

[0091]

[0092] By means of algebraic operation, r k , θ k and can be listed as:

[0093]

[0094] The obtained accurate and unambiguous estimate value is brought into the above formula, and the equation relationship is obtained:

[0095]

[0096] Except for the reference array element, the above equation can be listed as 2L, and the equation group is obtained:

[0097]

[0098] Among them, the two unknown quantities of the equation group are r k and r k sinθ k ; the coefficient matrix from the above formula is

[0099]

[0100] The unknown quantity vector is Δ f，k = [r k , r k sinθ k ] T ; according to the least square criterion, the solution of the unknown quantity vector is

[0101]

[0102] ​The expression of the azimuth angle accurate estimation value and the distance accurate estimation value is:

[0103]

[0104] wherein represents the first element of the unknown quantity vector, represents the second element of the unknown quantity vector.

[0105] Compared with the prior art, the above technical solution has the following beneficial effects:

[0106] The present application is based on the near-field source positioning problem of the accurate geometric propagation model, adopts a one-dimensional uniform COLD array to receive data signals, constructs a long matrix of element polarization joint cumulant, solves the matrix equation by using the least square method to estimate the position parameters, effectively utilizes all the element information that can be provided in the array, and has higher parameter estimation accuracy. Compared with the uniform linear array, the method can not only estimate the spatial parameters of the signal source, but also jointly estimate the polarization auxiliary angle and polarization phase difference parameters of the signal source, and has higher resolution capability. In addition, the present application is based on the accurate model, and adopts the accurate spatial amplitude phase factor. Compared with the Fresnel approximation, the method will not introduce system error, and has higher estimation accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0107] Figure 1 The figure is a structural schematic diagram of the one-dimensional uniform COLD array and the near-field accurate geometric propagation model adopted by the present application;

[0108] Figure 2 The figure is a spatial parameter scatter point matching result graph estimated by using the present application in the specific embodiment;

[0109] Figure 3 The figure is a polarization domain parameter scatter point matching result graph estimated by using the present application in the specific embodiment. DETAILED DESCRIPTION

[0110] In order to describe the technical content, structural features, achieved purposes and effects of the technical scheme in detail, the following will be described in detail in combination with specific embodiments and the accompanying drawings.

[0111] The present application proposes a multi-cumulative matrix rotation invariant near-field COLD accurate model direction finding method, and the specific steps are as follows:

[0112] Step 1: a near-field one-dimensional uniform COLD array model of the precise propagation model is established, array element position information is constructed, multiple narrowband near-field sources in space are incident to the array, a spatial amplitude phase factor in free space is obtained according to a geometric propagation relationship of the precise spherical wave front, the spatial amplitude phase factor constitutes a space domain steering vector, a polarization vector is obtained, a steering vector is obtained through the space domain steering vector and the polarization vector, the spatial amplitude phase factor includes two-dimensional parameter information of an azimuth angle and a distance, and the polarization vector includes two-dimensional parameter information of a polarization azimuth angle and a polarization phase difference.

[0113] Referring to Figure 1 Fig. 1, which is a schematic diagram of a precise spherical wave front geometric propagation model and a COLD array structure. This embodiment establishes a near-field one-dimensional uniform COLD array model of the precise propagation model, the COLD elements are uniformly placed on the x-axis, and the center element is taken as a reference element and a coordinate origin, the total number of elements is 2L+1, and the element spacing is d, d = λ / 2 (λ is the signal wavelength). It is assumed that K narrowband near-field sources in space are incident to the array, the position parameters of the kth signal are (θ k , r k ), the polarization parameters are (γ k , η k ), r k is the distance from the kth signal source to the reference element of the array, θ k is an azimuth angle, that is, the included angle between the incident direction of the kth signal source and the y-axis θ k ∈[-π / 2, π / 2], γ k is a polarization auxiliary angle of the kth signal source γ k ∈[0, π / 2], η k is a polarization phase difference of the kth signal source η k ∈[-π, π], and the distance from the kth signal source to the lth element is:

[0114]

[0115] wherein, l = -L, …, 0, …, L.

[0116] According to the geometric propagation relationship of the precise spherical wave front, the spatial amplitude phase factor received by the lth element from the kth signal source is:

[0117]

[0118] b(θ k , r k ) = [b -L，k , …, b l，k , …, b L,k ] Tdenotes the spatial steering vector of the kth signal source, with dimension (2L+1) x 1, and the exact delay phase τ l，k , the amplitude attenuation term ρ k，k .

[0119] The polarization vector of the kth signal source is denoted as:

[0120]

[0121] The steering vector of the kth signal source is:

[0122]

[0123] wherein denotes the Kronecker product, a(θ k , r k , γ k , η k ) has dimension 2(2L+1) x 1.

[0124] The received data vector of the array at time t is x(t),

[0125]

[0126] x(t) has dimension 2(2L+1) x 1, denotes the Khatri-Rao product, A = [a(θ1, r1, γ1, η1),..., a(θ k , r k , γ k , η k ),..., a(θ K , r K , γ k , η K )] is the array manifold matrix with dimension 2(2L+1) x K, B = [b(θ1, r1),..., b(θ k , r k ),..., b(θ K , r K )] is the spatial array manifold matrix, G = [g(γ1, η1),..., g(γ k , η k ),..., g(γ K , η K )] is the polarization domain array manifold matrix with dimension 2 x K, s(t) = [s1(t),..., s k (t),..., s K (t)] T is the signal vector with dimension K x 1, and n(t) is a Gaussian white noise vector with dimension 2(2L+1) x 1.

[0127] because The received data vector can be further represented as:

[0128]

[0129] Where, x1(t)=[x 1，-L (t), ..., x 1，0 (t), ..., x 1，L (t)] T It is a vector consisting of the first 2L+1 elements of x(t), x2(t) = [x 2，-L (t), ..., x 2，0 (t), ..., x 2，L (t)] T It is a vector consisting of the last 2L+1 elements of x(t), a diagonal matrix Diagonal matrix P2 = diag[cosγ1, ..., cosγ] k , ...,cosγ K Therefore, the guiding vector can be rewritten as a(θ). k r k γ k η k )=[a 1，-L，k , ..., a 1，L，k a 2，-L，k , ..., a 2，L,k ] T For ease of writing, we will use 'a' from now on. k to replace a(θ) k r k γ k η k ).

[0130] Step 2: Using the array element output data and array received data vector of the COLD array, first construct the fourth-order cumulant matrix of each array element, and then cascade them to obtain the joint cumulant long matrix.

[0131] Using the output data of the i-th array element and the array received data vector, a fourth-order cumulant matrix C can be constructed. l ,

[0132]

[0133] in,(.) * Indicates conjugation, (·) H Represents the conjugate transpose. The fourth-order cumulant matrix C l This can be further represented as C l =AΦ l C 4s A H, the dimension of the matrix is 2(2L+1) x 2(2L+1), wherein the diagonal matrix Diagonal elements of the matrix represents the cumulative quantity of the kth signal, the symbol cum(·) represents calculating the cumulative quantity, the diagonal matrix According to the construction method of the fourth-order cumulative quantity matrix described above, the fourth-order cumulative quantity matrix corresponding to each element is obtained, and there are 2L+1 elements, and the joint cumulative quantity long matrix Q with a dimension of 2(2L+1) 2 x 2(2L+1) is obtained,

[0134]

[0135] Wherein, Φ0 is a unit matrix.

[0136] Step 3: Calculate the covariance matrix of the joint cumulative quantity K matrix and perform eigenvalue decomposition, extract the signal subspace corresponding to the fourth-order cumulative quantity of each element according to the index order, and use the rotational invariance between each subspace to separate the spatial amplitude and phase factors, and obtain the array manifold estimation value of the space domain.

[0137] Calculate the covariance matrix R of the joint cumulative quantity long matrix Q, R=E{QQ H The dimension is 2(2L+1) 2 x 2(2L+1) 2 , and then perform eigenvalue decomposition on the matrix R, that is, Wherein, U s is a signal subspace composed of eigenvectors corresponding to K large eigenvalues, U N is a noise subspace composed of eigenvectors corresponding to 2(2L+1) 2 -K small eigenvalues, ∑ s is a diagonal matrix composed of K large eigenvalues, ∑ N is a diagonal matrix composed of 2(2L+1) 2 -K small eigenvalues.

[0138] The signal subspace U s can be decomposed into 2L+1 parts, the number of which corresponds to the number of elements, that is,

[0139]

[0140] Wherein The 2(2L+1)L+1th row to the 2(2L+1)L+2(2L+1)th row of the signal subspace U s is composed, which is represented as the signal subspace corresponding to the reference element cumulative quantity matrix.

[0141] Since the space spanned by the signal subspaces is the same as the space spanned by the manifold matrix, and the signal subspaces corresponding to the reference sensor element and the signal subspaces corresponding to the sensor elements are rotationally invariant, there exists a non-singular matrix such that the following relationship is established:

[0142]

[0143] Therefore, the signal subspaces and the signal subspace corresponding to the reference sensor element satisfy the relationship where ψ l = T -1 Φ l T, the matrix ψ l is similar to the diagonal matrix Φ l , and the diagonal elements of Φ l are the eigenvalues of ψ l . According to the least squares criterion, the matrix ψ l can be solved,

[0144]

[0145] where (·) + represents the pseudo-inverse of the matrix. The matrix ψ l solved is subjected to eigenvalue decomposition, and the diagonal matrix Φ l is obtained.

[0146] Then, the diagonal elements of the diagonal matrix Φ l obtained by eigenvalue decomposition are spanned into a row vector, and are concatenated in the order of the position index of the corresponding sensor element to obtain the manifold matrix estimate value of the spatial array

[0147]

[0148] where diag -1 (·) represents spanning the diagonal elements of the matrix into a row vector, and the spatial steering vector estimate value

[0149] Step 4: Further decomposition is performed on the signal subspaces corresponding to the fourth-order cumulants of each sensor element, and the polarization information can be separated from the signal subspaces containing the spatial polarization domain. The first half of the subspaces corresponding to the fourth-order cumulant matrices of each sensor element is further concatenated to obtain the signal subspace E h , and the second half of the subspaces corresponding to the fourth-order cumulant matrices of each sensor element is further concatenated to obtain the signal subspace E v . According to the rotational invariance between the subspaces, the matrix The polarization auxiliary angle estimation value and the polarization phase difference estimation value are obtained by characteristic decomposition and extraction of the features.

[0150] Since the matrix A = G o B, the fourth-order cumulant matrix C l may be decomposed as:

[0151]

[0152] Therefore, the signal subspace can be further decomposed as:

[0153]

[0154] wherein is composed of the first 2L+1 row vectors of the signal subspace , is composed of the last 2L+1 row vectors of the signal subspace .

[0155] Extracting from the signal subspace constructing the signal subspace E h ,

[0156]

[0157] Extracting from the signal subspace constructing the signal subspace E v ,

[0158]

[0159] According to the rotation invariance between the signal subspace corresponding to the fourth-order cumulant matrix of different array elements and the signal subspace corresponding to the fourth-order cumulant matrix of the reference array element, it can be known that

[0160]

[0161] According to the matrix and the matrix , the matrix may be calculated. h and E v are brought into the above formula and further simplified to obtain wherein The diagonal matrix P1 and P2 are brought into the above formula, and the matrix P can be calculated:

[0162]

[0163] Since P = TDT -1 , the matrix D is similar to the diagonal matrix P, and the diagonal elements of the matrix P are the eigenvalues of the matrix D, therefore, the estimation of the diagonal matrix P can be obtained by characteristic decomposition of the matrix D Therefore, the polarization auxiliary angle estimation value of the kth signal The polarization phase difference estimation value of the kth signal wherein represents a diagonal matrix The kth diagonal element.

[0164] Step 5: According to the spatial array manifold matrix estimation value, the spatial amplitude phase factor estimation value is extracted and the corresponding amplitude attenuation term is obtained. The matrix equations about the azimuth angle and the distance are solved, and the coarse estimation values of the azimuth angle and the distance are obtained by using the least square method.

[0165] Since the amplitude attenuation term ρ k，k is,

[0166]

[0167] After expanding and simplifying the above formula, we can obtain

[0168]

[0169] wherein d1 = ld represents the distance between the lth array element and the reference array element. The spatial amplitude phase factor estimation value obtained in step 3 is Take the modulus operation to obtain the estimation value of the amplitude attenuation term wherein abs(·) represents the modulus operation. The estimation value is brought into the above formula to obtain the equation relationship of the unknown quantity

[0170]

[0171] In addition to the reference array element, the above equation can be listed as 2L, and the equation group is as follows:

[0172]

[0173] wherein the two unknown quantities of the equation group are and r k sinθ k . The coefficient matrix of the equation group is

[0174]

[0175] and the unknown quantity vector is

[0176] The solution of the unknown quantity vector by using the least square method is

[0177]

[0178] The expressions of the distance estimation value and the azimuth angle estimation value are:

[0179]

[0180] in This represents the first element of the vector of unknowns. This represents the second element of the unknown vector.

[0181] Step 6: Based on the coarse estimates of the azimuth and distance from Step 5, an unambiguous coarse estimate of the precise delay phase can be calculated. Based on the spatial array manifold matrix estimate, the spatial amplitude phase factor estimate can be obtained, and the corresponding delay phase term can be extracted. Using the unambiguous coarse estimate of the precise delay phase to deambiguate, the precise delay phase estimate can be obtained. By simultaneously solving the matrix equations related to the azimuth and distance, and using the least squares method, the precise estimates of the azimuth and distance can be obtained.

[0182] The estimated spatial amplitude phase factor obtained in step 3 is used to perform angle calculations to obtain the estimated delay phase value. Here, `angle(·)` represents the angle calculation. It's important to note that the delayed phase obtained through this calculation is the principal argument value, not the true delayed phase; that is, the phase is somewhat ambiguous. This is because the delayed phase estimate... With the precise delay phase τ in step 1 l，k accurate delay phase estimate There exist integer multiples of 2πn, and a set can be constructed. Used for deblurring, accurate delayed phase estimation It is set S l Element.

[0183] Using the distance estimate obtained in step 5 and angle estimates Unambiguous coarse estimates of delayed phase can be constructed.

[0184]

[0185] in Precise delay phase estimate It is set S l The coarse estimate of the intermediate and delayed phases is unambiguous. The value with the smallest deviation, that is,

[0186]

[0187] To simplify the calculation, the accurate delay phase estimate obtained from the above formula is now presented. After normalizing to 2π / λ, the result is rewritten as...

[0188]

[0189] Delay term The angle and distance parameters of the source satisfy the relationship:

[0190]

[0191] By means of algebraic operation, the relationship of r k , θ k and can be expressed as:

[0192]

[0193] The obtained accurate and non-fuzzy estimation value is substituted into the above formula, and the equation relationship is obtained as:

[0194]

[0195] In addition to the reference array element, the above equation can be listed as 2L, and the equation group is obtained as:

[0196]

[0197] Among them, the two unknowns of the equation group are r k and r k sinθ k . The coefficient matrix can be listed from the above formula as:

[0198]

[0199] The unknown vector is Δ f，k = [r k , r k sinθ k ] T . According to the least square criterion, the solution of the unknown vector is :

[0200]

[0201] The expression of the accurate azimuth estimation value and the accurate distance estimation value is:

[0202]

[0203] Among them, φ represents the first element of the unknown vector, and θ represents the second element of the unknown vector.

[0204] The following provides examples to verify the effectiveness and accuracy of the multi-accumulation matrix rotation-invariant near-field COLD accurate model direction finding method of the present application.

[0205] To verify the effectiveness of the method of the present invention in near-field DOA parameter estimation, a simulation experiment was conducted on the algorithm steps. The specific simulation parameters are as follows:

[0206] The simulation of this invention assumes two incoherent narrowband near-field signal sources in space, incident on a uniform array of nine elements. The central element of the array serves as the reference element, and the element spacing is d = λ / 2, λ = 1. The incident azimuth angle, range, polarization auxiliary angle, and polarization phase difference parameters are (-10°, 5λ, 20°, -20°) and (30°, 6λ, 60°, 90°), respectively. The number of snapshots N = 1000, and the signal-to-noise ratio is 20 dB. (See also...) Figure 2 and Figure 3 The figures shown are the scatter plots of spatial domain parameters and the scatter plots of polarization domain parameters, respectively. Figures 2 to 3 Scatter plots of azimuth and range, polarization auxiliary angle, and polarization phase difference are presented. As can be seen from the figures, the root mean square errors of the azimuth and range parameters, as well as the polarization auxiliary angle and polarization phase difference, exhibit high accuracy even at a signal-to-noise ratio of 0 dB. Furthermore, the estimation accuracy increases synchronously with the increase of the signal-to-noise ratio, demonstrating good algorithm performance.

[0207] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Unless otherwise specified, an element defined by the phrase "comprising..." or "including..." does not exclude the presence of additional elements in the process, method, article, or terminal device that includes said element. Additionally, in this document, "greater than," "less than," "exceeding," etc., are understood to exclude the stated number; "above," "below," "within," etc., are understood to include the stated number.

[0208] Although the above embodiments have been described, those skilled in the art, once they understand the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the above descriptions are merely embodiments of the present invention and do not limit the scope of patent protection of the present invention. Any equivalent structural or procedural transformations made using the content of the present invention's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the scope of patent protection of the present invention.

Claims

1. A multi-accumulation-matrix rotation-invariant near-field COLD precise model direction-finding method, characterized in that, The method comprises the following steps: Step 1: a near-field one-dimensional uniform COLD array model of an accurate propagation model is established, position information of array elements is constructed, multiple narrow-band near-field sources in space are incident to the array, a spatial amplitude phase factor in free space is obtained according to a geometric propagation relationship of an accurate spherical wave front, the spatial amplitude phase factor constitutes a space domain steering vector, a polarization vector is obtained, a steering vector is obtained through the space domain steering vector and the polarization vector, the spatial amplitude phase factor comprises two-dimensional parameter information of an azimuth angle and a distance, and the polarization vector comprises two-dimensional parameter information of a polarization azimuth angle and a polarization phase difference; Step 2: using array element output data and array receiving data vectors of the COLD array, a fourth-order cumulative quantity matrix of each array element is first constructed, and then a joint cumulative quantity long matrix is obtained through cascading; Step 3: a covariance matrix of the joint cumulative quantity long matrix is calculated and is subjected to eigenvalue decomposition, a signal subspace corresponding to the fourth-order cumulative quantity of each array element is extracted in index order, and the spatial amplitude phase factor can be separated and obtained by using the rotational invariance between each subspace, and an array manifold estimation value of the space domain is obtained; Step 4: further decomposing the signal subspace corresponding to the fourth-order cumulant of each array element, the polarization information can be separated from the signal subspace containing the spatial polarization domain; the first half of the subspace corresponding to the fourth-order cumulant matrix of each array element is further concatenated to obtain the signal subspace , the second half of the subspace corresponding to the fourth-order cumulant matrix of each array element is further concatenated to obtain the signal subspace , according to the rotational invariance between each subspace, the matrix can be constructed, and by performing eigenvalue decomposition and extracting the eigenvalues, the polarization auxiliary angle estimation value and the polarization phase difference estimation value are obtained; Step 5: according to the array manifold matrix estimation value of the space domain, a spatial amplitude phase factor estimation value is extracted and a corresponding amplitude attenuation term is obtained, a matrix equation about the azimuth angle and the distance is simultaneously solved, a coarse estimation value of the azimuth angle and the distance is obtained by using a least square method; Step 6: according to the coarse estimation value of the azimuth angle and the distance in step 5, an accurate delay phase unambiguous coarse estimation value is calculated, according to the array manifold matrix estimation value of the space domain, a spatial amplitude phase factor estimation value is obtained, and a corresponding delay phase term is extracted, the delay phase term of the array manifold matrix estimation of the space domain is extracted, the accurate delay phase estimation value is obtained by using the accurate delay phase unambiguous coarse estimation value to solve ambiguity, and the accurate estimation value of the azimuth angle and the accurate estimation value of the distance are obtained by simultaneously solving the matrix equation about the azimuth angle and the distance and using the least square method; The step 6 specifically comprises the following steps: The estimated spatial amplitude phase factor obtained in step 3 is used to perform angle calculations to obtain the estimated delay phase value. , ,in This indicates the angle calculation; it's important to note that the delayed phase obtained through this calculation is the principal argument value, not the true delayed phase, meaning the phase is somewhat ambiguous; due to the delayed phase estimate... With the precise delayed phase in step 1 accurate delay phase estimate exist Integer multiples can be used to construct sets. Used for deblurring, accurate delayed phase estimation It is a set Element; Using the distance estimate obtained in step 5 and the angle estimate a delay-phase unambiguous coarse estimate can be constructed , wherein ; Precise delay phase estimate is the set with the delay phase unambiguous coarse estimate the value with the smallest deviation, i.e., To simplify the operation, the obtained accurate delay phase estimation value from the above formula is Normalized Processing, the result is recorded as , Delay term The angle and distance parameters of the signal source satisfy the relationship: By means of algebraic operation, the relationship of , And is: The obtained accurate non-fuzzy estimation value is brought into the above formula, and the equation relationship is obtained: In addition to the reference array element, the above equation can be listed Article, the equation group is obtained: where the two unknowns of the system of equations are and The coefficient matrix can be written from the above equations as The unknown vector is ; according to the least squares criterion, the solution of the unknown vector is , The expression for the azimuth fine estimate and the range fine estimate is: wherein denotes the first element of the unknown vector, denotes the second element of the unknown vector.

2. The multiple-accumulation-matrix, rotation-invariant, near-field COLD, precise model direction finding method of claim 1, wherein, The step 1 specifically comprises the following steps: A near-field one-dimensional uniform COLD array model is established to establish an accurate propagation model. This array places COLD elements uniformly. On the axis, with the central array element as the reference array element and the origin of the coordinate system, the total number of array elements is... The spacing between array elements is , , The wavelength is the signal wavelength; assuming there is in space A narrowband near-field source is incident on the array, the first... The position parameters of each signal are The polarization parameters are , It is the first The distance from the nth signal source to the array reference element, i.e., the nth signal source... The incident direction of each signal source and Angle between axes , It is the first Polarization auxiliary angle of each signal source , It is the first Polarization phase difference of each signal source , No. The signal source to the first The distance between each array element is: wherein ; According to the geometric propagation relation of the exact spherical wave front, the first element receives the spatial amplitude-phase factor from the first signal source. denotes the spatial steering vector of the ;​​​ The polarization vector of the first signal source is represented as: The steering vector of the first signal source is: wherein denotes the Kronecker product, has dimension ; The received data vector of the time array is , is of dimension , denotes the Khatri-Rao product, is an array manifold matrix of dimension , is an array manifold matrix for the spatial domain, is a polarized array manifold matrix of dimension , is a signal vector of dimension is a Gaussian white noise vector of dimension ​​ Due to , the received data vector can be further represented as: wherein, is a vector consisting of the first elements, is a vector consisting of the last elements, and is a diagonal matrix, ; thus, the steering vector can be rewritten as , where will be used in place of from now on.

3. The multiple-accumulation-matrix, rotation-invariant, near-field COLD, precise model direction finding method of claim 2, wherein, The step 2 specifically comprises the following steps: Using the first array element output data and the array received data vector, a fourth order cumulant matrix may be constructed, in, Indicates conjugate. Represents the conjugate transpose; fourth-order cumulant matrix It can be further expressed as The dimension of the matrix is , where the diagonal matrix diagonal elements of a matrix Indicates the first The cumulative amount of a signal, symbol This represents the calculation of cumulative amounts, a diagonal matrix. Based on the construction method of the fourth-order cumulant matrix described above, the fourth-order cumulant matrix corresponding to each array element can be obtained similarly. , resulting in a construction dimension of . Joint cumulative long matrix , wherein is the identity matrix.

4. The multiple-accumulation-matrix, rotation-invariant, near-field COLD, precise model direction finding method of claim 3 wherein, The step 3 specifically comprises the following steps: Computing joint cumulant long matrix covariance matrix , whose dimension is , then the matrix is decomposed into eigenvalues, i.e. where is the signal subspace composed of eigenvectors corresponding to large eigenvalues, is the noise subspace composed of eigenvectors corresponding to small eigenvalues, is a diagonal matrix composed of large eigenvalues, is a diagonal matrix composed of small eigenvalues. Signal subspace Decomposable into The number of parts corresponds to the number of array elements, that is wherein is a signal subspace of the first row to the row, denoted as a signal subspace corresponding to the reference array element cumulant matrix; Since the space spanned by the signal subspaces is the same as the space spanned by the manifold matrix, and the signal subspaces corresponding to the reference element accumulation matrix and the signal subspaces corresponding to each element accumulation matrix satisfy the rotational invariance, there is a non-singular matrix such that the following relationship holds: Therefore, the signal subspace corresponding to the reference array elements satisfies the relation, where the matrix is similar to a diagonal matrix , the diagonal elements of which are the eigenvalues of ; according to the least square criterion, the matrix , wherein denotes the pseudo-inverse of a matrix, and the matrix is decomposed into a diagonal matrix ; Then, the diagonal matrix obtained by the eigen decomposition The diagonal elements are zipped into a row vector, and concatenated in the order of the position index of the corresponding array elements, to obtain the spatial array manifold matrix estimate , where, denotes the diagonal elements of the matrix are spanned into a row vector, the spatial steering vector estimate .

5. The multiple-accumulation-matrix, rotation-invariant, near-field COLD, precise model direction finding method of claim 4 wherein, The step 4 specifically comprises the following steps: Because of the matrix , the fourth-order cumulant matrix can be decomposed as So the signal subspace can be further decomposed as: wherein is formed by the signal subspace before row vectors, is formed by the signal subspace after row vectors; extracting from the signal subspace constructing the signal subspace , extracting from the signal subspace constructing the signal subspace , According to the rotation invariance between the signal subspace corresponding to the fourth-order cumulant matrix of different array elements and the signal subspace corresponding to the fourth-order cumulant matrix of the reference array element, it is known that , According to the matrix and the matrix The matrix can be calculated; the and are brought into the above formula and further simplified to obtain , wherein ; the diagonal matrix is brought into the above formula, and the matrix can be calculated: Because, , the matrix is similar to a diagonal matrix , the diagonal elements of the matrix are the eigenvalues of the matrix , therefore, eigen decomposition of the matrix can get the estimation of the diagonal matrix ; so, the polarization auxiliary angle estimation value of the first signal , the polarization phase difference estimation value of the first signal , where represents the first diagonal element of the diagonal matrix .​ 6. The multiple-accumulation-matrix, matrix rotation-invariant, near-field COLD precise direction finding method of claim 5, wherein, The step 5 specifically comprises the following steps: Due to the amplitude attenuation term is, After expanding and simplifying the above equation, we have, where denotes the distance of the th array element from the reference array element; the spatial amplitude-phase factor estimate obtained in step 3 is subjected to a modulo operation to obtain an estimate of the amplitude attenuation term , where denotes a modulo operation; the estimate is substituted into the above equation to obtain an equation relationship for the unknown​ In addition to the reference elements, the above equation can be written as Equations can be obtained as follows: where two unknowns of the equation set are and ; the coefficient matrix can be obtained from the equation set, and the unknown vector is ; The solution of the unknown quantity vector is obtained by using the least square method as The expressions for the distance estimate and the azimuth angle estimate are: wherein denotes the first element of the unknown vector, denotes the second element of the unknown vector.

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