Polarization-sensitive mirror array polarization doa joint estimation method based on matrix reconstruction

Abstract

The application relates to the technical field of polarization-sensitive mirror-reflection-array signal processing, in particular to a polarization-DOA joint estimation method for a polarization-sensitive mirror-reflection array based on matrix reconstruction, which is characterized by the following steps: obtaining polarization-sensitive mirror-reflection-array receiving signals, constructing and-difference co, using the and-difference co idea to obtain the virtual array input of each subarray after the block representation of the covariance matrix, matrix reconstruction, performing eigenvalue decomposition to obtain a noise subspace, and then performing polarization-DOA joint estimation; under different signal-to-noise ratios, the spatial angle estimation precision of the polarization-DOA joint estimation method for the polarization-sensitive mirror-reflection array is higher than that of a traditional non-mirror array with the same number of array elements, and the estimation precision under a high signal-to-noise ratio is close to the estimation precision of a traditional array with the same number of virtual array elements.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of polarization-sensitive mirror-reflection-array signal processing, in particular to a polarization-DOA joint estimation method for a polarization-sensitive mirror-reflection array based on matrix reconstruction. BACKGROUND

[0002] Polarization-DOA joint estimation is committed to obtaining the direction of arrival and polarization information, and then providing the basis for emitter positioning, tracking and identification, and is often applied to the fields of communication, radar, sonar and the like. The subspace algorithm represented by multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) realizes the leap from traditional spatial spectrum estimation to super-resolution angle measurement, but the electromagnetic environment is gradually complex, and the signal form is various and complex. When the array spacing is less than half the wavelength, the performance of the classic algorithms such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) will be greatly reduced, and when the number of emitters is much larger than the number of array elements, the classic algorithms such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) will be invalid.

[0003] In the past decade, some scholars combined binary interferometer with Lloyd mirror interference to propose mirror image synthetic aperture technology. Compared with the array used in traditional synthetic aperture, the mirror image synthetic aperture forms a mirror reflection array by adding a reflecting surface in the traditional array, so that each array element can receive the direct incident signal and the reflected signal reflected by the reflecting surface. The acquisition of the two kinds of signals enables each pair of array elements to obtain the visibility function corresponding to the sum and difference sets of two baselines, thereby equivalently increasing the maximum baseline length of the array, enabling fewer array elements to be used in the mirror image synthetic aperture to achieve the same resolution as the traditional synthetic aperture.

[0004] Meanwhile, the existing super-resolution algorithm applied to the polarization-sensitive mirror-reflection array is based on compression sensing technology, which has high computational complexity and is not conducive to engineering implementation. Therefore, it is of great significance to research a polarization-DOA joint estimation algorithm with low complexity for the polarization-sensitive mirror-reflection array. SUMMARY

[0005] The application introduces the matrix reconstruction idea into the polarization-sensitive mirror-reflection array, deduces the polarization virtual array signal model based on matrix reconstruction, and further proposes a polarization-DOA joint estimation method for the polarization-sensitive mirror-reflection array based on matrix reconstruction, which greatly reduces the computational complexity and makes it possible to implement the polarization-DOA joint estimation based on the polarization-sensitive mirror-reflection array.

[0006] The application achieves the above-mentioned purposes by the following measures:

[0007] A polarization-sensitive mirror-reflection array polarization DOA joint estimation method based on matrix reconstruction, characterized in that the following steps are implemented:

[0008] Step 1: Obtain the polarization-sensitive mirror-reflection array receiving signal: consider K K far-field narrowband signals incident on a polarization-sensitive mirror-reflection array composed of N three orthogonal dipoles, N the positions of the three orthogonal dipoles are , and d 1>0. Since there is a reflecting surface, each dipole will simultaneously receive a direct incident signal and a reflected signal, and the array receiving signal is:

[0009]

[0010] wherein, is the direct incident signal, is the reflected signal, is the signal vector, N (t) is additive white Gaussian noise, is the receiving signal of each dipole, is the array manifold of the polarization-sensitive mirror-reflection array, is the steering vector of the polarization-sensitive mirror-reflection array, which can be written as the sum of the steering vector of the direct incident signal and the steering vector of the reflected signal:

[0011]

[0012] wherein, is the array spatial steering vector, denotes Kronecker multiplication, is the spatial response matrix of the three orthogonal dipoles:

[0013]

[0014] Further, the simplified mirror-reflection array steering vector can be written as:

[0015]

[0016] wherein, is the polarization domain steering vector of each dipole, is the simplified spatial steering vector of each dipole:

[0017]

[0018]

[0019] For convenience, instead of ;

[0020] Step 2: Constructing the sum-difference co-array:

[0021] Step 2-1: Utilizing the sum set sum-difference set Obtaining the sum-difference set:

[0022]

[0023] Step 2-2: To facilitate subsequent algorithms, remove the large amount of redundant elements and negative values in the sum-difference set, and obtain a sum-difference co-array with Q elements, and the elements in it represent the virtual array element positions:

[0024]

[0025] where unique(·) is to remove duplicate elements and negative values in the elements;

[0026] Step 3: After representing the covariance matrix in blocks, use the sum-difference co-array idea to obtain the virtual array outputs of each sub-array:

[0027] Step 3-1: Calculate the covariance matrix of the polarization-sensitive mirror reflection array received signal and represent it in blocks:

[0028]

[0029] where denotes the conjugate transpose, each sub-array corresponds to the covariance matrix between the dipoles,

[0030] Step 3-2: Vectorizing the sub-array in the covariance matrix R ij , i , j = X , Y , Z :

[0031]

[0032] where is the KR product, is the spatial array manifold of each dipole, is the noise power, is the power of the k th signal, is a column vector with only the first element being 1, where is the k th signal parallel to iThe polarization vector of the axis dipole;

[0033] Step 3-3: the vectorized data of each dipole is also divided into two parts corresponding to the difference co-array and the sum co-array according to the same rule:

[0034]

[0035] In the formula, when j is not Y , , otherwise , where:

[0036]

[0037]

[0038] At this time, according to the same rule, the vectorized data of each dipole is also divided into two parts corresponding to the difference co-array and the sum co-array:

[0039]

[0040] In the formula, is the difference co-array, is the sum co-array, and the selection of the ± sign is the same as in ;

[0041] Step 3-4: the virtual array output of each sub-array is obtained according to the linear relationship:

[0042]

[0043] In the formula, is the matrix describing the linear combination relationship, and is the conversion matrix, when j ≠ Y , any row has and 1 in the position index of H , and the rest of the elements are 0; when j = Y , any row has-1 in the position index of , and 1 in the position index of ;

[0044] Step 4: matrix reconstruction, including the following steps: Step 4-1: extending the virtual array output of each sub-array:

[0045]

[0046] In the formula, is ​The data matrix after the first element is removed and is flipped up and down, when i , j any of Y is , otherwise ;

[0047] Step 4-2: In the matrix, select the sub-matrix with elements in turn, output Q , and use Q sub-matrix to construct corresponding covariance matrix Q :

[0048]

[0049] Step 4-3: Use all to construct a 3 Q *3 Q dimensional covariance matrix :

[0050]

[0051] At this time, the steering vector corresponding to the reconstructed covariance matrix is:

[0052]

[0053] In the formula, is the spatial steering vector of each dipole after reconstruction:

[0054]

[0055]

[0056] wherein, is the th element in the matrix; i

[0057] Step 5: After the noise subspace is obtained by eigenvalue decomposition, polarization-DOA joint estimation is performed, including the following steps:

[0058] Step 5-1: Perform eigenvalue decomposition on to obtain the noise subspace:

[0059]

[0060] Step 5-2: Combine the rank loss principle to strip the spatial domain parameters and polarization domain parameters, and construct a spectral function containing only spatial domain parameters: ​​​

[0061] The spatial parameter of the signal can be obtained by searching the maximum value of , wherein The maximum value of Q corresponds to the direction of arrival of the signal , wherein

[0062] The direction of arrival of the signal is obtained Then, the equation is substituted into the following equation to search again Q The coordinates corresponding to the maximum value are the auxiliary angle of polarization of the signal:

[0063] .

[0064] The spatial angle estimation accuracy of the polarization-DOA joint estimation method of the polarization-sensitive mirror-reflection array proposed in the application is higher than that of the conventional non-mirror array with the same number of array elements under different signal-to-noise ratios, and the estimation accuracy under high signal-to-noise ratio is close to that of the conventional array with the same number of virtual array elements. BRIEF DESCRIPTION OF DRAWINGS

[0065] The accompanying drawings illustrate the present application. Figure 1 The accompanying drawings illustrate the present application.

[0066] The accompanying drawings illustrate the present application. Figure 2 When the array spacing is equal to half the wavelength, the spectrum peak function comparison diagram of the present application and the spatial smoothing method used in the polarization-sensitive mirror-reflection array and the rank loss dimension reduction MUISC algorithm used in the conventional non-mirror polarization-sensitive array is shown in the following figures, wherein Figure 2 (a) is a spatial angle spectrum peak comparison diagram, Figure 2 (b) is a spatial smoothing polarization spectrum peak diagram, Figure 2 (c) is a polarization spectrum peak diagram of the method proposed in the application, Figure 2 (d) is a polarization spectrum peak diagram of a 12-element conventional array.

[0067] The accompanying drawings illustrate the present application. Figure 3 When the array spacing is less than half the wavelength, the spectrum peak function comparison diagram of the present application and the spatial smoothing method used in the polarization-sensitive mirror-reflection array and the rank loss dimension reduction MUISC algorithm used in the conventional non-mirror polarization-sensitive array is shown in the following figures, wherein Figure 3 (a) is a spatial angle spectrum peak comparison diagram, Figure 3 (b) is a spatial smoothing polarization spectrum peak diagram, Figure 3 (c) is a polarization spectrum peak diagram of the method proposed in the application.

[0068] The accompanying drawings illustrate the present application. Figure 4The graph compares the performance of the root mean square error (RSME) of parameter estimation as a function of signal-to-noise ratio (SNR) for polarization-sensitive specular reflective arrays using the present invention and spatial smoothing method versus traditional non-mirror polarization-sensitive arrays using the rank-deficient dimension reduction MUISC algorithm. This is achieved when the array spacing is equal to half the wavelength. Figure 4 (a) is the space angle RSME diagram. Figure 4 (b) is the polarization secondary angle RSME diagram. Figure 4 (c) is the polarization phase angle RSME diagram. Detailed Implementation

[0069] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0070] As attached Figure 1 As shown, this invention proposes a joint polarization-DOA estimation method for polarization-sensitive mirror reflection arrays based on matrix reconstruction, as detailed below:

[0071] Step 1-1: Consider K A far-field narrowband signal is incident on by N A polarization-sensitive mirror reflection array composed of three orthogonal dipoles. N The positions of the triorthogonal dipoles are: ,in d Since 1>0, and due to the presence of a reflecting surface, each dipole will simultaneously receive both the directly incident signal and the reflected signal. Therefore, the array receives the following signal:

[0072]

[0073] in, For direct incident signals, For reflected signals, For signal vectors, N (t) represents additive white Gaussian noise. The received signals for each dipole. The array manifold of the polarization-sensitive mirror reflection array, The steering vector of the polarization-sensitive mirror reflection array can be written as the steering vector of the directly incident signal. and the steering vector of the reflected signal sum:

[0074]

[0075] In the formula,

[0076] For array spatial guidance vector, This indicates Kronecker multiplication. The spatial response matrix of a triorthogonal dipole:

[0077]

[0078] Further, the simplified specular reflection array steering vector can be written as:

[0079]

[0080] wherein, is the polarization domain steering vector of each dipole, is the simplified spatial domain steering vector of each dipole:

[0081]

[0082]

[0083] For convenience, the following is used instead of ;

[0084] Secondly, the sum-difference co-array is constructed, and the second step includes the following steps:

[0085] Step 2-1: the sum set and the difference set are obtained:

[0086]

[0087] Step 2-2: for the convenience of subsequent algorithms, after removing a large number of redundant elements and negative values in the sum-difference set, a sum-difference co-array with Q elements is obtained, and the elements in the sum-difference co-array represent the virtual array element positions:

[0088] wherein, unique(·) is used to remove the repeated elements and negative values in the elements.

[0089] Thirdly, the virtual array outputs of each sub-array are obtained by using the sum-difference co-array idea after the covariance matrix is represented in blocks, and the third step includes the following steps:

[0090] Step 3-1: the covariance matrix of the polarization-sensitive specular reflection array received signal is calculated and represented in blocks:

[0091]

[0092] wherein, denotes the conjugate transpose, and each sub-array corresponds to the covariance matrix between each dipole.

[0093] Step 3-2: the sub-arrays in the covariance matrix are vectorized R ij , i ,j = X , Y , Z :

[0094]

[0095] where, is the KR product, is the spatial array manifold of each dipole, is the noise power, is the power of the k th signal, is a column vector with only the first element being 1. where, is the polarization vector of the k th signal parallel to the i th axis dipole;

[0096] Step 3-3: decompose into two parts corresponding to the difference co-array and the sum co-array:

[0097]

[0098] where, j is not Y , , otherwise where:

[0099]

[0100]

[0101] At this time, according to the same rule, the vectorized data of each dipole is also divided into two parts corresponding to the difference co-array and the sum co-array:

[0102]

[0103] where, is the difference co-array, is the sum co-array, and the selection of the ± sign is the same as in ;

[0104] Step 3-4: obtain the virtual array output of each sub-array according to the linear relationship:

[0105]

[0106] where, is the matrix describing the linear combination relationship, which is the conversion matrix, when j ≠ Y , any row in and In H the position index of the row is 1, and the position index of the row is 1; j Y When any row is -1 in the position index of and 1 in the position index of ;

[0107] Fourth, matrix reconstruction, the fourth step includes the following steps: step 4-1: extending the virtual array output of each sub-array:

[0108]

[0109] In the formula, is After removing the first element and performing up-down flip, the data matrix is i , j any is Y when , otherwise ;

[0110] Step 4-2: select sub-array outputs with Q elements in Q , and use sub-arrays to construct Q corresponding covariance matrix :

[0111]

[0112] Step 4-3: use all to construct a 3 Q *3 Q dimensional covariance matrix :

[0113]

[0114] At this time, the steering vector corresponding to the reconstructed covariance matrix is:

[0115]

[0116] In the formula, is the spatial steering vector of each dipole after reconstruction:

[0117]

[0118]

[0119] ​​wherein, is the i th element in .

[0120] The fifth step, after eigenvalue decomposition to obtain the noise subspace, the polarization-DOA joint estimation is carried out, and the fifth step comprises the following steps:

[0121] Step 5-1: eigenvalue decomposition is carried out on to obtain the noise subspace:

[0122]

[0123] Step 5-2: combined with the rank loss principle, the spatial domain parameter and the polarization domain parameter are stripped, and the spectral function containing only the spatial domain parameter is constructed:

[0124] The spatial domain parameter of the signal can be obtained by searching the maximum value of , wherein The Q maximum value of is the direction of arrival of the signal.

[0125] After obtaining the direction of arrival of the signal , it is substituted into the following formula to search again, Q The coordinates corresponding to the maximum value are the polarization auxiliary angle of the signal:

[0126] Embodiment

[0127] The performance of the application can be illustrated by the following simulation:

[0128] 1. Simulation conditions:

[0129] Simulation 1: the mirror reflection array is composed of 12 biorthogonal dipoles, the traditional array has 12 and 25 biorthogonal dipoles respectively, the array form is a uniform linear array, the array element spacing is 6GHZ half wavelength, that is, 25mm; the number of snapshots is 200, the signal-to-noise ratio is 13dB, the center frequency of the three signals is 6GHZ, the incident angle is 40°, 50° and 75° respectively, and the polarization information is , and , under the mirror reflection array, the method and the polarization-DOA joint estimation method based on spatial smoothing are used, under the traditional array, the rank loss dimension reduction MUSIC algorithm is used, and the functional simulation results are shown in Figure 2 .

[0130] Simulation 2: The mirror surface reflecting array is composed of 12 double-quadrature dipoles, the traditional array has 12 and 25 double-quadrature dipoles respectively, the array form is a uniform linear array, the array element spacing is 6GHZ half wavelength, that is, 25mm; the number of snapshots is 200, the signal-to-noise ratio is 13dB, the center frequencies of the three signals are all 5GHZ, the incident angles are 40°, 50° and 75° respectively, and the polarization information is , and , the proposed method and the polarization-DOA joint estimation method based on spatial smoothing are used under the mirror surface reflecting array, the rank-deficient dimension reduction MUSIC algorithm is used under the traditional array, and the function simulation results are shown in Figure 3 .

[0131] Simulation 3: The mirror surface reflecting array is composed of 12 double-quadrature dipoles, the traditional array has 12 and 25 double-quadrature dipoles respectively, the array form is a uniform linear array, the array element spacing is 6GHZ half wavelength, that is, 25mm; the number of snapshots is 200, the signal-to-noise ratio is 3dB step, from 0dB to 21dB, the center frequencies of the three signals are all 6GHZ, the incident angles are 40°, 50° and 75° respectively, and the polarization information is , and , the proposed method and the polarization-DOA joint estimation method based on spatial smoothing are used under the mirror surface reflecting array, the rank-deficient dimension reduction MUSIC algorithm is used under the traditional array, 500 Monte Carlo experiments are carried out respectively, the root mean square error of each parameter estimation is taken as an evaluation index, and the simulation results are shown in Figure 4 .

[0132] 2. Simulation results

[0133] As can be seen from Figure 2 (a), when the array element spacing is equal to half wavelength, the traditional array and the polarization-sensitive mirror surface reflecting array can distinguish three targets, but the spectral peak height and the sharpness degree of the proposed method under the mirror surface array are obviously better than those of the traditional array. Comparing Figure 2 (b), (c) and (d), it can be seen that the polarization spectrum peak shape of the proposed method is also better than that of the traditional array, and the polarization phase angle cannot be estimated by the polarization-DOA joint estimation method based on spatial smoothing.

[0134] As can be seen from Figure 3 (a), when the array element spacing is less than half wavelength, the traditional array with 12 elements cannot distinguish two targets with an angle interval of 10°, but the proposed method under the mirror surface array can still successfully distinguish three targets, and the spectral peak height is better than that of the traditional array with 25 elements. And the polarization of the target can be correctly estimated, as Figure 3(c).

[0135] By Figure 4 (a) can be seen that the polarization-sensitive mirror-reflection array polarization-DOA joint estimation method proposed in the present application has higher spatial angle estimation accuracy than the conventional non-mirror array with the same number of array elements under different signal-to-noise ratios, and the estimation accuracy under high signal-to-noise ratio is similar to that of the conventional array with the same number of virtual array elements. It can be proved that the method proposed in the present application uses the characteristics of the mirror-reflection array to expand the array aperture, and at the same time uses fewer array elements to achieve the same spatial angle estimation accuracy as the conventional non-mirror array. However, as Figure 4 (b) and (c) show that the polarization parameter performance curves, the polarization parameter estimation accuracy of the method proposed in the present application is better than that of the conventional non-mirror array with the same number of physical array elements, and is similar to that of the conventional non-mirror array with 25 array elements. In summary, the algorithm proposed in the present application reduces the computational complexity of the algorithm while maintaining high parameter estimation accuracy compared with the polarization-DOA joint estimation algorithm based on spatial smoothing.

Claims

1. A polarization-sensitive mirror-reflection-array polarization DOA joint estimation method based on matrix reconstruction, characterized in that, The implementation is achieved by the following steps: Step 1: Obtain the polarization-sensitive mirror-reflection array receiving signal: considering K far-field narrow-band signals incident to a polarization-sensitive mirror-reflection array composed of N tri-orthogonal dipoles, the positions of the N tri-orthogonal dipoles are d = [d1, d2,..., dN], d1>0, since there is a reflecting surface, each dipole will simultaneously receive the direct incident signal and the reflected signal, and then the array receiving signal is: y = [y1, y2,..., yN]T= [a1, a2,..., aN]T+ [b1, b2,..., bN]T= [a1+b1, a2+b2,..., aN+bN]T= [x1, x2,..., xN]T N ] T wherein d1>0, since there is a reflecting surface, each dipole will simultaneously receive the direct incident signal and the reflected signal, then the array receiving signal is: where X d (t) is the direct incident signal, X r (t) is the reflected signal, S(t) = [s1, s2,..., s K ] T is the signal vector, N(t) is the additive white Gaussian noise, X p (t), p = X, Y, Z is the received signal of each dipole, A M = [a M (θ1, γ1, η1),..., a M (θ K , γ K , η K )] is the array manifold of the polarization-sensitive specular reflectarray, a M (θ k , γ k , η k ) is the steering vector of the polarization-sensitive specular reflectarray, which can be written as the sum of the steering vector of the direct incident signal a D (θ k , γ k , η k ) and the steering vector of the reflected signal a R (θ k , γ k , η k ). where a S (θ k ) = [exp(j2π / λ·d1sinθ k ),…,exp(j2π / λ·d N sinθ k )], k = 1, … K are array spatial steering vectors, denotes the Kronecker product, and Ξ(θ k ) is the spatial response matrix of a triorthogonal dipole: Further, the mirror reflection array steering vector after simplification can be written as: where ξ p , p = X, Y, Z are the polarized domain steering vectors of each dipole, d p (θ k ), p = X, Y, Z are the simplified spatial steering vectors of each dipole: For convenience, instead of substituting Step 2: Construct sum-difference co-array: Step 2-1: Obtain sum and difference sets {d i +d j} and {d i -d j} using the sum and difference sets {d i +d j} and {d i -d j} H = {d i +d j}∪{d i -d j} Step 2-2: For the convenience of subsequent algorithms, after removing a large number of redundant elements and negative values in the sum-difference set, a sum-difference co-array with Q elements can be obtained, and the elements in it represent the virtual array element positions: H' = unique(H) Where unique(·) is to remove duplicate elements and negative values in the elements; Step 3: After the covariance matrix is expressed in blocks, the virtual array outputs of each sub-array are obtained using the sum-difference co-array idea: Step 3-1: Calculate the covariance matrix of the polarization-sensitive mirror reflection array received signal and express it in blocks: where (·) H denotes the conjugate transpose, each submatrix corresponds to the covariance matrix between the dipoles, step 3-2: vectorization of the submatrix R in the covariance matrix ij i,j = X,Y,Z: In the formula, ⊙ represents the KR product. For the spatial array manifold of each dipole, For noise power, Let the power of the k-th signal be . It is a column vector where only the first element is 1. Where, ξ i,k Let be the polarization vector of the k-th signal parallel to the i-axis dipole; Step 3-3: The following is the result of the decomposition of the matrix A into the sum and difference coarrays: The decomposition into the sum and difference coarrays is given by: wherein when j is not Y, otherwise wherein: At this time, according to the same rule, the vectorized data of each dipole is also divided into two parts corresponding to the difference co-array and the sum co-array: In the formula, corresponding difference co-array, corresponding and co-array, the selection of the ± sign is the same as the same as in the middle; Step 3-4: According to the linear relationship, the virtual array outputs of each sub-array are obtained: z' ij = pinv(Pa ij ) · z ij wherein The matrix that describes the linear combination relationship is a conversion matrix. When j≠Y, Pa ij Any row has |b m -b n | and |b m +b n | is 1 in the position index of H, and the rest of the elements are 0; and when j=Y, Pa ij Any row has |b m -b n | in the position index of -1, and |b m +b n | in the position index of 1. Step 4: Matrix reconstruction, including the following steps: Step 4-1: Expand the virtual array outputs of each sub-array: Z ij = [±z ij , z ij ] T where z ij for z ij z ij The data matrix after flipping up and down after removing the first element, when i, j is any Y, the formula ±- when, otherwise, take +; Step 4-2: In Z ij Q sub-matrices with Q elements are selected in turn ij,q and Z ij is constructed using the Q sub-matrices Step 4-3: Utilize all R ij Construct a 3Q*3Q dimensional covariance matrix At this time, the steering vector corresponding to the reconstructed covariance matrix is: wherein is the spatial steering vector after reconfiguration of each dipole Where H'(i), i = 1, … Q is the ith element in H'; Step 5: Perform eigenvalue decomposition to obtain the noise subspace and then perform polarization-DOA joint estimation; Step 5 includes the following steps: Step 5-1: For Eigenvalue decomposition is performed to obtain the noise subspace: Step 5-2: Combine the rank loss principle to strip off the spatial domain parameters and the polarization domain parameters, and construct a spectral function containing only spatial domain parameters: The spatial parameter of the signal can be obtained by searching f θ (θ) for a maximum value, wherein f θ The θ corresponding to the Q maximum values of f (θ) is the DOA of the signal. Doppler shift Then, the result is substituted into the following equation to search again, and the coordinates corresponding to Q maximum values are the polarization auxiliary angles of the signal:

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