Co-prime array coherent DOA estimation method based on matrix filling
Through the coherent DOA estimation method of coherent arrays based on matrix fill, the complexity problem of coherent signal processing in sparse arrays is solved, and higher spatial freedom and positioning accuracy are achieved, more targets can be estimated, and computational complexity is reduced. , suitable for real-time processing.
Patent Information
- Application Number
- CN202411111865.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-08-14
- Publication Date
- 2024-12-10
AI Technical Summary
In the prior art, when processing coherent signals, especially for sparse arrays, it is difficult to effectively estimate the direction of multiple targets, and the calculation complexity is high, which cannot meet the needs of practical applications.
The coherent DOA estimation method of cohesive array based on matrix filling is adopted. By constructing the Toeplitz matrix and filling the zeros with the kernel norm minimization method, combined with the MUSIC method, the calculation complexity is reduced and the estimation performance is improved.
Effectively increase the array aperture, improve spatial freedom and positioning accuracy, can estimate more targets, and reduce the computational complexity, making it suitable for real-time processing.
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Figure CN119104978A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of array signal processing, and in particular relates to a coprime array coherent DOA estimation method based on matrix filling. Background Art
[0002] The problem of direction of arrival (DOA) estimation has been widely studied in various fields such as radar, sonar, navigation, and astronomy, mainly using antenna arrays to receive incident signals. Scholars have proposed a variety of DOA estimation algorithms using uniform linear arrays (ULAs), including multiple signal classification (MUSIC), estimation of signal parameters via rotation invariance techniques (ESPRIT), propagator method (PM), and parallel factor (PARAFAC) technology. However, these algorithms are developed under the assumption that the signals are incoherent. In actual application scenarios, due to factors such as multipath propagation, the signals are often coherent. Coherence brings about the rank deficiency problem of the covariance matrix of the received signal, making the above methods invalid.
[0003] Most of the methods for coherent signal processing are proposed for uniform linear arrays and cannot be directly applied to sparse arrays. Unlike uniform arrays, sparse arrays, such as coprime linear arrays (CA) and nested arrays (NA), have larger aperture arrays and thus greater design flexibility because the element spacing is not limited to half a wavelength. These sparse arrays offer stronger performance and a wider range of possibilities for DOA estimation. Summary of the invention
[0004] The technical problem to be solved by the present invention is to address the deficiencies of the above-mentioned prior art and to provide a coherent DOA estimation method for coprime arrays based on matrix filling, which can effectively increase the array aperture, have higher spatial degrees of freedom and positioning accuracy, and can estimate more targets. While ensuring the estimation performance, it reduces the computational complexity and is easy to process in real time.
[0005] In order to achieve the above technical objectives, the technical solution adopted by the present invention is:
[0006] The coherent DOA estimation method of coprime arrays based on matrix filling specifically includes the following steps:
[0007] Step 1: construct a coprime array coherent signal array model to obtain the covariance matrix of the signal;
[0008] Step 2: Select the first row of the covariance matrix to construct the Toeplitz matrix;
[0009] Step 3: Fill the zero part of the Toeplitz matrix by the nuclear norm minimization method to obtain the covariance matrix required for DOA estimation;
[0010] Step 4: Use the MUSIC method to solve the covariance matrix obtained in step 3 to obtain the DOA estimation result.
[0011] To optimize the above technical solutions, the specific measures taken also include:
[0012] In the coprime array coherent signal array model constructed in step 1 above, the number of array elements is M+N-1, which is obtained by forming two sparse uniform linear arrays, and the number of array elements of the two sparse uniform linear arrays is M and N respectively, and the array element spacing is Nd and Md respectively, d is the half wavelength of the signal, and M<N;
[0013] The leftmost array element of the two sparse uniform linear arrays is taken as the reference array element, and the array element position is represented by the set in is a set of integers:
[0014]
[0015] in
[0016] Assume that there are K far-field narrowband coherent signal sources incident on the above array, and their incident angle is θ = [θ 1 ,θ 2 ,…,θ K ] T , then the output signal of the array at time t is:
[0017]
[0018] Where A = [a(θ 1 ),a(θ 2 ),…,a(θ K )] is the direction matrix, represents the direction vector of the i-th signal, represents the i-th element in the coprime linear array, and l 1 =0, λ represents the wavelength of the signal, s(t) is the reference signal, α=[α 1 ,α 2 ,…,α K ] T is a non-zero complex-valued attenuation coefficient vector; n(t) represents zero-mean additive white Gaussian noise with a noise variance of σ 2 ;
[0019] Based on the idea of array interpolation, the interpolated uniform linear array is obtained by filling the holes of the coprime linear array with zeros. The corresponding physical array does not exist. The interpolated uniform linear array is recorded as in is a set of integers
[0020]
[0021] The output signal model of the uniform linear array obtained after zero padding is:
[0022]
[0023] in,<x(t)> l represents the value of the matrix x(t) at l, Indicates that element l belongs to the set But not in the collection Contains the signal received by the physical array element, and the received signal of the array element corresponding to the interpolation is recorded as zero;
[0024] The covariance matrix of the received signal is:
[0025]
[0026] in, R S =αs(t)s(t) H α H represents the covariance matrix of the source; is the identity matrix;
[0027] Formula (5) describes the covariance matrix of the signal when the number of snapshots is infinite. The sampling covariance matrix under the finite number of snapshots is:
[0028]
[0029] Among them, T represents the time period and t represents time.
[0030] The covariance matrix calculated by selecting formula (6) in step 2 above is Any row of constructs the following Toeplitz matrix:
[0031]
[0032] Among them, when there is no noise, Equivalent to the covariance matrix;
[0033]
[0034]
[0035] is a k-dimensional diagonal matrix;
[0036] when hour,
[0037] Step 3 above fills the Toeplitz matrix by solving the following rank minimization problem:
[0038]
[0039] Among them, the matrix is a binary matrix, The positions corresponding to the non-zero elements of are set to 1, otherwise they are set to 0;
[0040] Then, by minimizing the nuclear norm, formula (8) is transformed into:
[0041]
[0042] Formula (9) is the matrix filling model. yes The Toeplitz matrix obtained after filling is the covariance matrix required for DOA estimation;
[0043] where ||·|| * represents the nuclear norm of the matrix, Denotes the Hadamard product.
[0044] The above step 4 uses the MUSIC method to solve the DOA estimation model and obtain the DOA estimation result. The process is as follows:
[0045] right Perform eigenvalue decomposition and get:
[0046]
[0047] Among them, Σ S represents the eigenvalue corresponding to the signal subspace, U S Represents the eigenvector corresponding to the eigenvalue; Σ N represents the eigenvalue corresponding to the noise subspace, U N represents the eigenvector corresponding to the eigenvalue; Therefore, formula (10) changes to
[0048]
[0049] Simplified:
[0050]
[0051] Formula (12) can be further simplified to Then each column of the matrix A(θ) has an orthogonal relationship with the noise subspace, so
[0052]
[0053] Therefore, there is an orthogonal relationship between the noise feature vector and the signal vector, and the spatial spectrum function is:
[0054]
[0055] Formula (14) changes the value of θ to find the peak of the spatial spectrum and obtain the DOA value. The corresponding K maximum peaks are the DOAs of the incident K coherent signal sources.
[0056] The present invention has the following beneficial effects:
[0057] The Toeplitz matrix constructed by the present invention has a larger dimension, so its array aperture can be effectively improved after the matrix is restored, and more received data information can be used to improve the final estimation performance.
[0058] The present invention is no longer limited to coprime linear arrays, but can be applied to a variety of sparse arrays, and has a wider application range. BRIEF DESCRIPTION OF THE DRAWINGS
[0059] Figure 1 is a flow chart of the method of the present invention;
[0060] Figure 2 is the mutually prime array element position model;
[0061] Figure 3 Schematic diagram of root mean square error performance of the present invention and the traditional decorrelation method under different signal-to-noise ratios;
[0062] Figure 4 Schematic diagram of the root mean square error performance of the present invention and the traditional decorrelation method under different snapshot numbers. DETAILED DESCRIPTION
[0063] The embodiments of the present invention are further described in detail below in conjunction with the accompanying drawings.
[0064] In this paper, scalars, vectors, matrices, and sets are represented by lowercase a, bold lowercase a, bold uppercase A, and To represent the R-dimensional vector n, [n (1) ,n (2) ,...,n (R) ] T To represent, where n (r) Represents the rth element. A T , A *, A H They represent the transpose, conjugate, and conjugate transpose of matrix A, respectively. Tr[·] represents the trace operation of a matrix, diag(·) represents a diagonal matrix, whose diagonal elements are the elements in brackets, [A] i,j represents the jth element in the i-th row of matrix A. ||·|| * represents the nuclear norm of the matrix, represents the Hadamard product, <x> l Represents the value of matrix x at position l.
[0065] The present invention provides a coherent DOA estimation method for coprime arrays based on matrix filling, such as Figure 1 As shown, the specific steps include:
[0066] Step 1: construct a coprime array coherent signal array model to obtain the covariance matrix of the signal;
[0067] In the embodiment, the array element distribution model of the coprime array is as follows: Figure 2 As shown. The number of array elements is M+N-1, which is obtained by forming two sparse uniform linear arrays. The number of array elements of the two sparse uniform linear arrays is M and N respectively, and the array element spacing is Nd and Md respectively. d is the half wavelength of the signal, and M<N. Taking the leftmost array element of the two sparse uniform linear arrays as the reference array element, the array element position of this coprime linear array can be expressed as the set in is a set of integers defined as
[0068]
[0069] in d represents the half wavelength of the signal.
[0070] Assume that there are K far-field narrowband coherent signal sources incident on the above array, and their incident angle is θ = [θ 1 ,θ 2 ,…,θ K ] T , then the output signal of the array at time t can be written as:
[0071]
[0072] Where A = [a(θ 1 ),a(θ 2 ),…,a(θ K )] is the direction matrix. represents the i-th element in the coprime linear array, and l 1 =0, represents the direction vector of the i-th signal. λ represents the wavelength of the signal, s(t) is the reference signal, α=[α 1 ,α 2 ,…,α K ] T is a non-zero complex-valued attenuation coefficient vector. n(t) represents zero-mean additive white Gaussian noise with a noise variance of σ 2 .
[0073] Based on the idea of array interpolation, the interpolated uniform linear array can be obtained by filling the holes of the coprime linear array with zeros, and the corresponding physical array does not exist. The interpolated uniform linear array can be written as in is a set of integers
[0074]
[0075] The output signal model of the uniform linear array obtained after zero padding can be written as:
[0076]
[0077] in,<x(t)> l represents the value of the matrix x(t) at l, Indicates that element l belongs to the set But not in the collection It contains the signal received by the physical array element, and the received signal of the array element corresponding to the interpolation is recorded as zero.
[0078] The covariance matrix of the above received signal is defined as:
[0079]
[0080] in, R S =αs(t)s(t) H α H Represents the covariance matrix of the source.
[0081] Formula (5) describes the covariance matrix of the signal when the number of snapshots is infinite. In practical applications, the number of sampling snapshots is finite, so the sampling covariance matrix of formula (5) under the finite number of snapshots can be expressed as:
[0082]
[0083] That is, Equation (5) expresses the expected covariance when the number of sampling snapshots tends to infinity in an ideal situation. In practice, the number of samples is finite, and in simulation, finite snapshots are also used. It is an ideal situation Estimates.
[0084] Step 2: Select the first row of the covariance matrix to construct the Toeplitz matrix;
[0085] In the embodiment, the matrix calculated by formula (6) is selected Any row of constructs the following Toeplitz matrix
[0086]
[0087] Among them, when there is no noise, is equivalent to the covariance matrix, is a k-dimensional diagonal matrix. When
[0088] Step 3: Fill the zero part of the Toeplitz matrix by the nuclear norm minimization method to obtain the covariance matrix required for DOA estimation;
[0089] In the embodiment, because the interpolation is implemented by zero padding, the matrix contains some zero values, so the above construction matrix generally chooses the covariance matrix The first row of , where the zero values are the cross-correlation information that needs to be recovered.
[0090] Since the expected matrix It has a low rank property, so the Toeplitz matrix can be filled by solving the following rank minimization problem:
[0091]
[0092] Among them, the matrix is a binary matrix, The positions corresponding to the non-zero elements of are set to 1, otherwise they are set to 0.
[0093] By using nuclear norm minimization, the above NP-hard problem can be transformed into
[0094]
[0095] Formula (9) is the matrix filling model. yes The Toeplitz matrix obtained after filling is the covariance matrix required for DOA estimation;
[0096] where ||·|| * represents the nuclear norm of the matrix, Denotes the Hadamard product.
[0097] Formula (9) is a classic convex optimization problem, which can be effectively solved using the CVX toolbox of MATLAB. After recovery, the DOA of the coherent source can be solved by applying the MUSIC method.
[0098] Step 4: Use the MUSIC method to solve the covariance matrix obtained in step 3 to obtain the DOA estimation result.
[0099] In the embodiment, formula (7) contains zero values, so it cannot be directly used for DOA estimation. Formula (9) uses the low-rank recovery method to make up for it by finding the relationship between the rows and columns of the matrix. The zero value in is the filled matrix, which is required by the present invention. Therefore, the MUSIC method is used to solve the DOA problem, as follows:
[0100] It can be seen from formula (9) that is based on The relationship between rows and columns is obtained by filling in the missing parts of the matrix. By performing eigenvalue decomposition, we can obtain:
[0101]
[0102] Among them, Σ S represents the eigenvalue corresponding to the signal subspace, U S Represents the eigenvector corresponding to the eigenvalue; Σ N represents the eigenvalue corresponding to the noise subspace, U N represents the eigenvector corresponding to the eigenvalue; Therefore, formula (10) can be transformed into
[0103]
[0104] Simplified, we can see
[0105]
[0106] Since the source matrix R S is a full rank matrix, so it has an inverse matrix. So formula (12) can be further simplified to This shows that each column of the matrix A(θ) has an orthogonal relationship with the noise subspace, so
[0107]
[0108] Therefore, there is an orthogonal relationship between the noise feature vector and the signal vector, and the spatial spectrum function can be written as
[0109]
[0110] The above formula can be used to find the peak of the spatial spectrum and obtain the DOA value by changing the value of θ. The corresponding K maximum peaks are the DOA of the incident K coherent signal sources.
[0111] The matrix obtained in formula (9) The dimension is Therefore, the array aperture of the restored uniform linear array is also increased accordingly to This means that it utilizes more array information, thereby improving the final coherent DOA estimation performance. After rank recovery, the number of array elements is equivalent to The signal covariance matrix of the uniform linear array received by the invention is The degree of freedom is higher than that of traditional spatial smoothing decorrelation methods.
[0112] Consider a coprime linear array with two subarrays of M=3 and N=5 elements. Two far-field narrowband coherent signal sources are incident on the coprime linear array from 10° and 35°. The coherence coefficient is α=[1,0.9e j2π / 3 ] T , assuming that all noises are zero-mean additive white Gaussian noise. The performance estimation standard of the present invention is the root mean square error (RMSE) defined as:
[0113]
[0114] Among them, J and K represent the number of Monte Carlo simulations and the number of signal sources, respectively. represents the kth source θ in the jth simulation k In the following simulation experiment, the number of Monte Carlo simulations is set to J = 1000.
[0115] exist Figure 3 In the figure, the RMSE curves of the present invention, the nuclear norm minimization method, and the atomic norm minimization method are compared with the signal-to-noise ratio. In the simulation, the signal-to-noise ratio increases from -10dB to 20dB, the step is 5dB, and the number of sampling snapshots is 500. The simulation results show that the RMSE of the present invention is lower than that of other methods. When the signal-to-noise ratio is greater than -5dB, the present invention has the lowest RMSE among these methods, indicating that it has the lowest error and the best performance of coherent source DOA estimation. Moreover, its RMSE can continuously decrease with the increase of the signal-to-noise ratio, indicating that its coherent DOA estimation performance continues to improve with the increase of the signal-to-noise ratio.
[0116] exist Figure 4 In the figure, the RMSE curves of the present invention, the nuclear norm minimization method, and the atomic norm minimization method are compared with the snapshot number. In the simulation, the signal-to-noise ratio is fixed at 10dB, and the number of sampling snapshots increases from 100 to 600, with a step of 100. The simulation results show that with the continuous increase in the number of snapshots, the RMSE of these methods can continue to decrease, and the estimation performance continues to improve. However, the RMSE of the present invention is the lowest among them, and the downward trend is also relatively obvious, indicating that its estimation performance is the best.
[0117] In summary, from the analysis of the simulation effect diagram, it can be seen that the coherent DOA estimation method of a coprime array based on matrix filling proposed in the present invention can effectively increase the array aperture, has higher spatial degrees of freedom and positioning accuracy, and can estimate more targets. In addition, the method reduces the computational complexity.
[0118] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.< / x>
Claims
1. A coherent DOA estimation method for mutually prime arrays based on matrix filling, characterized in that: The specific steps include: Step 1: construct a coprime array coherent signal array model to obtain the covariance matrix of the signal; Step 2: Select the first row of the covariance matrix to construct the Toeplitz matrix; Step 3: Fill the zero part of the Toeplitz matrix by the nuclear norm minimization method to obtain the covariance matrix required for DOA estimation; Step 4: Use the MUSIC method to solve the covariance matrix obtained in step 3 to obtain the DOA estimation result.
2. The coherent DOA estimation method for mutually prime arrays based on matrix filling according to claim 1, characterized in that: In the coprime array coherent signal array model constructed in step 1, the number of array elements is M+N-1, which is obtained by forming two sparse uniform linear arrays, and the number of array elements of the two sparse uniform linear arrays is M and N respectively, and the array element spacings are Nd and Md respectively, d is half the wavelength of the signal, and M<N; The leftmost array element of the two sparse uniform linear arrays is taken as the reference array element, and the array element position is represented by the set in is a set of integers: in Assume that there are K far-field narrowband coherent signal sources incident on the above array, and their incident angles are θ = [θ1, θ2, …, θ K ] T , then the output signal of the array at time t is: Where A=[a(θ1),a(θ2),…,a(θ K )] is the direction matrix, represents the direction vector of the i-th signal, l i ,i=1,2,…, represents the i-th array element in the coprime linear array, and l1 = 0; λ represents the wavelength of the signal, s(t) is the reference signal, α = [α1, α2, …, α K ] T is a non-zero complex-valued attenuation coefficient vector; n(t) represents zero-mean additive white Gaussian noise with a noise variance of σ 2 ; Based on the idea of array interpolation, the interpolated uniform linear array is obtained by filling the holes of the coprime linear array with zeros. The corresponding physical array does not exist. The interpolated uniform linear array is recorded as in is a set of integers The output signal model of the uniform linear array obtained after zero padding is: in,<x(t)> l represents the value of the matrix x(t) at l, Indicates that element l belongs to the set But not in the collection Contains the signal received by the physical array element, and the received signal of the array element corresponding to the interpolation is recorded as zero; The covariance matrix of the received signal is: in, R S =αs(t)s(t) H α H represents the covariance matrix of the source; is the identity matrix; Formula (5) describes the covariance matrix of the signal with infinite snapshot number. The sampling covariance matrix with finite snapshot number is expressed as: Among them, T represents the time period and t represents time.
3. The coherent DOA estimation method for mutually prime arrays based on matrix filling according to claim 2, characterized in that: The step 2 selects the covariance matrix calculated by formula (6) Any row of constructs the following Toeplitz matrix: Among them, when there is no noise, Equivalent to the covariance matrix; is a k-dimensional diagonal matrix; when hour, 4. The coherent DOA estimation method for mutually prime arrays based on matrix filling according to claim 3, characterized in that: The step 3 fills the Toeplitz matrix by solving the following rank minimization problem: Among them, the matrix is a binary matrix, The positions corresponding to the non-zero elements of are set to 1, otherwise they are set to 0; Then, by minimizing the nuclear norm, formula (8) is transformed into: Formula (9) is the matrix filling model. yes The Toeplitz matrix obtained after filling is the covariance matrix required for DOA estimation; where ||·|| * represents the nuclear norm of the matrix, represents the Hadamard product.
5. The coherent DOA estimation method for mutually prime arrays based on matrix filling according to claim 4, characterized in that: The step 4 uses the MUSIC method to solve the DOA estimation model and obtain the DOA estimation result. The process is as follows: right Perform eigenvalue decomposition and get: Among them, Σ S represents the eigenvalue corresponding to the signal subspace, U S Represents the eigenvector corresponding to the eigenvalue; Σ N represents the eigenvalue corresponding to the noise subspace, U N represents the eigenvector corresponding to the eigenvalue; Therefore, formula (10) changes to Simplified: Formula (12) can be further simplified to Then each column of the matrix A(θ) has an orthogonal relationship with the noise subspace, so Therefore, there is an orthogonal relationship between the noise feature vector and the signal vector, and the spatial spectrum function is: Formula (14) changes the value of θ to find the peak of the spatial spectrum and obtain the DOA value. The corresponding K maximum peaks are the DOAs of the incident K coherent signal sources.