A meshless DOA estimation method based on L-shaped coprime array

Abstract

The application discloses a kind of meshless DOA estimation method based on L type coprime array.The application utilizes the conjugate symmetry of the correlation function of the signal received by physical array elements at different time intervals, realizes the conjugate augmentation of original coprime array, utilizes the idea of atom norm minimization in meshless method, interpolates the void in virtual conjugate augmented array, obtains the covariance matrix of virtual uniform linear array by solving the semi-definite programming problem, then carries out unitary transformation to the matrix, and obtains the estimation value of azimuth and elevation angle by ESPRIT method, then designs cost function by using the power difference of different signals, realizes the matching of azimuth and elevation angle.Compared with the traditional DOA estimation method, the present application improves the degree of freedom and array aperture of the array, and has superior estimation performance.The present application converts the complex value operation of matrix into real value operation, reduces the calculation complexity, and realizes the matching of azimuth and elevation angle.

Description

Technical Field

[0001] This invention belongs to the field of sensor array signal processing technology in the field of industrial Internet of Things information processing. It relates to a meshless DOA estimation method based on an L-shaped coprime array. Specifically, it is a method that uses the target signal data received by the sensor as a basis and transforms the measured data through conjugate augmentation, virtual array interpolation and unitary ESPRIT algorithm to achieve high-precision two-dimensional DOA estimation under an L-shaped coprime array. Background Technology

[0002] Array signal processing is an important branch of signal processing, widely used in communications, sonar, radar, and seismic exploration, playing a crucial role in these fields. Spatial spectrum estimation, or Direction of Arrival (DOA) estimation, is a key area of ​​array signal processing. Its basic research content involves determining the angle of the incident signal based on array received data through relevant algorithms. Traditional DOA estimation algorithms mainly include multiple signal classification (MUSIC) algorithms based on eigenvalue decomposition and subspace rotational invariance (ESPRIT) algorithms for estimating signal parameters. These algorithms are primarily applied to uniform linear array models, but their estimation performance degrades significantly under low signal-to-noise ratio, small snapshot number, and small angle intervals, and they cannot achieve two-dimensional angle matching. In recent years, with the rapid development of compressed sensing and sparse reconstruction theory, sparse representation-based direction-finding methods based on compressed sensing have attracted the attention of researchers. Among them, meshless DOA estimation algorithms have received more and more research due to their superior performance. However, most meshless DOA estimation algorithms are only applied to one-dimensional uniform linear arrays, and there are no good methods for handling some special arrays.

[0003] To address the aforementioned issues, Xue Zhang et al. proposed a method for DOA estimation of coprime arrays. This method utilizes virtual array interpolation and a meshless approach based on atomic norm minimization to construct a covariance matrix. Then, the MUSIC method is used to search for spectral peaks on the covariance matrix to achieve DOA estimation. However, this method has a limited number of virtual interpolation elements, and the spectral peak search results in a large computational load. Lin Xinping et al. proposed a DOA estimation method based on conjugate augmentation. This method utilizes the conjugate symmetry property of the cross-correlation function of the signals received by physical array elements at different time intervals to achieve virtual expansion of a one-dimensional linear array, increasing the array aperture and degrees of freedom of the original array. However, the virtual expanded array constructed by this algorithm has gaps that are not filled, and it is only applicable to one-dimensional arrays, unable to achieve two-dimensional angle matching. This patent addresses the problems of existing coprime array virtual extended array elements being limited, computationally complex, and unable to automatically match azimuth and elevation angles. It proposes a unitary meshless DOA estimation method based on an L-shaped coprime array. This method expands the array aperture, increases the array degrees of freedom, improves the accuracy and performance of two-dimensional DOA estimation, reduces computational complexity, and achieves angle matching. Summary of the Invention

[0004] This patent addresses the shortcomings of existing two-dimensional DOA estimation methods for L-shaped coprime arrays by proposing a meshless DOA estimation method based on L-shaped coprime arrays. This patented method utilizes the conjugate symmetry property of the cross-correlation functions of the signals received by physical array elements at different time intervals to achieve conjugate augmentation of the one-dimensional coprime array. Through a virtual array interpolation method based on atomic norm minimization, gaps are filled to obtain the covariance matrix corresponding to the interpolated virtual uniform linear array. Then, the covariance matrix is ​​transformed into a real-valued matrix through unitary transformation, and the azimuth and elevation angles are estimated using the ESPRIT method. Finally, matching of azimuth and elevation angles is achieved based on the power differences between different signals. This method improves the estimation accuracy of two-dimensional angles, expands the number of array-estimated signal angles, achieves matching of azimuth and elevation angles under L-shaped coprime arrays, reduces computational complexity, and has significant engineering application value.

[0005] A meshless DOA estimation method based on an L-shaped coprime array is proposed. This method is applicable to L-shaped coprime sensor arrays, which consist of two subarrays, subarray 1 and subarray 2, placed along the x-axis and z-axis of a coordinate system, respectively. Subarrays 1 and 2 have the same number of sensors and array structure, and the sensor at the origin is shared by both subarrays. Subarrays 1 and 2 are both coprime arrays composed of two nested uniform linear arrays with M and N sensors, respectively. The two uniform linear arrays share the first sensor, and the sensor spacing between the two uniform linear arrays is d1 = Nd and d2 = Md, respectively, where d = λ / 2, and λ is the signal wave amplitude. The length, M, and N are coprime numbers, and the total number of coprime array sensors is M+N-1. The first sensor of the coprime array is placed at the origin of the coordinate system. The positions of each sensor in the coprime array are L={md1|m=0,1,…,M-1}∪{nd2|n=1,2,…,N-1}, where “∪” represents the union of two sets. The number of sensors in subarray 1 and subarray 2 is P=M+N-1, and the total number of sensors in the L-shaped coprime sensor array is 2P-1=2M+2N-3. Taking the sensor at the origin as the reference point, the positions of the sensors in subarray 1 on the x-axis and subarray 2 on the z-axis are represented as l. xi ∈L and l zi ∈L, where l x1 <l x2 <...<l xP , l z1 <l z2 <...<l zP “∈” indicates that an element belongs to a set; the θ k ∈(0,π) represents the incident signal s k The angle between (t) and the x-axis is called the azimuth angle, φ. k ∈(0,π) represents the incident signal s k The angle between (t) and the z-axis is called the pitch angle, k = 1, 2, ..., K; Its characteristic is that DOA estimation includes the following steps:

[0006] Step 1: The array received data of subarray 1 on the x-axis and subarray 2 on the z-axis in the t-th snapshot is x(t) = A x (θ)s(t)+n x z(t), z(t) = A z (φ)s(t)+n z (t), where Let K represent the incident signal vector, K represent the number of incident signals, and the superscript "T" indicate the matrix transpose operation. Describes the set of m×n dimensional complex matrices. and Let A be the additive white Gaussian noise vectors received by subarray 1 and subarray 2, respectively, and let A be the noise vectors independent of the incident signal. x (θ)=[a x (θ1),a x (θ2),…,a x (θ K )] and A z (φ)=[a z (φ1),a z (φ2),…,a z (φ K )] represent the array manifold matrices of subarray 1 and subarray 2, respectively, where ,

[0007] Step 2: Calculate the data x received by each sensor in subarray 1 in Step 1. m Time cross-correlation function of x(t) and x1(t) Where τ and T represent time delay and number of snapshots, respectively, and τ≠0, m=1,2,…,P, the obtained Arrange them column-wise to obtain column vector r. x (τ), where Then solve for r. x * (-τ), where the superscript "*" indicates the conjugate operation, and r is selected. x The data from the second row to the last row of (τ) is represented as r. x' (τ), will Left-multiply by the permutation matrix J, then multiply r by... x' (τ) and Construct augmented column vectors by mutual association in Permutation matrix All elements on the secondary diagonal are 1s, and all other elements are 0s. Let τ represent the set of m×n dimensional real matrices, and then let τ take the values ​​of T in sequence. s 2T s ,…,N s T s T s and N s Representing the pseudo-sampling period and pseudo-snapshot number respectively, we obtain the pseudo-multi-snapshot received data R. x , where R x =[r xx (T s ),r xx (2T s ),…,r xx (N s T s )];

[0008] Step 3: Calculate the data z received by each sensor in subarray 2 in Step 1. m Time cross-correlation functions of z(t) and z1(t) Where τ and T represent time delay and number of snapshots, respectively, and τ≠0, m=1,2,…,P, the obtained Arrange them column-wise to obtain column vector r. z (τ), where Then solve for r. z * (-τ), choose r z (τ) and A z The data from the second row to the last row of (θ) is represented as r. z' (τ), will Left-multiply by the permutation matrix J, then multiply r by... z' (τ) and Construct augmented column vectors by mutual association in Next, let τ take T in sequence. s 2T s ,…,N s T s The pseudo-multi-shot received data R was obtained. z , where R z =[r zz (T s ),r zz (2T s ),…,r zz (N s T s )];

[0009] Step 4: Process the pseudo-multiple snapshot data R from Step 2 x Initialize each column vector in the data, that is... Where t = T s 2T s ,…,N s T s ,

[0010]

[0011]

[0012] Where max(L) s ) represents L s The largest sensor location in the middle, “<> l "This represents the data at position l of the array element." Let x represent an element that does not belong to a set, and then solve for x. xv The covariance matrix R of (t) xv,in The superscript "H" indicates the conjugate transpose, and then R is multiplied by a selection matrix P on the right to transform R. xv Extract the non-zero columns to obtain the matrix. Where R xv' =R xv P, Where e j Indicates the (l) sj A column vector where the value at position +(N-1)M+1 is 1 and the values ​​at all other positions are 0, j = 1, 2, ..., |L s |,|L s |=2P-1=2M+2N-3,|L v |=2(N-1)M+1;

[0013] Step 5: Use the CVX Convex Optimization Toolbox to optimize R in Step 4. xv' Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is

[0014]

[0015] in It is a Hermitian matrix. It is a complex matrix. Therefore The Hermitian, Toeplitz, and positive semi-definite complex matrices are the first column vectors. These matrix variables are set in the CVX Convex Optimization Toolbox. It is a real matrix whose coordinates are in relation to R. xv' The non-zero elements are set to 1 at their corresponding positions and 0 at other positions. "tr()" represents the trace of the matrix. Represents the Hadamard product, "|| || F “ denotes the Frobenius norm of the matrix, μ denotes the regularization parameter, “min” indicates taking the minimum value, and “≥” indicates positive semidefinite;

[0016] Step Six: The results obtained in Step Five... Take the conjugate and left-multiply by the permutation matrix J x Right multiply by the permutation matrix J x ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix on the left. conjugate transpose Right-multiplying unitary matrix Obtain the matrix after unitary transformation in I k J represents the k×k dimensional identity matrix. k Denotes a k×k dimensional permutation matrix, 0 m×n Represents an m×n dimensional matrix containing all zeros;

[0017] Step 7: For R xf Perform eigenvalue decomposition to obtain the corresponding eigenvalues. and eigenvectors Select the eigenvectors corresponding to the K largest eigenvalues ​​to form the signal subspace. make Then we get Ψ xr =(K1E xs ) + K2E xs Where “Re(·)” indicates taking the real part, Im(·) indicates taking the imaginary part, and the superscript “+” indicates the inverse of the matrix. xr Perform eigenvalue decomposition to obtain eigenvalues. This leads to an estimated value for the azimuth angle.

[0018]

[0019] Step 8: Process the pseudo-multiple snapshot data R from Step 3 z Initialize each column vector in the data, that is... Then solve covariance matrix R zv ,in Then, by right-multiplying the selection matrix P, R is... xv Extract the non-zero columns to obtain the matrix. Where R zv' =R zv P;

[0020] Step 9: Use the CVX Convex Optimization Toolbox to optimize R in Step 8. zv' Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is

[0021]

[0022] in It is a Hermitian matrix. It is a complex matrix. Therefore The first column vector is a Hermitian, positive semi-definite, and Toeplitz complex matrix. These variables are set in the CVX Convex Optimization Toolbox. It is a real matrix whose coordinates are in relation to R. zv' The non-zero element is 1 at the corresponding position and 0 at other positions;

[0023] Step 10: The results obtained in Step 9... Take the conjugate and left-multiply by the permutation matrix J z Right multiply by the permutation matrix J z ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix on the left. conjugate transpose Right-multiplying unitary matrix Obtain the matrix after unitary transformation

[0024] Step 11: For R zf Perform eigenvalue decomposition to obtain the corresponding eigenvalues. and eigenvectors Select the eigenvectors corresponding to the K largest eigenvalues ​​to form the signal subspace. Then we get Ψ zr =(K1E zs ) + K2E zs , to Ψ zr Perform eigenvalue decomposition to obtain eigenvalues. This may allow us to obtain an estimate of the pitch angle.

[0025]

[0026] Step Twelve: Based on the results obtained in Step Five calculate Based on the results obtained in step nine calculate in

[0027] A xv =[a xv (θ1),a xv (θ2),…,a xv (θ K )],

[0028]

[0029] A zv =[a zv (φ1),a zv (φ2),…,a zv (φ K )],

[0030]

[0031] Then solve the cost function. The column transformation matrix H is the minimum value, where Each row contains only one element that is 1, and the rest are 0. Therefore, [φ1, φ2, ..., φ K Multiplying by H on the right yields the azimuth angle [θ1, θ2, ..., θ]. K Matching pitch angles [φ1, φ2, ..., φ] K ]H.

[0032] Compared with the prior art, the technical solution of the present invention has the following technical effects:

[0033] (1) Based on the original array received data, this invention generates a virtual extended uniform linear array by using conjugate augmentation technology and the idea of ​​atomic norm minimization, which improves the estimation accuracy of two-dimensional DOA estimation and increases the array degrees of freedom.

[0034] (2) This invention separates the azimuth angle estimation from the elevation angle estimation, and converts the complex matrix into a real matrix through unitary transformation, which greatly reduces the computational complexity and computational cost.

[0035] (3) The present invention achieves the matching of azimuth and pitch angles. Attached Figure Description

[0036] Figure 1 This is a diagram of a one-dimensional coprime array model of this patent;

[0037] Figure 2 This is a diagram of the L-shaped coprime array model of this patent;

[0038] Figure 3 This is the two-dimensional DOA estimation result of the signal processing method of this patent under 8 signal sources;

[0039] Figure 4 This is the relationship curve between the root mean square error of the azimuth angle and the signal-to-noise ratio of the signal processing method of this patent;

[0040] Figure 5 This is the relationship curve between the root mean square error of the pitch angle and the signal-to-noise ratio of the signal processing method of this patent;

[0041] Figure 6 This is the relationship curve between the root mean square error of the azimuth angle and the number of snapshots in the signal processing method of this patent.

[0042] Figure 7 This is the relationship curve between the root mean square error of the pitch angle and the number of snapshots in the signal processing method of this patent.

[0043] Figure 8 This is a diagram showing the azimuth and elevation angle matching of the signal processing method of this patent. Detailed implementation method:

[0044] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0045] First embodiment: Figure 1 The array model of the coprime array used in this invention is given. Figure 2 The array model of the L-shaped coprime array used in this invention is given. The coprime array consists of two uniform linear arrays with M=3 and N=5 sensors respectively, and the total number of sensors in the coprime array is M+N-1=7. The sensor spacing between the two uniform linear arrays is d1=Nd and d2=Md respectively, d=0.25m. The x-axis coprime array and the y-axis coprime array on the L-shaped array are configured the same. The number of incident signals is K=8, and the azimuth and elevation angle pairs are (20,20),(40,50),(60,100),(80,80),(100,140),(130,60),(140,170),(150,150). The signal-to-noise ratio is 10dB, and the number of snapshots T=500.

[0046] The direction of arrival (DOA) estimation is performed using the above conditions, and the specific implementation process is as follows:

[0047] Step 1: The L-shaped coprime sensor array consists of two subarrays, subarray 1 and subarray 2, placed along the x-axis and z-axis of the coordinate system, respectively. Subarray 1 and subarray 2 have the same number and structure of sensors, and the sensor at the origin is shared by both subarrays. Subarray 1 and subarray 2 are both coprime arrays composed of two nested uniform linear arrays with sensor numbers of M=3 and N=5, respectively. The two uniform linear arrays share the first sensor, and the sensor spacing between the two uniform linear arrays is d1=5d and d2=3d, respectively. M and N are coprime numbers, and the total number of coprime array sensors is M + N - 1 = 7. The first sensor in the coprime array is placed at the origin of the coordinate system. The positions of each sensor in the coprime array are L = {0d, 3d, 5d, 6d, 9d, 10d, 12d}, where "∪" represents the union of two sets. The number of sensors in subarray 1 and subarray 2 is P = M + N - 1 = 7 each, and the total number of sensors in the L-shaped coprime sensor array is 2P - 1 = 2M + 2N - 3 = 13. Taking the sensor at the origin as the reference point, the positions of the sensors in subarray 1 on the x-axis and subarray 2 on the z-axis are represented as l. xi ∈L and l zi ∈L, i=1,2,…,7, where l x1 <l x2 <...<l x7 , l z1 <l z2 <...<lz7 “∈” indicates that an element belongs to a set. The data received by the arrays of subarray 1 and subarray 2 in the t-th beat is x(t) = A. x (θ)s(t)+n x z(t), z(t) = A z (φ)s(t)+n z x(t), where x(t) = [x1(t), x2(t), ..., x7(t)] T ,z(t)=[z1(t),z2(t),…,z7(t)] T , This represents the incident signal vector, K = 8 represents the number of incident signals, and the superscript "T" indicates the matrix transpose operation. and Let A represent the additive white Gaussian noise vectors received by subarray 1 and subarray 2, respectively, and let A be independent of the incident signal. x (θ) and A z (φ) represent the array manifold matrices of subarray 1 and subarray 2, respectively, A x (θ)=[a x (θ1),a x (θ2),…,a x (θ8)],A z (φ)=[a z (φ1),a z (φ2),…,a z (φ8)], where The θ k ∈(0,π) represents the incident signal s k The angle between (t) and the x-axis is called the azimuth angle, φ. k ∈(0,π) represents the incident signal s k The angle between (t) and the z-axis is called the pitch angle;

[0048] Step 2: Calculate the data x received by each sensor in subarray 1 in Step 1. m Time cross-correlation function of x(t) and x1(t) Where τ and T represent time delay and number of snapshots, respectively, and τ≠0, m=1,2,…,7, the obtained Arranged column-wise, we get a column vector expression r. x (τ), where Then solve for r. x * (-τ), where the superscript "*" indicates the conjugate operation, and r is selected. x The data from rows 2 to 7 of (τ) are represented as r. x' (τ), will Multiply by a permutation matrix J on the left, then multiply r by... x' (τ) and Construct augmented column vectors by mutual association in Permutation matrix It is a square matrix where all elements on the secondary diagonal are 1s, and all other elements are 0s. Let τ represent the set of m×n dimensional real matrices, and then let τ take the values ​​of T in sequence. s 2T s ,…,N s T s ,in and N s =T / 2=250 represents the pseudo-sampling period and the pseudo-snapshot number, respectively, to obtain the pseudo-multi-snapshot received data R. x , where R x =[r xx (T s ),r xx (2T s ),…,r xx (250T s )];

[0049] Step 3: Calculate the data z received by each sensor in subarray 2 in Step 1. m Time cross-correlation functions of z(t) and z1(t) Will Arrange them column-wise to obtain column vector r. z (τ), where Then solve for r. z * (-τ), choose r z (τ) and A z The data from rows 2 to 7 of (θ) are represented as r. z' (τ), will Left-multiply by a permutation matrix J, then multiply r by... z' (τ) and By combining them, we obtain an augmented column vector. in Next, let τ take T in sequence. s 2T s , ..., 250T s The pseudo-multi-shot received data R was obtained. z , where R z =[r zz (T s ),r zz (2T s ),…,r zz (250T s )];

[0050] Step 4: Process the pseudo-multiple snapshot data R from Step 2 x Initialize each column vector in the data, that is... Where t = T s 2T s ,…,250T s ,

[0051]

[0052]

[0053] Where max(L) s ) = 12 represents L s The largest sensor location in the middle, “<> l "This represents the data at position l of the array element." Let x represent an element that does not belong to a set, and then solve for x. xv The covariance matrix R of (t) xv ,in The superscript "H" indicates the conjugate transpose, and then R is multiplied by a selection matrix P on the right to transform R. xv Extract the non-zero columns to obtain the matrix. Where R xv' =R xv P, It is a selection matrix, where e i Indicates the (l) si A column vector where the value at position +12+1) is 1 and the values ​​at all other positions are 0, |L s |=2P-1=2M+2N-3=13,|L v |=2(N-1)M+1=25;

[0054] Step 5: Use the CVX Convex Optimization Toolbox to optimize R in Step 4. xv' Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is

[0055]

[0056] in It is a Hermitian matrix. It is a complex matrix. It is a The first column vector is a Hermitian, positive semi-definite, and Toeplitz complex matrix. These variables are set in the CVX Convex Optimization Toolbox. It is a real matrix whose coordinates are in relation to R. xv'The non-zero elements are set to 1 at their corresponding positions and 0 at other positions. "min" represents the minimum value, and "tr(·)" represents the trace of the matrix. Represents the Hadamard product, "|||| F “ denotes the Frobenius norm of the matrix, μ denotes the regularization parameter, u = 1, and “≥” indicates positive semidefinite;

[0057] Step Six: The results obtained in Step Five... Take the conjugate and left-multiply by the permutation matrix J x Right multiply by the permutation matrix J x ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix Q on the left. 25 conjugate transpose Right-multiplying unitary matrix Q 25 The matrix after unitary transformation is obtained. ,in I 12 J represents a 12×12 dimensional identity matrix. 12 Represents a 12×12 dimensional permutation matrix, 0 m×n Represents an m×n dimensional matrix containing all zeros;

[0058] Step 7: For R xf Perform eigenvalue decomposition to obtain the corresponding eigenvalues ​​λ. x1 ,λ x2 ,…,λ x25 and eigenvector v x1 ,v x2 ,…,v x25 Select the eigenvectors corresponding to the 8 largest eigenvalues ​​to form the signal subspace. Let C2 = [0 24×1 ,I 24×24 ], Then we get Ψ xr =(K1E xs ) + K2E xs Where “Re(·)” represents taking the real part and Im(·) represents taking the imaginary part, for Ψ xr Perform eigenvalue decomposition, where the superscript "+" indicates the inverse of the matrix, to obtain the eigenvalues. This may allow us to obtain an estimate of the azimuth angle.

[0059]

[0060] Step 8: Process the pseudo-multi-block data R from Step 3z Initialize each column vector in the data, that is... Then solve covariance matrix R zv ,in Then, by right-multiplying the selection matrix P, R is... xv Extract the non-zero columns to obtain the matrix. Where R zv' =R zv P;

[0061] Step 9: Use the CVX Convex Optimization Toolbox to optimize R in Step 4. zv' Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is

[0062]

[0063] in It is a Hermitian matrix. It is a complex matrix. It is a The first column vector is a Hermitian, positive semi-definite, and Toeplitz complex matrix. These variables are set in the CVX Convex Optimization Toolbox. It is a real matrix whose coordinates are in relation to R. zv' The non-zero element is 1 at the corresponding position and 0 at other positions, so u = 1;

[0064] Step 10: The results obtained in Step 9... Take the conjugate and left-multiply by a permutation matrix J z Right-multiply by a permutation matrix J z ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix Q on the left. 25 conjugate transpose Right-multiplying unitary matrix Q 25 The matrix after unitary transformation is obtained.

[0065] Step 11: For R zf Perform eigenvalue decomposition to obtain the corresponding eigenvalues ​​λ. z1 ,λ z2 ,…,λ z25 and eigenvector v z1 ,v z2 ,…,v z25Select the eigenvectors corresponding to the 8 largest eigenvalues ​​to form the signal subspace E. zS , and thus obtain Ψ zr =(K1E zs ) + K2E zs , to Ψ zr Perform eigenvalue decomposition to obtain eigenvalues. This may allow us to obtain an estimate of the pitch angle.

[0066]

[0067] Step Twelve: Based on the results obtained in Step Five calculate Based on the results obtained in step nine calculate in

[0068] A xv =[a xv (θ1),a xv (θ2),…,a xv (θ8)],

[0069]

[0070] A zv =[a zv (φ1),a zv (φ2),…,a zv (φ8)],

[0071]

[0072] Then solve the cost function. Obtain the optimized matrix H is a column transformation matrix, where each row of the matrix has only one element that is 1 and the rest are 0. Multiplying [φ1,φ2,…,φ8] by H on the right yields the elevation angle [φ1,φ2,…,φ8]H that matches the azimuth angle [θ1,θ2,…,θ8].

[0073] The method of this patent based on the above conditions was simulated using MATLAB simulation software, and the angle estimation result of this patent method is as follows: Figure 3 As shown, by Figure 3 The experimental results show that, under the conditions of this example, the method of this patent can estimate signal sources that exceed the number of its own physical array elements.

[0074] Second embodiment: The relationship curve between the root mean square error of azimuth angle and the signal-to-noise ratio of the signal processing method of this patent is shown in the figure below. Figure 4As shown, the performance curves of the virtual interpolation (NCA-VAI) DOA estimation method without conjugate augmentation are also presented. Both methods estimate the angle using the root-finding MUSIC method. The application conditions of the method of this invention are as follows:

[0075] The method of this invention is based on an array model of an L-shaped coprime array, as shown below. Figure 2 As shown, subarray 1 and subarray 2 along the x-axis and z-axis are both coprime arrays composed of two uniform linear arrays with M=3 and N=5 elements respectively. The total number of elements in the coprime array is M+N-1=7, and the element spacing is d1=Nd and d2=Md respectively. There are three incident signals with additive white Gaussian noise (GAD), which is independent of the incident signals. The azimuth angles of the three incident signals are (10°, 30°, 50°). A snapshot number T = 500 is used, and the signal-to-noise ratio (SNR) is varied from -10dB to 20dB in 5dB increments. 300 Monte Carlo simulations are performed, and the results are analyzed using MATLAB software. The performance comparison curves are shown below. Figure 4 As shown.

[0076] from Figure 4 It can be seen that the root mean square error of both methods decreases with the increase of signal-to-noise ratio. Among them, the root mean square error of the method of the present invention is smaller under the same conditions. From the overall effect, the method of the present invention performs better than the other method and has strong estimation performance at low signal-to-noise ratio. When the signal-to-noise ratio is 0dB, the root mean square error of the method of the present invention is 0.035° lower than that of the NCA-VAI method.

[0077] The third embodiment shows the relationship curve between the root mean square error of pitch angle and the signal-to-noise ratio of the signal processing method of this patent, and displays the accurate result of DOA estimation. The resulting diagram is shown below. Figure 5 As shown, the performance curves of the NCA-VAI estimation method are also presented. The application conditions of the method of this invention are as follows:

[0078] The method of this invention is based on an array model of an L-shaped coprime array, as shown below. Figure 2 As shown, subarrays 1 and 2 along the x and z axes are coprime arrays composed of two uniform linear arrays with M=3 and N=5 elements respectively. The total number of elements in the coprime arrays is M+N-1=7, with element spacing of d1=Nd and d2=Md, where d=0.25m. There are three incident signals with additive white Gaussian noise, independent of the incident signals. The elevation angles of the three incident signals are (20°, 40°, 60°). A snapshot number T=500 is used, with the signal-to-noise ratio starting from -10dB and increasing in 5dB increments up to 20dB. 300 independent Monte Carlo simulations are performed, and MATLAB simulation software is used for simulation analysis. The resulting performance analysis comparison curves are shown below. Figure 5 As shown.

[0079] from Figure 5 It can be seen that the root mean square error of both methods decreases with the increase of signal-to-noise ratio. Among them, the root mean square error of the method of the present invention is smaller. From the overall effect, the method of the present invention performs better than the other method. It also has strong estimation performance at low signal-to-noise ratio. When the signal-to-noise ratio is 0dB, the root mean square error of the method of the present invention is 0.049° lower than that of the NCA-VAI method.

[0080] Fourth embodiment: The relationship curve between the root mean square error of azimuth angle and the number of snapshots of the signal processing method of this patent, and the accurate result of DOA estimation are shown in the figure below. Figure 6 As shown, the performance curves of the NCA-VAI estimation method are also presented. The application conditions of the method of this invention are as follows:

[0081] The method of this invention is based on an array model of an L-shaped coprime array, as shown below. Figure 2 As shown, subarrays 1 and 2 along the x and z axes are coprime arrays composed of two uniform linear arrays with M=3 and N=5 elements respectively. The total number of elements in the coprime arrays is M+N-1=7, with element spacings of d1=Nd and d2=Md, where d=0.25m. There are three incident signals with additive white Gaussian noise, independent of the incident signals. The azimuth angles of the three incident signals are (10°, 30°, 50°). Using an SNR of 0dB, the number of snapshots T starts from 100 and increases in step size from 100 to 1000, performing 300 independent Monte Carlo simulations. The simulation analysis is conducted using MATLAB software, and the resulting performance comparison curves are shown below. Figure 6 As shown.

[0082] from Figure 6 It can be seen that the root mean square error of both methods decreases with the increase of the number of snapshots. Among them, the root mean square error of the method of the present invention is smaller. From the overall effect, the performance of the method of the present invention is better than that of the other method. It also has strong estimation performance under low signal-to-noise ratio. When the number of snapshots T is 500, the root mean square error of the method of the present invention is 0.113° lower than that of the NCA-VAI method.

[0083] Fifth embodiment: The relationship curve between the root mean square error of pitch angle and the number of snapshots in the signal processing method of this patent, and the accurate result of DOA estimation are shown in the figure below. Figure 7 As shown, the performance curves of the NCA-VAI estimation method are also presented. The application conditions of the method of this invention are as follows:

[0084] The method of this invention is based on an array model of an L-shaped coprime array, as shown below. Figure 2As shown, subarrays 1 and 2 along the x and z axes are coprime arrays composed of two uniform linear arrays with M=3 and N=5 elements respectively. The total number of elements in the coprime arrays is M+N-1=7, with element spacing of d1=Nd and d2=Md, where d=0.25m. There are three incident signals with additive white Gaussian noise, independent of the incident signals. The elevation angles of the three incident signals are (20°, 40°, 60°). Using an SNR of 0dB, the number of snapshots T starts from 100 and increases in step size from 100 to 500, performing 300 independent Monte Carlo simulations. The simulation analysis is conducted using MATLAB software, and the resulting performance comparison curves are shown below. Figure 7 As shown.

[0085] from Figure 7 It can be seen that the root mean square error of both methods decreases with the increase of the number of snapshots. Among them, the root mean square error of the method of the present invention is the smallest. From the overall effect, the method of the present invention performs better than the other method. It also has strong estimation performance under low signal-to-noise ratio. When the number of snapshots T is 500, the method of the present invention is better than the NCA-VAI method by 0.046° of root mean square error.

[0086] Sixth embodiment: The matching of azimuth and elevation angles in the signal processing method of this patent is shown in the following diagram. Figure 8 As shown. The application conditions of the method of the present invention are as follows:

[0087] The method of this invention is based on an array model of an L-shaped coprime array, as shown below. Figure 2 As shown, subarrays 1 and 2 along the x and z axes are coprime arrays composed of two uniform linear arrays with M=3 and N=5 elements respectively. The total number of elements in the coprime array is M+N-1=7, and the element spacing is d1=Nd and d2=Md, respectively, with d=0.25m. There are four incident signals, and the noise is additive white Gaussian noise, independent of the incident signals. The azimuth and elevation angles of the four incident signals are (10°, 10°), (20°, 35°), (30°, 20°), and (40°, 50°). One hundred independent Monte Carlo simulations were performed with an SNR of 5dB and a snapshot count of T=500. The simulation analysis was conducted using MATLAB software, and the resulting performance comparison curves are shown below. Figure 7 As shown.

[0088] from Figure 8 It can be seen that the estimated values ​​of the azimuth and elevation angles after matching are basically located next to the actual incident signal angles. The angle matching method of this invention can achieve the matching of azimuth and elevation angles under the conditions of this example, and the matching is good.

[0089] The specific examples described herein are merely illustrative of the invention. Those skilled in the art to which this invention pertains may make various modifications, additions, or similar substitutions to the described specific examples without departing from the invention or exceeding the scope defined by the appended claims.

Claims

1. A meshless DOA estimation method based on an L-shaped coprime array, applicable to L-shaped coprime sensor arrays, wherein the L-shaped coprime sensor array consists of two arrays placed in a coordinate system. Subarray 1 of the axis and placed in The axis is composed of two subarrays. Subarray 1 and subarray 2 have the same number of sensors and array structure, and the sensor at the origin is shared by both subarrays. Subarray 1 and subarray 2 each consist of a number of sensors respectively. and The two uniform linear arrays are nested together to form a coprime array; the two uniform linear arrays share the first sensor, and the sensor spacing between the two uniform linear arrays is respectively... and ,in , For the signal wavelength, and The total number of coprime array sensors is (coprime numbers). The first sensor in the coprime array is placed at the origin of the coordinate system, and the positions of the other sensors in the coprime array are as follows: ,in" "" represents the union of two sets; the number of sensors in subarray 1 and subarray 2 are both . The total number of sensors in the L-shaped coprime sensor array is Using the sensor at the origin as the reference point, located at Subarray 1 on the axis and The sensor positions of subarray 2 on the axis are represented as follows and , ,in , " " indicates that an element belongs to a set; the Indicates the incident signal and The angle between the axes is called the azimuth angle. Indicates the incident signal and The angle between the axes is called the pitch angle. Its characteristics are: The DOA estimation method includes the following steps: Step 1: In the first A quick snapshot Subarray 1 on the axis and The array receiving data of on-axis subarray 2 is , ,in , , Represents the incident signal vector. Indicates the number of incident signals, indicated by the superscript " "Indicates the matrix transpose operation," "express A set of complex matrices of dimension 1 and These are the additive white Gaussian noise vectors received by subarray 1 and subarray 2, respectively, and the noise is independent of the incident signal. and Let these represent the array manifold matrices of subarray 1 and subarray 2, respectively. , , ; Step 2: Calculate the data received by each sensor in subarray 1 in Step 1. and Time cross-correlation function ,in and These represent latency and snapshot count, respectively. , ,Will Arrange by columns to obtain column vectors ,in Then solve The superscript "*" indicates the conjugate operation, selecting... The data from the second row to the last row is represented as ,Will Left multiplication of permutation matrix Then and Construct augmented column vectors by mutual association ,in Permutation matrix All elements on the secondary diagonal are 1s, and all other elements are 0s. "express A set of real matrices of dimension, then let Take in sequence , , , ,in and Representing the pseudo-sampling period and pseudo-snapshot number respectively, we obtain pseudo-multi-snapshot received data. ,in ; Step 3: Calculate the data received by each sensor in subarray 2 in Step 1. and Time cross-correlation function ,in and These represent latency and snapshot count, respectively. , ,Will Arrange by columns to obtain column vectors ,in Then solve Select The data from the second row to the last row is represented as ,Will Left multiplication of permutation matrix Then and Construct augmented column vectors by mutual association ,in Then order Take in sequence , , , Obtain pseudo-multi-shot received data ,in ; Step 4: Process the pseudo-multiple snapshot data from Step 2 Initialize each column vector in the data, that is... ， in , , ， ， in express The largest sensor location in the middle, "Indicates the position of the array element" The data at the location, "" indicates that an element does not belong to a set, and then the solution is obtained. covariance matrix ,in The superscript "H" indicates the conjugate transpose, and then the selection matrix is ​​multiplied by the right. ,Will Extract the non-zero columns to obtain the matrix. ,in , ,in Indicates the first A column vector where the value at position 1 is equal to the value at the other positions and the value at the remaining positions is equal to the value at the first position. , , ; Step 5: Use the CVX Convex Optimization Toolbox to optimize the components from Step 4. Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is ， in It is a Hermitian matrix. It is a complex matrix. Therefore The Hermitian, Toeplitz, and positive semi-definite complex matrices are the first column vectors. These matrix variables are set in the CVX Convex Optimization Toolbox. It is a real matrix, which is in relation to... The non-zero element is 1 at the corresponding position and 0 at the other positions. "Represents the trace of the matrix," "Indicates the Hadamard product," " denotes the Frobenius norm of the matrix, This represents the regularization parameter, where "min" indicates taking the minimum value. "Indicates semi-positive definite; Step Six: The results obtained in Step Five... Take the conjugate and left-multiply by the permutation matrix. Right multiply by the permutation matrix ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix on the left. conjugate transpose Right multiplication of unitary matrix The matrix after unitary transformation is obtained. ,in When it is an odd number, ; When it is even, , express An identity matrix of dimension 1 express A permutation matrix of dimension 1 express A zero matrix of dimension; Step Seven: [Regarding...] Perform eigenvalue decomposition to obtain the corresponding eigenvalues. and eigenvectors Choose one The eigenvectors corresponding to the larger eigenvalues ​​form the signal subspace. ,make , , Then we get ,in" "Indicates taking the real part, This indicates taking the imaginary part, and the superscript "+" indicates the matrix inverse. Perform eigenvalue decomposition to obtain eigenvalues. , This allows us to obtain an estimate of the azimuth angle. ； Step 8: Process the pseudo-multiple snapshot data from Step 3 Initialize each column vector in the data, that is... Then solve covariance matrix ,in Then, by right multiplication of the selection matrix ,Will Extract the non-zero columns to obtain the matrix. ,in ; Step Nine: Use the CVX Convex Optimization Toolbox to optimize the components from Step Eight. Solving the semidefinite programming problem yields the optimized matrix. The semidefinite programming problem is ， in It is a Hermitian matrix. It is a complex matrix. Therefore The first column vector is a Hermitian, positive semi-definite, and Toeplitz complex matrix. These variables are set in the CVX Convex Optimization Toolbox. It is a real matrix, which is in relation to... The non-zero element is 1 at the corresponding position and 0 at other positions; Step 10: The results obtained in Step 9... Take the conjugate and left-multiply by the permutation matrix. Right multiply by the permutation matrix ,in Given a matrix whose secondary diagonal elements are all 1s and all other elements are all 0s, find the matrix... With the matrix after conjugate permutation Add them together, take the average, and then multiply it by the unitary matrix on the left. conjugate transpose Right multiplication of unitary matrix The matrix after unitary transformation is obtained. ; Step Eleven: [Regarding...] Perform eigenvalue decomposition to obtain the corresponding eigenvalues. and eigenvectors Choose one The eigenvectors corresponding to the larger eigenvalues ​​form the signal subspace. And thus obtain ,right Perform eigenvalue decomposition to obtain eigenvalues. , This allows us to obtain an estimate of the pitch angle. ； Step Twelve: Based on the results obtained in Step Five ,calculate According to the results obtained in step nine ,calculate ,in ， ， ， ， Then solve the cost function. The column transformation matrix at its minimum value ,in Each row contains only one element that is 1, and the rest are 0. Right multiplication and The obtained azimuth angle is the same as the estimated azimuth angle. Matching estimated pitch angle .

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