A meshless target parameter estimation method based on FDA-MIMO radar with single base station array

Abstract

The application provides a FDA-MIMO radar gridless target parameter estimation method based on a single base plane array, relates to the technical field of array signal processing, constructs a single base plane array FDA-MIMO radar system, and the transmitting end and the receiving end of the radar system are composed of a uniform plane array; a receiving signal model is established based on the radar system, the model is vectorized, a selection matrix is introduced to construct an angle-distance decoupling receiving signal model; the covariance matrix of the receiving signal model is calculated; a low-rank matrix reconstruction optimization problem suitable for the model is established according to the covariance matching criterion; the optimization problem is solved by the idea of alternating projection by using the properties of the covariance matrix; the information of the angle theta and the distance r is estimated by the multiple signal classification method according to the covariance matrix; the problems that the FDA-MIMO radar angle-distance decoupling model in the prior art is only suitable for a linear array, and the target parameter estimation method has low estimation accuracy and high calculation complexity can be solved.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, and in particular to a gridless target parameter estimation method for FDA-MIMO radar based on a monostatic surface array. Background Technology

[0002] Target parameter estimation is a major research area in the field of array signal processing technology, playing a crucial role in radar, sonar, voice, and wireless communication. Multiple Input Multiple Output (MIMO) radar has attracted widespread attention since its inception. However, during beam scanning, the beam pattern depends only on the angular direction, lacking selectivity in the range domain. Therefore, it cannot solve the problem of target localization that combines angle and range.

[0003] Frequency Diverse Array (FDA) radar adds a frequency increment at the carrier frequency of the array elements. Therefore, the beam pattern of FDA radar is a function of angle, range and time, which has the advantages of anti-clutter and anti-jamming. So it has wide applications in target parameter estimation.

[0004] Researchers have proposed combining FDA (Fast Array Multiple Input Multiple Output) and MIMO (Multiple Array Multiple Input Multiple Output) technologies to establish a novel frequency-diverse array multiple-input multiple-output (FDA-MIMO) radar system capable of joint angle-range target localization. However, studies based on linear array FDA-MIMO radar scenarios cannot simultaneously detect the target's azimuth and elevation angles. Extending the scenario to area array FDA-MIMO radar can acquire more target information and achieve higher estimation accuracy, but this leads to increased computational costs. While subspace-based methods offer high estimation accuracy, their performance degrades under harsh conditions. Although this difficulty can be overcome using CS (Computational Array) methods, most CS methods require dense grid partitioning to discretize the entire parameter space, resulting in significant computational costs. Therefore, finding target information in a monostatic area array FDA-MIMO radar without discretizing the target parameter space is crucial. Furthermore, in area array FDA-MIMO radar, the angle and range coupling problem also affects its estimation performance. Existing technology has problems with angle and range coupling in the system model, and the model is only applicable to linear arrays, resulting in low accuracy and high computational complexity. Therefore, decoupling the angle and range is also an urgent task to be solved. Summary of the Invention

[0005] The technical problem to be solved by this invention is to overcome the shortcomings of the prior art and provide a gridless target parameter estimation method for FDA-MIMO radar based on a single-base array. Compared with linear array-based FDA-MIMO radar, this method can acquire more target information, obtain higher estimation accuracy, estimate more signal sources with the same number of array elements, solve the optimization problem of low-rank matrix reconstruction established by the model using the gridless estimation method, and is not affected by grid mismatch effect. It can solve the problems of existing FDA-MIMO radar angle-range decoupling models that are only applicable to linear arrays, have low accuracy, and high computational complexity.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] A gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array, proposed according to the present invention, includes the following steps:

[0008] Step S1: The monostatic plane array FDA-MIMO radar system consists of a transmitter and a receiver. The transmitter and receiver respectively construct uniform planar arrays with M×M and N×N array elements, where M and N≥2.

[0009] Step S2: Based on the monostatic array FDA-MIMO radar system in Step S1, establish the received signal model of the system and vectorize it. Introduce a selection matrix into the vectorized received signal model to construct an angle-range decoupled received signal model. The angle-range decoupled received signal model is simply referred to as the received signal model.

[0010] Step S3: Calculate the covariance matrix based on the received signal model in step S2;

[0011] Step S4: Based on the covariance matrix in step S3, and according to the covariance matching criterion, establish a low-rank matrix reconstruction optimization problem applicable to the received signal model.

[0012] Step S5: Solve the optimization problem established in step S4 by using the properties of the covariance matrix through the alternating projection method;

[0013] Step S6: Based on the solution of the optimization problem in step S5, the covariance matrix T(u) is obtained. The angle θ is then obtained from the covariance matrix T(u) using the multiple signal classification method. The information is obtained, and the distance r is obtained, where θ and denoted as , and respectively as the angles between the target's direction relative to the reference array element and the x-axis and y-axis, respectively; and r as the distance from the target to the antenna.

[0014] As a further optimization of the gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array described in this invention, the monostatic array FDA-MIMO radar system is as follows: the transmitting end of the monostatic array FDA-MIMO radar system is a transmitting array with M×M elements composed of a uniform planar array, and the receiving end of the monostatic array FDA-MIMO radar system is a receiving array with N×N elements composed of a uniform planar array; wherein the element spacing of the transmitting array is d. t The element spacing of the receiving array is d. r Assume d t =d r =d, where d is half the wavelength.

[0015] As a further optimization of the FDA-MIMO radar gridless target parameter estimation method based on a monostatic array described in this invention, in step S2, the angle-range decoupled received signal model is as follows:

[0016]

[0017] Where X is the received signal when the receiving array receives L snapshots, X = [x(1), x(2), ..., x(L)], and x(l) is the received signal when the receiving array receives l snapshots, 1 ≤ l ≤ L, and the dimension of its received signal X is M. 2 N 2 ×L; Q is the selection matrix, with dimension M. 2 N 2 ×(M+N-1) 2 (2M-1), the structure of matrix Q is chosen to satisfy,

[0018]

[0019]

[0020]

[0021] Where the superscript T is the transpose operator, which divides Q into N parts by rows. z' This represents the z'th portion of Q, where Q is... z' Divide into N parts by row, Q z'y' Q represents z' The y'th portion will be Q z'y' Divide into M parts by row, Q z'y'j' Q represents z'y' The j'th copy, Q z'y'j'The position of the m'-th row with index (M+N-1)(2M-1)(z'+j'-2)+(2M-1)(y'+m'-2)+j+m-1 is 1, and all others are zero, z',y'=1,2,…,N, j',m'=1,2,…,M; θ and Let r be the angle between the target's direction relative to the reference array element and the x-axis and y-axis, r be the distance from the target to the antenna, and ⊙ denote the Khatri-Rao product. It is composed of vectors The matrix formed g1(θ k )and They respectively include and A vector of all elements. Represents the Kronecker product, a r (θ k )and θ represents the angles θ between the direction of the k-th target at the receiver relative to the reference array element and the x-axis and y-axis, respectively. k and The guiding vector, a t (θ k )and θ corresponding to the transmitter, respectively k and The steering vector; C(r) is the matrix composed of the range parts of the transmission steering vectors of K targets; the set of complex reflection coefficients P of the received signal: P = [p(1)p(2)...p(L)], with a dimension of K = L, p(l) is the complex reflection coefficient of the l-th snapshot, N is independent zero-mean complex Gaussian noise, N = [n(1)n(2)...n(L)], with a dimension of M 2 N 2 ×L,n(l) represents the noise of the l-th snapshot.

[0022] As a further optimization of the FDA-MIMO radar gridless target parameter estimation method based on a monostatic array described in this invention, in step S3, the covariance matrix is ​​calculated according to the received signal model, specifically, R = E[XX]. H ]=QT(u)Q H +σI, where X is the received signal model, R is the covariance matrix, Q is the selection matrix, and the noiseless covariance matrix T(u) is a matrix of dimension (M+N-1). 2 (2M-1)×(M+N-1) 2 The three-dimensional Toplitz matrix of (2M-1) is given by σ, where σ is the noise power, I is the identity matrix, H is the conjugate transpose operator, and E[·] is the covariance operator.

[0023] As a further optimization scheme for the gridless target parameter estimation method of FDA-MIMO radar based on a monostatic array described in this invention, in step S4, according to the covariance matrix in step S3 and the covariance matching criterion, a low-rank matrix reconstruction optimization problem suitable for the received signal model is established, as follows:

[0024]

[0025]

[0026] T(u)≥0

[0027] in, Let be the sampling covariance matrix of the received signal X, β be a user-defined threshold, rank[·] be the rank operator, T(u) be the noise-free covariance matrix, and ||·|| F Let u be the Frobenius norm, and u be a vector containing all information.

[0028] As a further optimization scheme for the gridless target parameter estimation method of FDA-MIMO radar based on a monostatic array described in this invention, in step S5, the optimization problem established in step S4 is solved by using the property of the covariance matrix through alternating projection. Specifically,

[0029] S51, the optimization problem in step S4 is equivalently represented as:

[0030]

[0031]

[0032] T(u)≥0

[0033] in, B indicates that the constraints of the optimization problem include a portion of the noiseless covariance matrix T(u). The remaining part of the constraints in the optimization problem is represented by σ, where σ is the noise power, Q is the selection matrix, and the noise-free covariance matrix T(u) has the following four properties: low rank, Topplitz, positive semidefinite, and...

[0034] The optimization problems of S52 and S51 are equivalent to:

[0035]

[0036] Where matrices Y and Z belong to R respectively Y R Z ;R Y Let R be a set of complex-valued matrices with rank no greater than K. Y ={Y:rank(Y)≤K};RZ Let be a set of complex-valued matrices that satisfy three conditions: Toplitz and positive semidefinite are denoted as

[0037] S53. Construct the following update criteria.

[0038]

[0039] in,(·) (μ) 、(·) (μ+1) The results of the μth and μ+1th iterations of (·) are represented by δ1 and δ2, where δ1 and δ2 are the step sizes and K represents the number of targets. To set R respectively Y Set R Z The projection of the iteration stops under the condition that the iteration is... ε is a user-defined parameter, Y (μ) Let Y be the result of the μth iteration. (μ+1) Z is the result of the (μ+1)th iteration of Y. (μ) Let Z be the result of the μth iteration of Z. (μ+1) This is the result of the (μ+1)th iteration of Z;

[0040] S54. Solve the optimization problem in step S52 according to the update rule established in step S53 until it converges or reaches the maximum number of iterations, and obtain the noiseless covariance matrix T(u).

[0041] As a further optimization scheme for the FDA-MIMO radar gridless target parameter estimation method based on a monostatic array described in this invention, in step S53, To set R Y The projection, specifically,

[0042] Let D = Y (μ) -δ1(Y (μ) μZ (μ) For matrix D, its position in set R Y Projection on Represented as,

[0043]

[0044] Where, Σ K It is a diagonal matrix whose diagonal elements are the K largest singular values ​​in matrix D, U K and V K These are the corresponding left singular matrix and right singular matrix, respectively.

[0045] As a further optimization scheme for the FDA-MIMO radar gridless target parameter estimation method based on a monostatic array described in this invention, in step S53, To set R Z The projection, specifically, is to let A = Z (μ) -δ2(Z (μ) -Y (μ+1) For matrix A, its position in set R is... Z Projection on The projection is represented as three sequential combinations of projections: P A (A), P T (A') and P ι (A"), P A (A), P T (A') and P ι (A) represent the matrix A in set R. Z The first, second, and third projections in sequence on the screen, i.e.

[0046] Projection P A (A) projects matrix A onto the constraints. Above, that is, the center of the circle is The interior or surface of a sphere with a radius of a custom threshold β, where, Then we obtain matrix A': Where Q is the selection matrix. Let P be the sampling covariance matrix of the received signal X. C (A) represents matrix C under constraints. Projection on:

[0047] Projection P T (A') is obtained by projecting matrix A' onto the set of three-dimensional Toplitz matrices, resulting in matrix A" = P. T (A');

[0048] Projection P ι (A) represents the projection of matrix A" onto a set of positive semi-definite matrices. Then, by performing eigenvalue decomposition on matrix A", the projection P is obtained. ι (A) is represented as: P ι (A")=Πdiag(ι + )Π H , among which, ι + Let A be the set of eigenvalues ​​to which all negative eigenvalues ​​of matrix A" are set to zero, and let Π be the eigenvectors corresponding to the eigenvalues ​​of matrix A".

[0049] As a further optimization of the FDA-MIMO radar gridless target parameter estimation method based on a monostatic surface array described in this invention, projection P T (A') is the projection of matrix A' onto the set of three-dimensional Topplitz matrices, specifically,

[0050] S531, Assumption Let A' be a matrix of dimension YMN×YMN, where Y is the dimension of the first dimension of matrix A'. Then, matrix A' is divided into the following submatrices:

[0051]

[0052] in, Let A' be the subarray in row n1 and column n2. for The subarray in row m1 and column m2 has dimensions Y×Y, n1,n2=1,…,N,m1,m2=1,…,M;

[0053] S532, Order

[0054]

[0055]

[0056]

[0057] Where mean{·} represents the mean of the elements in the set. Let A' be the mean of a block matrix of dimension MY×MY along the i-th diagonal (the main diagonal is defined as the zeroth diagonal, with the diagonals of the upper triangle numbered sequentially and the diagonals of the lower triangle numbered sequentially). The dimension is MY×MY. The dimension is Y×Y, and it represents a matrix. The block matrix in row m1 and column m2 of the given matrix. for The mean of the block matrix of dimension Y×Y on the j-th diagonal. Representation matrix The element in the m1-th row and m2-th column of the array. Representation matrix The mean of the elements on the y-th diagonal, a vector containing information about the first dimension of the matrix. Depend on Composition, 1-Y≤y≤Y-1, then the vector Represented as,

[0058]

[0059] Depend on The vector consisting of dimensions (2Y-1)(2M-1) containing information from the second-dimensional matrix. Represented as,

[0060]

[0061] in, For a vector containing information of the first dimension matrix, 1-M≤o≤M-1;

[0062] S533, Let the vector The column vector formed Given a vector containing information from the second-dimensional matrix, 0 ≤ n' ≤ N-1, construct the three-dimensional Toplitz matrix based on the structure of the noise-free covariance matrix T(u) from step S3. Its dimensions are (M+N-1) 2 (2M-1)×(M+N-1) 2 (2M-1), the three-dimensional Topplitz matrix That is, projection P T (A').

[0063] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0064] The gridless target parameter estimation method for FDA-MIMO radar based on monostatic arrays of the present invention, on the one hand, extends the application scenario to area array FDA-MIMO radar, which can acquire more target information and obtain higher estimation accuracy; on the other hand, it proposes a model suitable for the decoupling of angle and range of area arrays, and uses a gridless estimation method to solve the optimization problem of low-rank matrix reconstruction established by the model, which is not affected by grid mismatch effect, and can achieve high-resolution target parameter estimation while reducing computational complexity. Attached Figure Description

[0065] Figure 1 This is a flowchart illustrating the gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array according to an embodiment of the present invention.

[0066] Figure 2 This is a schematic diagram illustrating the monostatic array FDA-MIMO radar system in the embodiment.

[0067] Figure 3 This is a comparative diagram showing the changes in angle estimation error, range estimation error, and runtime with signal-to-noise ratio between the gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array in the embodiment and existing methods; wherein, (a) is a comparative diagram showing the changes in angle θ estimation error with signal-to-noise ratio between the embodiment method and existing methods, and (b) is a comparative diagram showing the changes in angle θ estimation error between the embodiment method and existing methods. (c) is a comparative diagram showing the difference between the distance estimation error of the embodiment method and the existing method as the signal-to-noise ratio changes; (d) is a comparative diagram showing the difference between the running time of the embodiment method and the existing method as the signal-to-noise ratio changes.

[0068] Figure 4 This is a comparative diagram showing the changes in angle estimation error, range estimation error, and running time with snapshots for the gridless target parameter estimation method of the FDA-MIMO radar based on a monostatic array in the embodiment, and existing methods. Specifically, (a) is a comparative diagram showing the changes in angle θ estimation error with snapshots for the embodiment method and existing methods, and (b) is a comparative diagram showing the changes in angle θ estimation error with snapshots for the embodiment method and existing methods. (c) is a comparative diagram showing the change of distance estimation error with snapshots between the embodiment method and the existing method, and (d) is a comparative diagram showing the change of running time with snapshots between the embodiment method and the existing method. Detailed Implementation

[0069] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0070] A gridless target parameter estimation method based on a monostatic surface array for FDA-MIMO radar, such as... Figure 1 This includes the following steps:

[0071] S1. The transmitter and receiver are constructed using a transmitter array with M×M elements and a receiver array with N×N elements, respectively. Both the transmitter and receiver arrays are uniform planar arrays (UPA), resulting in a monostatic planar array FDA-MIMO radar system; where M and N ≥ 2.

[0072] In step S1, the following is obtained: Figure 2 The monostatic area array FDA-MIMO radar system shown is characterized by a transmitting array of M×M elements composed of UPAs at the transmitting end and a receiving array of N×N elements composed of UPAs at the receiving end; wherein the element spacing of the transmitting array is d. t The element spacing of the receiving array is d. r For ease of analysis, assume d. t =d r =d, where d is half the wavelength;

[0073] S2. Based on the monostatic area array FDA-MIMO radar system in step S1, establish the received signal model of this system and vectorize it. Introduce a selection matrix into the vectorized received signal model to construct an angle-range decoupled received signal model, which is simply referred to as the received signal model; specifically:

[0074] S21. For L snapshots and K far-field targets, after passing through the matched filter, the output signal of the l-th snapshot is expressed as:

[0075]

[0076] Where X(l) has dimension M 2 ×N 2 r k Let θ be the distance from the k-th target to the antenna. k and Let p be the angle between the direction of the k-th target relative to the reference array element and the x-axis and y-axis, respectively, where k = 1, 2, ..., K, l = 1, 2, ..., L, and p are the angles between the direction of the k-th target relative to the reference array element and the x-axis and y-axis, respectively. k (l) represents the complex reflection coefficient of the k-th target in the l-th snapshot, and the receiving steering vector of the k-th target is... a r (θ k )and θ represents the angles θ between the direction of the k-th target at the receiver relative to the reference array element and the x-axis and y-axis, respectively. k and The guidance vector of the k-th target is . Let θ be the angle component of the launch steering vector of the k-th target and the range component related to angle θ. Parts and Angles The relevant distance part is and,

[0077]

[0078]

[0079]

[0080] Where Δf is the frequency increment and c0 is the speed of light. Indicates the Kronecker product. This represents the Hadamarda unit, (·). T For the transpose operator, N(l) is a variable of dimension M. 2 ×N 2 Independent zero-mean complex Gaussian noise;

[0081] S22. Vectorize the output signal X(l) of the l-th snapshot to obtain:

[0082]

[0083] Where vec(·) represents the vectorization operator, p k(l) represents the complex reflection coefficient of the k-th target in the l-th snapshot. Let x(l) represent the Kronecker product, with dimension M. 2 N 2 ×1, n(l) = vec(N(l)), whose dimension is M 2 N 2 ×1;

[0084] x(l) can also be expressed as,

[0085]

[0086] in For N, all elements are 1 2 Let diag(A) be a column vector formed by extracting all the diagonal elements of matrix A. The essence of x(l) and n(l) is that they are composed of four consecutive Kronecker products. They can also be written as follows:

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093] Where, x z'y'j' (l),n z'y'j' (l) has a dimension of M×1, z',y'=1,2,…,N,j'=1,2,…,M;

[0094] S23, Definition It contains All elements in It contains All elements, angle-guided vector Also defined It includes If x contains all elements, then x zyj (l) can be represented as,

[0095]

[0096] Where z', y' = 1, 2, ..., N, j' = 1, 2, ..., M, q z'y'j'(A) is the extraction operator, representing the extraction from dimension (2M-1)×(M+N-1). 2 M elements are extracted from matrix A to form a column vector, where (j'+m'-1,(M+N-1)(z'+j'-2)+y'+m'-1) represents the coordinates of the m'-th extracted element, where m'=1,2,…,M,n z'y'j' (l) represents noise.

[0097] The purpose of introducing the selection matrix is ​​to separate angle and distance, transforming the coupled angle and distance model into a decoupled one. The dimension is defined as M×(M+N-1). 2 The selection matrix Q of (2M-1) z'y'j' To represent the extraction operator q zyj (·), Q z'y'j' The position of the m'-th row with index (M+N-1)(2M-1)(z'+j'-2)+(2M-1)(y'+m'-2)+j'+m'-1 is 1, and all others are zero. z',y'=1,2,…,N, j',m'=1,2,…,M, then x z'y'j' (l) can be represented as,

[0098]

[0099] Then x(l) is represented as,

[0100]

[0101] Where the dimension is M 2 N 2 ×(M+N-1) 2 The (2M-1) choice matrix Q satisfies,

[0102]

[0103]

[0104]

[0105] Where z', y' = 1, 2, ..., N, j' = 1, 2, ..., M, and

[0106]

[0107] C(r) = [c(r1), c(r2), ..., c(r)] K )]

[0108] p(l)=[p1(l),p2(l),…,p K (l)] T ,

[0109] in, The dimension is (M+N-1). 2 ×K, C(r) has a dimension of (2M-1)×K, and p(l) has a dimension of K×1;

[0110] S24. Establish a decoupled model for FDA-MIMO radar angle and range:

[0111]

[0112] in,

[0113] X = [x(1), x(2), ..., x(L)]

[0114] P = [p(1), p(2), ..., p(L)]

[0115] N = [n(1),n(2),…,n(L)]

[0116] X represents the received signal when the receiver array receives L snapshots, Q is the selection matrix, and θ and Let r be the angle between the target's direction relative to the reference array element and the x-axis and y-axis, r be the distance from the target to the antenna, and ⊙ denote the Khatri-Rao product. It is composed of vectors The matrix formed g1(θ k )and Each contains and A vector of all elements. Represents the Kronecker product, a r (θ k )and θ represents the angles θ between the direction of the k-th target at the receiver relative to the reference array element and the x-axis and y-axis, respectively. k and The guiding vector, a t (θ k )and θ corresponding to the transmitter, respectively k and The steering vector; C(r) is the matrix composed of the range parts of the transmission steering vectors of K targets; the set of complex reflection coefficients P of the received signal: P = [p(1)p(2)...p(L)], with a dimension of K×L, p(l) is the complex reflection coefficient of the l-th snapshot, N is independent zero-mean complex Gaussian noise, N = [n(1)n(2)...n(L)], with a dimension of M 2 N 2 ×L,n(l) represents the noise of the l-th snapshot.

[0117] It can be seen that the angle and distance of the received signal have been decoupled.

[0118] S3. Calculate the covariance matrix based on the received signal model in step S2; calculate the covariance matrix R of the received signal X from the receiving array, specifically:

[0119]

[0120] Where Q is the selection matrix, (·) H This is the conjugate transpose operator. in, It contains all steering vector information for the received signal. It is composed of vectors The matrix formed by these components, C(r), represents the range portion of the launch steering vectors of the K targets, c(r). k A matrix composed of ) and R S =E[PP H ] = diag(ρ), where P is the set of complex reflection coefficients of the received signal, R S Let ρ be the covariance matrix of the set of complex reflection coefficients P of the received signal, where ρ = [ρ1ρ2…ρ K ] T Let σ be the signal power, σ be the noise power, and I be the identity matrix. The noise-free covariance matrix is... For a 3D Toplitz matrix, generally, the structure of a d-dimensional Toplitz matrix is ​​as follows:

[0121]

[0122] in It is a d-dimensional vector. It is a d-dimensional Topulitz matrix, n d =(n d ,n d-1 If ,…,n1), then u d-1 (i)=u d (i,:,:,…,:), where i=1-n d ,2-n d ,…,n d -1. When d = 1 and n1 = (n1), This can be expressed as,

[0123]

[0124] In this embodiment, T(u) = T3(u3), n3 = (M+N-1, M+N-1, 2M-1).

[0125] S4. Based on the covariance matrix in step S3 and according to the covariance matching criterion, establish a low-rank matrix reconstruction optimization problem suitable for the received signal model:

[0126]

[0127]

[0128] T(u)≥0

[0129] in, Let be the sampling covariance matrix of the received signal X, β be a user-defined threshold, rank[·] be the rank operator, T(u) be the noise-free covariance matrix, and ||·|| F Let u be the Frobenius norm, and u be a vector containing all information.

[0130] S5. Solve the optimization problem established in step S4 by using the properties of the covariance matrix through alternating projection. Using alternating projection to solve the optimization problem can reduce complexity and ensure estimation accuracy.

[0131] S51, the optimization problem in step S4 is equivalently represented as:

[0132]

[0133]

[0134] T(u)≥0

[0135] Where β is a custom threshold, Where B represents the portion of the constraints of the optimization problem that includes the noiseless covariance matrix T(u). This represents the other parts of the constraints in the optimization problem. Let σ be the sampling covariance matrix of the received signal X, σ be the noise power, and Q be the selection matrix. It can be seen that the noise-free covariance matrix T(u) has the following four constraints: low rank, Topplitz, positive semi-definite, and...

[0136] The optimization problems of S52 and S51 are equivalent to:

[0137]

[0138] Where matrices Y and Z belong to R respectively Y R Z ;R Y Let R be a set of complex-valued matrices with rank no greater than K. Y ={Y:rank(Y)≤K};R Z Let be a set of complex-valued matrices that satisfy three conditions: Toplitz and positive semidefinite are denoted as

[0139] S53. Construct the following update criteria.

[0140]

[0141] in,(·) (μ) 、(·) (μ+1) The results of the μth and μ+1th iterations of (·) are represented by δ1 and δ2, where δ1 and δ2 are the step sizes and K represents the estimated number of targets. To set R respectively Y Set R Z The projection of the iteration stops under the condition that the iteration is... ε is a user-defined parameter, Y (μ) Let Y be the result of the μth iteration. (μ+1) Z is the result of the (μ+1)th iteration of Y. (μ) Let Z be the result of the μth iteration of Z. (μ+1) This is the result of the (μ+1)th iteration of Z;

[0142] In step S53, To set R Y The projection, specifically,

[0143] Let D = Y (μ) -δ1(Y (μ) -Z (μ) For matrix D, its position in set R Y Projection on Represented as,

[0144]

[0145] Where, Σ K It is a diagonal matrix whose diagonal elements are the K largest singular values ​​in matrix D, U K and V K These are the corresponding left singular matrix and right singular matrix, respectively.

[0146] In step S53, To set R Z The projection, specifically, is to let A = Z (μ) -δ2(Z (μ) -Y (μ+1) For matrix A, its position in set R is... Z Projection on The projection is represented as three sequential combinations of projections: P A (A), P T (A') and P ι (A"), P A (A), PT (A') and P ι (A) represent the matrix A in set R. Z The first, second, and third projections in sequence on the screen, i.e.

[0147] Projection P A (A) projects matrix A onto the constraints. Above, that is, the center of the circle is The interior or surface of a sphere with a radius of a custom threshold β, where, Then we obtain matrix A': Where Q is the selection matrix. Let P be the sampling covariance matrix of the received signal X. C (A) represents matrix C under constraints. Projection on:

[0148] Projection P T (A') is obtained by projecting matrix A' onto the set of three-dimensional Toplitz matrices, resulting in matrix A" = P. T (A');

[0149] S531, Assumption Let A' be a matrix of dimension YMN×YMN, where Y is the dimension of the first dimension of matrix A'. Then, matrix A' is divided into the following submatrices:

[0150]

[0151]

[0152] in, Let A' be the subarray in row n1 and column n2. for The subarray in row m1 and column m2 has dimensions Y×Y, n1,n2=1,…,N,m1,m2=1,…,M.

[0153] S532, Order

[0154]

[0155]

[0156]

[0157] Where mean{·} represents the mean of the elements in the set. Let A' be the mean of a block matrix of dimension MY×MY along the i-th diagonal (the main diagonal is defined as the zeroth diagonal, with the diagonals of the upper triangle numbered sequentially and the diagonals of the lower triangle numbered sequentially). The dimension is MY×MY. The dimension is Y×Y, and it represents a matrix. The block matrix in row m1 and column m2 of the given matrix. for The mean of the block matrix of dimension Y×Y on the j-th diagonal. Representation matrix The element in the m1-th row and m2-th column of the array. Representation matrix The mean of the elements on the y-th diagonal, a vector containing information about the first dimension of the matrix. Depend on Composition, 1-Y≤y≤Y-1, then the vector Represented as,

[0158]

[0159] Depend on The vector consisting of dimensions (2Y-1)(2M-1) containing information from the second-dimensional matrix. Represented as,

[0160]

[0161] in, Let be a vector containing information of the first dimension matrix, 1-M≤o≤M-1. S533, Let the vector... The column vector formed Given a vector containing information from the second-dimensional matrix, 0 ≤ n' ≤ N-1, and based on the structure of the noise-free covariance matrix T(u) in step S3, a vector with dimension (M+N-1) can be constructed. 2 (2M-1)×(M+N-1) 2 (2M-1) three-dimensional Topulitz matrix The three-dimensional Toplitz matrix That is, projection P T (A').

[0162] Projection P ι (A) represents the projection of matrix A" onto a set of positive semi-definite matrices. Then, by performing eigenvalue decomposition on matrix A", the projection P is obtained. ι (A) is represented as: P ι (A")=Πdiag(ι + )Π H , among which, ι + Let A be the set of eigenvalues ​​to which all negative eigenvalues ​​of matrix A" are set to zero, and let Π be the eigenvectors corresponding to the eigenvalues ​​of matrix A".

[0163] S54. Solve the optimization problem in step S52 according to the update rule established in step S53 until it converges or reaches the maximum number of iterations, and obtain the noiseless covariance matrix T(u).

[0164] S6. Based on the solution of the optimization problem in step S5, the covariance matrix T(u) is obtained. The angle θ and [other parameters] are then obtained from the covariance matrix T(u) using the multiple signal classification method. The information is obtained, and the distance r is obtained, where θ and denoted as , and respectively as the angles between the target's direction relative to the reference array element and the x-axis and y-axis, respectively; and r as the distance from the target to the antenna.

[0165] In step S6, the noiseless covariance matrix T(u) is solved using Vandermonde decomposition or subspace class methods to obtain angle and distance information.

[0166] The noiseless covariance matrix T(u) can be expressed as,

[0167]

[0168] Where, ρ k Let the signal power of the k-th target be . g1(θ k )and Each contains and A vector containing all elements, c(r) k ) contains All elements in the matrix. From the above equation, it can be seen that T(u) is a positive semi-definite matrix of rank K, and contains information about all incident signals, i.e., angle information. and distance information The required angle and distance can be obtained by performing Vandermonde decomposition on T(u).

[0169] This gridless target parameter estimation method for monostatic area array FDA-MIMO radar has two main advantages. First, it extends the application scenario to area array FDA-MIMO radar, enabling the acquisition of more target information and achieving higher estimation accuracy. Second, it proposes a model suitable for the decoupling of angle and range in area arrays. The gridless estimation method solves the optimization problem of low-rank matrix reconstruction established by this model, unaffected by grid mismatch effects, thus reducing computational complexity while maintaining estimation accuracy. This gridless target parameter estimation method for monostatic area array FDA-MIMO radar is applicable to the gridless estimation of angle and range in area array monostatic FDA-MIMO radar.

[0170] The simulation example verifies the gridless target parameter estimation method of FDA-MIMO radar based on a monostatic array as follows:

[0171] Simulation Example 1: A 2×2 UPA is used for the transmitting antenna, and a 3×3 UPA is used for the receiving antenna. The element spacing is half a wavelength. The reference element carrier frequency f1 = 10 GHz, and the frequency increment Δf = 5 kHz. The parameters of the optimization problem in step S5 are set using the following MATLAB statement: β = chi2inv(1 - 0.0001, L_x * L_z), where L_x is the number of transmitting elements and L_z is the vector. The length of the baseband signal used by the transmitting array is a randomly generated complex Gaussian signal, and the noise is independent zero-mean complex Gaussian noise. The snapshot length is L = 400. The experiment is set with two targets, with angles θ = [-30°, 50°] and [...]. The distance is r = [4km, 5km]. The 3D-MUSIC algorithm reduces the angle and distance search step size in the 3D search as the signal-to-noise ratio increases. The estimation performance is verified through 200 independent experiments.

[0172] Figure 3 This is a schematic diagram comparing the angle estimation error, range estimation error, and runtime variation with signal-to-noise ratio of the FDA-MIMO radar parameter estimation method based on a monostatic array in this embodiment with existing methods 3D-MUSIC and 3D-ANM. Figure 3 As can be seen from (a), (b), (c), and (d), the estimation performance of the method proposed in the embodiments is better than that of existing methods, and the running time is shorter than that of existing methods. The above results demonstrate the high accuracy and effectiveness of the method proposed in the embodiments.

[0173] Simulation Example 2: A 2×2 UPA is used for the transmitting antenna, and a 3×3 UPA is used for the receiving antenna. The element spacing is half a wavelength. The reference element carrier frequency f1 = 10 GHz, and the frequency increment Δf = 5 kHz. The parameters of the optimization problem in step S5 are set using the following MATLAB statement: β = chi2inv(1 - 0.0001, L_x * L_z), where L_x is the number of transmitting elements and L_z is the vector. The length of the baseband signal used by the transmitting array is a randomly generated complex Gaussian signal, and the noise is independent zero-mean complex Gaussian noise with a fixed signal-to-noise ratio of 0dB. The experiment was set with two targets, with angles of θ = [-30°, 50°] and... The distance is r = [4km, 5km]. The number of snapshots increases from 100 to 500 at equal intervals of 50, and the estimated performance is verified through 400 independent experiments.

[0174] Figure 4This is a schematic diagram comparing the angle estimation error, range estimation error, and runtime variations of the FDA-MIMO radar without a grid based on a monostatic array, as well as existing methods such as 3D-MUSIC and 3D-ANM, with the changes in snapshot time. Figure 4 As can be seen from (a), (b), (c), and (d), the estimation performance of the method proposed in the embodiments is better than that of existing methods, and the running time is shorter than that of existing methods. The above results demonstrate the high accuracy and computational efficiency of the results obtained by the method proposed in this invention.

[0175] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array, characterized in that, Includes the following steps: Step S1: The monostatic array FDA-MIMO radar system consists of a transmitter and a receiver. The transmitter and receiver each construct an array of [number of elements to be filled in]. , A uniform planar array, wherein, ; Step S2: Based on the monostatic array FDA-MIMO radar system in Step S1, establish the received signal model of the system and vectorize it. Introduce a selection matrix into the vectorized received signal model to construct an angle-range decoupled received signal model. Step S3: Calculate the covariance matrix of the received signal model based on the angle-distance decoupling from step S2; Step S4: Based on the covariance matrix in step S3, and according to the covariance matching criterion, establish a low-rank matrix reconstruction optimization problem applicable to the received signal model. Step S5: Solve the optimization problem established in step S4 by using the properties of the covariance matrix through the alternating projection method; Step S6: Obtain the covariance matrix based on the solution to the optimization problem in step S5. The covariance matrix is ​​used in the multi-signal classification method. Obtaining the angle and The information, and the distance. The information, among which, and The directions of the target relative to the reference array elements are respectively: shaft and The included angle of the axis, The distance from the target to the antenna; In step S2, the angle-range decoupled received signal model is as follows: ； in, For the receiving array to receive The received signal during a quick snapshot , For the receiving array to receive The received signal during a quick snapshot Its received signal The dimension is ; To select a matrix, its dimension is Select matrix The structure satisfies, ； Where the superscript T is the transpose operator, which will... Divided by row share, express The The portion will Divided by row share, express The The portion will Divided by row share, express The share, The The row index value is The position is 1, and all others are zero. , ; and The direction of the target relative to the reference element and shaft and The included angle of the axis, The distance from the target to the antenna. Represents the Khatri-Rao product. It is composed of vectors The matrix formed , and They respectively include and A vector of all elements. Indicates the Kronecker product. and These are respectively the first corresponding to the receiving end. The direction of the target relative to the reference element and shaft and Angle between axes and The guide vector, and These are respectively the transmitter's corresponding and The guide vector; for A matrix consisting of the range portion of the transmission steering vector of each target; a set of complex reflection coefficients of the received signal. : Its dimensions are , For the first The complex reflection coefficient of a snapshot, It is independent, zero-mean complex Gaussian noise. Its dimensions are , For the first A quick snapshot of noise.

2. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 1, characterized in that, The monostatic FDA-MIMO radar system is characterized by its transmitter being a uniform planar array with a number of elements. The transmitting array of the monostatic planar array FDA-MIMO radar system is a receiver composed of a uniform planar array with the following number of elements: The receiving array; The element spacing of the transmitting array is The element spacing of the receiving array is Assuming ,in It is half a wavelength.

3. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 1, characterized in that, In step S3, the covariance matrix is ​​calculated based on the angle-distance decoupled received signal model. Specifically, ,in, For angle-range decoupled received signal model, Let covariance matrix be the variance matrix. To select the matrix, the noise-free covariance matrix It is a dimension of The three-dimensional Topletz matrix, For noise power, It is the identity matrix, and the superscript H is the conjugate transpose operator. This is the covariance operator.

4. A gridless target parameter estimation method for FDA-MIMO radar based on a monostatic array as described in any one of claims 1-3, characterized in that, In step S4, based on the covariance matrix from step S3 and according to the covariance matching criterion, a low-rank matrix reconstruction optimization problem suitable for the angle-range decoupled received signal model is established, as follows: ； in, To receive signals The sampling covariance matrix, For custom thresholds, For rank operators, It is a noise-free covariance matrix. It is the Frobenius norm. It is a vector containing all the information.

5. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 4, characterized in that, In step S5, the optimization problem established in step S4 is solved using the properties of the covariance matrix through an alternating projection method. Specifically, S51, the optimization problem in step S4 is equivalently represented as: ； in, , , The constraints of the optimization problem include a noiseless covariance matrix. Part of This represents the other parts of the constraints in the optimization problem, where, For noise power, To select the matrix, the noise-free covariance matrix It has the following four properties: low rank, Topelitz, positive semidefinite, and ; The optimization problems of S52 and S51 are equivalent to: ； Among them, matrix , Belonging to , ; For a rank not greater than The set of complex-valued matrices, ; Let be a set of complex-valued matrices that satisfy three conditions: Toplitz and semidefinite are denoted as ; S53. Construct the following update criteria. ； in, , express The , The result of the second iteration and Step size, Indicates the number of targets. , To set respectively ,gather The projection of the iteration stops under the condition that the iteration is... , These are custom parameters. for The The result of the second iteration for The The result of the second iteration for The The result of the second iteration for The The result of the next iteration; S54. Solve the optimization problem in step S52 according to the update rule established in step S53 until it converges or reaches the maximum number of iterations, and obtain the noise-free covariance matrix. .

6. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 5, characterized in that, In step S53, To set The projection, specifically, make For matrix D, its position in set Projection on Represented as, ； in, It is a diagonal matrix whose diagonal elements are the largest in matrix D. A singular value, and These are the corresponding left singular matrix and right singular matrix, respectively.

7. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 5, characterized in that, In step S53, To set The projection, specifically, is to let For matrix In the set Projection on The projection is represented by three sequential combinations of projections: , and , , and Represent matrices respectively In the set The first, second, and third projections in sequence on the screen, i.e. ; projection It is a matrix Projection to constraints Above, that is, the center of the circle is Radius is a custom threshold The interior or surface of the sphere, where, Then the matrix is ​​obtained. : ,in, To select a matrix, To receive signals The sampling covariance matrix, For matrix Under constraints Projection on: ; projection It is a matrix Projecting onto a set of three-dimensional Toplitz matrices yields the matrix. ; projection It is a matrix Projecting the matrix into a set of positive semi-definite matrices will... Performing eigenvalue decomposition, then the projection Represented as: ,in, To make the matrix The set of eigenvalues ​​for which all negative eigenvalues ​​are set to zero. For matrix The eigenvectors corresponding to the eigenvalues.

8. The method for estimating gridless target parameters of FDA-MIMO radar based on a monostatic array as described in claim 7, characterized in that, projection It is a matrix Projected onto the set of three-dimensional Toplitz matrices, specifically, S531, Assumption , The dimension is Y is The dimension of the first dimension of the matrix, will the matrix Divide into the following subarrays, ； in, for No. OK Subarrays of columns, for The OK The subarray of the column has a dimension of , , ; S532, Order ； in, This represents the mean of the elements in the set. For matrix The The dimension on the diagonal is The mean of the block matrix is ​​given, and the main diagonal is defined as the zeroth diagonal. The diagonal numbers of the upper triangle increase sequentially, while the diagonal numbers of the lower triangle decrease sequentially. The dimension is , The dimension is Its representation matrix The Middle Line number Block matrix of columns, for The The dimension on the diagonal is The mean of the block matrix, Representation matrix The Middle Line number Column elements, Representation matrix The The mean of the elements on each diagonal, a vector containing information about the first dimension of the matrix. Depend on composition, Then the vector Represented as, ； Depend on The dimensions of the composition are A vector containing information of the second-dimensional matrix. Represented as, ； in, It is a vector containing information from the first dimension matrix. ; S533, Let the vector The column vector formed : , It is a vector containing information from the second-dimensional matrix. The noise-free covariance matrix in step S3 The structure is used to establish a three-dimensional Toplitz matrix. Its dimensions are The three-dimensional Toplitz matrix That is, projection .

Images