Method for two-dimensional direction of arrival estimation based on recursive root-seeking and multi-parallel sparse linear array decoupling

Abstract

The application discloses a kind of based on recursive root-seeking's multi-parallel sparse linear array decoupling two-dimensional direction-of-arrival estimation method, to solve the deficiency of high computational complexity and freedom degree limited use etc. of existing method in high-dimensional spectrum peak search.It is that the implementation steps are: multi-parallel sparse linear array signal modeling;Multi-parallel sparse linear array covariance matrix is reconstructed into multi-parallel uniform linear array covariance matrix;Two-dimensional direction-of-arrival estimation is decoupled into polynomial root-seeking problem by dimension reduction;Based on recursive root-seeking polynomial coefficient is solved;Based on polynomial coefficient and Rayleigh-Ritz theorem, two-dimensional direction-of-arrival closed-form estimation is realized.The application has significant advantages in computing efficiency, signal recognition and scope of application, and is suitable for target detection and passive positioning in radar, sonar and wireless communication fields.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, and particularly relates to a direction-of-arrival estimation method for parallel sparse linear arrays. Specifically, it is a two-dimensional direction-of-arrival estimation method for decoupled multi-parallel sparse linear arrays based on recursive root-finding, which can be used for target detection and passive localization. Background Technology

[0002] Two-dimensional direction-of-arrival (DOA) estimation techniques are widely used in radar, sonar, wireless communication, and other fields. DOA estimation based on sparse arrays has attracted significant attention due to its higher array degrees of freedom (DoFs). Common sparse arrays include sparse rectangular arrays, open-box arrays, hourglass arrays, L-shaped sparse arrays, V-shaped sparse arrays, and multi-parallel sparse linear arrays. Among these configurations, multi-parallel sparse linear arrays offer significant advantages due to their structural flexibility, allowing for flexible adjustments to the number, position, and element count of subarrays. This flexibility makes multi-parallel sparse linear arrays suitable for conformal designs and helps reduce aerodynamic drag, space occupation, and radar cross-section.

[0003] In multi-parallel sparse linear arrays, all subarrays have the same structure. Based on the (cross-)covariance matrices of these subarrays, the covariance matrix of the corresponding uniform array can be reconstructed, thus achieving two-dimensional direction-of-arrival (DOA) estimation based on subspace. Although the subspace rotation method is well-known for its high computational efficiency, it requires the array to possess two-dimensional rotation invariance, which does not hold true in some multi-parallel sparse linear arrays, thus limiting its application. Another common approach is DOA estimation based on two-dimensional multi-signal classification methods. Two-dimensional multi-signal classification methods have significant advantages in terms of asymptotic variance and signal discriminability, but their drawback is the need for computationally complex two-dimensional spectral peak searches, which imposes a significant computational burden in practical applications. Summary of the Invention

[0004] To overcome the high computational complexity of the aforementioned two-dimensional multi-signal classification methods, this invention proposes a decoupled two-dimensional direction-of-arrival (DOA) estimation method based on recursive root-finding using multiple parallel sparse linear arrays. This method fully utilizes the parallel structure of the multiple parallel sparse linear arrays, maintaining the same configuration for each subarray, and derives the polynomial coefficients recursively for the first time, thus efficiently achieving two-dimensional ODA estimation. The method uses two independent one-dimensional ODA estimation subroutines, significantly reducing computational complexity. The first subroutine employs a recursive root-finding method, while the second subroutine is based on the Rayleigh-Ritz theorem, avoiding the computational burden of two-dimensional spectral peak searching. This method can flexibly adjust the number of subarrays while maintaining a fixed number of sensors, thereby synergistically utilizing apertures in multiple dimensions, significantly improving the accuracy and discriminability of two-dimensional ODA estimation while maintaining low computational complexity. Furthermore, this invention derives closed-form solutions for two-dimensional ODA estimation using sparse parallel planar arrays and sparse parallel cylindrical arrays as examples, and analyzes the computational complexity, further assisting in selecting an appropriate number of subarrays in practical applications.

[0005] The objective of this invention is achieved through the following technical solution: a two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupling of multi-parallel sparse linear arrays, comprising the following steps:

[0006] (1) Modeling of signals from multi-parallel sparse linear arrays:

[0007] There are M far-field narrowband uncorrelated signals, incident on L x A sparse array consisting of ≥2 parallel sparse linear arrays, each subarray containing L y One sensor unit; the subarray extends parallel to the y-axis;

[0008] For a planar sparse array, each subarray is uniformly and parallelly arranged along the x-axis; for a cylindrical sparse array, adjacent subarrays have the same angle between their projections onto the xoz plane.

[0009] The direction of arrival of each signal is determined by the azimuth angle. and pitch angle The angle between the incident signal and the y-axis is denoted as . Let the angle between the incident signal and the x-axis be denoted as .

[0010] For a planar sparse array, the subarray arranged along the y-axis is used as the reference subarray; for a cylindrical sparse array, a virtual sparse linear array is constructed along the centerline of the cylinder as the reference subarray; the guiding vector a of the reference subarray is obtained. 0,m ;

[0011] Steering vector of sparse array Where β mThis is the subarray steering vector along the xoz plane;

[0012] (2) Reconstruct the covariance matrix of the multi-parallel sparse linear array into the covariance matrix of the multi-parallel uniform linear array:

[0013] The covariance matrix R of the output of a multi-parallel sparse linear array xx Represented as:

[0014]

[0015] in and σ 2 Let I represent the power of the m-th signal and the noise power, respectively. n Represents an n×n identity matrix; matrix R xx Block representation is Each dimension is L y ×L y The submatrices; using the Hermitian property of each submatrix, the covariance matrix of the virtual multi-parallel uniform linear matrix is ​​obtained.

[0016]

[0017] in This represents the total number of sensor units in each virtual uniform linear array. This represents the guiding vector for the virtual reconstruction of the uniform reference subarray;

[0018] (3) The two-dimensional direction of arrival estimation is reduced in dimension and decoupled into a polynomial root-finding problem:

[0019] Using the principle of multiple signal classification, we can obtain:

[0020]

[0021] Where P = UU H , express The corresponding noise subspace; let in z = e j2πdcosα / λ α represents the angle between the incident signal to be estimated and the y-axis, which can be estimated using the M roots of the following equation:

[0022] det[D(z -1 )PD(z)]=det(Η)=0,

[0023] According to the rank reduction principle, the M roots can be calculated using the following formula:

[0024]

[0025] here, p,q=1,2,…,L x Where {a:b} represents {a,a+1,…,b}, and a and b are integers; and They can be rewritten as and

[0026] (4) Calculate polynomial coefficients based on the recursive root-finding method:

[0027] dimensional coefficient vector c p,q p,q=1,2,…,L x Calculated in the following way:

[0028]

[0029] in Indicates along matrix P p,q The Sum of the elements along the diagonal; we can obtain:

[0030]

[0031] in dimensional polynomial coefficient vector The derivation method is as follows:

[0032] When L x When c = 2, (2) =c 1,1 *c 2,2 -c 2,1 *c 1,2 ;

[0033] When L x When ≥3, Expand by the cofactor of the selected column or row, and calculate recursively, that is:

[0034]

[0035] in p = 1, 2, ..., L x Let H be the coefficient vector of the cofactor corresponding to the (p,1)th element of H;

[0036] (5) Solve using closed-form based on Rayleigh-Ritz theorem:

[0037] Calculate polynomials There are M roots, and the M roots that are inside the unit circle and closest to the unit circle are selected. The angles of M signals The estimation is performed in the following manner:

[0038] α m =arccos[λarg(z m ) / (2πd)],

[0039] Where arg(·) represents the phase angle of the complex number within the parentheses, d represents the minimum sensor spacing between each subarray, and λ represents the signal wavelength; based on the principle of multiple signal classification and β m From the expression, we can see that:

[0040]

[0041] Using Rayleigh-Ritz theorem and It can be seen that β m and The eigenvector b corresponding to the smallest eigenvalue m Collinear;

[0042] Based on the geometric structure and eigenvector b of the sparse array m Estimated angle via {θ m ,φ m} and {α m ,β m The relationship between} is obtained The estimate.

[0043] Furthermore, in step (1), a method of dynamically adjusting the subarray configuration is adopted to flexibly configure the number and position of subarrays according to the specific application scenario; under the condition that the total number of sensors is fixed, the array degrees of freedom are maximized by increasing the number of subarrays or adjusting the sparse configuration of subarrays. For specific application scenarios, sparse parallel planar arrays, sparse parallel cylindrical arrays, or other multi-parallel sparse linear array configurations can be flexibly selected.

[0044] Furthermore, the reference subarray guiding vector a 0,m Represented as:

[0045]

[0046] in l y =1,2,…,L y Indicates the lth y The distance between each sensor and the x-axis.

[0047] Furthermore, for a sparse array in planar form, the corresponding subarray steering vector along the xoz plane is:

[0048]

[0049] Where dx This represents the distance between adjacent subarrays.

[0050] Furthermore, for a sparse array in cylindrical form, the corresponding subarray steering vector along the xoz plane is:

[0051]

[0052] Where k m =-(cosβ) m ,cosα m ,cosφ m ) T It is the propagation vector associated with the m-th signal. l r =1,2,…,L x Represents the lth cylindrical base located on the xoz plane r The position vector of each sensor; d r L represents the radius of the cylindrical base; Δ represents the angle between the projections of adjacent subarrays onto the xoz plane. x ≤2π / Δ.

[0053] Furthermore, in step (5), for a sparse array in planar form, the angle... The estimation is performed in the following manner:

[0054]

[0055] in Represents the eigenvector b m The nth x Each element.

[0056] Furthermore, in step (5), for a sparse array in cylindrical form, the angle... The estimation is performed by deriving the following from the geometry of the cylindrical array:

[0057] cosβ m {sin(l r Δ)-sin[(l r -1)Δ]}+cosφ m

[0058]

[0059] Among them l r For 1, 2, ..., L x -1, when l r When ≥2, the following formula applies:

[0060] cosβ m {sin(l r Δ)-sin[(lr -2)Δ]}+cosφ m

[0061]

[0062] Compared with the prior art, the present invention has the following advantages:

[0063] (1) This invention is applicable to various parallel sparse linear arrays, including sparse parallel planar arrays and sparse parallel cylindrical arrays. These arrays are flexibly configured, and the number, position and number of elements of each subarray can be adjusted to suit different scenario requirements. They can also make coordinated use of apertures in multiple dimensions, thereby effectively improving the array's degree of freedom.

[0064] (2) This invention can realize two-dimensional direction of arrival closed estimation with relatively low computational complexity. The polynomial coefficients are derived for the first time through recursion, and efficient closed estimation is completed based on the Rayleigh-Ritz theorem. No spectral peak search is required, which significantly reduces the computational burden and has better efficiency than existing two-dimensional multi-signal classification methods.

[0065] (3) Compared with traditional uniform arrays, the method of the present invention can make full use of multi-dimensional apertures by flexibly adjusting the number and position of sub-arrays under a fixed number of sensors, which greatly improves the degree of freedom of the array and enhances the identification and estimation accuracy of the signal. Attached Figure Description

[0066] Figure 1 This is the overall flowchart of the present invention.

[0067] Figure 2 This is a schematic diagram of the sparse parallel planar linear array structure proposed in this invention.

[0068] Figure 3 This is a schematic diagram of the sparse parallel cylindrical linear array structure proposed in this invention.

[0069] Figure 4 This is a scatter plot of the two-dimensional direction of arrival estimation using the method proposed in this invention under a sparse parallel planar linear array.

[0070] Figure 5 This is a scatter plot of the two-dimensional direction of arrival estimation using the method proposed in this invention on a sparse parallel cylindrical linear array.

[0071] Figure 6 This is a graph showing the variation of the two-dimensional root mean square error of the direction of arrival with the signal-to-noise ratio for the method proposed in this invention.

[0072] Figure 7 This is a graph showing the variation of the two-dimensional root mean square error of the direction of arrival with the number of snapshots for the method proposed in this invention. Detailed Implementation

[0073] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings.

[0074] To address the problems of high computational complexity in traditional two-dimensional multi-signal classification methods (such as two-dimensional peak search), strict limitations on array structure imposed by subspace rotation methods, and the inability of polynomial root-finding methods to effectively utilize multidimensional apertures, this invention provides a decoupled two-dimensional direction-of-arrival (DOA) estimation method based on recursive root-finding using multi-parallel sparse linear arrays, achieving joint estimation of the two-dimensional DOA. (Refer to...) Figure 1 The implementation steps of this invention are as follows:

[0075] Step 1: Modeling of multi-parallel sparse linear array signals:

[0076] There are M far-field narrowband uncorrelated signals, incident on L x A sparse array consisting of ≥2 parallel sparse linear arrays, where each sparse linear array (subarray) is fully augmentable, and each subarray contains L y One sensor unit. The subarray extends parallel to the y-axis.

[0077] For a planar sparse array, the subarrays are arranged uniformly and parallelly along the x-axis. For a cylindrical sparse array, adjacent subarrays have the same angle between their projections onto the xoz plane. Figure 2 In this array, each subarray is an identical nested array, and the position of each subarray sensor element on the y-axis is represented as (0, 1, 2, 5)d, where d represents the minimum sensor spacing of each subarray. x This represents the distance between adjacent subarrays. Figure 3 In the middle, d r L represents the radius of the cylindrical base, Δ represents the angle between the projections of adjacent subarrays onto the xoz plane, and L represents the radius of the cylindrical base. x ≤2π / Δ.

[0078] The direction of arrival of each signal is determined by the azimuth angle. and pitch angle The angle between the incident signal and the y-axis is denoted as . Let the angle between the incident signal and the x-axis be denoted as . The following relationship exists:

[0079] sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m ,

[0080] For sparse arrays in planar form, such as Figure 2As shown, taking the subarray arranged along the y-axis as the reference subarray, its steering vector (associated with the m-th signal) can be expressed as:

[0081]

[0082] in l y =1,2,…,L y Indicates the lth y The distance between each sensor and the x-axis, where λ represents the signal wavelength, (·) T This represents the transpose operator.

[0083] For sparse arrays in cylindrical form, such as Figure 3 As shown, a virtual sparse linear array is constructed along the centerline of the cylinder as a reference subarray, and its guiding vector can be similarly expressed as:

[0084]

[0085] The guiding vector a of the sparse array m Represented as:

[0086]

[0087] Where β m The subarray steering vector along the xoz plane. Represents the Kronecker product.

[0088] For a sparse array in planar form, its corresponding β m Recorded as:

[0089]

[0090] Where d x This represents the distance between adjacent subarrays.

[0091] For a sparse array in cylindrical form, its corresponding β m Recorded as:

[0092]

[0093] Where k m =-(cosβ) m ,cosα m ,cosφ m ) T It is the propagation vector associated with the m-th signal. l r =1,2,…,L x Represents the lth cylindrical base located on the xoz plane r The position vector of each sensor.

[0094] Step 2: Reconstruct the covariance matrix of the multi-parallel sparse linear array into the covariance matrix of the multi-parallel uniform linear array:

[0095] The covariance matrix R of the output of a multi-parallel sparse linear array xx Represented as:

[0096]

[0097] in and σ 2 Let represent the power of the m-th signal and the noise power, respectively. H I represents the conjugate transpose operator. n Let represent an n×n identity matrix. Through analysis... and The data structure allows us to discover that matrix R... xx Block representation is Each dimension is L y ×L y The submatrices. Therefore, by utilizing the Hermitian property of each submatrix, the covariance matrix of the virtual multi-parallel uniform linear array is obtained.

[0098]

[0099] in This represents the total number of sensor units in each virtual uniform linear array. This represents the guiding vector for the virtual reconstruction of the uniform reference subarray.

[0100] Step 3: Reduce the dimensionality of the two-dimensional direction of arrival estimation and decouple it into a polynomial root-finding problem:

[0101] Using the principle of multiple signal classification, we can obtain:

[0102]

[0103] Where P = UU H , express The corresponding noise subspace. Let... in And z = e j2πdcosα / λ α represents the angle between the incident signal to be estimated and the y-axis, which can be estimated using the M roots of the following equation:

[0104] det[D(z -1 )PD(z)]=det(Η)=0,

[0105] Where det(·) represents the determinant operator. According to the rank reduction principle, the M roots of this equation can be calculated using the following formula:

[0106]

[0107] here, p,q=1,2,…,L x , where {a:b} represents {a,a+1,…,b}, and a and b are integers. and They can be rewritten as and

[0108] Step 4: Calculate the polynomial coefficients based on the recursive root-finding method:

[0109] dimensional coefficient vector c p,q (p,q=1,2,…,L x It can be calculated in the following ways:

[0110]

[0111] in Indicates along matrix P p,q The The sum of the elements on the diagonal. Therefore, we can obtain:

[0112]

[0113] in dimensional polynomial coefficient vector The derivation method is as follows:

[0114] When L x When c = 2, (2) =c 1,1 *c 2,2 -c 2,1 *c 1,2 , where * is the convolution operator.

[0115] When L x When =3, it is as follows:

[0116] c (3) =c 1,1 (c 2,2 *c 3,3 -c 2,3 *c 3,2 )

[0117] -c 2,1 (c 1,2 *c 3,3 -c 1,3 *c 3,2 )

[0118] +c 3,1 (c 1,2 *c 2,3 -c 1,3 *c 2,2 )

[0119] In addition, when L x When ≥3, It can be expanded using the cofactors of the selected columns or rows and calculated recursively, that is:

[0120]

[0121] in p = 1, 2, ..., L x Let H be the coefficient vector of the cofactor corresponding to the (p,1)th element of H.

[0122] Step 5: Solve using closed-form equations based on the Rayleigh-Ritz theorem:

[0123] Calculate polynomials There are M roots, and the M roots that are inside the unit circle and closest to the unit circle are selected. Furthermore, the angles of the M signals The following methods can be used for estimation:

[0124] α m =arccos[λarg(z m ) / (2πd)],

[0125] Where arg(·) represents the phase angle of the complex number within the parentheses. Based on the principle of multiple signal classification and β... m From the expression, we can see that:

[0126]

[0127] Therefore, through Rayleigh-Ritz theorem and It can be seen that β m and The eigenvector b corresponding to the smallest eigenvalue m Collinear, i.e., b m =b m β m , where b m It is a non-zero constant.

[0128] Therefore, for a sparse array in planar form, the angle The following methods can be used for estimation:

[0129]

[0130] in Represents the eigenvector b m The nth x Each element.

[0131] For a sparse array in cylindrical form, the angle The following methods can be used for estimation:

[0132] Based on the geometry of the cylindrical array, it can be deduced that:

[0133] cosβ m {sin(l r Δ)-sin[(l r -1)Δ]}+cosφ m

[0134]

[0135] Among them l r For 1, 2, ..., L x -1, when l r When ≥2, the following formula applies:

[0136] cosβ m {sin(l r Δ)-sin[(l r -2)Δ]}+cosφ m

[0137]

[0138] Furthermore, the relationship between {θ,φ} and {α,β} can be obtained... The estimate.

[0139] The effects of the present invention will be further described below with reference to simulation examples.

[0140] Simulation Examples: To verify the advantages of the method in terms of enhanced degrees of freedom and improved performance, a comparative analysis was conducted between this method and the subspace rotation joint diagonalization method, the two-dimensional multi-signal classification method, a specifically parameterized decoupled root multi-signal classification method, and the Cramer-Rhodes lower bound. Specific parameters were set in the simulation experiments, and multiple scenarios were designed for verification, including verification of enhanced degrees of freedom, and analysis of the statistical performance and computational complexity based on two-dimensional direction-of-arrival estimation. The experiments consisted of 10,000 independent trials.

[0141] First, an underdetermined scenario is simulated, where L is the number of subarrays in a planar sparse array. x The number of subarrays L in the cylindrical sparse array is set to 4. x Set to 3, Δ is set to 2arcsin[λ / (5d)] r[ ], ensuring that the distance between two adjacent sensors does not exceed half a wavelength. For a sparse parallel planar linear array, assuming 12 signals, where The range is [5°, 25°, 45°, 65°, 85°, 7°, 27°, 47°, 65°, 75°, 2°, 77°]. [5°, 25°, 45°, 65°, 85°, 40°, 60°, 80°, 20°, 40°, 55°, 72°]. For a sparse parallel cylindrical linear array, assuming 9 signals... The values ​​are (25°, 23°), (45°, 43°), (65°, 63°), (85°, 83°), (10°, 35°), (30°, 55°), (80°, 45°), (2°, 65°), and (70°, 85°). Figure 4 and Figure 5 The results of two-dimensional direction-of-arrival (DOA) estimation based on 100 independent trials are presented, with a signal-to-noise ratio of 10 dB and a sample size of 3,000,000. Figure 4 and Figure 5 As can be seen, the decoupled root multi-signal classification method proposed in this invention successfully identified 12 and 9 signals in sparse parallel planar linear array signal parameter estimation and sparse parallel cylindrical linear array signal parameter estimation, respectively. However, within the framework of the proposed method, when using a multi-parallel uniform linear array with the same number of subarrays and sensors as the sparse parallel planar linear array signal parameter estimation and sparse parallel cylindrical linear array signal parameter estimation, the maximum number of recognizable signals drops to 8 and 6, respectively. Furthermore, in the case of the subspace rotation joint diagonalization direction matrix method, the maximum number of recognizable signals is 12 and 8, respectively. Therefore, simulation results verify the enhanced signal recognition capability of the proposed method.

[0142] Then, for a given scenario, the statistical estimation performance of different methods in terms of signal-to-noise ratio and number of snapshots is compared. Let L... x =4, and the sensor position along the y-axis is set to (0,1,4,6)d. To evaluate the statistical performance of the proposed decoupled root multi-signal classification method under different conditions, two other configurations were also studied, namely, assuming L... x =3 and L x =2, and their sensor positions along the y-axis are configured as (0,1,2,5,8,11)d and (0,1,2,3,7,11,15,19)d, respectively. The total number of sensors is similar in all three cases. The two-dimensional directions of arrival of the two signals... They are (50°, 15°) and (78°, 65°) respectively. Figure 6 and Figure 7The statistical estimation comparison results show that the proposed method outperforms the subspace rotation joint diagonalization method, at the cost of higher computational burden. These advantages are particularly evident under conditions of high signal-to-noise ratio and large snapshot number. Furthermore, the decoupled root multi-signal classification method based on multi-parallel sparse linear arrays outperforms the decoupled root multi-signal classification method based on multi-parallel uniform linear arrays, thanks to the increased array degrees of freedom. Among the three sensor configurations, the larger L... x The value makes the decoupled root multi-signal classification method more efficient in two-dimensional apertures, thus resulting in better performance. However, when the signal-to-noise ratio exceeds 6dB, a larger L... x The decoupled root multi-signal classification method exhibits relatively poor performance, mainly due to its high sensitivity to covariance matrix reconstruction errors. The two-dimensional multi-signal classification method slightly outperforms the decoupled root multi-signal classification method under high SNR and large snapshot number conditions, but its performance deteriorates significantly at low SNR and small snapshot number, and its threshold characteristics are inferior to those of the decoupled root multi-signal classification method. This phenomenon is attributed to the synergistic effect of the decoupled root multi-signal classification method in aperture utilization and the constraints of the two-dimensional multi-signal classification method in search. Furthermore, the two-dimensional multi-signal classification method also introduces a significant computational burden.

[0143] Finally, the computational complexity of each method was compared by average CPU runtime. Experiments were conducted on a Windows 1020a machine equipped with 32.0GB RAM and an Intel(R) Core(TM) i7-10875H 2.30GHz processor using the MATLAB 2.30a platform. Table 1 summarizes the average computation time required for each method per trial, where the number of two-dimensional search grids is (0°–90°) / 0.2° = 450. Notably, among the compared methods, two-dimensional multi-signal classification exhibits the highest computational time, while the proposed method has a slightly higher computational burden compared to the subspace rotation joint diagonalization method. As can be seen from Table 1, the decoupled root multi-signal classification based on a multi-parallel uniform linear array exhibits the lowest computational complexity. This can be attributed to its elimination of the subarray data fitting process and the dimensionality reduction of matrices and vectors required in the algorithm's computational flow.

[0144] Table 1. Comparison of average computation time for different algorithms with different numbers of snapshots

[0145]

[0146] However, the estimation accuracy of decoupled root multi-signal classification based on multi-parallel uniform linear arrays is not as good as that based on multi-parallel sparse linear arrays.

[0147] In summary, this invention introduces an efficient two-dimensional direction-of-arrival estimation method for multi-parallel sparse linear arrays with enhanced array degrees of freedom, termed the recursive root-finding decoupled multi-signal classification method. This method provides, for the first time, an efficient closed-form solution for sparse parallel planar arrays and sparse parallel cylindrical arrays, where the polynomial coefficients are derived recursively, making it applicable to various scenarios with different subarrays.

[0148] The specific embodiments described herein are merely illustrative of the spirit of the invention. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific embodiments or use similar methods to substitute them, without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

Claims

1. A two-dimensional direction-of-arrival estimation method based on recursive root-finding using a multi-parallel sparse linear array for decoupling, characterized in that... Includes the following steps: (1) Modeling of signals from multi-parallel sparse linear arrays: There are M far-field narrowband uncorrelated signals, incident on... A sparse array consisting of parallel sparse linear arrays, each subarray containing One sensor unit; the subarray extends parallel to the y-axis; For a planar sparse array, each subarray is arranged uniformly and parallelly along the x-axis; for a cylindrical sparse array, adjacent subarrays have the same angle between their projections onto the xoz plane. The direction of arrival of each signal is determined by the azimuth angle. and pitch angle The angle between the incident signal and the y-axis is denoted as . Let the angle between the incident signal and the x-axis be denoted as ; For a planar sparse array, the subarray arranged along the y-axis is used as the reference subarray; for a cylindrical sparse array, a virtual sparse linear array is constructed along the center line of the cylinder as the reference subarray. Obtain the reference subarray steering vector ; Steering vector of sparse array ,in This is the subarray steering vector along the xoz plane; (2) Reconstruct the covariance matrix of the multi-parallel sparse linear array into the covariance matrix of the multi-parallel uniform linear array: Covariance matrix of output from multiple parallel sparse linear arrays Represented as: , in and Let these represent the power of the m-th signal and the noise power, respectively. express identity matrix; matrix Block representation is Each dimension is The submatrices; using the Hermitian property of each submatrix, the covariance matrix of the virtual multi-parallel uniform linear matrix is ​​obtained. : , in This represents the total number of sensor units in each virtual uniform linear array. This represents the guiding vector for the virtual reconstruction of the uniform reference subarray; (3) The two-dimensional direction of arrival estimation is reduced in dimension and decoupled into a polynomial root-finding problem: Using the principle of multiple signal classification, we can obtain: , in , express The corresponding noise subspace; let ,in , , The angle between the incident signal to be estimated and the y-axis can be expressed by the following equation: Root estimation: , According to the principle of demotion The root can be calculated using the following formula: , here, , ,in express a and b are integers; , , , , , , , , , , , ,and They can be rewritten as , , , , , , , , , , , ,and , ; (4) Calculate polynomial coefficients based on the recursive root-finding method: dimensional coefficient vector , Calculated in the following way: , in , Indicates along the matrix The Sum of the elements along the diagonal; we can obtain: , in dimensional polynomial coefficient vector The derivation method is as follows: when hour, ; when hour, Expand by the cofactor of the selected column or row, and calculate recursively, that is: , in , express The Each element corresponds to the coefficient vector of the cofactor; (5) Solve using closed-form based on Rayleigh-Ritz theorem: Calculate polynomials 1. Select M roots that are inside the unit circle and closest to the unit circle. ; Angles of M signals The estimation is performed in the following manner: , in This represents the phase angle of the complex number within the parentheses. This represents the minimum sensor spacing between each subarray. Indicates the signal wavelength; based on the principle of multiple signal classification and From the expression, we can see that: , Using Rayleigh-Ritz theorem and It can be seen that, and Eigenvector corresponding to the smallest eigenvalue Collinear; Based on the geometry and eigenvectors of the sparse array Estimated angle ;pass and The relationship was obtained The estimate.

2. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 1, characterized in that, In step (1), a method of dynamically adjusting the subarray configuration is adopted to flexibly configure the number and position of subarrays according to the specific application scenario; under the condition that the total number of sensors is fixed, the array degree of freedom is maximized by increasing the number of subarrays or adjusting the subarray sparsity configuration.

3. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 1, characterized in that, The reference subarray guiding vector Represented as: , in Indicates the first The distance between each sensor and the x-axis.

4. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 1, characterized in that, For a sparse array in planar form, the corresponding subarray steering vector along the xoz plane is: , in This represents the distance between adjacent subarrays.

5. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 1, characterized in that, For a sparse array in cylindrical form, the corresponding subarray steering vector along the xoz plane is: in It is the propagation vector associated with the m-th signal. Describes the cylindrical base located on the xoz plane. Position vectors of each sensor; Indicates the radius of the cylindrical base; This represents the angle between the projections of adjacent subarrays onto the xoz plane. .

6. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 5, is characterized in that, In step (5), for a sparse array in planar form, the angle The estimation is performed in the following manner: , in Representing the eigenvector The One element, This represents the distance between adjacent subarrays.

7. The two-dimensional direction-of-arrival estimation method based on recursive root-finding and decoupled multi-parallel sparse linear arrays according to claim 6, is characterized in that, In step (5), for a sparse array in cylindrical form, the angle... The estimation is performed by deriving the following from the geometry of the cylindrical array: in for ,when At that time, we have the following formula: 。

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