Space-domain and Polarization-domain Joint Estimation Method Based on Unitary Transformation and Subspace Weighting

Abstract

The present invention relates to a spatial - polarization domain joint estimation method based on unitary transformation and subspace weighting, comprising: obtaining the array covariance matrix of the received data of the polarization - sensitive array according to the steering vector of the electromagnetic signal received by the polarization - sensitive array; using unitary transformation processing to convert the array covariance matrix into a real - number covariance matrix and determining the noise subspace; using the subspace weighting method to perform weighting processing on the noise subspace, and constructing a spatial - polarization domain spectral peak search function using the weighted noise subspace; performing dimensionality reduction processing on the spatial - polarization domain spectral peak search function and estimating the estimation result of the two - dimensional spatial parameters; according to the estimation result of the two - dimensional spatial parameters, performing dimensionality reduction processing on the spatial - polarization domain spectral peak search function and estimating the estimation result of the two - dimensional polarization domain parameters. The method of the present invention improves the problem of eigenvalue divergence and can maintain good parameter estimation performance when the signal - to - noise ratio is low or the number of snapshots is small.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar signal processing, and particularly relates to a method for joint estimation of spatial domain - polarization domain based on unitary transformation and subspace weighting. Background Technique

[0002] A polarization - sensitive array consists of multiple electromagnetic vector sensors, which can effectively detect the spatial domain information and polarization domain information of electromagnetic signals. Using the above - mentioned information, the spatial domain parameters and polarization domain parameters of electromagnetic signals can be estimated simultaneously. The spatial domain - polarization domain joint estimation algorithm based on a polarization - sensitive array can effectively improve the radar detection performance and avoid the parameter matching problem of traditional arrays, which is an important research content in the current field of electronic countermeasures.

[0003] The electromagnetic vector sensors in a polarization - sensitive array can be divided into full - electromagnetic vector sensors and incomplete - electromagnetic vector sensors. Among them, significant mutual coupling effects will occur between full - electromagnetic vector sensors. Therefore, it is difficult to widely promote the use of full - electromagnetic vector sensors; while incomplete - electromagnetic vector sensors have lower costs and smaller mutual coupling effects. Therefore, current research on spatial domain - polarization domain joint estimation mainly focuses on incomplete - electromagnetic vector sensors.

[0004] The classical spatial domain - polarization domain joint estimation algorithm is the long - vector MUSIC algorithm proposed by Yuan.X. This algorithm estimates the spatial domain parameters and polarization domain parameters of signals through a spectral peak search function. The estimation performance of the algorithm is good. However, in actual situations, the spatial search range and polarization domain search range are large, and multi - dimensional spectral peak search is required. Therefore, when using the long - vector MUSIC algorithm to estimate the spatial domain parameters and polarization domain parameters of signals, the computational complexity is extremely high, and the real - time processing ability is insufficient; in addition, in the case of low signal - to - noise ratio and small number of snapshots, the estimation accuracy of the long - vector MUSIC algorithm is also greatly affected. Summary of the Invention

[0005] In order to solve the above problems existing in the prior art, the present invention provides a method for joint estimation of spatial domain - polarization domain based on unitary transformation and subspace weighting. The technical problems to be solved by the present invention are realized through the following technical solutions:

[0006] The present invention provides a method for joint estimation of spatial domain - polarization domain based on unitary transformation and subspace weighting, including:

[0007] Step 1: Obtain the array covariance matrix of the received data of the polarization - sensitive array according to the steering vector of the electromagnetic signals received by the polarization - sensitive array;

[0008] Step 2: Use unitary transformation to process the array covariance matrix to convert it into a real - number covariance matrix, and determine the noise subspace;

[0009] Step 3: Use the subspace weighting method to weight the noise subspace, and construct a spatial-polarization domain spectral peak search function using the weighted noise subspace;

[0010] Step 4: Perform dimensionality reduction on the spatial-polarization domain spectral peak search function, and estimate the estimation results of the two-dimensional spatial parameters;

[0011] Step 5: According to the estimation results of the two-dimensional spatial parameters, perform dimensionality reduction on the spatial-polarization domain spectral peak search function, and estimate the estimation results of the two-dimensional polarization domain parameters.

[0012] In an embodiment of the present invention, the said Step 1 includes:

[0013] Step 1.1: Construct a signal vector of the electromagnetic signal according to the steering vector of the electromagnetic signal received by the polarization sensitive array;

[0014] Step 1.2: According to the signal vector, use the polarization bidirectional averaging method to obtain the array covariance matrix of the data received by the polarization sensitive array.

[0015] In an embodiment of the present invention, the said Step 1.1 includes:

[0016] Step 1.1.1: Obtain the steering vector of the electromagnetic signal received by the polarization sensitive array as:

[0017]

[0018] where A represents the steering vector of the electromagnetic signal, A s represents the spatial steering vector, A p represents the spatial-polarization domain steering vector, represents the direct product, θ represents the elevation angle of the electromagnetic signal, φ represents the azimuth angle of the electromagnetic signal, γ represents the polarization auxiliary angle of the electromagnetic signal, η represents the polarization phase difference of the electromagnetic signal, k represents the kth electromagnetic signal, and K represents the number of electromagnetic signals;

[0019] Step 1.1.2: Set the permutation matrix According to the permutation matrix and the steering vector of the electromagnetic signal, construct the signal vector of the electromagnetic signal as:

[0020]

[0021] where y(t) represents the signal vector of the electromagnetic signal, W represents an M×M all-ones matrix, M represents the number of array elements, I2 represents a 2×2 identity matrix, s(t) represents the electromagnetic wave signal, represents the noise matrix, and x(t) represents the array received signal matrix, Denote the steering vector after reconstruction.

[0022] In an embodiment of the present invention, step 1.2 obtains:

[0023] Step 1.2.1: Obtain the covariance matrix of the conjugate symmetric model signal according to the signal vector as:

[0024] R y = E[y(t)y H (t)];

[0025] In the formula, R y represents the covariance matrix of the conjugate symmetric model signal, H represents the conjugate transpose, and E[] represents the statistical average;

[0026] Step 1.2.2: According to the covariance matrix of the conjugate symmetric model signal, use the polarization bidirectional averaging method to obtain the array covariance matrix of the polarization sensitive array received data as:

[0027]

[0028] In the formula, R FB represents the array covariance matrix, J M is a matrix with all anti-diagonal elements equal to 1, and * represents the conjugate.

[0029] In an embodiment of the present invention, step 2 includes:

[0030] Step 2.1: Construct the unitary transformation matrix Q as:

[0031]

[0032] In the formula, I M represents the M×M dimensional identity matrix, I M-1 represents the (M - 1)×(M - 1) dimensional identity matrix, represents the M×M dimensional permutation matrix, represents the (M - 1)×(M - 1) dimensional permutation matrix, and T represents the transpose;

[0033] Step 2.2: Perform unitary transformation on the array covariance matrix according to the unitary transformation matrix Q to obtain the real covariance matrix as:

[0034] R RV = Q H R FB Q;

[0035] In the formula, R RV represents the real covariance matrix;

[0036] Step 2.3: Perform eigenvalue decomposition on the real covariance matrix to obtain the corresponding eigenvalue matrix and eigenvector matrix. Arrange the eigenvalues in the eigenvalue matrix in descending order. The first K eigenvalues form the eigenvalue matrix of the electromagnetic signal, and the remaining M - K eigenvalues form the eigenvalue matrix of the noise subspace;

[0037] Among them, the eigenvalue decomposition process is as follows:

[0038]

[0039] In the formula, Γ represents the eigenvalue matrix of R RV , V represents the eigenvector matrix, Γ S represents the eigenvalue matrix of the electromagnetic signal, V S represents the eigenvector matrix of the electromagnetic signal, Γ N represents the eigenvalue matrix of the noise subspace, V N represents the eigenvector matrix of the noise subspace.

[0040] In an embodiment of the present invention, the said Step 3 includes:

[0041] Step 3.1: Obtain the weighted correction value of the noise subspace as:

[0042]

[0043] In the formula, α i represents the i-th correction value after weighting the noise subspace, Y(i) represents the M linear equation sampling points from 1 to 2, Γ N，1 represents the first eigenvalue of the eigenvalue matrix of the noise subspace, Γ N,i represents the i-th eigenvalue of the eigenvalue matrix of the noise subspace;

[0044] Step 3.2: Correct the eigenvalues of the noise subspace according to the weighted correction value of the noise subspace to obtain the weighted noise subspace:

[0045]

[0046]

[0047] Among them, G N represents the weighted noise subspace, represents the eigenvalue matrix of the corrected noise subspace, represents the i-th eigenvalue of the eigenvalue matrix of the corrected noise subspace;

[0048] Step 3.3: Construct the spatial - polarization domain spectral peak search function according to the weighted noise subspace as:

[0049]

[0050] Wherein, F(θ, φ, γ, η) represents the spatial - polarization domain spectral peak search function, represents the search steering vector.

[0051] In one embodiment of the present invention, step 4 includes:

[0052] Step 4.1: Decompose and transform the denominator of the spatial - polarization domain spectral peak search function into a mathematical expression including a spatial domain parameter vector and a polarization domain parameter vector:

[0053]

[0054] Wherein, h(γ, η) represents the polarization domain parameter vector, and H(θ, φ) represents the spatial domain parameter vector;

[0055] Step 4.2: During the search process, when the search steering vector correctly points to the spatial angle of the incident signal, perform dimensionality reduction processing on the spatial - polarization domain spectral peak search function to obtain a new spatial domain spectral peak search function as:

[0056]

[0057] Wherein, F RV (θ, φ) represents the new spatial domain spectral peak search function;

[0058] Step 4.3: Perform spectral peak search on the new spatial domain spectral peak search function to obtain the estimated results of the two - dimensional spatial domain parameters, including the azimuth angle and elevation angle of the electromagnetic signal.

[0059] In one embodiment of the present invention, step 5 includes:

[0060] Step 5.1: Decouple the estimated results of the two - dimensional spatial domain parameters from the unknown polarization parameters. The decoupled spatial domain parameter components are expressed as:

[0061]

[0062] Wherein, represents the spatial domain estimated steering vector matrix containing only spatial domain information obtained by using the estimated results of the two - dimensional spatial domain parameters of the k - th electromagnetic signal;

[0063] Step 5.2: According to the decoupled spatial domain parameter components, obtain a new spatial - polarization domain spectral peak search function expressed as:

[0064]

[0065] Wherein, denotes the spatial domain parameter matrix obtained by using the spatial domain estimation steering vector matrix and the eigenvector matrix of the noise subspace;

[0066] Step 5.3: According to the new spatial domain - polarization domain spectrum peak search function, transform the problem of solving any non - zero polarization domain parameter vector into the solution of a quadratic optimization problem, and the quadratic optimization problem is expressed as:

[0067]

[0068] Step 5.4: Solve the quadratic optimization problem to obtain the estimation results of the two - dimensional parameters in the polarization domain, including the polarization auxiliary angle and the polarization phase difference of the electromagnetic signal, where:

[0069]

[0070]

[0071] In the formula, denotes the estimation result of the polarization auxiliary angle of the k - th electromagnetic signal, denotes the estimation result of the polarization phase difference of the k - th electromagnetic signal, denotes the matrix pencil The eigenvector matrix composed of the generalized eigenvectors corresponding to the minimum generalized eigenvalue of, and i represents The i - th element within.

[0072] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0073] 1. In the spatial domain - polarization domain joint estimation method based on unitary transformation and subspace weighting of the present invention, when estimating the spatial domain - polarization domain parameters of electromagnetic signals, compared with the long - vector MUSIC algorithm, while greatly reducing the amount of computation, the estimation accuracy of multi - dimensional parameters does not decrease significantly. The method of the present invention converts the covariance matrix into a real - number matrix through unitary transformation, so the amount of computation in the process of eigenvalue decomposition and spectrum peak search is greatly reduced, and at the same time, a dimension - reduction algorithm is used to simplify the computation of parameter estimation.

[0074] 2. The proposed method for joint spatial and polarization domain estimation based on unitary transformation and subspace weighting in the present invention improves the estimation performance of the algorithm without increasing the computational complexity compared to the unitary transformation MUSIC algorithm in the case of low signal-to-noise ratio and small number of snapshots. It also alleviates the problem of eigenvalue divergence and ensures the robustness of the signal subspace. Due to the severe divergence of eigenvalues in the case of low signal-to-noise ratio and small number of snapshots, the estimation results of multi-dimensional parameters will have large errors. By using the subspace weighting method, the eigenvalues can be converged, and the influence of noise and the number of snapshots on the estimation results of multi-dimensional parameters can be reduced. When performing subspace weighting, the correction value is a real number, and the robustness of the signal subspace will not be changed during the weighting process of the real matrix. In addition, the subspace weighting method weights the subspace through multiplication, which will not lead to an exponential increase in computational complexity. Since the problem of eigenvalue divergence is alleviated, the resolution of the algorithm is improved.

[0075] The above description is only an overview of the technical solution of the present invention. In order to understand the technical means of the present invention more clearly, it can be implemented according to the content of the specification. In order to make the above and other purposes, features, and advantages of the present invention more obvious and understandable, the following preferred embodiments are specifically given and described in detail in conjunction with the accompanying drawings. Brief Description of the Drawings

[0076] Figure 1 is a flowchart of a method for joint spatial and polarization domain estimation based on unitary transformation and subspace weighting provided by an embodiment of the present invention;

[0077] Figure 2 is a curve of the root mean square error of the elevation angle estimation result varying with the signal-to-noise ratio in the simulation experiment provided by an embodiment of the present invention;

[0078] Figure 3 is a curve of the root mean square error of the azimuth angle estimation result varying with the signal-to-noise ratio in the simulation experiment provided by an embodiment of the present invention;

[0079] Figure 4 is a curve of the root mean square error of the polarization phase difference estimation result varying with the signal-to-noise ratio in the simulation experiment provided by an embodiment of the present invention;

[0080] Figure 5 is a curve of the root mean square error of the polarization auxiliary angle estimation result varying with the signal-to-noise ratio in the simulation experiment provided by an embodiment of the present invention;

[0081] Figure 6 is a curve of the root mean square error of the elevation angle estimation result varying with the number of snapshots in the simulation experiment provided by an embodiment of the present invention;

[0082] Figure 7 is a curve of the root mean square error of the azimuth angle estimation result varying with the number of snapshots in the simulation experiment provided by an embodiment of the present invention;

[0083] Figure 8 is the root mean square error change curve of the polarization phase difference estimation result with the change of the number of snapshots in the simulation experiment provided by the embodiment of the present invention;

[0084] Figure 9 is the root mean square error change curve of the polarization-assisted angle estimation result with the change of the number of snapshots in the simulation experiment provided by the embodiment of the present invention. Detailed implementation manners

[0085] In order to further elaborate on the technical means and effects adopted by the present invention to achieve the predetermined invention purpose, the following combines the accompanying drawings and specific implementation manners to detail a spatial-polarization domain joint estimation method based on unitary transformation and subspace weighting proposed according to the present invention.

[0086] The foregoing and other technical contents, features, and effects of the present invention can be clearly presented in the following detailed description in conjunction with the accompanying drawings. Through the description of the specific implementation manners, a more in-depth and specific understanding of the technical means and effects adopted by the present invention to achieve the predetermined purpose can be obtained. However, the accompanying drawings are only provided for reference and illustration, and are not used to limit the technical solution of the present invention.

[0087] Embodiment 1

[0088] Please refer to Figure 1 , Figure 1 is a flowchart of a spatial-polarization domain joint estimation method based on unitary transformation and subspace weighting provided by the embodiment of the present invention. As shown in the figure, the method includes:

[0089] Step 1: Obtain the array covariance matrix of the received data of the polarization-sensitive array according to the steering vector of the electromagnetic signal received by the polarization-sensitive array;

[0090] In an optional implementation manner, Step 1 includes:

[0091] Step 1.1: Construct the signal vector of the electromagnetic signal according to the steering vector of the electromagnetic signal received by the polarization-sensitive array;

[0092] Optionally, Step 1.1 includes:

[0093] Step 1.1.1: Obtain the steering vector of the electromagnetic signal received by the polarization-sensitive array as:

[0094]

[0095] wherein, A represents the steering vector of the electromagnetic signal, A s represents the spatial steering vector, and A p represents the spatial-polarization domain steering vector, Denotes the direct product, θ denotes the elevation angle of the electromagnetic signal, φ denotes the azimuth angle of the electromagnetic signal, γ denotes the polarization auxiliary angle of the electromagnetic signal, η denotes the polarization phase difference of the electromagnetic signal, k denotes the k-th electromagnetic signal, and K denotes the number of electromagnetic signals.

[0096] Among them, the expressions of the spatial domain steering vector and the spatial-polarization domain steering vector are:

[0097]

[0098]

[0099] In the formula, λ denotes the wavelength of the received electromagnetic signal, μ k =[sinφ k cosθ k ,sinφ k sinθ k ,cosφ k T , which represents the unit vector of the direction of arrival of the k-th electromagnetic signal.

[0100] Step 1.1.2: Set the permutation matrix According to the permutation matrix and the steering vector of the electromagnetic signal, the signal vector of the electromagnetic signal is constructed as:

[0101]

[0102] In the formula, y(t) represents the signal vector of the electromagnetic signal, W represents an M×M all-ones matrix, M represents the number of array elements, I2 represents a 2×2 identity matrix, s(t) represents the electromagnetic wave signal, represents the noise matrix, x(t) represents the array received signal matrix, represents the reconstructed steering vector.

[0103] Among them, the reconstructed steering vector is expressed as:

[0104]

[0105]

[0106] Step 1.2: According to the signal vector, use the polarization bidirectional averaging method to obtain the array covariance matrix of the polarization sensitive array received data.

[0107] Optionally, Step 1.2 includes:

[0108] Step 1.2.1: The covariance matrix of the conjugate symmetric model signal obtained according to the signal vector is: ​

[0109] R y = E[y(t)y H (t)](7);

[0110] Wherein, R y represents the covariance matrix of the conjugate symmetric model signal, H represents the conjugate transpose, and E[] represents the statistical average;

[0111] Step 1.2.2: According to the covariance matrix of the conjugate symmetric model signal, use the polarization two-way averaging method to obtain the array covariance matrix of the polarization-sensitive array received data as:

[0112]

[0113] Wherein, R FB represents the array covariance matrix, J M is a matrix with all anti-diagonal elements equal to 1, and * represents the conjugate.

[0114] Step 2: Use unitary transformation to process the array covariance matrix into a real covariance matrix and determine the noise subspace;

[0115] In an optional implementation manner, Step 2 includes:

[0116] Step 2.1: Construct the unitary transformation matrix Q as:

[0117]

[0118] Wherein, I M represents the M×M dimensional identity matrix, I M-1 represents the (M - 1)×(M - 1) dimensional identity matrix, represents the M×M dimensional permutation matrix, represents the (M - 1)×(M - 1) dimensional permutation matrix, and T represents the transpose;

[0119] Step 2.2: Perform unitary transformation on the array covariance matrix according to the unitary transformation matrix Q to obtain the real covariance matrix as:

[0120] R RV = Q H R FB Q (10);

[0121] Wherein, R RV represents the real covariance matrix;

[0122] Step 2.3: Perform eigenvalue decomposition on the real covariance matrix to obtain the corresponding eigenvalue matrix and eigenvector matrix. Arrange the eigenvalues in the eigenvalue matrix in descending order. The first K eigenvalues form the eigenvalue matrix of the electromagnetic signal, and the remaining M - K eigenvalues form the eigenvalue matrix of the noise subspace;

[0123] Among them, the eigenvalue decomposition process is as follows:

[0124]

[0125] In the formula, Γ represents the eigenvalue matrix of R RV , V = [v1, v2,... v M , represents the eigenvector matrix, Γ S represents the eigenvalue matrix of the electromagnetic signal, V S represents the eigenvector matrix of the electromagnetic signal, Γ N represents the eigenvalue matrix of the noise subspace, V N represents the eigenvector matrix of the noise subspace.

[0126] Step 3: Use the subspace weighting method to weight the noise subspace, and use the weighted noise subspace to construct a spatial - polarization domain spectral peak search function;

[0127] In an optional implementation manner, Step 3 includes:

[0128] Step 3.1: Obtain the corrected value after weighting the noise subspace as:

[0129]

[0130] In the formula, α i represents the i - th corrected value after weighting the noise subspace, Y(i) represents the M linear equation sampling points from 1 to 2, Γ N，1 represents the first eigenvalue of the eigenvalue matrix of the noise subspace, Γ N,i represents the i - th eigenvalue of the eigenvalue matrix of the noise subspace;

[0131] Step 3.2: According to the corrected value after weighting the noise subspace, correct the eigenvalues of the noise subspace to obtain the weighted noise subspace:

[0132]

[0133]

[0134] Among them, G N represents the weighted noise subspace, represents the eigenvalue matrix of the corrected noise subspace, denotes the i-th eigenvalue of the eigenvalue matrix of the corrected noise subspace;

[0135] Step 3.3: According to the weighted noise subspace, construct the spatial-polarization domain spectral peak search function as:

[0136]

[0137] In the formula, F(θ, φ, γ, η) represents the spatial-polarization domain spectral peak search function, denotes the search steering vector.

[0138] Step 4: Perform dimensionality reduction on the spatial-polarization domain spectral peak search function, and estimate the estimation results of the two-dimensional spatial parameters;

[0139] In an optional implementation manner, Step 4 includes:

[0140] Step 4.1: Decompose and transform the denominator of the spatial-polarization domain spectral peak search function into a mathematical expression including a spatial parameter vector and a polarization domain parameter vector:

[0141]

[0142] In the formula, h(γ, η) represents the polarization domain parameter vector, and H(θ, φ) represents the spatial parameter vector;

[0143] Among them, the spatial parameter vector is expressed as:

[0144]

[0145]

[0146] In the formula, D(θ, φ) represents the spatial steering vector matrix containing only spatial information, denotes the search spatial steering vector, denotes the search spatial-polarization domain steering vector containing only spatial information.

[0147] Step 4.2: During the search process, when the search steering vector correctly points to the spatial angle of the incident signal, perform dimensionality reduction on the spatial-polarization domain spectral peak search function to obtain a new spatial spectral peak search function as:

[0148]

[0149] Among them, F RV (θ, φ) represents the new spatial spectral peak search function.

[0150] In this embodiment, during the search process when the search steering vector correctly points to the spatial angle of the incident signal,

[0151]

[0152] h H (γ k ,η k )H(θ k ,φ k )h(γ k ,η k ) = 0 (21);

[0153] Multiply both sides of Equation (21) on the left by D H (θ k ,φ k )G N to obtain:

[0154] H(θ k ,φ k )h(γ k ,η k ) = 0 (22);

[0155] Thereby, a new spatial domain spectral peak search function F RV (θ, φ) is obtained.

[0156] Step 4.3: Perform spectral peak search on the new spatial domain spectral peak search function to obtain the estimated results of the two-dimensional spatial domain parameters, including the azimuth angle and elevation angle of the electromagnetic signal.

[0157] In this embodiment, an existing spectral peak search method is used to perform spectral peak search on the new spatial domain spectral peak search function to obtain the estimated results of the azimuth angle and elevation angle of the electromagnetic signal. The specific process will not be elaborated here.

[0158] Step 5: According to the estimated results of the two-dimensional spatial domain parameters, perform dimensionality reduction processing on the spatial domain - polarization domain spectral peak search function, and estimate the estimated results of the two-dimensional polarization domain parameters.

[0159] In an alternative embodiment, Step 5 includes:

[0160] Step 5.1: Decouple the estimated results of the two-dimensional spatial domain parameters from the unknown polarization parameters. The decoupled spatial domain parameter components are expressed as:

[0161]

[0162] wherein, represents the spatial domain estimated steering vector matrix containing only spatial domain information obtained from the estimated results of the two-dimensional spatial domain parameters of the k-th electromagnetic signal;

[0163] wherein, The expression is as follows:

[0164]

[0165] When M is an even number, H1 represents the first M / 2 rows of the matrix; when M is an odd number, H1 represents the first (M - 1) / 2 rows of the matrix.

[0166] Step 5.2: According to the decoupled spatial domain parameter components, obtain a new spatial domain - polarization domain peak search function expressed as:

[0167]

[0168] In the formula, represents the spatial domain parameter matrix obtained by using the spatial domain estimation steering vector matrix and the eigenvector matrix of the noise subspace;

[0169] Step 5.3: According to the new spatial domain - polarization domain peak search function, transform the problem of solving any non - zero polarization domain parameter vector into the solution of a quadratic optimization problem;

[0170] In this embodiment, for the new spatial domain - polarization domain peak search function, for any non - zero polarization parameter vector h(γ,η), there must exist

[0171]

[0172] Therefore, the formula (26) can be transformed into the solution of a quadratic optimization problem, and its minimum variance is solved using linear constraint conditions. The quadratic optimization problem is expressed as:

[0173]

[0174] Step 5.4: Solve the quadratic optimization problem to obtain the estimation results of the two - dimensional polarization domain parameters, including the polarization auxiliary angle and the polarization phase difference of the electromagnetic signal.

[0175] Specifically, first, construct the cost function as:

[0176] L(h(γ,η),λ) = h H (γ,η)H z (θ,φ)h(γ,η) + λ(1 - h H (γ,η)D H (θ,φ)D(θ,φ)h(γ,η)) (28);

[0177] where λ is the Lagrange multiplier. Next, take the partial derivatives of h(γ,η) and λ and set the results to zero to obtain:

[0178]

[0179] Using Multiply the left - hand side of formula (29) by h H (γ,η) to obtain:

[0180] h H (γ,η)H z (θ,φ)h(γ,η) = λ (30);

[0181] Then, the polarization auxiliary angle and polarization phase difference of the electromagnetic signal can be expressed as:

[0182]

[0183]

[0184] In the formula, represents the estimation result of the polarization auxiliary angle of the k-th electromagnetic signal, represents the estimation result of the polarization phase difference of the k-th electromagnetic signal, represents the matrix pencil The eigenvector matrix composed of the generalized eigenvectors corresponding to the minimum generalized eigenvalue of, i represents The i-th element within.

[0185] In the embodiment of the present invention, the joint spatial-polarization domain estimation method based on unitary transformation and subspace weighting greatly reduces the amount of computation compared with the long-vector MUSIC algorithm, while the estimation accuracy of multi-dimensional parameters does not decrease significantly. Moreover, in the case of low signal-to-noise ratio and small number of snapshots, compared with the unitary transformation MUSIC algorithm, it improves the estimation performance of the algorithm without increasing the amount of computation, improves the problem of eigenvalue divergence, and ensures the robustness of the signal subspace.

[0186] Embodiment 2

[0187] In this embodiment, through simulation experiments and comparative experiments with the unitary transformation MUSIC algorithm and the long-vector MUSIC algorithm, the effect of the joint spatial-polarization domain estimation method based on unitary transformation and subspace weighting in Embodiment 1 is described.

[0188] 1. Simulation conditions

[0189] The simulation parameters are set as follows: the azimuth angle search is θ ∈ [0, 2π), the elevation angle range is φ ∈ [0, π / 2), the polarization auxiliary range is γ ∈ [0, π / 2), the polarization phase difference range is η ∈ [-π, π), and the search step size is set to 0.1°. When conducting the root mean square error experiment, the root mean square error experiments of the three algorithms are statistically obtained from 100 independent Monte-Carlo experiments. The angle measurement performances of the long-vector MUSIC algorithm, the unitary transformation MUSIC algorithm, and the unitary transformation-subspace weighting method proposed in the present invention are compared and analyzed.

[0190] 2. Simulation content

[0191] Simulation Experiment 1: In this experiment, the root mean square error of parameter estimation is analyzed by varying the signal-to-noise ratio. The simulation parameters are set as follows: the array structure has 64 array elements, the element spacing is half a wavelength, the number of snapshots is 200, the signal arrival direction is set to (0°, 0°), the polarization mode is horizontal linear polarization, i.e., γ = 0°, η = 0°, the interference-to-noise ratio step interval is 5 dB, increasing gradually from 0 dB to 20 dB.

[0192] Simulation Experiment 2: In this experiment, the root mean square error of parameter estimation is analyzed by varying the number of snapshots. The simulation parameters are set as follows: the number of array elements is 64, the element spacing is half a wavelength, the arrival direction is set to (0°, 0°), the polarization mode is left-handed circular polarization, i.e., γ = 45°, η = 90°, the interference-to-noise ratio is 5 dB, and the number of snapshots step interval is 50, increasing gradually from 50 to 500.

[0193] 3. Analysis of Simulation Results

[0194] Please refer to Figures 2 - 9 From Figures 2 - 5 it can be seen that the root mean square error of the estimation results of the three algorithms all decreases with the increase of the signal-to-noise ratio. The method of the embodiment of the present invention has better estimation performance than the unitary transformation MUSIC algorithm in the case of low signal-to-noise ratio and approaches the long vector MUSIC algorithm. When the signal-to-noise ratio is large, the estimation performances of the three algorithms tend to be the same. In summary, the estimation performances of the three methods all decline with the increase of the signal-to-noise ratio. In the case of low signal-to-noise ratio, the root mean square error of the estimation method of the embodiment of the present invention is better than that of the unitary transformation MUSIC algorithm and approaches the long vector MUSCI algorithm.

[0195] From Figures 6 - 9 it can be seen that the root mean square error of the estimation results of the three algorithms all decreases with the increase of the number of snapshots. The method of the embodiment of the present invention has better estimation performance than the unitary transformation MUSIC algorithm in the case of small number of snapshots and approaches the long vector MUSIC algorithm. When the number of snapshots is large, the estimation performances of the three algorithms tend to be the same. In summary, the estimation performances of the three methods all decline with the increase of the number of snapshots. In the case of small number of snapshots, the root mean square error of the estimation method of the embodiment of the present invention is better than that of the unitary transformation MUSIC algorithm and approaches the long vector MUSCI algorithm.

[0196] In addition, when the long vector MUSIC algorithm performs eigenvalue decomposition, the covariance matrix is a complex matrix, while the method of the embodiment of the present invention converts the covariance matrix into a real matrix; when performing spectral peak search, the long vector MUSIC algorithm first needs to perform a four-dimensional spectral peak search and the noise subspace is complex, while the estimation algorithm of the embodiment of the present invention uses the idea of dimension reduction to reduce the four-dimensional spectral peak search function to two-dimensional and the noise subspace is real. In summary, the computational complexity of the estimation algorithm of the embodiment of the present invention is much lower than that of the long vector MUSIC algorithm.

[0197] When the unitary transform MUSIC algorithm performs covariance matrix transformation, it increases the difference between eigenvalues, resulting in a larger divergence degree of the noise subspace at low signal-to-noise ratio and small number of snapshots. When performing spectral peak search, the orthogonality between the search steering vector and the noise subspace becomes worse, leading to a large error in the estimation results of the spatial and polarization domain parameters of the electromagnetic signal. The method of the embodiment of the present invention improves the divergence problem of eigenvalues through the method of subspace weighting and enhances the orthogonality between the search steering vector and the noise subspace. Therefore, in the case of low signal-to-noise ratio and small number of snapshots, without increasing the computational complexity and while maintaining the robustness of the signal subspace, the resolution and estimation performance of the algorithm are improved.

[0198] It should be noted that in this article, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variant is intended to cover non-exclusive inclusion, so that an article or device including a series of elements not only includes those elements, but also includes other elements not expressly listed. Without further limitation, an element defined by the statement "including one..." does not exclude the existence of another identical element in the article or device including the said element.

[0199] The above content is a further detailed description of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is only limited to these descriptions. For those of ordinary skill in the technical field to which the present invention pertains, without departing from the concept of the present invention, several simple deductions or substitutions can be made, and all should be regarded as belonging to the protection scope of the present invention.

Claims

1. A joint spatial-polarization domain estimation method based on unitary transformation and subspace weighting, characterized in that Including: Step 1: Obtain the array covariance matrix of the received data of the polarization-sensitive array according to the steering vector of the electromagnetic signal received by the polarization-sensitive array; Step 2: Use unitary transformation to process the array covariance matrix into a real covariance matrix and determine the noise subspace; Step 3: Use the subspace weighting method to weight the noise subspace, and use the weighted noise subspace to construct a spatial-polarization domain peak search function; Step 3 includes: Step 3.1: Obtain the corrected value after weighting the noise subspace as: where α i represents the i-th correction value after weighting of the noise subspace, Y(i) represents the sampling points of M linear equations from 1 to 2, Γ N，1 represents the first eigenvalue of the eigenvalue matrix of the noise subspace, Γ N,i represents the i-th eigenvalue of the eigenvalue matrix of the noise subspace; M - K is the number of eigenvalues of the eigenvalue matrix of the noise subspace; Step 3.2: According to the corrected value after weighting the noise subspace, correct the eigenvalues of the noise subspace to obtain the weighted noise subspace: Among them, G N represents the weighted noise subspace, V N represents the eigenvector matrix of the noise subspace, represents the eigenvalue matrix of the corrected noise subspace, represents the i-th eigenvalue of the eigenvalue matrix of the corrected noise subspace; Step 3.3: According to the weighted noise subspace, construct the spatial-polarization domain peak search function as: In the formula, F(θ, φ, γ, η) represents the spatial-polarization domain spectral peak search function, represents the search steering vector; θ represents the elevation angle of the electromagnetic signal, φ represents the azimuth angle of the electromagnetic signal, γ represents the polarization auxiliary angle of the electromagnetic signal, η represents the polarization phase difference of the electromagnetic signal, and H represents the conjugate transpose; Step 4: Perform dimensionality reduction on the spatial-polarization domain peak search function and estimate the estimation result of the two-dimensional spatial parameters; Step 5: According to the estimation result of the two-dimensional spatial parameters, perform dimensionality reduction on the spatial-polarization domain peak search function and estimate the estimation result of the two-dimensional polarization domain parameters.

2. The method for joint estimation in spatial domain - polarization domain based on unitary transformation and subspace weighting according to claim 1, wherein Step 1 includes: Step 1.1: Construct the signal vector of the electromagnetic signal according to the steering vector of the electromagnetic signal received by the polarization-sensitive array; Step 1.2: According to the signal vector, use the polarization bidirectional averaging method to obtain the array covariance matrix of the received data of the polarization-sensitive array.

3. The method for joint estimation in spatial domain and polarization domain based on unitary transformation and subspace weighting according to claim 2, wherein Step 1.1 includes: Step 1.1.1: Obtain the steering vector of the electromagnetic signal received by the polarization-sensitive array as: In the formula, A represents the steering vector of the electromagnetic signal, A s represents the spatial steering vector, A p represents the spatial-polarization domain steering vector, represents the direct product, θ represents the elevation angle of the electromagnetic signal, φ represents the azimuth angle of the electromagnetic signal, γ represents the polarization auxiliary angle of the electromagnetic signal, η represents the polarization phase difference of the electromagnetic signal, k represents the k-th electromagnetic signal, and K represents the number of electromagnetic signals; Step 1.1.2: Set the permutation matrix According to the permutation matrix and the steering vector of the electromagnetic signal, the signal vector of the electromagnetic signal is constructed as follows: where \(y(t)\) represents the signal vector of the electromagnetic signal, \(W\) represents an \(M\times M\) all-ones matrix, \(M\) represents the number of array elements, \(I_2\) represents a \(2\times2\) identity matrix, and \(s(t)\) represents the electromagnetic wave signal, represents the noise matrix, and \(x(t)\) represents the array received signal matrix, represents the reconstructed steering vector.

4. The joint estimation method in the spatial domain - polarization domain based on unitary transformation and subspace weighting according to claim 3, characterized in that Step 1.2 obtains: Step 1.2.1: The covariance matrix of the conjugate symmetric model signal obtained according to the signal vector is: R y = E[y(t)y H (t)]; where, R y represents the covariance matrix of the conjugate symmetric model signal, H represents the conjugate transpose, and E[] represents the statistical average; Step 1.2.2: According to the covariance matrix of the conjugate symmetric model signal, use the polarization bidirectional averaging method to obtain the array covariance matrix of the received data of the polarization-sensitive array as: where, R FB represents the array covariance matrix, J M is a matrix with all anti-diagonal elements equal to 1, and * represents conjugate.

5. The joint estimation method in the spatial domain - polarization domain based on unitary transformation and subspace weighting according to claim 4, characterized in that, Step 2 includes: Step 2.1: Construct the unitary transformation matrix Q as: where I M represents an M×M identity matrix, and I M-1 represents an (M−1)×(M−1) identity matrix, represents an M×M permutation matrix, represents an (M−1)×(M−1) permutation matrix, and T represents the transpose; Step 2.2: Perform unitary transformation on the array covariance matrix according to the unitary transformation matrix Q to obtain the real covariance matrix as: R RV = Q H R FB Q; where, R RV represents a real covariance matrix; Step 2.3: Perform eigenvalue decomposition on the real covariance matrix to obtain the corresponding eigenvalue matrix and eigenvector matrix. Arrange the eigenvalues in the eigenvalue matrix in descending order. The first K eigenvalues form the eigenvalue matrix of the electromagnetic signal, and the remaining M - K eigenvalues form the eigenvalue matrix of the noise subspace; Among them, the eigenvalue decomposition process is as follows: where Γ represents the eigenvalue matrix of R RV , V represents the eigenvector matrix, Γ S represents the eigenvalue matrix of the electromagnetic signal, and V S represents the eigenvector matrix of the electromagnetic signal, Γ N represents the eigenvalue matrix of the noise subspace, and V N represents the eigenvector matrix of the noise subspace.

6. The method for joint spatial domain - polarization domain estimation based on unitary transformation and subspace weighting according to claim 5, wherein Step 4 includes: Step 4.1: Decompose and transform the denominator of the spatial-polarization domain peak search function into a mathematical expression including a spatial parameter vector and a polarization domain parameter vector: In the formula, h(γ,η) represents the polarization domain parameter vector, and H(θ,φ) represents the spatial parameter vector; Step 4.2: During the search process, when the search steering vector correctly points to the spatial angle of the incident signal, perform dimensionality reduction on the spatial-polarization domain peak search function to obtain a new spatial peak search function as: Among them, F RV (θ, φ) represents a new spatial domain spectral peak search function; Step 4.3: Perform peak search on the new spatial spectrum peak search function to obtain the estimation results of two-dimensional spatial parameters, including the azimuth angle and elevation angle of the electromagnetic signal.

7. The method for joint spatial domain - polarization domain estimation based on unitary transformation and subspace weighting according to claim 6, wherein The said Step 5 includes: Step 5.1: Decouple the estimation results of the two-dimensional spatial parameters from the unknown polarization parameters. The decoupled spatial parameter components are expressed as: In the formula, represents an airspace estimation steering vector matrix that only contains airspace information obtained from the estimation result of the two-dimensional airspace parameters of the k-th electromagnetic signal; Step 5.2: According to the decoupled spatial parameter components, obtain a new spatial-polarization domain spectrum peak search function expressed as: In the formula, represents the spatial domain parameter matrix obtained by using the spatial domain estimation steering vector matrix and the noise subspace eigenvector matrix; Step 5.3: According to the new spatial-polarization domain spectrum peak search function, transform the problem of solving any non-zero polarization domain parameter vector into the solution of a quadratic optimization problem, and the quadratic optimization problem is expressed as: Step 5.4: Solve the quadratic optimization problem to obtain the estimation results of two-dimensional polarization domain parameters, including the polarization auxiliary angle and polarization phase difference of the electromagnetic signal, where: In the formula, represents the polarization auxiliary angle estimation result of the k-th electromagnetic signal, represents the polarization phase difference estimation result of the k-th electromagnetic signal, represents the matrix pencil The eigenvector matrix composed of the generalized eigenvectors corresponding to the minimum generalized eigenvalue of, i represents The i-th element within.

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