A high-precision two-dimensional DOA estimation method based on optimal redundant linear array

Abstract

The application relates to the technical field of antennas and discloses a high-precision two-dimensional DOA estimation method based on optimal redundant linear arrays, 1) input: array receiving data y i (t), optimal redundant parallel array two-subarray element position coordinates P S1 and P S2 ; 2) parameter calculation ① calculating array autocovariance matrices of two optimal redundant linear arrays according to a formula; ② calculating array cross-covariance matrices between the optimal redundant linear arrays according to a formula. The high-precision two-dimensional DOA estimation method based on optimal redundant linear arrays proposes an extended covariance matrix reconstruction algorithm, the algorithm constructs an extended covariance matrix containing autocovariance matrices and cross-covariance matrices, further constructs an extended covariance matrix reconstruction model, simultaneously realizes high-precision estimation of initial autocovariance matrices and initial cross-covariance matrices, and based on the initial autocovariance matrices and the initial cross-covariance matrices, utilizes a covariance vector sparse representation and a total least squares method in sequence to realize high-precision two-dimensional DOA estimation.

Description

Technical Field

[0001] This invention relates to the field of antenna technology, specifically to a high-precision two-dimensional DOA estimation method based on an optimal redundant linear array. Background Technology

[0002] With the continuous development of radar countermeasure equipment, the single-beam scanning amplitude comparison method is commonly used in the mainstream radar countermeasure reconnaissance equipment currently in service. The accuracy and angular resolution of this method depend on the beamwidth, making it difficult to achieve a high level.

[0003] To change this situation, phase interferometers have been gradually used to replace amplitude comparison method for direction finding in equipment development. This can effectively improve the estimation accuracy, but it cannot distinguish multiple signals at the same time.

[0004] In comparison, the array direction finding system, which is widely adopted in many fields such as radar, communication, and sonar, has the ability to simultaneously find directions from multiple signals, has high direction finding accuracy, and strong anti-interference capabilities.

[0005] However, for drone platforms, the uniform array antennas commonly used in radar are not well adapted to the application scenario of drone platforms and have poor performance.

[0006] Chinese Patent 2021113243853 discloses an array antenna, a sparse rectangular array, and an antenna design method. This patent discloses an optimal redundant parallel array. When using this array, a two-dimensional DOA estimation method is required. However, the traditional two-dimensional DOA estimation has low accuracy. Therefore, a high-precision two-dimensional DOA estimation method based on an optimal redundant linear array is proposed to solve the above-mentioned problems. Summary of the Invention

[0007] (a) Technical problems to be solved

[0008] To address the shortcomings of existing technologies, this invention provides a high-precision two-dimensional DOA estimation method based on an optimal redundant linear array, which has advantages such as high precision and solves the problem of low accuracy.

[0009] (II) Technical Solution

[0010] The technical solution of this invention to solve the above-mentioned technical problems is as follows: A high-precision two-dimensional DOA estimation method based on an optimal redundant linear array, comprising the following steps:

[0011] 1) Input: Array receives data The coordinates of the elements of the two subarrays in the optimal redundant parallel array. and ;

[0012] 2) Parameter Calculation

[0013] ①According to the formula Calculate the array autocovariance matrix of two optimal redundant linear arrays;

[0014] ②According to the formula Calculate the array cross-covariance matrix between optimal redundant linear arrays;

[0015] ③Based on and , Calculate the sparse choice matrix and And then according to the formula structure ;

[0016] 3): Calculate the noiseless extended covariance matrix

[0017] ①Based on formula Constructing a matrix And then according to the formula Calculate the parameter matrix ;

[0018] ② Furthermore, based on the formula Construct parameter matrix ;

[0019] ③According to the formula Calculate the extended covariance matrix ;

[0020] 4): Angle estimate

[0021] ① Grid division A complete dictionary set has been constructed. ;

[0022] ②Based on the extended covariance matrix The initial covariance vector is obtained. ;

[0023] ③Based on and The included angle is calculated using the covariance vector sparse representation algorithm. The estimated value;

[0024] 5): Angle β estimation

[0025] ①Based on extended covariance matrix Japanese style The covariance vector is calculated. ;

[0026] ②Based on the included angle According to the formula calculate And then according to the formula Obtain an estimated value for the included angle β;

[0027] 6): Adjust the included angle The included angle β is combined in the calculated order to obtain a two-dimensional DOA estimate. .

[0028] The beneficial effects of this invention are:

[0029] This high-precision two-dimensional DOA estimation method based on an optimal redundant linear array proposes an extended covariance matrix reconstruction algorithm. This algorithm constructs an extended covariance matrix containing the autocovariance matrix and the cross-covariance matrix, and then builds an extended covariance matrix reconstruction model. A fast calculation formula for the extended covariance matrix is ​​derived, and high-precision estimation of the initial autocovariance matrix and the initial cross-covariance matrix is ​​achieved. Finally, based on the initial autocovariance matrix and the initial cross-covariance matrix, high-precision two-dimensional DOA estimation is achieved by successively using the sparse representation of the covariance vector and the overall least squares method.

[0030] Based on the above technical solution, the present invention can be further improved as follows.

[0031] Furthermore, the optimal redundant parallel array is a type of sparse parallel array. A sparse parallel array consists of two sparse linear arrays placed side-by-side with a spacing of d, and the element positions of the two sparse linear arrays are represented as follows:

[0032] .

[0033] Furthermore, the optimal redundant parallel array is divided into a unidirectional optimal redundant parallel array and a bidirectional optimal redundant parallel array. In a unidirectional optimal redundant parallel array, the two optimal redundant linear arrays are completely identical, while in a bidirectional optimal redundant parallel array, the two optimal redundant linear arrays are symmetrical in structure.

[0034] Furthermore, the spacing between the two parallel subarrays in the unidirectional optimal redundancy parallel array is... The positions of the array elements of the two subarrays are respectively

[0035] .

[0036] Furthermore, in the bidirectional optimal redundancy parallel array, the element positions of the two optimal redundancy linear arrays are respectively...

[0037] .

[0038] Furthermore, the sparse selection matrix of the sparse parallel array Depending on the positions of the elements in the sparse parallel array, it can be directly determined by the sparse selection matrix of the two sparse linear arrays. and Direct composition, as follows:

[0039]

[0040] The sparse selection matrix depends on the positions of the sparse array elements. The In the line, only The corresponding position is 1, and all other positions are 0.

[0041]

[0042] in For the first A unit vector with one element being 1 and the rest being 0. For the initial uniform linear array, the first The position coordinates of each array element For the sparse linear array The position coordinates of each array element.

[0043] Furthermore, the coupling coefficient matrix of the sparse parallel array includes not only the coupling effect between elements within two subarrays, but also the coupling effect between adjacent elements between two subarrays. Let the coupling coefficient matrix of the sparse parallel array be... Assume the coupling coefficient matrix within the two subarrays is and The coupling coefficient matrix between the elements of the two subarrays is Then the coupling coefficient matrix of the sparse parallel array can be expressed as:

[0044]

[0045] Unlike sparse linear arrays, sparse parallel arrays are difference virtual arrays. It is a set of two-dimensional coordinates, specifically represented as

[0046] Attached Figure Description

[0047] Figure 1 These are schematic diagrams of four simplified sparse parallel array structures of the present invention;

[0048] Figure 2 These are schematic diagrams of three existing sparse parallel array structures of the present invention;

[0049] Figure 3 This is a diagram showing the correspondence between the sparse selection matrix and the distribution of elements in the sparse parallel array of this invention.

[0050] Figure 4This is a schematic diagram of the difference virtual array of the sparse parallel array of the present invention;

[0051] Figure 5 This is a diagram of the unidirectional optimal redundancy parallel array and the bidirectional optimal redundancy parallel array of the present invention;

[0052] Figure 6 This is a distribution diagram of the two-dimensional difference virtual array of the present invention. Detailed Implementation

[0053] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0054] In this embodiment, a high-precision two-dimensional DOA estimation method based on an optimal redundant linear array includes the following steps:

[0055] 1) Input: Array receives data The coordinates of the elements of the two subarrays in the optimal redundant parallel array. and ;

[0056] 2) Parameter Calculation

[0057] ①According to the formula Calculate the array autocovariance matrix of two optimal redundant linear arrays;

[0058] ②According to the formula Calculate the array cross-covariance matrix between optimal redundant linear arrays;

[0059] ③Based on and , Calculate the sparse choice matrix and And then according to the formula structure ;

[0060] 3): Calculate the noiseless extended covariance matrix

[0061] ①Based on formula Constructing a matrix And then according to the formula Calculate the parameter matrix ;

[0062] ② Furthermore, based on the formula Construct parameter matrix ;

[0063] ③According to the formula Calculate the extended covariance matrix ;

[0064] 4): Angle estimate

[0065] ① Grid division A complete dictionary set has been constructed. ;

[0066] ②Based on the extended covariance matrix The initial covariance vector is obtained. ;

[0067] ③Based on and The included angle is calculated using the covariance vector sparse representation algorithm. The estimated value;

[0068] 5): Angle β estimation

[0069] ①Based on extended covariance matrix Japanese style The covariance vector is calculated. ;

[0070] ②Based on the included angle According to the formula calculate And then according to the formula Obtain an estimated value for the included angle β;

[0071] 6): Adjust the included angle The included angle β is combined in the calculated order to obtain a two-dimensional DOA estimate. .

[0072] Since the direction finding accuracy mainly depends on the array aperture size, and is limited by the characteristics of the parallel array structure, the estimation accuracy of the included angle in the two-dimensional DOA estimation is much lower than that of the included angle. Therefore, the two-dimensional DOA estimation accuracy mainly depends on the estimation accuracy of the included angle, and the estimation accuracy of the cross-covariance matrix determines the included angle. Therefore, how to improve the estimation accuracy of the cross-covariance matrix has become the key to improving the two-dimensional DOA estimation accuracy of the optimal redundant parallel array.

[0073] Since the eigenvalues ​​of the cross-covariance matrix are complex numbers and do not have positive semi-definite properties, the cross-covariance matrix can be directly reconstructed and estimated. In order to improve the estimation accuracy of the cross-covariance matrix, this section constructs a noiseless extended covariance matrix containing the initial cross-covariance matrix. The eigenvalues ​​of this noiseless extended covariance matrix are the signal power and the noise power, so it satisfies the positive semi-definite constraint and can be reconstructed and estimated.

[0074] Based on the self-covariance matrix and cross-covariance matrix of two subarrays in the initial uniform parallel array, the following extended covariance matrix can be constructed, which is expressed as:

[0075]

[0076] Among them, matrix It is a noiseless extended covariance matrix.

[0077] Based on the Toeplitz property of the autocovariance matrix and crosscovariance matrix of a uniform linear array, the noiseless extended covariance matrix can be expressed as:

[0078]

[0079] Similarly, a covariance matrix can be constructed based on the autocovariance matrix and crosscovariance matrix of two optimal redundant linear arrays. Its relationship with the noiseless extended covariance matrix is ​​as follows:

[0080]

[0081] Unlike the cross-covariance matrix of the two submatrices, the noiseless extended covariance matrix is ​​a positive semi-definite matrix, so it can be reconstructed and estimated using a norm minimization model.

[0082] Based on the optimal solution conditions in convex optimization, the analytical expression for the noiseless extended covariance matrix estimate was obtained, thus realizing fast and high-precision cross-covariance matrix estimation. The specific derivation process is shown below.

[0083] This section transforms the problem of solving the noiseless extended covariance matrix into the following optimization problem.

[0084]

[0085] in,

[0086]

[0087] Then, using the trace norm for convex relaxation, it is transformed into

[0088]

[0089] The aforementioned convex optimization model based on trace norm minimization can be solved directly using the CVX toolbox, but the computational cost is very high. The computation time for a single DOA estimation based on an optimal redundant parallel array of 20 elements exceeds 180 seconds. To reduce the computational complexity of matrix reconstruction, the matrix analytical calculation formula is derived by solving the KKT optimal solution condition equation, which greatly improves the computational efficiency of matrix reconstruction.

[0090] First, by introducing Lagrange multipliers into the error constraint inequality, the trace norm minimization model described above can be transformed into a LASSO model, i.e.

[0091]

[0092] Furthermore, the formula can be Transform into

[0093]

[0094] Based on the KKT conditional optimality theory in convex optimization, the optimal solution to the above problem satisfies the equation...

[0095]

[0096] Block Toeplitz (Toeplitz matrix) compression It is a matrix Compressed into vector C (6M-3)×1 The operation.

[0097] definition ,Right now

[0098]

[0099] The formula can then be simplified to

[0100]

[0101] in It can be further simplified to

[0102]

[0103] in Toeplitz compression It is a matrix structure based on the Toeplitz matrix. Compressed into vector C (2M-1)×1 The operation.

[0104] Furthermore, it can be Simplified to

[0105]

[0106] Where vector ,vector , , parameter matrix .

[0107] Parameter matrix and The calculation formula is:

[0108]

[0109] in, Represented by matrix set The new matrix formed by the corresponding columns, Represented by matrix Remove sets The new matrix formed by the corresponding rows and the remaining rows, set , , ,matrix .

[0110] Based on The formula can be Transform into

[0111] For ease of calculation, it is transformed into the following system of real linear equations.

[0112]

[0113] in, Represents a complex vector The real part, Represents a complex vector The imaginary part.

[0114] Due to the matrix Since it is a non-singular matrix, its pseudo-inverse can be directly calculated, yielding the equation.

[0115]

[0116] You can get Based on the formula It can be obtained through estimation The corresponding initial autocovariance matrix and initial cross-covariance matrix for denoising are extracted respectively, and then two-dimensional DOA estimation is performed using the covariance vector sparse representation algorithm and the overall least squares method.

[0117] The two-dimensional DOA estimation problem of optimal redundant parallel arrays can be transformed into two one-dimensional DOA estimation problems, as follows.

[0118] First, the included angle is determined based on the sparse representation of the initial autocovariance vector. The estimate.

[0119] In the extended covariance matrix reconstruction algorithm, the estimates of the two self-covariance matrices are completely identical, therefore only one sparse reconstruction is needed. The initial self-covariance matrix vector is constructed below. .vector elements in It can be represented as

[0120]

[0121] Based on the obtained vector An initial autocovariance vector can be constructed. for

[0122]

[0123] By discretizing the spatial domain into a grid, the initial autocovariance vector can be sparsely represented.

[0124]

[0125] Where the estimation error vector The emergence of overcomplete dictionaries stems from the finite number of signal samples. From vector Composition, without considering mesh mismatch, direction vector With supercomplete dictionary Then the relation is satisfied. Considering The non-convexity of the norm makes optimization computation difficult, therefore the model selected... Norm as the objective function. Threshold. This indicates the allowable range of error between the actual value and the theoretical value of the model.

[0126]

[0127] in, .

[0128] Secondly, the included angle is determined based on the initial cross-covariance matrix. The estimate.

[0129] R is obtained based on the extended covariance matrix. xx The included angle can be obtained. .

[0130] Preferably, the optimal redundant parallel array is a type of sparse parallel array. The sparse parallel array consists of two sparse linear arrays placed side-by-side with a spacing of d, and the element positions of the two sparse linear arrays are represented as follows:

[0131] .

[0132] based on Figure 6 The array evaluation metrics in this section can be used to compare and analyze the performance of existing coprime parallel arrays, generalized coprime parallel arrays, nested parallel arrays, and the unidirectional optimal redundant parallel arrays and bidirectional optimal redundant parallel arrays proposed in this section.

[0133] Performance Comparison Table of Sparse Parallel Arrays

[0134] Array degrees of freedom Array scalability Array sparsity 1st order array coupling Coprime parallel array 19 41.2% 2.6 14 Generalized coprime parallel array 21 68.6% 2.6 14 Nested parallel arrays 29 100% 3.0 18 One-way optimal redundant parallel matrix 35 100% 3.6 12 Bidirectional optimal redundant parallel matrix 35 83% 3.6 4

[0135] In summary, we can conclude that:

[0136] 1. Under the same physical array element number, the coprime parallel array has the smallest array degree of freedom, while the two optimal redundant parallel arrays have the largest array degree of freedom.

[0137] 2. The scalability of coprime parallel arrays, generalized coprime parallel arrays, and bidirectional minimum redundancy parallel arrays is less than 100%, and they belong to partially scalable arrays.

[0138] 3. Coprime parallel arrays and generalized coprime parallel arrays have the lowest array sparsity, while the two optimal redundant parallel arrays have the highest array sparsity.

[0139] 4. Nested parallel arrays have the largest array coupling effect. Compared with coprime parallel arrays and generalized coprime parallel arrays, unidirectional optimal redundant parallel arrays have a smaller array coupling effect, while bidirectional optimal redundant parallel arrays have the smallest array coupling degree.

[0140] Preferably, the optimal redundancy parallel array is divided into a unidirectional optimal redundancy parallel array and a bidirectional optimal redundancy parallel array. In the unidirectional optimal redundancy parallel array, the two optimal redundancy linear arrays are completely identical, while in the bidirectional optimal redundancy parallel array, the two optimal redundancy linear arrays are symmetrical in structure.

[0141] Preferably, the spacing between the two parallel subarrays in the unidirectional optimal redundancy parallel array is . The positions of the array elements of the two subarrays are respectively

[0142]

[0143] pass Figure 5 In a unidirectional optimal redundancy parallel array, the two optimal redundancy linear arrays are the same. The dashed circle represents the number of array element pairs with an element spacing of d between the two optimal redundancy linear arrays, which can reflect the magnitude of the element coupling effect between the two optimal redundancy linear arrays.

[0144] Preferably, in the bidirectional optimal redundancy parallel array, the element positions of the two optimal redundancy linear arrays are respectively...

[0145] ;

[0146] Preferably, the sparse selection matrix of the sparse parallel array Depending on the positions of the elements in the sparse parallel array, it can be directly determined by the sparse selection matrix of the two sparse linear arrays. and Direct composition, as follows:

[0147]

[0148] The sparse selection matrix depends on the positions of the sparse array elements. The In the line, only The corresponding position is 1, and all other positions are 0.

[0149]

[0150] in For the first A unit vector with one element being 1 and the rest being 0. For the initial uniform linear array, the first The position coordinates of each array element For the sparse linear array The position coordinates of each array element, assuming Then the sparse choice matrix is ​​as follows Figure 3 As shown;

[0151] Preferably, the coupling coefficient matrix of the sparse parallel array includes not only the coupling effect between elements within two subarrays, but also the coupling effect between adjacent elements between two subarrays. Let the coupling coefficient matrix of the sparse parallel array be... Assume the coupling coefficient matrix within the two subarrays is and The coupling coefficient matrix between the elements of the two subarrays is Then the coupling coefficient matrix of the sparse parallel array can be expressed as:

[0152]

[0153] Unlike sparse linear arrays, sparse parallel arrays are difference virtual arrays. It is a set of two-dimensional coordinates, specifically represented as

[0154]

[0155] pass Figure 4 As shown, when Then its difference virtual array is ,in That is, a continuous, void-free, three-parallel uniform array with a aperture of 6d.

[0156] Working principle:

[0157] This high-precision two-dimensional DOA estimation method based on an optimal redundant linear array proposes an extended covariance matrix reconstruction algorithm. This algorithm constructs an extended covariance matrix containing the autocovariance matrix and the cross-covariance matrix, and then builds an extended covariance matrix reconstruction model. A fast calculation formula for the extended covariance matrix is ​​derived, and high-precision estimation of the initial autocovariance matrix and the initial cross-covariance matrix is ​​achieved. Finally, based on the initial autocovariance matrix and the initial cross-covariance matrix, high-precision two-dimensional DOA estimation is achieved by successively using the sparse representation of the covariance vector and the overall least squares method.

[0158] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0159] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A high-precision two-dimensional DOA estimation method based on an optimal redundant linear array, comprising the following steps: 1) Input: Array receives data The coordinates of the elements of the two subarrays in the optimal redundant parallel array. and ; 2) Parameter Calculation ①According to the formula Calculate the array autocovariance matrix of two optimal redundant linear arrays; ②According to the formula Calculate the array cross-covariance matrix between optimal redundant linear arrays; ③Based on and , Calculate the sparse choice matrix and And then according to the formula structure ; in For the first A unit vector with one element being 1 and the rest being 0. For the initial uniform linear array, the first The position coordinates of each array element For the sparse linear array The position coordinates of each array element; 3): Calculate the noiseless extended covariance matrix , ①Based on formula Constructing a matrix And then according to the formula Calculate the parameter matrix ; ② Furthermore, based on the formula Construct parameter matrix ; ③According to the formula Calculate the extended covariance matrix ; 4): Angle estimate ① Grid division A complete dictionary set has been constructed. ; ②Based on the extended covariance matrix The initial covariance vector is obtained. ; ③Based on and The included angle is calculated using the covariance vector sparse representation algorithm. The estimated value; 5): Angle estimate ①Based on extended covariance matrix Japanese style The covariance vector is calculated. ; ②Based on the included angle According to the formula calculate And then according to the formula Obtain the included angle The estimated value; 6): Adjust the included angle and included angle Combine the calculated values ​​in the correct order to obtain a two-dimensional DOA estimate. .

2. The high-precision two-dimensional DOA estimation method based on an optimal redundant linear array according to claim 1, characterized in that: The optimal redundant parallel array is a type of sparse parallel array. A sparse parallel array consists of two sparse linear arrays placed side-by-side with a spacing of d, and the element positions of the two sparse linear arrays are represented as follows: 。 3. The high-precision two-dimensional DOA estimation method based on an optimal redundant linear array according to claim 1, characterized in that: The optimal redundancy parallel array is divided into unidirectional optimal redundancy parallel array and bidirectional optimal redundancy parallel array. In the unidirectional optimal redundancy parallel array, the two optimal redundancy linear arrays are completely identical, while in the bidirectional optimal redundancy parallel array, the two optimal redundancy linear arrays are symmetrical.

4. The high-precision two-dimensional DOA estimation method based on an optimal redundant linear array according to claim 3, characterized in that: The spacing between the two parallel subarrays in the unidirectional optimal redundancy parallel array is: The positions of the array elements of the two subarrays are respectively 。 5. The high-precision two-dimensional DOA estimation method based on an optimal redundant linear array according to claim 3, characterized in that: In the bidirectional optimal redundancy parallel array, the element positions of the two optimal redundancy linear arrays are respectively... 。 6. The high-precision two-dimensional DOA estimation method based on an optimal redundant linear array according to claim 2, characterized in that: The coupling coefficient matrix of the sparse parallel array includes not only the coupling effect between elements within two subarrays, but also the coupling effect between adjacent elements between two subarrays. Let the coupling coefficient matrix of the sparse parallel array be... Assume the coupling coefficient matrix within the two subarrays is and The coupling coefficient matrix between the elements of the two subarrays is Then the coupling coefficient matrix of the sparse parallel array can be expressed as: ， Unlike sparse linear arrays, sparse parallel arrays are difference virtual arrays. It is a set of two-dimensional coordinates, specifically represented as 。

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