Two-dimensional super-resolution direction of arrival estimation method compatible with arbitrary geometric planar sensor array

Abstract

The two-dimensional super-resolution direction of arrival estimation method compatible with any geometric shape planar sensor array comprises the following steps: 1) arranging a planar sensor array; 2) based on two-dimensional inverse Fourier transform, a covariance equation between intensity signals generated by a signal source at any two sensors is established; 3) using the planar sensor array, a measured signal P ★ is obtained, and a covariance vector c ★ is calculated; 4) using the covariance equation and the covariance vector c ★ , a measured covariance matrix c0 ★ is obtained, and a measured signal covariance deinterference mathematical model is established based on an atomic norm minimization method; 5) the measured signal covariance deinterference mathematical model is solved to obtain an alpha auxiliary matrix and a beta auxiliary matrix; 6) based on a matrix pencil and a pairing method, the alpha auxiliary matrix and the beta auxiliary matrix are used to obtain an estimation result of the signal source DOA. The method can realize two-dimensional super-resolution DOA estimation, is not limited by the geometric shape of the planar array, and can accurately estimate the source DOA with high probability when the interference is not too strong.

Description

Technical Field

[0001] This invention relates to the field of sensors, specifically a two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometric shapes. Background Technology

[0002] Meshless compressed sensing, based on compressed sensing theory and treating the target source region as a continuum, is a novel direction-of-arrival (DOA) estimation method for extracting source information from microphone array measurement data. Two-dimensional meshless compressed sensing methods based on planar sensor arrays hold promise for achieving source super-resolution DOA estimation in the hemispherical space in front of the array. However, existing two-dimensional meshless compressed sensing methods are not well-compatible with arbitrary planar arrays, which limits their widespread application. Summary of the Invention

[0003] The purpose of this invention is to propose a two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry, overcoming the limitation of existing two-dimensional meshless compressed sensing methods that cannot be well compatible with arbitrary planar arrays.

[0004] The technical solution adopted to achieve the purpose of this invention is as follows: a two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometric shapes, comprising the following steps:

[0005] 1) Arrange a planar sensor array.

[0006] 2) Based on the two-dimensional inverse Fourier transform, establish the covariance equation between the intensity signals generated by the signal source at any two sensors.

[0007] 3) Using a planar sensor array, the signal from the signal source at the sensor is measured to obtain the measured signal P. ★ And calculate the covariance matrix C ★ And covariance vector c ★ .

[0008] 4) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ Furthermore, a mathematical model for removing interference from the covariance of the measured signal was established based on the atomic norm minimization method.

[0009] 5) Solve the mathematical model for interference removal based on the covariance of the measured signal to obtain the α auxiliary matrix. and β auxiliary matrix

[0010] 6) Based on the matrix bundle and pairing method, using the α auxiliary matrix and β auxiliary matrix The estimated DOA of the signal source is obtained.

[0011] Furthermore, in step 2), the step of establishing the covariance equation between the intensity signals generated by the signal source at any two sensors based on the two-dimensional inverse Fourier transform includes:

[0012] 2.1) Establish Cartesian coordinates with the center of the planar sensor array as the origin, and define the Cartesian coordinates of the sensors in the planar sensor array as (x... m ,y m ,0), m=1,2,…,M are sensor indices, and the DOA of any signal source in space is (θ i ,φ i ), θ i φ is the angle between the positive z-axis of the Cartesian coordinate system and the line connecting the origin to the signal source. i Let be the angle between the positive x-axis and the projection of the line connecting the origin to the signal source onto the xy-plane. Let i = 1, 2, ..., I be the index of the signal source.

[0013] Any signal source in space at the sensor (x) m ,y m The intensity signal p generated at (,0) m As shown below:

[0014]

[0015] In the formula, s i Let be the strength of the i-th signal source. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0016] The covariance of the signals generated by the signal source at any two sensors is shown below:

[0017]

[0018] In the formula, C mm' Let m be the covariance of the signals generated by the signal source at any two sensors, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, and m ≠ m'. p represents expectation m The signal generated by the m signal source at the sensor is called m. Let m' be the conjugate of the signal generated by the signal source at the sensor. Let s be the mean square intensity of the i-th signal source. iLet Δx be the intensity of the i-th signal source, and let “|·|” denote the modulus. mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i 。(θ) i ,φ i ) represents the DOA of any signal source in space.

[0019] 2.2) Construct an nth-order Bessel function of the first kind, J, with z as the independent variable. n (z), numerical simulation of the function |J| under different values ​​of z and n n (z)|, fit the result that makes |J n The smallest positive order N for (z)|≈0 is shown below:

[0020] N=f(z) (3)

[0021] In the formula, f(z) represents the expression for different z corresponding to |J n The smallest positive order expression for (z)|≈0.

[0022] 2.3) The Jacobi-Anger identity is shown below:

[0023]

[0024] Therefore, formula (2) can be rewritten as:

[0025]

[0026] In the formula, N x =f(2πΔx) mm' / λ), N y =f(2πΔy) mm' / λ) represent the values ​​of N in the x and y dimensions, respectively. These are the nth-order Bessel functions of the first kind in the x and y dimensions, respectively. x n y Let n be the value of n in the x and y dimensions, respectively.

[0027] 2.4) The two-dimensional inverse Fourier transform of the source strength.

[0028] 2.5) Construct matrix J x and J y .

[0029]

[0030]

[0031] In the formula, N1=f(2π[Δx] mm' ] max / λ), N2=f(2π[Δy mm' ] max / λ), [Δx mm '] max 、[Δy mm '] max For all Δx mm' Δy mm' The maximum value in Δx, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, m ≠ m', Δx mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor, λ is the wavelength, and f(·) represents the values ​​of different independent variables corresponding to |J n The smallest positive order expression for (·)|≈0. n (·) is the first-order Bessel function of order n. It is the set of real numbers.

[0032] 2.6) Let vector Among them, C mm' Let be the covariance of the signals generated by the signal source at any two sensors, with the superscript "T" indicating transpose. Given a complex set of signals, the covariance equations of the signals generated by the signal source at any two sensors are shown below:

[0033]

[0034] In the formula, diag(·) represents the vector formed by the diagonal elements of the matrix within the brackets. Let s be the mean square intensity of the i-th signal source. iLet represent the intensity of the i-th signal source, and let "|·|" denote the modulo operation. Represents expectation. Vector d(α) i ) and vector d(β) i As shown in formulas (9) and (10):

[0035]

[0036]

[0037] In the formula, the parameter sinα i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0038] Furthermore, in step 3), record Let L be the matrix composed of the measured signals from the sensor, where L is the total number of data snapshots.

[0039] The covariance matrix C of the signal source measured at different sensors ★ As shown below:

[0040]

[0041] In the formula, Let L be the matrix of signals generated at the sensor, and L be the total number of data snapshots. The superscript "H" indicates transpose / conjugate.

[0042] Covariance vector Let C be the covariance matrix. ★ A vector formed by stacking columns.

[0043] Furthermore, in step 4), the atomic norm-based minimization method is constrained by the sparsity of the source distribution within the continuous domain.

[0044] Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ The steps for establishing a mathematical model for removing interference from the covariance of the measured signal based on the atomic norm minimization method include:

[0045] 4.1) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ As shown below:

[0046]

[0047] In the formula, This is the interference vector.

[0048] 4.2) Setting According to convex geometry theory, the atomic norm of B is defined as:

[0049]

[0050] In the formula, A collection of atoms Let be the set of positive real numbers, and inf(·) is the function that takes the lower bound.

[0051] 4.3) Minimize the parameters by solving for the atomic norm Remove the measured covariance matrix c0 ★ The interference in the data is as follows:

[0052]

[0053] In the formula, ε is the disturbance control parameter, and ||·||2 represents the l2 norm. arg min a(x)subject to b(x) represents the value of x when a(x) takes its minimum value, provided that b(x) holds.

[0054] Furthermore, the interference vector includes: interference signals during actual sensor measurements, interference caused by replacing the expected value with the mean of a finite snapshot, and interference caused by Bessel function truncation.

[0055] Furthermore, in step 5), the mathematical model for solving the covariance of the measured signal to remove interference is shown below:

[0056]

[0057]

[0058] In the formula, T α and T β All are Hermitian Toeplitz matrices, tr(·) denotes the trace of the matrix, and the symbol "≥0" indicates positive semi-definite. Let α be the auxiliary matrix. Let β be the auxiliary matrix.

[0059] Furthermore, the solution tool for the covariance interference removal mathematical model includes the SDPT3 solver in the CVX toolbox.

[0060] Furthermore, in step 6), based on the matrix bundle and pairing method, the α auxiliary matrix is ​​used. and β auxiliary matrix The steps to obtain the DOA estimation result of the signal source include:

[0061] 6.1) Estimate the number of signal sources Retention Matrix C ★ For features greater than the threshold, delete matrix C. ★ The number of feature values ​​less than the threshold is used as the number of signal sources.

[0062] 6.2) From the matrix Extract From the matrix Extract And based on the multi-signal classification method, pairing and

[0063] 6.3) Calculate the signal source And quantify the source strength.

[0064] Furthermore, in step 6.2), from the matrix Extract From the matrix Extract And based on the multi-signal classification method, pairing and The steps are as follows:

[0065] 6.2.1) Eigenvalue decomposition.

[0066]

[0067] In the formula, U1 is a matrix The unitary matrix formed by the eigenvectors of , where Λ1 is a matrix U2 is a diagonal matrix formed by the eigenvalues ​​of . The unitary matrix formed by the eigenvectors of , where Λ2 is a matrix. A diagonal matrix formed by the eigenvalues ​​of .

[0068] 6.2.2) Note For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A matrix consisting of the eigenvectors corresponding to the retained eigenvalues. For matrix of Let the matrix consisting of the eigenvectors corresponding to the retained eigenvalues ​​be... Delete the last row of Y Delete the last row of Y' Delete the first row of Y Delete the first row of Y' Computation matrix bundle (Y d ,Y u The generalized characteristics of ) are worth Calculate the matrix bundle (Y') d ,Y' u The generalized characteristics of ) are worth u and v are respectively matrix bundles (Y) d ,Y u ), matrix bundle (Y' d ,Y' u The generalized eigenvalue ordinal number of ).

[0069] 6.2.3) Matrix bundle (Y) d ,Y u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get Matrix bundle (Y' d ,Y' u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get

[0070] 6.2.4) Using multi-signal classification method for Find the paired v, as shown below:

[0071]

[0072] In the formula, g(u) is a mapping function, the independent variable is u, and the function value is v paired with u. From matrix C ★ of The eigenvectors corresponding to the deleted eigenvalues ​​are used to construct the vectors, with the superscript "H" indicating transpose and conjugate. (x m ,y m ) represents the sensor coordinates, and λ represents the wavelength.

[0073] 6.2.5) According to And the matched v, will correspond to Write into the same vector set to obtain

[0074] Furthermore, in step 6.3), the signal source is calculated. The steps for quantifying the source strength are as follows:

[0075] 6.3.1) According to sinα≡sinθcosφ, sinβ≡sinθsinφ, from... Calculated from

[0076] 6.3.2) The source strength is shown below:

[0077]

[0078] In the formula, The matrix consists of signals generated by the signal source at the sensor. The sensor matrix is ​​calculated based on the estimated signal source DOA. The superscript "+" indicates the pseudo-inverse. The source strength is quantified.

[0079] The technical effectiveness of this invention is undeniable. Within the framework of two-dimensional source DOA estimation for planar sensor arrays, this invention utilizes the Jacobi-Anger identity and the properties of the Bessel function to express the covariance of the signal generated by the source at any sensor as a related form of the two-dimensional inverse Fourier transform of the source strength. Based on this form, a mathematical model for interference removal using the measured signal covariance, constrained by the sparsity of the source distribution, is established in the continuous domain. This model is then transformed into a decoupled positive semidefinite programming solution. Using the solution as input, the source DOA is estimated through matrix bundles and pairing methods, ultimately forming a meshless compressed sensing method compatible with planar sensor arrays of arbitrary geometric shapes. Simulation examples and Monte Carlo simulations demonstrate that the method of this invention can achieve two-dimensional super-resolution DOA estimation without being limited by the geometry of the planar array. When the interference is not too strong, the method of this invention can accurately estimate the source DOA with high probability. Attached Figure Description

[0080] Figure 1 For different values ​​of z and n, |J n (z)|;Figure (1a)|J n (z)|Imaging image; Figure (1b) shows the relationship between N and z.

[0081] Figure 2 Figure 2a shows the distribution of four planar array sensors and the source DOA estimation results of the method of the present invention when using the corresponding arrays; Figure 2b shows the distribution of the circular array sensors; Figure 2c shows the distribution of the star array sensors; Figure 2d shows the distribution of the fan-shaped wheel array sensors; Figure 2e shows the source DOA estimation results of the method of the present invention when using a circular array; Figure 2f shows the source DOA estimation results of the method of the present invention when using a star array; Figure 2g shows the source DOA estimation results of the method of the present invention when using a wheel array; Figure 2h shows the source DOA estimation results of the method of the present invention when using a fan-shaped wheel array.

[0082] Figure 3 The root mean square error is obtained from 100 Monte Carlo simulations using the method of this invention under different signal-to-noise ratios and total number of data snapshots. Detailed Implementation

[0083] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0084] Example 1:

[0085] See Figures 1 to 3 A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry includes the following steps:

[0086] 1) Arrange a planar sensor array.

[0087] 2) Based on the two-dimensional inverse Fourier transform, establish the covariance equation between the intensity signals generated by the signal source at any two sensors.

[0088] 3) Using a planar sensor array, the signal from the signal source at the sensor is measured to obtain the measured signal P. ★ And calculate the covariance matrix C ★ And covariance vector c ★ .

[0089] 4) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ Furthermore, a mathematical model for removing interference from the covariance of the measured signal was established based on the atomic norm minimization method.

[0090] 5) Solve the mathematical model for interference removal based on the covariance of the measured signal to obtain the α auxiliary matrix. and β auxiliary matrix

[0091] 6) Based on the matrix bundle and pairing method, using the α auxiliary matrix and β auxiliary matrix The estimated DOA of the signal source is obtained.

[0092] In step 2), the steps of establishing the covariance equation between the intensity signals generated by the signal source at any two sensors based on the two-dimensional inverse Fourier transform include:

[0093] 2.1) Establish Cartesian coordinates with the center of the planar sensor array as the origin, and define the Cartesian coordinates of the sensors in the planar sensor array as (x... m ,y m ,0), m=1,2,…,M are sensor indices, and the DOA of any signal source in space is (θ i ,φ i ), θ iφ is the angle between the positive z-axis of the Cartesian coordinate system and the line connecting the origin to the signal source. i Let be the angle between the positive x-axis and the projection of the line connecting the origin to the signal source onto the xy-plane. Let i = 1, 2, ..., I be the index of the signal source.

[0094] Any signal source in space at the sensor (x) m ,y m The intensity signal p generated at (,0) m As shown below:

[0095]

[0096] In the formula, s i Let be the strength of the i-th signal source. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0097] The covariance of the signals generated by the signal source at any two sensors is shown below:

[0098]

[0099] In the formula, C mm' Let m be the covariance of the signals generated by the signal source at any two sensors, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, and m ≠ m'. p represents expectation m The signal generated by the m signal source at the sensor is called m. Let m' be the conjugate of the signal generated by the signal source at the sensor. Let s be the mean square intensity of the i-th signal source. i Let Δx be the intensity of the i-th signal source, and let “|·|” denote the modulus. mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i, parameter sinβ i ≡sinθ i sinφ i 。(θ) i ,φ i ) represents the DOA of any signal source in space.

[0100] 2.2) Construct an nth-order Bessel function of the first kind, J, with z as the independent variable. n (z), numerical simulation of the function |J| under different values ​​of z and n n (z)|, the result is as follows Figure 1 (a) shows. Clearly, for a specific z, n is either large enough or small enough, |J n (z)|≈0. Let N be the value that makes |J|≈0. n The smallest positive order of (z)|≈0 (less than 10⁻⁴) is due to J -n (z)=(-1) n J n (z), then when |n|≥N, |J n (z)|≈0. From Figure 1 (a) searches for different values ​​of N corresponding to z, and the results are as follows: Figure 1 (b) indicates that the fitted value is |J n The smallest positive order N for (z)|≈0 is shown below:

[0101] N=f(z) (3)

[0102] In the formula, f(z) represents the expression for different z corresponding to |J n The smallest positive order expression for (z)|≈0. Equation (3) corresponds to Figure 1 (b) Curve.

[0103] 2.3) The Jacobi-Anger identity is shown below:

[0104]

[0105] Therefore, formula (2) can be rewritten as:

[0106]

[0107] In the formula, N x =f(2πΔx) mm' / λ), N y =f(2πΔy) mm' / λ) represent the values ​​of N in the x and y dimensions, respectively. These are the nth-order Bessel functions of the first kind in the x and y dimensions, respectively. x n y Let n be the value of n in the x and y dimensions, respectively.

[0108] 2.4) The two-dimensional inverse Fourier transform of the source strength.

[0109] 2.5) Construct matrix J x and J y .

[0110]

[0111]

[0112] In the formula, N1=f(2π[Δx] mm' ] max / λ), N2=f(2π[Δy mm' ] max / λ), [Δx mm' ] max 、[Δy mm' ] max For all Δx mm' Δy mm' The maximum value in Δx, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, m ≠ m', Δx mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor, λ is the wavelength, and f(·) represents the values ​​of different independent variables corresponding to |J n The smallest positive order expression for (·)|≈0. n (·) is the first-order Bessel function of order n. It is the set of real numbers.

[0113] 2.6) Let vector Among them, C mm' Let be the covariance of the signals generated by the signal source at any two sensors, with the superscript "T" indicating transpose. Given a complex set of signals, the covariance equations of the signals generated by the signal source at any two sensors are shown below:

[0114]

[0115] In the formula, diag(·) represents the vector formed by the diagonal elements of the matrix within the brackets. Let s be the mean square intensity of the i-th signal source. i Let represent the intensity of the i-th signal source, and let "|·|" denote the modulo operation. Represents expectation. Vector d(α) i ) and vector d(β) i As shown in formulas (9) and (10):

[0116]

[0117]

[0118] In the formula, the parameter sinα i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0119] In step 3), record Let L be the matrix composed of the measured signals from the sensor, where L is the total number of data snapshots.

[0120] The covariance matrix C of the signal source measured at different sensors ★ As shown below:

[0121]

[0122] In the formula, Let L be the matrix of signals generated at the sensor, and L be the total number of data snapshots. The superscript "H" indicates transpose / conjugate.

[0123] Covariance vector Let C be the covariance matrix. ★ A vector formed by stacking columns.

[0124] In step 4), the atomic norm-based minimization method is constrained by the sparsity of the source distribution within the continuous domain.

[0125] Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ The steps for establishing a mathematical model for removing interference from the covariance of the measured signal based on the atomic norm minimization method include:

[0126] 4.1) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ As shown below:

[0127]

[0128] In the formula, This is the interference vector.

[0129] 4.2) Setting According to convex geometry theory, the atomic norm of B is defined as:

[0130]

[0131] In the formula, A collection of atoms Let be the set of positive real numbers, and inf(·) is the function that takes the lower bound.

[0132] 4.3) Minimize the parameters by solving for the atomic norm Remove the measured covariance matrix c0 ★ The interference in the data is as follows:

[0133]

[0134] In the formula, ε is the disturbance control parameter, and ||·||2 represents the l2 norm. arg min a(x)subject to b(x) represents the value of x when a(x) takes its minimum value, provided that b(x) holds.

[0135] The interference vector includes: interference signals during actual sensor measurements, interference caused by replacing the expected value with the mean of a finite snapshot, and interference caused by Bessel function truncation.

[0136] In step 5), the mathematical model for solving the covariance of the measured signal to remove interference is shown below:

[0137]

[0138]

[0139] In the formula, T α and T β All are Hermitian Toeplitz matrices, tr(·) denotes the trace of the matrix, and the symbol "≥0" indicates positive semi-definite. Let α be the auxiliary matrix. Let β be the auxiliary matrix. Equation (15) shows a positive semidefinite programming problem that is a standard convex optimization problem.

[0140] The solution tools for the covariance interference removal mathematical model include the SDPT3 solver in the CVX toolbox.

[0141] In step 6), based on the matrix bundle and pairing method, the α auxiliary matrix is ​​used. and β auxiliary matrix The steps to obtain the DOA estimation result of the signal source include:

[0142] 6.1) Estimate the number of signal sources Retention Matrix C ★For features greater than the threshold, delete matrix C. ★ The number of feature values ​​less than the threshold is used as the number of signal sources.

[0143] 6.2) From the matrix Extract From the matrix Extract And based on the multi-signal classification method, pairing and

[0144] 6.3) Calculate the signal source And quantify the source strength.

[0145] In step 6.2), from the matrix Extract From the matrix Extract And based on the multi-signal classification method, pairing and The steps are as follows:

[0146] 6.2.1) Eigenvalue decomposition.

[0147]

[0148] In the formula, U1 is a matrix The unitary matrix formed by the eigenvectors of , where Λ1 is a matrix U2 is a diagonal matrix formed by the eigenvalues ​​of . The unitary matrix formed by the eigenvectors of , where Λ2 is a matrix. A diagonal matrix formed by the eigenvalues ​​of .

[0149] 6.2.2) Note For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A matrix consisting of the eigenvectors corresponding to the retained eigenvalues. For matrix of Let the matrix consisting of the eigenvectors corresponding to the retained eigenvalues ​​be... Delete the last row of Y Delete the last row of Y' Delete the first row of Y Delete the first row of Y' Computation matrix bundle (Y d ,Y u The generalized characteristics of ) are worth Calculate the matrix bundle (Y') d ,Y' u The generalized characteristics of ) are worth u and v are respectively matrix bundles (Y) d ,Y u ), matrix bundle (Y' d ,Y' u The generalized eigenvalue ordinal number of ).

[0150] 6.2.3) Matrix bundle (Y) d ,Y u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get Matrix bundle (Y' d ,Y' u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get

[0151] 6.2.4) Using multi-signal classification method for Find the paired v, as shown below:

[0152]

[0153] In the formula, g(u) is a mapping function, the independent variable is u, and the function value is v paired with u. From matrix C ★ of The eigenvectors corresponding to the deleted eigenvalues ​​are used to construct the vectors, with the superscript "H" indicating transpose and conjugate. (x m ,y m ) represents the sensor coordinates, and λ represents the wavelength.

[0154] 6.2.5) According to And the matched v, will correspond to Write into the same vector set to obtain

[0155] In step 6.3), the signal source is calculated. The steps for quantifying the source strength are as follows:

[0156] 6.3.1) According to sinα≡sinθcosφ, sinβ≡sinθsinφ, from... Calculated from

[0157] 6.3.2) The source strength is shown below:

[0158]

[0159] In the formula, The matrix consists of signals generated by the signal source at the sensor. The sensor matrix is ​​calculated based on the estimated signal source DOA. The superscript "+" indicates the pseudo-inverse. The source strength is quantified.

[0160] Example 2:

[0161] See Figures 1 to 3 A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry includes the following steps:

[0162] 1) Arrange a planar sensor array.

[0163] 2) Based on the two-dimensional inverse Fourier transform, establish the covariance equation between the intensity signals generated by the signal source at any two sensors.

[0164] 3) Using a planar sensor array, the signal from the signal source at the sensor is measured to obtain the measured signal P. ★ And calculate the covariance matrix C ★ And covariance vector c ★ .

[0165] 4) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ Furthermore, a mathematical model for removing interference from the covariance of the measured signal was established based on the atomic norm minimization method.

[0166] 5) Solve the mathematical model for interference removal based on the covariance of the measured signal to obtain the α auxiliary matrix. and β auxiliary matrix

[0167] 6) Based on the matrix bundle and pairing method, using the α auxiliary matrix and β auxiliary matrix The estimated DOA of the signal source is obtained.

[0168] Example 3:

[0169] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 2. In step 2), the step of establishing the covariance equation between the intensity signals generated by the signal source at any two sensors based on the two-dimensional inverse Fourier transform includes:

[0170] 2.1) Establish Cartesian coordinates with the center of the planar sensor array as the origin, and define the Cartesian coordinates of the sensors in the planar sensor array as (x... m ,y m,0), m=1,2,…,M are sensor indices, and the DOA of any signal source in space is (θ i ,φ i ), θ i φ is the angle between the positive z-axis of the Cartesian coordinate system and the line connecting the origin to the signal source. i Let be the angle between the positive x-axis and the projection of the line connecting the origin to the signal source onto the xy-plane. Let i = 1, 2, ..., I be the index of the signal source.

[0171] Any signal source in space at the sensor (x) m ,y m The intensity signal p generated at (,0) m As shown below:

[0172]

[0173] In the formula, s i Let be the strength of the i-th signal source. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0174] The covariance of the signals generated by the signal source at any two sensors is shown below:

[0175]

[0176] In the formula, C mm' Let m be the covariance of the signals generated by the signal source at any two sensors, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, and m ≠ m'. p represents expectation m The signal generated by the m signal source at the sensor is called m. Let m' be the conjugate of the signal generated by the signal source at the sensor. Let s be the mean square intensity of the i-th signal source. i Let Δx be the intensity of the i-th signal source, and let “|·|” denote the modulus. mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor. The imaginary unit is λ, where λ is the wavelength, and the parameter sinα is... i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i 。(θ) i ,φ i ) represents the DOA of any signal source in space.

[0177] 2.2) Construct an nth-order Bessel function of the first kind, J, with z as the independent variable. n (z), numerical simulation of the function |J| under different values ​​of z and n n (z)|, the result is as follows Figure 1 (a) shows. Clearly, for a specific z, n is either large enough or small enough, |J n (z)|≈0. Let N be the value that makes |J|≈0. n The smallest positive order of (z)|≈0 (less than 10⁻⁴) is due to J -n (z)=(-1) n J n (z), then when |n|≥N, |J n (z)|≈0. From Figure 1 (a) searches for different values ​​of N corresponding to z, and the results are as follows: Figure 1 (b) indicates that the fitted value is |J n The smallest positive order N for (z)|≈0 is shown below:

[0178] N=f(z) (3)

[0179] In the formula, f(z) represents the expression for different z corresponding to |J n The smallest positive order expression for (z)|≈0. Equation (3) corresponds to Figure 1 (b) Curve.

[0180] 2.3) The Jacobi-Anger identity is shown below:

[0181]

[0182] Therefore, formula (2) can be rewritten as:

[0183]

[0184] In the formula, N x =f(2πΔx) mm' / λ), N y =f(2πΔy) mm' / λ) represent the values ​​of N in the x and y dimensions, respectively. These are the nth-order Bessel functions of the first kind in the x and y dimensions, respectively. x n y Let n be the value of n in the x and y dimensions, respectively.

[0185] 2.4) The two-dimensional inverse Fourier transform of the source strength.

[0186] 2.5) Construct matrix J x and J y .

[0187]

[0188]

[0189] In the formula, N1=f(2π[Δx] mm' ] max / λ), N2=f(2π[Δy mm' ] max / λ), [Δx mm' ] max 、[Δy mm' ] max For all Δx mm' Δy mm' The maximum value in Δx, where m and m' are sensor indices, m = 1, 2, ..., M, m' = 1, 2, ..., M, m ≠ m', Δx mm' =x m -x m' Δy mm' =y m -y m' x m and x m' All are the Cartesian x-coordinates of the sensor, y m and y m' All are Cartesian ordinates of the sensor, λ is the wavelength, and f(·) represents the values ​​of different independent variables corresponding to |J n The smallest positive order expression for (·)|≈0. n (·) is the first-order Bessel function of order n. It is the set of real numbers.

[0190] 2.6) Let vector Among them, C mm' Let be the covariance of the signals generated by the signal source at any two sensors, with the superscript "T" indicating transpose. Given a complex set of signals, the covariance equations of the signals generated by the signal source at any two sensors are shown below:

[0191]

[0192] In the formula, diag(·) represents the vector formed by the diagonal elements of the matrix within the brackets. Let s be the mean square intensity of the i-th signal source. i Let represent the intensity of the i-th signal source, and let "|·|" denote the modulo operation. Represents expectation. Vector d(α) i ) and vector d(β) i As shown in formulas (9) and (10):

[0193]

[0194]

[0195] In the formula, the parameter sinα i ≡sinθ i cosφ i , parameter sinβ i ≡sinθ i sinφ i .

[0196] Example 4:

[0197] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 2. In step 3), [the following is noted:] Let L be the matrix composed of the measured signals from the sensor, where L is the total number of data snapshots.

[0198] The covariance matrix C of the signal source measured at different sensors ★ As shown below:

[0199]

[0200] In the formula, Let L be the matrix of signals generated at the sensor, and L be the total number of data snapshots. The superscript "H" indicates transpose / conjugate.

[0201] Covariance vector Let C be the covariance matrix. ★ A vector formed by stacking columns.

[0202] Example 5:

[0203] The two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 2. In step 4), the atomic norm-based minimization method is constrained by the sparsity of the source distribution in the continuous domain.

[0204] Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★The steps for establishing a mathematical model for removing interference from the covariance of the measured signal based on the atomic norm minimization method include:

[0205] 4.1) Using the covariance equation and covariance vector c ★ The measured covariance matrix c0 is obtained. ★ As shown below:

[0206]

[0207] In the formula, This is the interference vector.

[0208] 4.2) Setting According to convex geometry theory, the atomic norm of B is defined as:

[0209]

[0210] In the formula, A collection of atoms Let be the set of positive real numbers, and inf(·) is the function that takes the lower bound.

[0211] 4.3) Minimize the parameters by solving for the atomic norm Remove the measured covariance matrix c0 ★ The interference in the data is as follows:

[0212]

[0213] In the formula, ε is the disturbance control parameter, and ||·||2 represents the l2 norm. arg min a(x)subject to b(x) represents the value of x when a(x) takes its minimum value, provided that b(x) holds.

[0214] Example 6:

[0215] The two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry has the main steps described in Example 5. The interference vector includes: interference signals during actual sensor measurements, interference caused by replacing the expected value with the mean of finite snapshots, and interference caused by Bessel function truncation.

[0216] Example 7:

[0217] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 2. In step 5), the mathematical model for solving the covariance of the measured signal to remove interference is shown below:

[0218]

[0219]

[0220] In the formula, T α and T β All are Hermitian Toeplitz matrices, tr(·) denotes the trace of the matrix, and the symbol "≥0" indicates positive semi-definite. Let α be the auxiliary matrix. Let β be the auxiliary matrix. Equation (15) shows a positive semidefinite programming problem that is a standard convex optimization problem.

[0221] Example 8:

[0222] The main steps of the two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry are described in Example 7. The solution tool for the covariance interference removal mathematical model includes the SDPT3 solver in the CVX toolbox.

[0223] Example 9:

[0224] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 2. In step 6), based on the matrix bundle and pairing method, an α-auxiliary matrix is ​​used... and β auxiliary matrix The steps to obtain the DOA estimation result of the signal source include:

[0225] 6.1) Estimate the number of signal sources Retention Matrix C ★ For features greater than the threshold, delete matrix C. ★ The number of feature values ​​less than the threshold is used as the number of signal sources.

[0226] 6.2) From the matrix Extract From the matrix Extract And based on the multi-signal classification method, pairing and

[0227] 6.3) Calculate the signal source And quantify the source strength.

[0228] Example 10:

[0229] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 9, specifically in step 6.2), where the matrix... Extract From the matrix Extract And based on the multi-signal classification method, pairing and The steps are as follows:

[0230] 6.2.1) Eigenvalue decomposition.

[0231]

[0232] In the formula, U1 is a matrix The unitary matrix formed by the eigenvectors of , where Λ1 is a matrix U2 is a diagonal matrix formed by the eigenvalues ​​of . The unitary matrix formed by the eigenvectors of , where Λ2 is a matrix. A diagonal matrix formed by the eigenvalues ​​of .

[0233] 6.2.2) Note For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A matrix consisting of the eigenvectors corresponding to the retained eigenvalues. For matrix of Let the matrix consisting of the eigenvectors corresponding to the retained eigenvalues ​​be... Delete the last row of Y Delete the last row of Y' Delete the first row of Y Delete the first row of Y' Computation matrix bundle (Y d ,Y u The generalized characteristics of ) are worth Calculate the matrix bundle (Y') d ,Y' u The generalized characteristics of ) are worth u and v are respectively matrix bundles (Y) d ,Y u ), matrix bundle (Y' d ,Y' u The generalized eigenvalue ordinal number of ).

[0234] 6.2.3) Matrix bundle (Y) d ,Y u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get Matrix bundle (Y' d ,Y' u generalized eigenvalues ​​of ) Take the empty part and remove the duplicate root to get

[0235] 6.2.4) Using multi-signal classification method for Find the paired v, as shown below:

[0236]

[0237] In the formula, g(u) is a mapping function, the independent variable is u, and the function value is v paired with u. From matrix C ★ of The eigenvectors corresponding to the deleted eigenvalues ​​are used to construct the vectors, with the superscript "H" indicating transpose and conjugate. (x m ,y m ) represents the sensor coordinates, and λ represents the wavelength.

[0238] 6.2.5) According to And the matched v, will correspond to Write into the same vector set to obtain

[0239] Example 11:

[0240] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is described in Example 9, specifically in step 6.3, where the signal source is calculated. The steps for quantifying the source strength are as follows:

[0241] 6.3.1) According to sinα≡sinθcosφ, sinβ≡sinθsinφ, from... Calculated from

[0242] 6.3.2) The source strength is shown below:

[0243]

[0244] In the formula, The matrix consists of signals generated by the signal source at the sensor. The sensor matrix is ​​calculated based on the estimated signal source DOA. The superscript "+" indicates the pseudo-inverse. The source strength is quantified.

[0245] Example 12:

[0246] A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometry is proposed in this invention, comprising the following steps:

[0247] Step 1: Express the covariance of the signal generated by the source at the sensor as the correlation form of the two-dimensional inverse Fourier transform of the source strength.

[0248] The DOA of any source in space can be represented by (θ, φ), where θ ∈ [0, π / 2] is the elevation angle, i.e., the angle between the positive z-axis of the Cartesian coordinate system and the line connecting the origin to the source, and φ ∈ [0, 2π) is the azimuth angle, i.e., the angle between the positive x-axis and the projection of the line connecting the origin to the source onto the xy-plane. Let sinα ≡ sinθcosφ and sinβ ≡ sinθsinφ. The source in Cartesian coordinates is (x m ,y m The signal generated at the sensor with ,0) can be represented as:

[0249]

[0250] Where m = 1, 2, ..., M are sensor indices, i = 1, 2, ..., I are source indices, and s i The intensity of source i, sinα is the imaginary unit, λ is the wavelength, and sinα is the infinity symbol. i and sinβ i DOA(θ) of source i i ,φ i ) is calculated. Let m' = 1, 2, ..., M also be sensor indices. When the sources are mutually independent, p m The covariance between pm and pm' can be expressed as:

[0251]

[0252] in, To indicate expectation, the superscript "*" indicates conjugate. Let Δx be the mean square intensity of source i, and let "|·|" denote the modulus. mm' =x m -x m' Δy mm' =y m -y m' .

[0253] J n (z) is the nth-order Bessel function of the first kind with z as the independent variable. Numerical simulation is performed for |J| under different values ​​of z and n. n (z)|, the result is as follows Figure 1 (a) shows. Clearly, for a specific z, n is either large enough or small enough, |J n (z)|≈0. Let N be the value that makes |J|≈0. n The smallest positive order of (z)|≈0 (less than 10⁻⁴) is due to J -n (z)=(-1) n J n (z), then when |n|≥N, |J n (z)|≈0. From Figure 1 (a) searches for different values ​​of N corresponding to z, and the results are as follows: Figure 1 (b) indicates the relationship between N and z, which is obtained by fitting the equation:

[0254]

[0255] Among them, symbols This means rounding the value to the nearest integer in the direction of positive infinity. Equation (3) corresponds to Figure 1 (b) The curve. Accordingly, the Jacobi-Anger identity can be written as:

[0256]

[0257] According to equation (4), equation (2) can be written as:

[0258]

[0259] Where, N x =f(2πΔx) mm' / λ) and N y =f(2πΔy) mm' / λ) represent the values ​​of N in the x and y dimensions, respectively. It can be viewed as the two-dimensional inverse Fourier transform of the source strength.

[0260] Let N1 = f(2π[Δx] mm' ] max / λ), N2=f(2π[Δy mm' ] max / λ), where [Δx mm' ] max and [Δy mm' ] max For all Δx mm' and Δy mm' Find the maximum value in the matrix. Construct the matrix.

[0261]

[0262]

[0263] in, Given the set of real numbers. Construct vectors. and The superscript "T" indicates transpose. Let be the set of complex numbers. According to equation (5), the following relationship holds:

[0264]

[0265] Here, diag(·) represents the vector formed by the diagonal elements of the matrix within the brackets.

[0266] Step 2: Remove interference from the covariance of the measured signal based on decoupling atomic norm minimization.

[0267] remember Let L be the matrix composed of the measured signals from the sensor, where L is the total number of data snapshots. The matrix composed of the covariances among all measured signals can be calculated as follows: The superscript "H" indicates transpose conjugate. C ★ A vector formed by stacking columns. According to equation (8), c ★ It can be represented as:

[0268]

[0269] in, It can be called the interference vector, which is caused by interference signals in the sensor's measured signals, replacing the expectation with the mean of a finite snapshot, and Bessel function truncation, among other factors.

[0270] make According to convex geometry theory, the atomic norm of B can be defined as:

[0271]

[0272] in, A collection of atoms It is the set of positive real numbers. It can be used to measure the sparsity of source distributions in a continuous domain. c can be removed by solving the following atomic norm minimization problem. ★ Interference in:

[0273]

[0274] Where ε is the disturbance control parameter, and ||·||2 represents the l2 norm.

[0275] The minimization problem shown in equation (11) can be transformed into a decoupled semidefinite programming problem to be solved as follows:

[0276]

[0277]

[0278] Among them, T α and T β All are Hermitian Toeplitz matrices, tr(·) represents the trace of the matrix, and the symbol “≥0” indicates positive semidefinite. The positive semidefinite programming problem shown in Equation (12) is a standard convex optimization problem, which can be solved using the SDPT3 solver in the CVX toolbox.

[0279] Step 3: Source DOA estimation based on matrix bundle and pairing method

[0280] Solved using equation (12) and The source DOA information can be extracted using the following matrix bundle and pairing steps:

[0281] 1) Estimate the number of sources Use C ★ The number of larger eigenvalues ​​is taken as the number of sources.

[0282] 2) From Extract First, eigenvalue decomposition

[0283]

[0284] in, for The unitary matrix formed by the eigenvectors of , for A diagonal matrix formed by the eigenvalues ​​of . The number of larger eigenvalues ​​is twice the number of sources. Let... for A diagonal matrix formed by the square roots of the larger eigenvalues. for Let the matrix consisting of the eigenvectors corresponding to the larger eigenvalues ​​be... Then, delete the last row of Y. Delete the first row of Y Computation matrix bundle (Y d ,Y u The generalized characteristics of ) are worth Finally, by taking the imaginary part and removing the duplicate roots, we obtain...

[0285] 3) From Extract The specific method is similar to step 2), and the final result is...

[0286] 4) Pairing and Based on the multi-signal classification method, calculate the following functions in order: Find the matching v:

[0287]

[0288] in, By C ★ of The eigenvectors are constructed from the eigenvalues ​​corresponding to the smaller eigenvalues. The final result is... Abbreviated as

[0289] 5) Computational source Based on the relationship between (sinα,sinβ) and (θ,φ), from It can be calculated from After obtaining the source DOA, the source strength can be quantified as follows:

[0290]

[0291] in, The perceptual matrix is ​​calculated based on the estimated source DOA, with the superscript "+" indicating the pseudo-inverse.

[0292] The basic execution steps of the two-dimensional super-resolution DOA estimation method for planar sensor arrays compatible with arbitrary geometric shapes proposed in this invention are as follows:

[0293] Input: Sensor measured signal Sensor coordinates, wavelength λ, interference control parameter ε.

[0294] 1. By P ★ Calculate the covariance matrix C ★ sum vector c ★ .

[0295] 2. Using the sensor coordinates and λ as input, calculate the highest order N1 and N2 of the Bessel function according to equation (3), and calculate J according to equations (6) and (7). x and J y J2.

[0296] 3. Using c ★ J x J y Given ε as input, solving equation (12) shows a positive semidefinite programming problem, yielding... and

[0297] 4. Using C ★ , and As input, obtain the source DOA by following the procedure shown in step 3.

[0298] 5. Quantize the source strength according to equation (15).

[0299] Example 13:

[0300] A two-dimensional super-resolution direction-of-arrival (DOA) estimation method compatible with planar sensor arrays of arbitrary geometry was proposed. To analyze and verify the DOA estimation performance of the proposed method, simulation experiments were conducted.

[0301] Assume three sources with DOAs of (15°, 45°), (40°, 235°), and (70°, 300°) respectively, and intensities (reference maximum values) of 0dB, -3dB, and -6dB respectively, with a radiation signal wavelength of 0.17m. The following methods are used sequentially... Figure 2 Measurements were performed on four planar arrays shown in (a)-(d), with a total of 100 data snapshots and an interference signal with a signal-to-noise ratio of 20dB added. Figure 2 (e)-(h) are the source DOA estimation results. Obviously, the source DOA is accurately estimated regardless of the geometric shape of the planar array used.

[0302] When performing Monte Carlo simulations, the root mean square error (RMSE) is defined as follows to measure the accuracy of the source DOA estimation:

[0303]

[0304] Where, ΔΩ Si,d Let [ΔΩ] be the angular distance between the estimated and true values ​​of the DOA of source i in the d-th Monte Carlo calculation. Si,d |ΔΩ Si,d ≤T] represents all ΔΩ not greater than T. Si,d The resulting column vector, where size(·) represents the number of elements contained within the parentheses. Condition ΔΩ Si,d The use of ≤T is to convert ΔΩ Si,d Significantly high levels of exceptional cases are removed. These cases have a very low probability of occurrence but significantly increase the RMSE, leading to an unfair and unobjective assessment of the DOA estimation error. ΔΩ Si,d The defining formula is:

[0305]

[0306] in, and (θ) Si,d ,φ Si,d Let be the estimated and actual DOA values ​​of source i in the d-th Monte Carlo calculation, respectively. In each calculation, the estimated source and the actual source are paired according to the closest DOA principle: if... The estimated total number of sources is not less than I (the actual total number of sources). The top I estimated strong sources are paired with the actual sources one by one. If the value is less than I, the source will be estimated. Each real source is paired one by one, and the remaining ones are... The source is lost, corresponding to ΔΩ Si,d It is ∞.

[0307] Monte Carlo calculations are performed by changing the signal-to-noise ratio and the total number of data snapshots. 100 calculations are performed for each pair of signal-to-noise ratio and total number of data snapshots. Source strength and interference signals are randomly generated for each snapshot in each calculation. Figure 2 (a)-(d) show the RMSE results for the four geometric planar matrices, which are similar. To save space, the results are not presented here. Figure 3 by Figure 2 (d) Taking a sector-shaped planar array as an example, the RMSE histogram is presented. Obviously, when the signal-to-noise ratio is greater than or equal to 15dB, the RMSE is very low, with a maximum of no more than 1.3°, and the RMSE becomes smaller as the signal-to-noise ratio and the total number of data snapshots increase. This proves that the method of the present invention can accurately estimate the source DOA with high probability when the interference is not too strong.

Claims

1. A two-dimensional super-resolution direction-of-arrival estimation method compatible with planar sensor arrays of arbitrary geometric shapes, characterized in that, Includes the following steps: 1) Arrange a planar sensor array; 2) Based on the two-dimensional inverse Fourier transform, establish the covariance equation between the intensity signals generated by the signal source at any two sensors; 3) Using a planar sensor array, the signal from the signal source at the sensor is measured to obtain the measured signal. And calculate the covariance matrix. Sum of covariance vectors ; 4) Using the covariance equation and covariance vector The measured covariance matrix was obtained. Furthermore, a mathematical model for removing interference from the covariance of the measured signal was established based on the atomic norm minimization method. 5) Solve the mathematical model for interference removal based on the covariance of the measured signal, and obtain... Auxiliary matrix and Auxiliary matrix ; 6) Based on matrix bundles and pairing methods, utilizing... Auxiliary matrix and Auxiliary matrix The estimated DOA of the signal source is obtained.

2. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 1, characterized in that, In step 2), the step of establishing the covariance equation between the intensity signals generated by the signal source at any two sensors based on the two-dimensional inverse Fourier transform includes: 2.1) Establish Cartesian coordinates with the center of the planar sensor array as the origin, and set the Cartesian coordinates of the sensors in the planar sensor array as follows: , For sensor indexing, the DOA of any signal source in space is: , For the positive Cartesian coordinate system The angle between the axis and the line connecting the origin to the signal source. positive The line connecting the axis and the origin to the signal source is in The angle between planar projections; For signal source index, Any signal source in space is detected by the sensor. The intensity signal p generated at the location m As shown below: 　　　　　（1） In the formula, For the first The strength of each signal source, The imaginary unit, For wavelength, parameters ,parameter ; The covariance of the signals generated by the signal source at any two sensors is shown below: 　　（2） In the formula, Let be the covariance of the signals generated by the signal source at any two sensors. and All are sensor indexes. , , , p represents expectation m for The signal generated by the signal source at the sensor, for The conjugate of the signal generated by the signal source at the sensor. For the first The mean square intensity of each signal source For the first The strength of a signal source, symbol " " indicates taking the modulus, , ; and All are Cartesian x-coordinates of the sensor. and All are Cartesian ordinates of the sensor. The imaginary unit, For wavelength, parameters ,parameter ; For any signal source in space, the DOA is denoted as . 2.2) Constructing with as independent variable First-order Bessel function Numerical simulations are different and Functions under the value Fitting out The smallest positive order N is shown below: 　　　　　　　　　　（3） In the formula, For different Corresponding to The smallest positive order expression; 2.3) The Jacobi-Anger identity is as follows: 　　　　　（4） Therefore, formula (2) can be rewritten as: 　　（5） In the formula, , They are respectively , Wei Shang The value of , , They are respectively , On the dimensional Bessel functions of the first kind, , They are respectively , The possible values ​​of n in dimension; 2.4) Two-dimensional inverse Fourier transform of the source strength; 2.5) Constructing the matrix and ; （6） （7） In the formula, , , , Each for all , The maximum value in, and All are sensor indexes. , , , , ; and All are Cartesian x-coordinates of the sensor. and All are Cartesian ordinates of the sensor. For wavelength, For different independent variables, make The smallest positive order expression; for Bessel functions of the first kind; It is the set of real numbers; 2.6) Let vector ,in, Let $\mathbf{a}$ be the covariance of the signals generated by the signal source at any two sensors, marked with a superscript "\mathbf{a}$. " indicates transpose, Given a complex set of signals, the covariance equations of the signals generated by the signal source at any two sensors are shown below: 　　　　　（8） In the formula, Represents the vector formed by the diagonal elements of the matrix within the brackets; For the first The mean square intensity of each signal source For the first The strength of a signal source, symbol " " indicates taking the modulus, Represents expectation; vector sum vector As shown in formulas (9) and (10): 　　　　（9） 　　（10） In the formula, the parameters ,parameter .

3. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 1, characterized in that, In step 3), record The matrix is ​​composed of the sensor's measured signals, where, The total number of data snapshots; Covariance matrix of the signal source measured at different sensors As shown below: 　　　　　　　（11） In the formula, The matrix consists of signals generated by the source at the sensor. This represents the total number of data snapshots; superscript " " indicates transpose and conjugate; Covariance vector Covariance matrix A vector formed by stacking columns.

4. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 1, characterized in that, In step 4), the atomic norm-based minimization method is constrained by the sparsity of the source distribution within the continuous domain; Using covariance equation and covariance vector The measured covariance matrix was obtained. The steps for establishing a mathematical model for removing interference from the covariance of the measured signal based on the atomic norm minimization method include: 4.1) Using the covariance equation and covariance vector The measured covariance matrix was obtained. As shown below: （12） In the formula, This is the interference vector; 4.2) Setting According to convex geometry theory, The atomic norm is defined as: 　　　　（13） In the formula, A collection of atoms For the set of positive real numbers, To take the lower bound function; 4.3) Minimize the parameters by solving for the atomic norm Remove the measured covariance matrix The interference in the data is as follows: 　　（14） In the formula, For disturbance control parameters, express Norm; Indicates in If it is established, The value of x when it reaches its minimum value.

5. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 4, characterized in that, The interference vector includes: interference signals during actual sensor measurements, interference caused by replacing the expected value with the mean of a finite snapshot, and interference caused by Bessel function truncation.

6. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 1, characterized in that, In step 5), the mathematical model for solving the covariance of the measured signal to remove interference is shown below: 　　（15） In the formula, and All are Hermitian Toeplitz matrices. The trace of a matrix is ​​represented by the symbol "". "Indicates semi-positive definite, for Auxiliary matrix, for Auxiliary matrix.

7. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 6, characterized in that, The solution tools for the covariance interference removal mathematical model include the SDPT3 solver in the CVX toolbox.

8. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 1, characterized in that, In step 6), based on the matrix bundle and pairing method, using Auxiliary matrix and Auxiliary matrix The steps to obtain the DOA estimation result of the signal source include: 6.1) Estimate the number of signal sources Preserving the matrix Delete matrix features that are greater than the threshold. The number of feature values ​​less than the threshold is used as the number of signal sources; 6.2) From the matrix Extract From the matrix Extract And according to the multi-signal classification method, pairing and ; 6.3) Calculate the DOA of the signal source And quantify the source strength.

9. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 8, characterized in that, In step 6.2), from the matrix Extract From the matrix Extract And according to the multi-signal classification method, pairing and The steps are as follows: 6.2.1) Eigenvalue decomposition; 　　　　　（16） In the formula, For matrix The unitary matrix formed by the eigenvectors of , For matrix A diagonal matrix composed of the eigenvalues; For matrix The unitary matrix formed by the eigenvectors of , For matrix A diagonal matrix composed of the eigenvalues; 6.2.2) Note For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A diagonal matrix composed of the square roots of the retained eigenvalues. For matrix of A matrix consisting of the eigenvectors corresponding to the retained eigenvalues. For matrix of Let the matrix consisting of the eigenvectors corresponding to the retained eigenvalues ​​be... , ;delete The last line ,delete The last line ,delete The first line ,delete The first line ; Calculate matrix bundle The generalized features are worth Calculate matrix bundle The generalized features are worth u and v are matrix bundles, respectively. Matrix bundle The generalized eigenvalue ordinal number; 6.2.3) Matrix bundle generalized eigenvalues Take the empty part and remove the duplicate root to get Matrix bundle generalized eigenvalues Take the empty part and remove the duplicate root to get ; 6.2.4) Using multi-signal classification method for Find a match As shown below: （17） In the formula, Let be a mapping function, and let be the independent variable. The function value is the same as Paired , From the matrix of The eigenvectors corresponding to the deleted eigenvalues ​​are represented by the superscript "". " indicates transpose and conjugate; For sensor coordinates, Wavelength; 6.2.5) According to and matching , will the corresponding Write into the same vector set to obtain .

10. The two-dimensional super-resolution direction-of-arrival estimation method for planar sensor arrays compatible with arbitrary geometric shapes according to claim 8, characterized in that, In step 6.3), the signal source DOA is calculated. The steps for quantifying the source strength are as follows: 6.3.1) According to , ,from Calculated from ; 6.3.2) The source strength is shown below: 　　　　　　　　　（18） In the formula, The matrix consists of signals generated by the signal source at the sensor. The superscript "" represents the sensing matrix calculated based on the estimated DOA of the signal source. "Indicates false rebellion, The source strength is quantified.

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