Multi-parallel sparse linear array decoupling two-dimensional direction-of-arrival estimation method based on recursive root seeking

Abstract

The invention discloses a multi-parallel sparse linear array decoupling two-dimensional direction-of-arrival estimation method based on recursive root search, and aims to solve the defects of high calculation complexity, limited utilization of degree of freedom and the like caused by high-dimensional spectral peak search in the existing method. The method comprises the following steps of: modeling multiple parallel sparse linear array signals; reconstructing the multi-parallel sparse linear array covariance matrix into a multi-parallel uniform linear array covariance matrix; the two-dimensional direction of arrival estimation is subjected to dimensionality reduction, decoupling and transformation into a polynomial rooting problem; solving a polynomial coefficient based on recursive root seeking; and realizing two-dimensional direction-of-arrival closed estimation based on the polynomial coefficient and the Rayleigh-Ritz theorem. The method has the remarkable advantages in calculation efficiency, signal identification and application range, and is suitable for target detection and passive positioning in the fields of radar, sonar, wireless communication and the like.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and particularly relates to a method for estimating the direction of arrival (DOA) for a parallel sparse linear array, specifically a method for decoupling two-dimensional DOA estimation for multiple parallel sparse linear arrays based on recursive root finding, which can be used for target detection and passive positioning. Background Art

[0002] Two-dimensional direction-of-arrival (DOA) estimation technology is widely used in multiple fields such as radar, sonar, and wireless communication. Implementing two-dimensional DOA estimation based on a sparse array has a higher array degree of freedom (DoFs) and has received extensive attention. Common sparse arrays include sparse rectangular arrays, unpacked arrays, hourglass arrays, L-shaped sparse arrays, V-shaped sparse arrays, and multiple parallel sparse linear arrays, etc. Among these configurations, the multiple parallel sparse linear array has significant advantages due to the flexibility of its structure, which can flexibly adjust the number of subarrays, their positions, and the number of elements in each subarray. This flexibility makes the multiple parallel sparse linear array suitable for conformal design and helps reduce aerodynamic drag, occupied space, and radar cross-section, etc.

[0003] In a multiple parallel sparse linear array, all subarrays have the same structure. Based on the (cross) covariance matrices of these subarrays, the covariance matrix of the corresponding uniform array of the multiple parallel sparse linear array can be reconstructed, thereby realizing two-dimensional DOA estimation based on the subspace. Although the subspace rotation method is well-known for its high computational efficiency, it requires the array to have two-dimensional rotational invariance, which does not hold in some multiple parallel sparse linear arrays, thus limiting its application scope. Another common method is DOA estimation based on the two-dimensional multiple signal classification (MUSIC) method. The two-dimensional MUSIC method has significant advantages in terms of asymptotic variance and signal identifiability, but its disadvantage is that it requires a two-dimensional spectral peak search with high computational complexity, which brings a large computational burden in practical applications. Summary of the Invention

[0004] To break through the limitation of high computational complexity of the above-mentioned two-dimensional multiple signal classification method, the present invention proposes a decoupled two-dimensional direction of arrival (DOA) estimation method for multi-parallel sparse linear arrays based on recursive root finding. This method makes full use of the parallel structure of the multi-parallel sparse linear arrays, maintains the same configuration for each sub-array, and first derives the polynomial coefficients through a recursive manner, thereby efficiently realizing two-dimensional DOA estimation. This method performs estimation through two independent one-dimensional DOA estimation subroutines, significantly reducing the computational complexity. The first subroutine adopts the recursive root finding method, and the second subroutine is based on the Rayleigh-Ritz theorem, avoiding the computational burden brought by two-dimensional spectral peak searching. This method can flexibly adjust the number of sub-arrays on the basis of a fixed number of sensors, thereby synergistically utilizing the apertures in multiple dimensions, significantly improving the accuracy and identifiability of two-dimensional DOA estimation, while maintaining a low computational complexity. In addition, the present invention takes a sparse parallel planar array and a sparse parallel cylindrical array as examples, derives the closed-form solution of two-dimensional DOA estimation, and analyzes the computational complexity, further helping to select the appropriate number of sub-arrays in practical applications.

[0005] The object of the present invention is achieved through the following technical solutions: A decoupled two-dimensional direction of arrival (DOA) estimation method for multi-parallel sparse linear arrays based on recursive root finding, the method comprising the following steps:

[0006] (1) Signal modeling of multi-parallel sparse linear arrays:

[0007] There are M far-field narrowband uncorrelated signals incident on a sparse array composed of L x ≥2 parallel sparse linear arrays, and each sub-array contains L y sensor units; the extension direction of the sub-array is parallel to the y-axis;

[0008] For the sparse array in planar form, each sub-array is uniformly arranged in parallel along the x-axis; for the sparse array in cylindrical form, the projection angles of adjacent sub-arrays in the x-o-z plane are the same;

[0009] The direction of arrival of each signal is represented by the azimuth angle and the elevation angle ; the angle between the incident signal and the y-axis is denoted as The angle between the incident signal and the x-axis is denoted as

[0010] For the sparse array in planar form, the sub-array arranged along the y-axis is used as the reference sub-array; for the sparse array in cylindrical form, a virtual sparse linear array along the center line of the cylinder is constructed as the reference sub-array; the steering vector a 0,m of the reference sub-array is obtained;

[0011] The steering vector of the sparse array where β mis the subarray steering vector along the x-o-z 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 output by the multi-parallel sparse linear array xx is expressed as:

[0014]

[0015] where and σ 2 represent the power of the m-th signal and the noise power respectively, and I n represents the n×n identity matrix; the matrix R xx can be block-represented as submatrices of dimension L y ×L y ; using the Hermitian property of each submatrix, the covariance matrix of the virtual multi-parallel uniform linear array is obtained

[0016]

[0017] where represents the total number of sensor elements of each virtual uniform linear array, represents the steering vector of the virtual reconstructed uniform reference subarray;

[0018] (3) Decouple the two-dimensional direction-of-arrival estimation into a polynomial root-finding problem:

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

[0020]

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

[0022] det[D(z -1 )PD(z)] = det(Η) = 0,

[0023] According to the reduced-rank principle, the M roots can be calculated according to 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 can be rewritten as and

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

[0027] The coefficient vector c of dimension p,q , p, q = 1, 2, …, L x is calculated as follows:

[0028]

[0029] where represents the sum of elements along the p,q -th diagonal of matrix P ; we get:

[0030]

[0031] where the coefficient vector of the -dimensional polynomial is derived as follows:

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

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

[0034]

[0035] where p = 1, 2, …, L x represents the coefficient vector of the cofactor corresponding to the (p, 1)-th element of Η;

[0036] (5) Conduct a closed - form solution based on the Rayleigh - Ritz theorem:

[0037] Calculate the roots of the polynomial and select M roots that are inside the unit circle and closest to the unit circle The angles of the M signals It is estimated by the following method:

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

[0039] where arg(·) represents the phase angle of the complex number in the brackets, d represents the minimum sensor spacing of each subarray, and λ represents the signal wavelength; According to the principle of multiple signal classification and the expression of β m it can be known that:

[0040]

[0041] Through the Rayleigh-Ritz theorem and it can be known that β m and the eigenvector b corresponding to the minimum eigenvalue m are collinear;

[0042] According to the geometric structure of the sparse array and the eigenvector b m , estimate the angle through the relationship between {θ m , φ m} and {α m , β m} to obtain the estimate of .

[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 specific application scenarios; Under the condition that the total number of sensors is fixed, maximize the degrees of freedom of the array by increasing the number of subarrays or adjusting the subarray sparsification configuration method. For specific application scenarios, sparse parallel plane arrays, sparse parallel cylindrical arrays or other multi-parallel sparse linear array configurations can be flexibly selected.

[0044] Furthermore, the reference subarray steering vector a 0,m is expressed as:

[0045]

[0046] where l y = 1, 2,..., L y represents the distance between the l y th sensor and the x-axis.

[0047] Furthermore, for the sparse array in planar form, its corresponding subarray steering vector along the x - o - z plane is:

[0048]

[0049] where dx Indicates the distance between adjacent sub-arrays.

[0050] Furthermore, for a cylindrical sparse array, the sub-array steering vector corresponding to the x-o-z plane is:

[0051]

[0052] where k m = -(cosβ m , cosα m , cosφ m ) T is the propagation vector related to the m-th signal, l r = 1, 2, …, L x represents the position vector of the l-th sensor on the cylindrical base located on the x-o-z plane; d r represents the radius of the cylindrical base; Δ represents the projection angle of adjacent sub-arrays on the x-o-z plane, and L r ≤ 2π / Δ. x

[0053] Furthermore, in step (5), for a planar sparse array, the angle is estimated by the following method:

[0054]

[0055] where represents the n-th m element of the eigenvector b x .

[0056] Furthermore, in step (5), for a cylindrical sparse array, the angle is estimated by the following method: Derived from the geometric structure of the cylindrical array:

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

[0058]

[0059] where l r is 1, 2, …, L x - 1, and when l r ≥ 2, the following formula holds:

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

[0061]

[0062] The present invention has the following advantages compared with the prior art:

[0063] (1) The present invention can be applied to a variety of parallel sparse linear arrays, including sparse parallel planar arrays and sparse parallel cylindrical arrays. These arrays have flexible configurations, and the number, position of sub-arrays, and the number of array elements in each sub-array can be adjusted, which are suitable for different scenario requirements and can synergistically utilize apertures in multiple dimensions, thereby effectively improving the array degrees of freedom;

[0064] (2) The present invention can achieve closed-form estimation of two-dimensional direction of arrival, with relatively low computational complexity. By recursively deriving polynomial coefficients for the first time and completing efficient closed-form estimation based on the Rayleigh-Ritz theorem, without spectral peak searching, the computational burden is significantly reduced, and it has better efficiency compared with the existing two-dimensional multiple signal classification methods;

[0065] (3) Compared with traditional uniform planar 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, greatly improving the array degrees of freedom and enhancing signal discrimination and estimation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 is the overall flow block diagram of the present invention.

[0067] Figure 2 is the schematic diagram of the sparse parallel planar linear array structure proposed by the present invention.

[0068] Figure 3 is the schematic diagram of the sparse parallel cylindrical linear array structure proposed by the present invention.

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

[0070] Figure 5 is the scatter plot of two-dimensional direction of arrival estimation of the method proposed by the present invention under a sparse parallel cylindrical linear array.

[0071] Figure 6 is the curve graph of the root mean square error of two-dimensional direction of arrival of the method proposed by the present invention varying with the signal-to-noise ratio.

[0072] Figure 7 is the curve graph of the root mean square error of two-dimensional direction of arrival of the method proposed by the present invention varying with the number of snapshots. DETAILED DESCRIPTION OF THE INVENTION

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

[0074] To solve the problems that the traditional two-dimensional multiple signal classification method requires two-dimensional spectral peak search with high computational complexity, the subspace rotation method has strict restrictions on the array structure, and the polynomial root seeking method cannot effectively utilize the multi-dimensional aperture, etc., the present invention provides a decoupled two-dimensional direction of arrival estimation method for multi-parallel sparse linear arrays based on recursive root seeking to achieve the joint estimation of the two-dimensional direction of arrival. Refer to Figure 1 The implementation steps of the present invention are as follows:

[0075] Step 1: Signal modeling of multi-parallel sparse linear arrays:

[0076] There are M far-field narrowband uncorrelated signals incident on a sparse array composed of L x ≥2 parallel sparse linear arrays, and each sparse linear array (sub-array) is fully augmentable, and each sub-array contains L y sensor units. The extension direction of the sub-array is parallel to the y-axis.

[0077] For the sparse array in planar form, each sub-array is arranged uniformly and parallel along the x-axis. For the sparse array in cylindrical form, the projection angles of adjacent sub-arrays in the x-o-z plane are the same. Figure 2 In, each sub-array is the same nested array, and the positions of the sensor units of each sub-array on the y-axis are expressed as (0, 1, 2, 5)d, where d represents the minimum sensor interval of each sub-array, and d x represents the distance between adjacent sub-arrays. Figure 3 In, d r represents the radius of the cylindrical base, Δ represents the projection angle of adjacent sub-arrays in the x-o-z plane, and L x ≤2π / Δ.

[0078] The direction of arrival of each signal is represented by the azimuth angle and the elevation angle . Denote the angle between the incident signal and the y-axis as Denote the angle between the incident signal and the x-axis as There is the following relationship:

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

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

[0081]

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

[0083] For a cylindrical sparse array, as Figure 3 shown, construct a virtual sparse linear array along the cylindrical center line as the reference subarray, and its steering vector can similarly be expressed as:

[0084]

[0085] The steering vector a of the sparse array m is expressed as:

[0086]

[0087] where β m is the steering vector of the subarray along the x - o - z plane, represents the Kronecker product.

[0088] For a planar sparse array, its corresponding β m is denoted as:

[0089]

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

[0091] For a cylindrical sparse array, its corresponding β m is denoted as:

[0092]

[0093] where k m = -(cosβ m , cosα m , cosφ m ) T is the propagation vector associated with the m-th signal, l r = 1, 2, …, L x represents the position vector of the l-th sensor on the cylindrical base located in the x - o - z plane. r ​​

[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 \(R\) output by the multi-parallel sparse linear array xx is expressed as:

[0096]

[0097] where and \(\sigma\) 2 respectively represent the power of the \(m\)-th signal and the noise power, \((\cdot)^{H}\) H represents the conjugate transpose operator, and \(I\) n represents the \(n\times n\) identity matrix. By analyzing and 's data structure, it can be found that the matrix \(R\) xx can be block-represented as sub-matrices with dimensions of \(L\) y × \(L\) y . Therefore, using the Hermitian property of each sub-matrix, the covariance matrix of the virtual multi-parallel uniform linear array

[0098]

[0099] where represents the total number of sensor elements of each virtual uniform linear array, represents the steering vector of the virtual reconstructed uniform reference sub-array.

[0100] Step 3: Decouple the two-dimensional direction-of-arrival estimation into a polynomial root-finding problem:

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

[0102]

[0103] where \(P = UU^{H}\) H , represents 's corresponding noise subspace. Let where and \(z = e^{j2\pi f\tau}\), \(\alpha\) represents the angle between the incident signal to be estimated and the \(y\)-axis, and can be estimated by the \(M\) roots of the following equation: j2πdcosα / λ

[0104] \(\text{det}[D(z^{j2\pi f\tau})PD(z)]=\text{det}(\varXi)=0\), -1

[0105] where \(\text{det}(\cdot)\) represents the determinant operator. According to the reduced-rank principle, the \(M\) roots of this equation can be calculated according to 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 can be rewritten as and

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

[0109] The coefficient vector c of dimension p,q (p, q = 1, 2, …, L x ) can be calculated as follows:

[0110]

[0111] where represents the sum of the elements along the p,q th diagonal of the matrix P . Therefore, we can obtain:

[0112]

[0113] where the coefficient vector of the dimensional polynomial

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

[0115] When L x = 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 ≥ 3, it can be expanded by the cofactor of the selected column or row and calculated recursively, i.e.:

[0120]

[0121] where p = 1, 2, …, L x represents the coefficient vector of the cofactor corresponding to the (p, 1)-th element of Η.

[0122] Step Five: Closed-form solution based on the Rayleigh-Ritz theorem:

[0123] Calculate the roots of the polynomial and select M roots that are inside the unit circle and closest to the unit circle Furthermore, the angles of the M signals can be estimated in the following way:

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

[0125] where arg(·) represents the phase angle of the complex number inside the parentheses. According to the principle of multiple signal classification and the expression of β m it can be known that:

[0126]

[0127] Therefore, through the Rayleigh-Ritz theorem and it can be known that β m is collinear with the eigenvector b corresponding to the minimum eigenvalue, i.e., b m = b m β m where b m is a non-zero constant. m

[0128] Therefore, for a sparse array in planar form, the angle can be estimated in the following way:

[0129]

[0130] where ​Denote the eigenvector b m the n x -th element.

[0131] For a cylindrical sparse array, the angle can be estimated as follows:

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

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

[0134]

[0135] where l r is 1, 2, …, L x -1, and when l r ≥2, the following equation holds:

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

[0137]

[0138] Furthermore, the estimation of can be obtained through the relationship between {θ, φ} and {α, β}.

[0139] Next, the effects of the present invention are further described with reference to simulation examples.

[0140] Simulation example: To verify the advantages of the method in terms of enhanced degrees of freedom and improved performance, the differences between this method and the subspace rotation joint diagonalization method, two-dimensional multiple signal classification method, a specific parameterized decoupled root multiple signal classification method, and the Cramér-Rao lower bound were compared and analyzed. 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 were carried out with 10,000 independent trials.

[0141] First, an underdetermined scenario was simulated. The number of subarrays L x of the planar sparse array was set to 4, and the number of subarrays L x of the cylindrical sparse array was set to 3. Δ was set to 2arcsin[λ / (5d r)], such that the distance between two adjacent sensors does not exceed half a wavelength. For a sparse parallel planar linear array, assume 12 signals, where are [5°, 25°, 45°, 65°, 85°, 7°, 27°, 47°, 65°, 75°, 2°, 77°], are [5°, 25°, 45°, 65°, 85°, 40°, 60°, 80°, 20°, 40°, 55°, 72°]. For a sparse parallel cylindrical linear array, assume 9 signals, 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 show the two-dimensional direction-of-arrival estimation results based on 100 independent trials, where the signal-to-noise ratio is 10 dB and the number of sampling snapshots is 3,000,000. From Figure 4 and Figure 5 , it can be seen that the decoupled root multiple signal classification method proposed by the present invention successfully identifies 12 and 9 signals in the sparse parallel planar linear array signal parameter estimation and the 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 subarray and sensor numbers as those in the sparse parallel planar linear array signal parameter estimation and the sparse parallel cylindrical linear array signal parameter estimation configurations, the maximum number of identifiable signals is reduced to 8 and 6, respectively. In addition, in the case of the subspace rotation joint diagonalization direction matrix method, the maximum number of identifiable signals is 12 and 8, respectively. Therefore, the simulation results verify the enhanced effect of the proposed method in signal identification ability.

[0142] Then, for a determined scenario, compare the statistical estimation performances of different methods in terms of signal-to-noise ratio and number of snapshots. Let L x = 4, and the positions of the sensors along the y-axis are set to (0, 1, 4, 6)d. To evaluate the statistical performance of the proposed decoupled root multiple signal classification method under different conditions, two other configurations are also studied, that is, assume L x = 3 and L x = 2, respectively, and their sensor position configurations along the y-axis are (0, 1, 2, 5, 8, 11)d and (0, 1, 2, 3, 7, 11, 15, 19)d, respectively. The total number of sensors in the three cases is similar. The two-dimensional directions-of-arrival of the two signals are (50°, 15°) and (78°, 65°), respectively. Figure 6 and Figure 7The comparison results of the presented statistical estimations show that the performance of the proposed method is superior to that of the subspace rotation joint diagonalization method at the cost of a higher computational burden. The above advantages are particularly obvious under high signal-to-noise ratio and large number of snapshots conditions. In addition, the performance of the decoupled root multiple signal classification method based on multi-parallel sparse linear arrays is superior to that of the decoupled root multiple signal classification method based on multi-parallel uniform linear arrays, which benefits from the increase in the degrees of freedom of the array. Among the three sensor configurations, a larger L x value makes the utilization of the decoupled root multiple signal classification method more efficient on the two-dimensional aperture, resulting in better performance. However, when the signal-to-noise ratio exceeds 6 dB, the decoupled root multiple signal classification method corresponding to a larger L x value shows relatively poor performance, mainly due to the high sensitivity of the proposed method to the covariance matrix reconstruction error. The performance of the two-dimensional multiple signal classification method is slightly better than that of the decoupled root multiple signal classification method under high signal-to-noise ratio and large number of snapshots conditions, but its performance significantly degrades under low signal-to-noise ratio and small number of snapshots, and its threshold characteristic is not as good as that of the decoupled root multiple signal classification method. This phenomenon is attributed to the cooperative effect of the decoupled root multiple signal classification method in aperture utilization and the constraints of the two-dimensional multiple signal classification method in search. In addition, the two-dimensional multiple signal classification method also brings a significant computational burden.

[0143] Finally, the computational complexity of each method is compared by the average CPU running time. The experiments were carried out on a Windows 10 20a machine equipped with 32.0 GB RAM and an Intel(R) Core(TM) i7-10875H 2.30 GHz processor using the MATLAB 2.30a platform. Table 1 summarizes the average calculation time required for each method per trial, where the number of two-dimensional search grids is (0° - 90°) / 0.2° = 450. It is worth noting that among the compared methods, the two-dimensional multiple signal classification shows the highest calculation time, and the computational burden of the proposed method is slightly higher compared with the subspace rotation joint diagonalization method. As can be seen from Table 1, using the decoupled root multiple signal classification based on multi-parallel uniform linear arrays shows the lowest computational complexity. This can be attributed to the fact that it does not require the subarray data fitting process and the dimensionality reduction of the matrices and vectors required in the algorithm calculation process.

[0144] Table 1. Comparison of the average calculation time of different algorithms for different numbers of snapshots

[0145]

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

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

[0148] The specific embodiments described herein are merely illustrative of the spirit of the present invention. Those skilled in the art to which the present invention pertains may make various modifications or supplements to the described specific embodiments or use similar means for substitution, but will not deviate from the spirit of the present invention or exceed the scope defined by the appended claims.

Claims

1. A decoupled two-dimensional direction of arrival estimation method for multi-parallel sparse linear arrays based on recursive root finding, characterized in that It includes the following steps: (1) Modeling of multi-parallel sparse linear arrays: There are M far-field narrowband uncorrelated signals incident on a sparse array composed of L x ≥ 2 parallel sparse linear arrays, and each sub-array contains L y sensor units; the extension direction of the sub-array is parallel to the y-axis; For a sparse array in planar form, each sub-array is uniformly arranged in parallel along the x-axis; for a sparse array in cylindrical form, the projection angles of adjacent sub-arrays in the x-o-z plane are the same. The direction of arrival of each signal is represented by the azimuth angle and the elevation angle ; Denote the angle between the incident signal and the y-axis as Denote the angle between the incident signal and the x-axis as For a sparse array in planar form, the sub-array arranged along the y-axis is taken as the reference sub-array; for a sparse array in cylindrical form, a virtual sparse linear array along the cylindrical center line is constructed as the reference sub-array. Obtain the reference subarray steering vector a 0,m ; Steering vector of sparse array where β m is the steering vector of the subarray along the x-o-z plane; (2) Reconstructing the covariance matrix of multi-parallel sparse linear arrays into the covariance matrix of multi-parallel uniform linear arrays: Covariance matrix R of multi-parallel sparse linear array output xx It is expressed as: where and σ 2 represent the power of the m-th signal and the noise power respectively, and I n represents the n×n identity matrix; the matrix R xx can be block-represented as sub-matrices of dimension L y ×L y ; using the Hermitian property of each sub-matrix, the covariance matrix of the virtual multi-parallel uniform linear array is obtained Among them represents the total number of sensor units of each virtual uniform linear array represents the steering vector of the virtual reconstructed uniform reference subarray (3) Transforming the two-dimensional direction-of-arrival estimation into a polynomial root-finding problem by dimensionality reduction and decoupling: Using the principle of multiple signal classification, we can obtain: where \(P = UU\) H , denotes the corresponding noise subspace; Let where \(z = e\) j2πdcosα / λ , \(\alpha\) represents the angle between the incident signal to be estimated and the y-axis, and can be estimated by the M roots of the following equation: det[D(z -1 )PD(z)] = det(Η) = 0, According to the rank reduction principle, M roots can be calculated according to the following formula: Here, where {a:b} represents {a, a + 1, …, b}, and a and b are integers; and can be rewritten as and (4) Calculating the polynomial coefficients based on a recursive root-finding method: dimensional coefficient vector c p,q , where p, q = 1, 2, …, L x is calculated by the following method: Among them represents the sum of elements along the p,q th diagonal of matrix P; it can be obtained that: Among them Coefficient vector of the d - dimensional polynomial is derived as follows: When L x = 2, c (2) = c 1,1 * c 2,2 - c 2,1 * c 1,2 ; When L x ≥ 3, Expand by the cofactors of the selected column or row and calculate recursively, i.e.: Among them represents the coefficient vector of the cofactor corresponding to the (p,1)-th element of Η; (5) Conducting a closed-form solution based on the Rayleigh-Ritz theorem: Calculate the roots of the polynomial and select the M roots that are inside the unit circle and closest to the unit circle angles of the M signals Estimate them in the following way: α m = arccos[λarg(z m ) / (2πd)], where arg(·) represents the phase angle of the complex number in the parentheses, d represents the minimum sensor interval of each subarray, and λ represents the signal wavelength; according to the principle of multiple signal classification and the expression of β m it can be known from the expression of According to the Rayleigh-Ritz theorem and it can be seen that β m is collinear with the eigenvector b corresponding to the minimum eigenvalue m ; Estimate the angle based on the geometric structure of the sparse array and the eigenvector b m , and estimate the angle Obtain the estimate of m through the relationship between {θ m , φ m} and {α m , β .

2. The method for decoupled two-dimensional direction of arrival estimation of a multi-parallel sparse linear array based on recursive root finding according to claim 1, wherein, In step (1), a method of dynamically adjusting the sub-array configuration is adopted to flexibly configure the number and position of sub-arrays according to specific application scenarios; under the condition that the total number of sensors is fixed, the maximum array degrees of freedom are achieved by increasing the number of sub-arrays or adjusting the sub-array sparsification configuration method.

3. The multi-parallel sparse linear array decoupling two-dimensional direction of arrival estimation method based on recursive root finding according to claim 1, characterized in that, The reference subarray steering vector a 0,m is expressed as: Among them represents the distance between the l-th y sensor and the x-axis.

4. The method for two-dimensional direction of arrival estimation of multi-parallel sparse linear arrays based on recursive root finding and decoupling according to claim 1, wherein For a sparse array in planar form, the corresponding steering vector of the sub-array in the x-o-z plane is: where d x represents the distance between adjacent sub-arrays.

5. The method for decoupling two-dimensional direction of arrival estimation of a multi-parallel sparse linear array based on recursive root finding according to claim 1, wherein For a sparse array in cylindrical form, the corresponding steering vector of the sub-array in the x-o-z plane is: where k m = -(cosβ m , cosα m , cosφ m ) T is the propagation vector associated with the m-th signal, represents the position vector of the l-th r sensor on the cylindrical base located in the x - o - z plane; d r represents the radius of the cylindrical base; Δ represents the projection angle of adjacent sub-arrays in the x - o - z plane, and L x ≤ 2π / Δ.

6. The method for decoupling two-dimensional direction of arrival estimation of a multi-parallel sparse linear array based on recursive root finding according to claim 1, characterized in that In step (5), for a sparse array in planar form, the angle is estimated by the following method: wherein represents the n m -th element of the eigenvector b x .

7. The method for decoupling two-dimensional direction of arrival estimation of a multi-parallel sparse linear array based on recursive root finding according to claim 1, wherein In step (5), for a cylindrical sparse array, the angle is estimated as follows: Based on the geometric structure of the cylindrical array, it is deduced that: where l r is 1, 2, …, L x -1, when l r ≥ 2, there is the following formula:

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