A Robust Adaptive Beamforming Method Based on Coprime Array Structure

Abstract

The present invention discloses a robust adaptive beamforming method based on a co-prime array structure. First, a co-prime array is constructed to calculate the sample covariance matrix; then, after vectorization, redundancy removal, element rearrangement, and interpolation, a virtual covariance matrix with holes is correspondingly obtained, and then the completed virtual covariance matrix is obtained; the incident angle of the desired signal is obtained by searching for the spectral peak in the desired signal angular sector, and the zero points of the constructed indicator function are solved to obtain the orthogonal basis; a projection matrix is constructed using the orthogonal basis to project the completed virtual covariance matrix, and the interference and noise covariance matrix of the co-prime array is correspondingly obtained, and the inverse of the interference and noise covariance matrix is approximately calculated; the spatial subspace where the desired signal is located is solved to obtain the estimated steering vector of the desired signal; the weight vector of the beamformer is calculated using the inverse of the interference and noise covariance matrix and the estimated steering vector of the desired signal, and a beamformer with robust adaptability is obtained.

Description

Technical Field

[0001] The present invention relates to the field of beamforming research in the field of array signal processing. Robust adaptive beamforming is carried out by using a non-uniform co-prime array structure. Especially in non-ideal cases with different matching errors, after processing, a more accurate steering vector of the desired signal estimation and the inverse of the interference plus noise covariance matrix can be obtained for adaptive beamforming, which can better improve the robustness of the beamformer under various errors. Background Art

[0002] Adaptive beamforming is a data-driven beamforming technique. It can adjust the beamforming weight vector according to the received multi-channel signals, so that the signal-to-interference-plus-noise ratio of the output signal is greatly improved compared with the original received signal, realizing spatial domain filtering. However, practice has proved that when there is an error in the desired signal steering vector and the sampling covariance matrix contains the desired signal, the beamformer will suppress the desired signal as interference, that is, the "self-cancellation" phenomenon, resulting in a decrease in the robustness of the beamformer.

[0003] For the traditional method using a uniform linear array, some scholars have proposed a method of spatial power spectrum sampling. By using the selectivity of the steering vector, a similar sampling operation (which can also be regarded as a projection) is performed on the sampling covariance matrix to obtain the interference plus noise covariance matrix. This can avoid searching for the power spectrum peak of the interference power in the angle domain. In other words, only knowing the angle region of the desired signal can estimate the interference plus noise covariance matrix, and it avoids the high computational complexity brought by the traditional integral reconstruction algorithm. However, this type of method can obtain better performance only when the number of samples is large, that is, when the number of array elements is large enough.

[0004] In order to improve the actual performance of the beamformer, research on the array structure is also constantly underway. The traditional uniform linear array can only expand the degrees of freedom and obtain a larger array aperture by increasing the number of array elements, while the co-prime array can be realized with fewer array elements through array difference. And due to the sparse arrangement of the array elements in the co-prime array, the element spacing becomes larger, making the mutual coupling effect between the elements lower. Benefiting from the estimation of the direction of arrival of the co-prime array, its research on adaptive beamforming has gradually developed. However, most algorithms rely on the continuous part of the virtual array elements to process the received signals, while ignoring the part with holes. For this, in recent years, the application of the covariance matrix completion algorithm based on interpolation technology in the direction of arrival estimation can make full use of the data of the virtual array elements, thereby improving the estimation accuracy. Similar to the direction of arrival estimation, in beamforming, higher-dimensional information can also be obtained by using the completed array structure to improve the performance of the beamformer.

[0005] Based on the above analysis, it is possible to consider applying the method of spatial power spectrum sampling to the co-prime array to study a new robust method to improve the robustness of the beamformer. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to overcome the deficiencies of the prior art and provide a robust adaptive beamforming method based on the co-prime array structure. By using the matrix completion technology, the holes in the virtual array elements corresponding to the co-prime array are interpolated and filled to obtain a completed covariance matrix corresponding to an array structure with more virtual array elements. Then, by using the information in a higher dimension and performing spatial sampling operations, a high-dimensional interference plus noise covariance matrix is obtained, which contains all the information of the interference plus noise covariance matrix that can be obtained from the real physical array elements. In addition, the steering vector of the desired signal is corrected to obtain an estimated steering vector of the desired signal, thereby realizing the robustness of the beamformer under different mismatch errors and further improving the performance of the beamformer.

[0007] The object of the present invention is achieved by the following technical solutions: A robust adaptive beamforming method based on the co-prime array structure, including the following contents:

[0008] Step 1: Select a pair of co-prime numbers to construct a co-prime array to receive the incident signal, and obtain the sampling covariance matrix of the incident signal;

[0009] Step 2: Vectorize, remove redundancy, and rearrange the elements of the sampling covariance matrix obtained in Step 1 to obtain a virtual received data vector; Insert zero values into the virtual received data vector to obtain a virtual covariance matrix with holes; Complete the virtual covariance matrix with holes to obtain a completed virtual covariance matrix corresponding to the completed virtual array;

[0010] Step 3: Search for the incident angle of the desired signal at the spectral peak in the desired signal angular sector; Obtain the virtual steering vector of the desired signal according to the structure of the completed virtual array, and construct an indicator function; Solve the zero point of the indicator function to obtain a set of spatial orthogonal bases;

[0011] Step 4: Use the set of spatial orthogonal bases obtained in Step 3 to construct a sampling matrix, project the completed virtual covariance matrix obtained in Step 2 to obtain a virtual interference plus noise covariance matrix; Further obtain the interference plus noise covariance matrix of the co-prime array; Calculate the inverse of the interference noise covariance matrix;

[0012] Step 5: Use the desired signal angular sector and the sampling covariance matrix based on Step 1 to obtain the subspace where the desired signal is located, and use the incident angle of the desired signal in Step 3 to obtain an estimated steering vector of the desired signal;

[0013] Step 6: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference plus noise covariance matrix obtained in Step 4, calculate the weight vector of the beamformer and perform robust beamforming on the array received data.

[0014] Further, the specific implementation of Step 1 is as follows:

[0015] Step 11: Select a pair of relatively prime numbers M and N, where M < N. The co-prime array A1 consists of two sparse uniform sub-arrays A11 and A12. The sub-array A11 is a sparse uniform linear array composed of 2M elements with an inter-element spacing of Nd; the sub-array A12 is a sparse uniform linear array composed of N elements with an inter-element spacing of Md, where d is the half-wavelength. Except for the common reference element, there are no overlapping elements between the sub-arrays A11 and A12. The co-prime array A1 has a total of M′ = 2M + N - 1 elements, and the element positions are denoted as: r i d, r i is the following set of integers elements of:

[0016]

[0017] If there are L + 1 < M′ far-field narrowband uncorrelated signal sources including a desired signal and L interferences incident on the co-prime array A1, then the sampling covariance matrix of the incident signals is calculated as:

[0018]

[0019] where x(k) represents the data vector received by the co-prime array A1 at time k, K represents the number of snapshots, and H represents the conjugate transpose operator.

[0020] Further, the specific implementation of Step 2 is as follows:

[0021] Step 21: Vectorize the sampling covariance matrix to obtain the following vector:

[0022]

[0023] where vec(·) represents the vectorization operation operator. Then, perform redundancy removal and element reordering on the vector y to obtain the virtual received data vector y of the virtual difference array A2 derived from the co-prime array A1 v , as follows:

[0024]

[0025] where q represents the index value of the element position of the virtual difference array A2, that is is the set of positions where the elements of the virtual difference array A2 are located; the set Denote the set of pairs with the difference of array element position indices being q, i.e., |·| represents the number of elements in the set; Denote the value at the position where the difference of array element position indices is n1 - n2; <y v > q Denote the value at the index position q of y v i.e., <y v > q Arrange in ascending order according to the array element position indices to obtain the virtual received data vector of the virtual difference array A2

[0026] Step 22, Fill all the data between the elements with discontinuous array element position indices in the vector y v with 0 to obtain a higher-dimensional vector i.e.:

[0027]

[0028] where is regarded as the maximum virtual received data vector of the maximum virtual array A3, p represents the index value of the array element position of the maximum virtual array A3, and the integer set is the set of positions where the array elements of the maximum virtual array A3 are located, V = 2MN - N + 1; Using has a Hermitian structure, i.e., to obtain the following Toeplitz matrix:

[0029]

[0030] Denote as the virtual covariance matrix with holes;

[0031] Step 23, Denote as a binary matrix, where the positions of the 1 elements correspond to the positions of the non-zero elements in; the positions of the 0 elements correspond to the positions of the zero elements in, and complete the virtual covariance matrix with holes to obtain the completed matrix as

[0032]

[0033]

[0034] where represents the Hadamard product, The operation of finding the Frobenius norm of a matrix is denoted as, the operation of finding the trace of a matrix is denoted as Tr(·), μ is the regularization parameter, and ≥ represents the positive semi - definite symbol; the maximum virtual array A3 forms a complemented virtual array A4 with the total number of array elements being V from the virtual array elements at positions 0 to V - 1. The obtained matrix is regarded as the received data covariance matrix of the complemented virtual array A4 and is called the complemented virtual covariance matrix.

[0035] Furthermore, the specific implementation of step 3 is as follows:

[0036] Step 31: Construct a Capon spectrum in the desired signal angular sector Θ:

[0037]

[0038] where a v (θ) represents the virtual steering vector of the complemented virtual array A4 corresponding to the angle θ, and a v (θ)=[1, e -jπsinθ , …, e -jπ(V-1)sinθ T . Conduct a Capon spectrum peak search to obtain the angle estimation value of the desired signal Thus, the virtual steering vector of the desired signal is obtained

[0039] Step 32: Construct the following indicator function:

[0040]

[0041] Let Then the indicator function is further expressed as:

[0042]

[0043] where is the angle estimation value of the desired signal, and V is the number of array elements of the complemented virtual array A4.

[0044] Step 33: f(g) has a total of V - 1 zeros, which are integers in the set except 0, denoted as g i , i = 1, 2, …, V - 1. Find the corresponding θ i for g i as:

[0045]

[0046] According to the obtained and the structure of the complemented virtual array A4, obtain the corresponding virtual steering vector a v (θ i ​), i = 1, 2, …, V - 1, and then obtain A set of spatially orthogonal bases in space

[0047] Furthermore, the specific implementation of step 4 is as follows:

[0048] Step 41: According to the set of spatially orthogonal bases obtained in step 3 Construct the sampling matrix as:

[0049]

[0050] where Θ represents the desired signal angular sector, is the complement of Θ, and then sample the completed virtual covariance matrix obtained in step 2 in the non-signal angular sector to obtain the virtual interference plus noise covariance matrix as:

[0051]

[0052] Step 42: The completed virtual array A4 has consecutive array elements from element 0 to element V - 1 = 2MN - N, including all the array elements of the true co-prime array A1. Using the obtained virtual interference plus noise covariance matrix correspondingly express it in the interference plus noise covariance matrix of the co-prime array A1 as:

[0053]

[0054] where,

[0055] Step 43: Perform eigenvalue decomposition on the interference plus noise covariance matrix as:

[0056]

[0057] where, The two diagonal matrices contain the first L large eigenvalues and the remaining M' - L small eigenvalues of, and U1 and U2 are matrices composed of the corresponding eigenvectors; Calculate the inverse of the reconstructed interference plus noise covariance matrix as:

[0058]

[0059] where, represents the estimated noise power.

[0060] Furthermore, the specific implementation of step 5 is as follows:

[0061] Step 51: Perform the following discrete summation operation in the angular sector Θ of the desired signal to obtain the summation matrix C:

[0062]

[0063] where a(θ j ) represents the steering vector of the co-prime array A1 corresponding to the angle θ j , that is J represents the number of sampling points in the desired signal region Θ; perform eigen-decomposition on the matrix C to obtain the eigenvector c corresponding to the largest eigenvalue s ;

[0064] Step 52: Perform eigen-decomposition on the sampling covariance matrix obtained in Step 1, which is:

[0065]

[0066] where the two diagonal matrices contain the first L + 1 eigenvalues arranged in descending order and the remaining M' - L - 1 eigenvalues, and U s , U n are matrices composed of the corresponding eigenvectors, and the space where the desired signal is located is Span{E s} = Span{[U s , c s}, Span{·} represents the column space spanned by the matrix, and E s = [U s , c s represents the matrix formed by splicing U s and c s ;

[0067] Step 53: Use the angle estimate value of the desired signal to obtain the steering vector of the desired signal, and then perform correction to obtain the estimated steering vector

[0068]

[0069] Furthermore, Step 6 includes the following steps:

[0070] Step 61: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference plus noise covariance matrix obtained in Step 4, calculate the weight vector according to the following formula:

[0071]

[0072] where is the inverse of the interference plus noise covariance matrix calculated in step 4, is the estimated steering vector obtained in step 5, and w is obtained as the weight vector of the beamformer;

[0073] Step 62: Weight the weight vector w with the data x(k) received by the co-prime array A1 at time k to obtain the output signal z(k) = w H x(k), realizing robust adaptive beamforming based on the co-prime array structure.

[0074] The advantages of the present invention compared with the prior art are as follows:

[0075] (1) The present invention uses matrix completion technology to interpolate and fill the holes in the virtual array elements corresponding to the co-prime array to obtain a completed covariance matrix corresponding to an array structure with more virtual array elements, and then uses the information of higher dimensions to perform spatial sampling operations to obtain a high-dimensional interference plus noise covariance matrix, which contains all the information of the interference plus noise covariance matrix that can be obtained by the real physical array elements. In addition, the steering vector of the desired signal is corrected to obtain the estimated steering vector of the desired signal, thereby realizing the robustness of the beamformer under different mismatch errors and further improving the performance of the beamformer.

[0076] (2) The difference between the present invention and the prior art lies in: adopting a co-prime array structure, expanding the aperture of the array, having higher degrees of freedom and accuracy; different estimation methods for the interference plus noise covariance matrix, the present invention uses the method of spatial spectrum sampling, without performing spectral peak search for interference, reducing the computational complexity; correcting the steering vector of the desired signal, having stronger robustness in the case of low signal-to-noise ratio.

[0077] (3) The present invention uses the array structure of the improved co-prime array to interpolate and fill the holes in the discontinuous parts that appear in the corresponding virtual array elements to obtain a completed covariance matrix corresponding to an array structure with more virtual array elements, and uses the array structure with higher degrees of freedom to estimate the incident angle of the desired signal, having higher accuracy.

[0078] (4) For the array structure with higher degrees of freedom, a selection function with more zeros can be constructed to obtain an orthogonal basis in a higher-dimensional complex space, thereby constructing a projection matrix to obtain a high-dimensional interference plus noise covariance matrix, which contains all the information of the interference plus noise covariance matrix that can be obtained by the real physical array elements, avoiding spectral peak search in the interference region, not requiring prior information of interference, and reducing the high computational complexity brought by spectral peak search.

[0079] (5) In addition, the present invention corrects the steering vector of the desired signal. Since in the case of low signal-to-noise ratio, the signal subspace obtained by sample covariance eigenvalue decomposition may not contain the component of the desired signal. Therefore, a space where the desired signal is located is constructed to obtain the estimated steering vector of the desired signal. The estimated steering vector can enable the beamformer to cope with various mismatch errors and further enhance the robustness of the beamformer. Description of the Drawings

[0080] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0081] Figure 1 Flowchart of a robust adaptive beamforming method based on a co-prime array structure of the present invention;

[0082] Figure 2 Co-prime array structure provided by an embodiment of the present invention;

[0083] Figure 3 Curve of the output signal-to-interference-plus-noise ratio varying with the input signal-to-noise ratio of the beamformer provided by an embodiment of the present invention under the condition of direction-of-arrival error;

[0084] Figure 4 Curve of the output signal-to-interference-plus-noise ratio varying with the input signal-to-noise ratio of the beamformer provided by an embodiment of the present invention under the condition of array element position error. Detailed Embodiments

[0085] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. The proportional relationship of the number of components may change during actual implementation. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the protection scope of the present invention.

[0086] As Figure 1As shown in the figure, an embodiment of the present invention provides a robust adaptive beamforming method based on a co-prime array structure, constructs a co-prime array to calculate the sampling covariance matrix, performs vectorization, redundancy removal, element rearrangement, and interpolation to obtain a virtual covariance matrix with holes, and thus obtains a completed virtual covariance matrix. Search for the incident angle of the desired signal at the spectral peak in the desired signal angle sector, solve the zero points of the constructed indicator function to obtain the orthogonal basis, project the completed virtual covariance matrix, and correspondingly obtain the interference noise covariance matrix of the physical array, and approximately calculate the inverse of the matrix. Solve the space where the desired signal is located to obtain the estimated steering vector of the desired signal. Calculate the weight vector of the beamformer to achieve beamforming for any matching error.

[0087] The method provided by the present invention is applicable to an improved co-prime array configuration. For the convenience of description, it is assumed that L + 1 far-field narrowband uncorrelated signal sources are received, including one desired signal and L interferences. Then, the data received by the array at time k can be expressed as:

[0088]

[0089] where s0(k) represents the waveform of the desired signal at time k, a0 represents the steering vector of the desired signal, s l (k), l = 1, 2,, L represents the waveform of the l-th interference source at time k, a l represents the steering vector of the l-th interference source, and x n (k) represents the noise received by the array at time k. It is assumed that the means of the desired signal, interference, and noise are all zero.

[0090] It mainly includes the following steps:

[0091] Step 1: Select a pair of co-prime numbers to construct a co-prime array to receive the incident signal, and obtain the sampling covariance matrix of the incident signal.

[0092] The said Step 1 includes the following steps:

[0093] Step 11: Select a pair of co-prime numbers M = 3 and N = 5. The co-prime array A1 is composed of two sparse uniform sub-arrays. The sub-array A11: a sparse uniform linear array composed of 2M = 6 array elements with a spacing of Nd = 5d; the sub-array A12: a sparse uniform linear array composed of N array elements with a spacing of Md = 3d, where d is the half-wavelength. Except for the common reference array element, there are no overlapping array elements between the sub-arrays A11 and A12. The co-prime array A1 has a total of M' = 2M + N - 1 = 10 array elements, and the array element positions can be recorded as: r i d, r i is the following set of integers of elements:

[0094]

[0095] There are L + 1 = 3 far - field narrow - band uncorrelated signal sources, including one desired signal and 2 interferences, incident on array A1. Then, the sampling covariance matrix of the received data can be calculated as follows:

[0096]

[0097] where \(x(k)\) represents the data vector received by the co - prime array A1 at time k, and \(K = 300\) represents the number of snapshots, H represents the conjugate transpose operator.

[0098] Step 2: Vectorize, remove redundancy, and re - arrange the elements of the sampling covariance matrix obtained in Step 1 to obtain a virtual received data vector; insert zero values into the virtual received data vector to obtain a virtual covariance matrix with holes; complete the virtual covariance matrix with holes to obtain a completed virtual covariance matrix corresponding to the completed virtual array.

[0099] The said Step 2 includes the following steps:

[0100] Step 21: Vectorize the sampling covariance matrix to obtain the following vector:

[0101]

[0102] where \(vec(\cdot)\) represents the vectorization operation operator. Then, remove redundancy and re - arrange the elements of the vector \(y\) to obtain the virtual received data vector \(y\) of the virtual difference array A2 derived from the co - prime array A1 v , as follows:

[0103]

[0104] where \(q\) represents the index value of the element position of the virtual difference array A2, that is is the set of positions where the elements of the virtual difference array A2 are located; the set represents the set of pairs with the difference of element position index values being \(q\), that is \(\vert\cdot\vert\) represents the number of elements in the set; represents the value at the position where the difference of the corresponding element position index values is \(n1 - n2\); \(\lt y v \gt q represents the value of \(y\) at the index position \(q\), that is, \(y v values, that is, \(y vq arranged in ascending order of element position index values to obtain the virtual received data vector of the virtual difference array A2

[0105] Step 22: For the vector \(y\)v All the data between the elements with discontinuous middle element position indices are filled with 0 to obtain a higher-dimensional vector That is:

[0106]

[0107] Among them, can be regarded as the maximum virtual received data vector of the maximum virtual array A3. p represents the index value of the element position of the maximum virtual array A3, and the integer set is the set of positions where the elements of the maximum virtual array A3 are located, and V = 2MN - N + 1 = 26. Using has a Hermitian structure, that is the following Toeplitz matrix is obtained:

[0108]

[0109] It is called the virtual covariance matrix with holes.

[0110] Step 23, denote as a binary matrix, where the positions of the 1 elements correspond to the positions of the non-zero elements in; the positions of the 0 elements correspond to the positions of the zero elements in. Solving the following problem can complete the virtual covariance matrix with holes to obtain the completed matrix as

[0111]

[0112]

[0113] Among them, represents the Hadamard product, represents the operation of finding the Frobenius norm of the matrix, Tr(·) represents the operation of finding the trace of the matrix, μ is the regularization parameter, and represents the positive semi-definite symbol. The virtual elements at positions from 0 to V - 1 of the maximum virtual array A3 form a completed virtual array A4 with a total number of array elements of V. The obtained matrix can be regarded as the received data covariance matrix of the completed virtual array A4, and is called the completed virtual covariance matrix.

[0114] In step 2 of the present invention, by vectorizing the sampling covariance, removing redundancy, reordering elements, and interpolating and completing, the sampling covariance matrix of the co-prime array A1 is theoretically transformed into the completed virtual covariance matrix of the completed virtual array A4. The problem is considered from the dimension M' = 2M + N - 1 of the co-prime array A1 to the dimension V = 2MN - N + 1 of the completed virtual array A4.

[0115] Step 3: Search for the incident angle of the desired signal in the spectral peak of the desired signal angular sector; obtain the virtual steering vector of the desired signal according to the structure of the complemented virtual array, and construct an indicator function; solve the zeros of the indicator function to obtain a set of spatial orthogonal bases.

[0116] The said Step 3 includes the following steps:

[0117] Step 31: Construct a Capon spectrum in the desired signal angular sector Θ:

[0118]

[0119] where, a v (θ) represents the virtual steering vector of the complemented virtual array A4 corresponding to the angle θ, a v (θ) = [1, e -jπsinθ ,, e -jπ(V-1)sinθ T , perform Capon spectral peak search to obtain the angle estimation value of the desired signal Thus, obtain the virtual steering vector of the desired signal

[0120] Step 32: Construct the following indicator function:

[0121]

[0122] where, is the angle estimation value of the desired signal, and V is the number of array elements of the complemented virtual array A4;

[0123] Step 33: f(g) has a total of V - 1 zeros, which are integers in the set except 0, denoted as g i , i = 1, 2,, V - 1, find the corresponding θ i for g i as:

[0124]

[0125] According to the obtained and the structure of the complemented virtual array A4, obtain the corresponding virtual steering vector a v (θ i ), i = 1, 2,, V - 1, and further obtain a set of spatial orthogonal bases in space

[0126] In Step 3 of the present invention, by utilizing the higher degrees of freedom of the complemented virtual array A4, the obtained angle estimation value is more accurate, and the obtained virtual steering vector​ With a higher dimension, the constructed indicator function can obtain a set of spatially orthogonal bases in a higher-dimensional space, which indicates that the indicator function has better unbiasedness and selectivity.

[0127] Step 4: Construct a sampling matrix using the set of spatially orthogonal bases obtained in Step 3, project the completed virtual covariance matrix obtained in Step 2 to obtain a virtual interference plus noise covariance matrix; and then obtain the interference plus noise covariance matrix of the co-prime array; calculate the inverse of the interference noise covariance matrix.

[0128] The said Step 4 includes the following steps:

[0129] Step 41: According to the set of spatially orthogonal bases obtained in Step 3 construct the sampling matrix as:

[0130]

[0131] where Θ represents the desired signal angular sector, is the complement of Θ, and then sample the completed virtual covariance matrix obtained in Step 2 in the non-signal angular sector to obtain the virtual interference plus noise covariance matrix as:

[0132]

[0133] Step 42: The completed virtual array A4 has consecutive array elements from element 0 to element V - 1 = 2MN - N, which contains all the array elements of the true co-prime array A1. Use the obtained virtual interference plus noise covariance matrix to express the interference plus noise covariance matrix of the co-prime array A1 one by one as:

[0134]

[0135] where,

[0136] Step 43: Perform eigenvalue decomposition on the interference plus noise covariance matrix as:

[0137]

[0138] where, the two diagonal matrices contain the first L large eigenvalues and the remaining M' - L small eigenvalues of, and U1, U2 are matrices composed of the corresponding eigenvectors; calculate the inverse of the reconstructed interference plus noise covariance matrix as:

[0139]

[0140] Among them, represents the estimated noise power.

[0141] In step 4 of the present invention, a sampling matrix B is constructed through a set of spatial orthogonal bases, and the virtual interference plus noise covariance matrix The interference plus noise covariance matrix of the co-prime array A1 is directly obtained through the corresponding relationship between the co-prime array A1 and the completed virtual array A4 Calculate the inverse of the interference plus noise covariance matrix through the above formula It is equivalent to that the interference component is completely suppressed, and the performance of the beamformer is further improved.

[0142] Step 5: Use the desired signal angular sector and the sampling covariance matrix obtained based on step 1 to obtain the subspace where the desired signal is located, and use the incident angle of the desired signal in step 3 to obtain the estimated steering vector of the desired signal.

[0143] The said step 5 includes the following steps:

[0144] Step 51: Perform the following discrete summation operation in the angular sector Θ of the desired signal to obtain the summation matrix C:

[0145]

[0146] Among them, a(θ j ) represents the steering vector of the co-prime array A1 corresponding to the angle θ j That is, J represents the number of sampling points in the desired signal region Θ; perform eigenvalue decomposition on the matrix C to obtain the eigenvector c corresponding to the largest eigenvalue s ;

[0147] Step 52: Perform eigenvalue decomposition on the sampling covariance matrix obtained in step 1 For:

[0148]

[0149] Among them, The two diagonal matrices contain The first L + 1 eigenvalues arranged in descending order and the remaining M' - L - 1 eigenvalues, U s , U n Are matrices composed of corresponding eigenvectors, and the space where the desired signal is located is Span{E s} = Span{[U s , c s}, Span{·} represents the column space spanned by the matrix, E s = [U s , cs represents U s and c s to form a matrix;

[0150] Step 53: Using the angle estimate value of the desired signal to obtain the steering vector of the desired signal and then perform correction to obtain the estimated steering vector of the desired signal

[0151]

[0152] The eigenvector c corresponding to the maximum eigenvalue of the matrix summation matrix C obtained in Step 5 of the present invention s contains more information of the steering vector of the desired signal, U obtained by eigen - decomposition s is the signal subspace, so the matrix E formed s spans a space that covers the steering vector of the desired signal, and the estimated steering vector of the desired signal can be accurately corrected

[0153] Step 6: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference - plus - noise covariance matrix obtained in Step 4, calculate the weight vector of the beamformer, and perform robust beamforming on the array - received data.

[0154] The said Step 6 includes the following steps:

[0155] Step 61: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference - plus - noise covariance matrix obtained in Step 4 calculate the weight vector according to the following formula:

[0156]

[0157] where, is the inverse of the interference - plus - noise covariance matrix calculated in Step 4, is the estimated steering vector obtained in Step 5, and w is obtained as the weight vector of the beamformer;

[0158] Step 62: Weight the weight vector w with the data x(k) received by the co - prime array A1 at time k to obtain the output signal z(k)=w H x(k), to achieve robust adaptive beamforming based on the co - prime array structure.

[0159] The co-prime array with M = 3 and N = 5, a total of 10 array elements, provided by the embodiment of the present invention. The desired signal comes from θ0 = 2°, the interference sources come from -32° and θ2 = 18°, and there is an angular estimation error of 2°. The angular sector of the desired signal Θ = [θ0 - 5°, θ0 + 5°]. The comparison methods are respectively: the sum covariance matrix reconstruction method based on the co-prime array (sum reconstruction method), the covariance matrix reconstruction method based on the maximum entropy spectrum (maximum entropy spectrum method), and the beamforming method based on the optimal worst performance (worst performance optimal method).

[0160] As Figure 3 shown, under the condition of the interference-to-noise ratio of 20 dB, the direction-of-arrival error follows a uniform distribution on [-4°, 4°], the number of snapshots is fixed at 300, and the performance curves of the output signal-to-interference-plus-noise ratio of different methods versus the signal-to-noise ratio under 200 Monte Carlo experiments.

[0161] As Figure 4 shown, under the condition of the interference-to-noise ratio of 20 dB, the array position error follows a uniform distribution on [-0.1λ, 0.1λ], the number of snapshots is fixed at 300, and the performance curves of the output signal-to-interference-plus-noise ratio of different methods versus the signal-to-noise ratio under 200 Monte Carlo experiments.

[0162] It can be seen from the figure that the method proposed by the present invention can obtain obvious performance improvement. The output signal-to-interference-plus-noise ratio under the direction-of-arrival error is almost the same as that of the theoretical optimal beamformer. The output signal-to-interference-plus-noise ratio under the array position error is the highest, which shows the effectiveness of the method proposed by the present invention. In addition, the method of the present invention does not involve the solution of any convex optimization problems and does not involve the spectral peak search problem of interference. Therefore, the computational complexity is relatively low, and the present invention shows good robustness under different array errors.

[0163] Through the description of the above embodiments, those skilled in the art can clearly understand that the above embodiments can be implemented by software, or can be implemented by means of software plus a necessary general hardware platform. Based on such an understanding, the technical solutions of the above embodiments can be embodied in the form of a software product, and the software product can be stored in a non-volatile storage medium (which can be a CD-ROM, a USB flash drive, a mobile hard disk, etc.), including several instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in various embodiments of the present invention.

[0164] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered by the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.

Claims

1. A robust adaptive beamforming method based on a co-prime array structure, characterized in that It includes the following steps: Step 1: Select a pair of relatively prime numbers to construct a coprime array to receive the incident signal, and obtain the sampling covariance matrix of the incident signal; Step 2: Vectorize, remove redundancy, and rearrange elements of the sampling covariance matrix obtained in Step 1 to obtain a virtual received data vector; Insert zero values into the virtual received data vector to obtain a virtual covariance matrix with holes; Complete the virtual covariance matrix with holes to obtain a completed virtual covariance matrix corresponding to the completed virtual array; Step 3: Search for the incident angle of the desired signal at the spectral peak in the desired signal angular sector; Obtain the virtual steering vector of the desired signal according to the structure of the completed virtual array, and construct an indicator function; Solve the zero points of the indicator function to obtain a set of spatial orthogonal bases; Step 4: Use the set of spatial orthogonal bases obtained in Step 3 to construct a sampling matrix, project the completed virtual covariance matrix obtained in Step 2 to obtain a virtual interference plus noise covariance matrix; Furthermore, obtain the interference plus noise covariance matrix of the coprime array; Calculate the inverse of the interference noise covariance matrix; Step 5: Use the desired signal angular sector and the sampling covariance matrix based on Step 1 to obtain the subspace where the desired signal is located, and use the incident angle of the desired signal in Step 3 to obtain the estimated steering vector of the desired signal; Step 6: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference plus noise covariance matrix obtained in Step 4, calculate the weight vector of the beamformer and perform robust beamforming on the array received data; The specific implementation of Step 5 is as follows: Step 51, in the angular sector of the desired signal Perform the following discrete summation operation to obtain a summation matrix : Among them, represents the steering vector of the co-prime array A1 corresponding to the angle , that is , , , represents the number of sampling points in the desired signal region ; perform eigenvalue decomposition on the matrix to obtain the eigenvector corresponding to the largest eigenvalue ; Step 52: Perform eigenvalue decomposition on the sampled covariance matrix obtained in Step 1 which is: Among them, , The two diagonal matrices contain The first eigenvalues arranged in descending order and the remaining eigenvalues, , are matrices composed of the corresponding eigenvectors. The space where the desired signal is located is , represents the column space spanned by the matrix, represents and assembled matrix; Step 53: Using the angle estimation value of the desired signal , obtain the steering vector of the desired signal , and then perform correction to obtain the estimated steering vector of the desired signal : ； The specific implementation of Step 1 is as follows: Step 11: Select a pair of relatively prime numbers , , , the co-prime array A1 is composed of two sparse uniform sub-arrays A11 and A12. The sub-array A11 is a sparse uniform linear array with a spacing of formed by array elements; the sub-array A12 is a sparse uniform linear array with a spacing of formed by array elements, where is the half-wavelength; except for the common reference element, there are no overlapping elements between the sub-arrays A11 and A12. The co-prime array A1 has a total of array elements, and the element positions are denoted as: , are elements of the following integer set : There is far-field narrowband uncorrelated signal sources, including a desired signal and interference incident on the co-prime array A1. Then, the sampling covariance matrix of the incident signals is calculated as follows: Among them, represents the data vector received by the co-prime array A1 at time , represents the number of snapshots, represents the conjugate transpose operator.

2. The robust adaptive beamforming method based on the co-prime array structure according to claim 1, characterized in that: The specific implementation of Step 2 is as follows: Step 21: Vectorize the sampling covariance matrix to obtain the following vector: Among them, represents a vectorization operation operator, and then performs redundancy removal and element rearrangement on the said vector to obtain the virtual received data vector of the virtual difference matrix A2 derived from the coprime matrix A1 , as follows: Among them, represents the index value of the element position of the virtual difference array A2, that is , , , is the set of positions where the elements of the virtual difference array A2 are located; the set represents the set of binary tuples with the difference of element position index values being , that is ; represents finding the number of elements in the set; represents the value corresponding to the difference of element position index values being ; represents the value at the index position of , that is the value, that is arranged in ascending order of the element position index values to obtain the virtual received data vector of the virtual difference array A2; Step 22, fill all the data between the elements with discontinuous array element position index values in the vector with 0 to obtain a higher-dimensional vector , that is: Among them, is regarded as the maximum virtual received data vector of the maximum virtual array A3, represents the index value of the array element position of the maximum virtual array A3, and the integer set is the set of positions where the array elements of the maximum virtual array A3 are located, ; By using which has a Hermitian structure, that is, , the following Toeplitz matrix is obtained: Called a virtual covariance matrix with holes; Step 23, denote as a binary matrix, where the positions of the 1 elements correspond to the positions of the non-zero elements in; the positions of the 0 elements correspond to the positions of the zero elements in, and complete the virtual covariance matrix with holes to obtain the completed matrix as : Among them, represents the Hadamard product, represents the operation of finding the Frobenius norm of a matrix, represents the operation of finding the trace of a matrix, is the regularization parameter, represents the positive semi - definite symbol; the maximum virtual array A3 ranges from 0 to The virtual array elements at the position form a completed virtual array A4 with the total number of array elements being The obtained matrix is regarded as the received data covariance matrix of the completed virtual array A4, and is called the completed virtual covariance matrix.

3. A robust adaptive beamforming method based on a co-prime array structure according to claim 2, characterized in that: The specific implementation of Step 3 is as follows: Step 31, in the desired signal angular sector Construct a Capon spectrum: Among them, represents the virtual steering vector corresponding to the complemented virtual array A4 at the angle . Perform Capon spectral peak search to obtain the angle estimation value of the desired signal, and thus obtain the virtual steering vector of the desired signal ; Step 32: Construct the following indicator function: ， Let , then the indicator function is further expressed as: Among them, is the angle estimation value of the desired signal, is the number of array elements of the complemented virtual array A4; Step 33, A total of zeros, for the set Integers other than 0, denoted as , find out The corresponding is: According to the obtained and the structure of the complemented virtual array A4, the corresponding virtual steering vector is obtained, and then a set of spatially orthogonal bases in space is obtained.

4. A robust adaptive beamforming method based on a co-prime array structure according to claim 3, characterized in that: The specific implementation of Step 4 is as follows: Step 41: Based on the set of spatially orthogonal bases obtained in Step 3 , construct the sampling matrix as follows: Among them, represents the desired signal angular sector, is the complement set of, and then for the completed virtual covariance matrix obtained in step 2, sample in the non-signal angular sector to obtain the virtual interference plus noise covariance matrix as: Step 42, complete the virtual array A4 with consecutive array elements from array element 0 to array element , including all array elements of the real co-prime array A1, and use the obtained virtual interference plus noise covariance matrix to correspond one by one to the interference plus noise covariance matrix of the co-prime array A1 , which is expressed as: Among them, , ; Step 43, perform eigen - decomposition on the interference - plus - noise covariance matrix as follows: Among them, and the two diagonal matrices contain the first large eigenvalues of and the remaining small eigenvalues. and are matrices composed of corresponding eigenvectors; The inverse of the reconstructed interference plus noise covariance matrix is calculated as: Among them, represents the estimated noise power.

5. A robust adaptive beamforming method based on a co-prime array structure according to claim 1, characterized in that: Step 6 includes the following steps: Step 61: Based on the estimated steering vector obtained in Step 5 and the inverse of the interference plus noise covariance matrix obtained in Step 4 , calculate the weight vector according to the following formula: ； Among them, is the inverse of the interference plus noise covariance matrix calculated in step 4, is the estimated steering vector obtained in step 5, and is the weight vector of the beamformer; Step 62: Weight the weight vector and the co-prime matrix A1 at time with the received data to obtain the output signal of the beamformer , and implement robust adaptive beamforming based on the co-prime array structure.

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