Amplitude-phase error and direction-of-arrival joint estimation method based on Gaussian Newton method

Abstract

The invention relates to a Gaussian Newton method-based amplitude-phase error and direction-of-arrival joint estimation method, which comprises the following steps of: constructing a likelihood function about array element amplitude error, array element phase error and direction-of-arrival according to an array observation model, and calculating a cost function according to the likelihood function, converting a parameter estimation problem of the likelihood function into a parameter optimization problem to obtain a target function; calculating an initial value of a direction of arrival in the target function by adopting a MUSIC algorithm, and setting an initial value of an array element amplitude error and an initial value of an array element phase error; and according to the initial value of the direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, jointly estimating the array element amplitude error, the array element phase error and the direction of arrival by adopting a Gauss-Newton method to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error and an estimated value of the direction of arrival. According to the method, the influence of the array amplitude-phase error on the DOA estimation precision is reduced, and the DOA estimation performance is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and in particular relates to a method for jointly estimating amplitude and phase errors and a direction of arrival based on a Gauss-Newton method. Background Art

[0002] Direction of arrival estimation plays a vital role in radar, sonar and wireless communication systems. Accurate direction of arrival estimation can significantly improve the performance of the system, especially in target detection and tracking. Many direction of arrival estimation algorithms, such as the MUSIC (Multiple Signal Classification) algorithm and the sparse reconstruction algorithm, perform well in practical applications, but their effectiveness depends on the accurate known array manifold. However, due to the difficulty in ensuring that the array elements are completely consistent during the manufacturing process and the aging of electronic components in the antenna array over time, there will be differences in amplitude and phase between array elements. In order to obtain good direction of arrival estimation results, it is necessary to seek a direction of arrival estimation method in the presence of array errors to reduce the impact of array errors on direction of arrival estimation. Summary of the invention

[0003] In order to solve the above problems existing in the prior art, the present invention provides a method for jointly estimating amplitude-phase error and direction of arrival based on the Gauss-Newton method. The technical problem to be solved by the present invention is achieved by the following technical solutions:

[0004] The embodiment of the present invention provides a method for jointly estimating amplitude and phase errors and direction of arrival based on the Gauss-Newton method, comprising the steps of:

[0005] Constructing a likelihood function of array element amplitude error, array element phase error and wave direction of arrival according to the array observation model, and calculating a cost function according to the likelihood function, transforming the parameter estimation problem of the likelihood function into a parameter optimization problem, and obtaining an objective function;

[0006] The MUSIC algorithm is used to calculate the initial value of the direction of arrival in the objective function, and the initial value of the array element amplitude error and the initial value of the array element phase error are set;

[0007] According to the initial value of the direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error and the direction of arrival to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error and a estimated value of the direction of arrival.

[0008] In one embodiment of the present invention, a likelihood function of array element amplitude error, array element phase error and direction of arrival is constructed according to the array observation model, and a cost function is calculated according to the likelihood function, and the parameter estimation problem of the likelihood function is transformed into a parameter optimization problem to obtain an objective function, including:

[0009] Constructing a likelihood function of array element amplitude error, array element phase error and wave direction of arrival based on the array received signal of the uniform linear array in the array observation model;

[0010] Taking the negative logarithm of the likelihood function and removing the constant term to obtain a cost function regarding array element amplitude error, array element phase error and direction of arrival;

[0011] Based on the cost function, the parameter estimation problem of the likelihood function is transformed into a parameter optimization problem to obtain an objective function.

[0012] In one embodiment of the present invention, the array receives the signal:

[0013] x(t)=ΓA(θ)s(t)+v(t),t=1,2,…,N;

[0014] Where x(t) represents the array receiving signal at time t, A(θ) = [a(θ1), a(θ2), …, a(θ K )] represents the array manifold, j represents the imaginary unit, d represents the array element spacing, λ represents the wavelength, θ k represents the angle of the kth source, θ=[θ1,θ2,…,θ K ] T represents the direction of arrival, K represents the number of signal sources, M represents the number of array elements, s(t)=[s1(t),s2(t),…,s K (t)] T , represents the kth signal at time t, a tk represents the amplitude of the kth incoming signal at time t, φ tk represents the phase of the kth incoming signal at time t, v(t) = [v1(t), v2(t), …, v M (t)] T represents noise, v(t)~CN(0,Q v ) means the mean is zero and the covariance matrix is ​​Q v The complex circularly symmetric Gaussian white noise is N, N represents the number of snapshots, Γ represents the array amplitude and phase error matrix; taking the amplitude and phase error of the last array element as a reference, the array amplitude and phase error matrix is ​​expressed as a diagonal matrix: ρ m It represents the ratio of the amplitude error of the mth array element to the Mth array element, It represents the difference between the phase error of the mth array element and the Mth array element.

[0015] In one embodiment of the present invention, the likelihood function is:

[0016]

[0017] in, ρ=[ρ1,ρ2,…,ρ M-1 ] T represents the array element amplitude error, represents the array element phase error, I N represents the N-order identity matrix, represents the Kronecker product, and

[0018] The cost function is:

[0019]

[0020] in,

[0021] The objective function is:

[0022]

[0023] in, They represent the estimated value of direction of arrival, the estimated value of array element amplitude error, and the estimated value of array element phase error respectively.

[0024] In one embodiment of the present invention, the MUSIC algorithm is used to calculate the initial value of the direction of arrival in the objective function, including:

[0025] The covariance matrix is ​​obtained by calculating the autocorrelation of the array element receiving signal matrix, and the covariance is subjected to singular value decomposition, and the covariance matrix after the singular value decomposition is divided into the form of the sum of two matrices according to the number of information sources, and the eigenvectors corresponding to MK eigenvalues ​​are obtained, where M represents the number of array elements and K represents the number of information sources;

[0026] Constructing a space spectrum function using the eigenvectors corresponding to the MK eigenvalues;

[0027] According to the spatial spectrum function, the angles corresponding to the first K larger peaks of the spatial spectrum are used as initial values ​​of the direction of arrival.

[0028] In one embodiment of the present invention, the covariance matrix divided into the sum of two matrices is:

[0029] R XX =U SΣ S U S H +U V Σ V U V H ;

[0030] Among them, Σ S is the diagonal matrix consisting of the first K larger eigenvalues ​​of Σ, U S is the eigenvector corresponding to the K largest eigenvalues, Σ V is the diagonal matrix consisting of the last MK smaller eigenvalues ​​of Σ, U V is the eigenvector corresponding to the MK smaller eigenvalues, Σ is the diagonal matrix formed by the M eigenvalues ​​obtained by performing singular value decomposition on the covariance matrix;

[0031] The spatial spectrum function is:

[0032]

[0033] Among them, P MUSIC (θ) represents the spatial spectrum, j represents the imaginary unit, d represents the array element spacing, λ represents the wavelength, and M represents the number of array elements;

[0034] The initial value of the direction of arrival is:

[0035] θ (0) =[θ1 (0) ,θ2 (0) ,…,θ K (0) ] T ;

[0036] Wherein, K represents the number of information sources.

[0037] In one embodiment of the present invention, setting an array element amplitude error initial value and an array element phase error initial value includes:

[0038] Set the initial value of the array element amplitude error to: ρ (0) =[ρ1 (0) ,ρ2 (0) ,…,ρ M-1 (0) ] T =[1,1,...,1] T , where M represents the number of array elements;

[0039] Set the initial value of the array element phase error to:

[0040] In one embodiment of the present invention, according to the initial value of the direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error and the direction of arrival to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error and an estimated value of the direction of arrival, including:

[0041] The Gauss-Newton method is used to obtain the iterative formulas for the direction of arrival and amplitude and phase errors;

[0042] Based on the iterative formula, iteration is performed using the initial value of the direction of arrival, the initial value of the array element amplitude error, and the initial value of the array element phase error. When the difference between the cost functions of two adjacent iterations is less than a preset accuracy, the iteration is stopped to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error, and an estimated value of the direction of arrival.

[0043] In one embodiment of the present invention, the iterative formula is:

[0044]

[0045] in,(·) (q) represents the value after q iterations, Re(·) means taking the real part, I N represents the N-order identity matrix, represents the Kronecker product, Q v represents the covariance matrix of complex-valued circularly symmetric Gaussian white noise,

[0046] Jac(α) is The Jacobian matrix of α is:

[0047]

[0048] in,

[0049]

[0050] The array element amplitude error estimation value, array element phase error estimation value and wave direction of arrival estimation value are:

[0051]

[0052] in, They represent the estimated value of direction of arrival, the estimated value of array element amplitude error, and the estimated value of array element phase error respectively.

[0053] Compared with the prior art, the present invention has the following beneficial effects:

[0054] The present invention first constructs a likelihood function and transforms the parameter estimation problem of the likelihood function into a parameter optimization problem. Then, the MUSIC algorithm is used to obtain the initial value of the direction of arrival, and the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error and the direction of arrival. The calculation complexity is greatly reduced and the method has good convergence. The influence of the array amplitude and phase errors on the accuracy of the direction of arrival estimation is reduced, and the direction of arrival estimation performance is improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 A schematic flow chart of a method for jointly estimating amplitude and phase error and direction of arrival based on the Gauss-Newton method provided in an embodiment of the present invention;

[0056] Figure 2 The figure is a comparison of the results of using the MUSIC algorithm to estimate the direction of arrival directly without compensating the array error when the signal-to-noise ratio is 20 dB, and using the method of this embodiment to jointly estimate the array error and the direction of arrival;

[0057] Figure 3 This is a performance result diagram of single target direction of arrival estimation under different signal-to-noise ratios according to an embodiment of the present invention;

[0058] Figure 4 This is a diagram showing the performance results of estimating the direction of arrival of a new target at different signal-to-noise ratios after the array error is compensated according to an embodiment of the present invention. DETAILED DESCRIPTION

[0059] The present invention is further described in detail below with reference to specific embodiments, but the embodiments of the present invention are not limited thereto.

[0060] Embodiment 1

[0061] See also Figure 1 , Figure 1 A flowchart of a method for jointly estimating amplitude, phase error and direction of arrival based on the Gauss-Newton method is provided in an embodiment of the present invention. The method for jointly estimating amplitude, phase error and direction of arrival based on the Gauss-Newton method comprises the following steps:

[0062] S1, constructing a likelihood function about array element amplitude error, array element phase error and wave direction of arrival according to the array observation model, and calculating a cost function according to the likelihood function, transforming the parameter estimation problem of the likelihood function into a parameter optimization problem, and obtaining an objective function. Step S1 includes the following steps:

[0063] S11. Construct a likelihood function about array element amplitude error, array element phase error and wave arrival direction based on the array receiving signal of the uniform linear array in the array observation model.

[0064] Specifically, consider a uniform linear array with M array elements, the number of signal sources is K, the number of snapshots is N, and the array receiving signal x(t) at time t can be expressed as:

[0065] x(t)=ΓA(θ)s(t)+v(t), t=1,2,…,N (1)

[0066] Where x(t) represents the array receiving signal at time t, A(θ) = [a(θ1), a(θ2), …, a(θ K )] represents the array manifold, j represents the imaginary unit, d represents the array element spacing, λ represents the wavelength, θ k represents the angle of the kth source, θ=[θ1,θ2,…,θ K ] T represents the direction of arrival, K represents the number of signal sources, M represents the number of array elements, s(t)=[s1(t),s2(t),…,s K (t)] T , represents the kth signal at time t, a tk represents the amplitude of the kth incoming signal at time t, φ tk represents the phase of the kth incoming signal at time t, v(t) = [v1(t), v2(t), …, v M (t)] T represents noise, v(t)~CN(0,Q v ) means the mean is zero and the covariance matrix is ​​Q v The complex circularly symmetric Gaussian white noise is N, N represents the number of snapshots, and Γ represents the array amplitude and phase error matrix. Taking the amplitude and phase error of the last array element as a reference, the array amplitude and phase error matrix is ​​expressed as a diagonal matrix:

[0067]

[0068] Among them, ρ m It represents the ratio of the amplitude error of the mth array element to the Mth array element, It represents the difference between the phase error of the mth array element and the Mth array element.

[0069] Let ρ = [ρ1, ρ2, …, ρ M-1 ] T represents the array element amplitude error, represents the array element phase error, and Then formula (1) can be rewritten as:

[0070]

[0071] Therefore, the likelihood function of array element amplitude error, array element phase error and wave direction of arrival can be obtained:

[0072]

[0073] in, I N represents the N-order identity matrix, represents the Kronecker product.

[0074] S12. Take the negative logarithm of the likelihood function and remove the constant term to obtain a cost function of array element amplitude error, array element phase error and direction of arrival.

[0075] Specifically, taking the negative logarithm of both sides of formula (4) yields:

[0076]

[0077] Removing the constant term in formula (5), the cost function can be expressed as:

[0078]

[0079] in,

[0080] S13. Based on the cost function, the parameter estimation problem of the likelihood function is transformed into a parameter optimization problem to obtain the objective function.

[0081] Specifically, the parameter estimation problem of the likelihood function of formula (4) can be obtained by finding the maximum value. Since formula (5) takes the negative logarithm of formula (4), the parameter estimation problem is transformed into a parameter optimization problem, that is, finding the minimum value of the cost function, thereby obtaining the objective function:

[0082]

[0083] in, They represent the estimated value of direction of arrival, the estimated value of array element amplitude error, and the estimated value of array element phase error respectively.

[0084] S2, using the MUSIC algorithm to calculate the initial value of the direction of arrival in the objective function, and setting the initial value of the array element amplitude error and the initial value of the array element phase error. Step S2 specifically includes:

[0085] S21. Calculate the autocorrelation of the array element receiving signal matrix to obtain a covariance matrix, perform singular value decomposition on the covariance, and divide the decomposed covariance matrix into the form of the sum of two matrices according to the number of information sources.

[0086] First, the autocorrelation of the received signal matrix X of the array element is obtained to obtain the covariance matrix R XX :

[0087] R XX =E{XX H} (8)

[0088] Where X = [x(1), x(2), …, x(N)] T .

[0089] Perform singular value decomposition on the covariance matrix to obtain M eigenvalues ​​and the eigenvectors corresponding to the eigenvalues:

[0090] R XX =UΣU H (9)

[0091] Among them, Σ represents the diagonal matrix formed by M eigenvalues, and U represents the eigenvector corresponding to the eigenvalue.

[0092] The covariance matrix R after decomposing the singular value according to the number of sources XX Partitioned into the form of the sum of two matrices:

[0093] R XX =U S Σ S U S H +U V Σ V U V H (10)

[0094] Among them, Σ S is the diagonal matrix consisting of the first K larger eigenvalues ​​of Σ, U S is the eigenvector corresponding to the K largest eigenvalues, Σ V is the diagonal matrix consisting of the last MK smaller eigenvalues ​​of Σ, U V is the eigenvector corresponding to the MK smaller eigenvalues.

[0095] S22, using the eigenvector U corresponding to MK eigenvalues V Construct a spatial spectrum function:

[0096]

[0097] Among them, P MUSIC (θ) represents the spatial spectrum,

[0098] S23. According to the spatial spectrum function, the angles corresponding to the first K larger peaks of the spatial spectrum are taken as the initial values ​​of the direction of arrival: θ (0) =[θ1 (0) ,θ2 (0) ,…,θ K (0) ] T.

[0099] Furthermore, the initial value of the array element amplitude error is set to: (0) =[ρ1 (0) ,ρ2 (0) ,…,ρ M-1 (0) ] T =[1,1,...,1] T , set the initial value of the array element phase error to:

[0100] S3, according to the initial value of the direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error and the direction of arrival to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error and an estimated value of the direction of arrival. Step S3 specifically includes:

[0101] S31. Use the Gauss-Newton method to obtain the iterative formula for the wave arrival direction and amplitude and phase errors.

[0102] In array signal processing, the above joint estimation problem can be transformed into an optimization problem by constructing a likelihood function, that is, finding the array amplitude phase error and wave arrival direction when the likelihood function reaches the maximum. When solving optimization problems, commonly used algorithms include gradient descent method and Newton method. The gradient descent method has first-order convergence and a slow convergence speed. Although the Newton method has second-order convergence, it needs to calculate and store more complex second-order derivatives, and there are numerical stability problems when the Hessian matrix is ​​inverted. The Gauss-Newton method decomposes the original problem into a linear least squares problem for solution. Although its convergence rate is generally linear, it uses the structural information of the nonlinear least squares problem, which is usually better than the gradient descent method and can more effectively approach the optimal solution. Compared with the Newton method, the Gauss-Newton method only needs to calculate and process the first-order derivative (Jacobian matrix), so it has higher computational efficiency in many practical applications and better numerical stability. Therefore, this embodiment uses the Gauss-Newton method to find the optimal solution.

[0103] First, for the cost function Perform a first-order expansion:

[0104]

[0105] in, Jac(α) is The Jacobian matrix of α is:

[0106]

[0107]

[0108] in,

[0109] Substituting formula (12) into formula (6) yields:

[0110]

[0111] The gradient of formula (18) is:

[0112]

[0113] Among them, Re(·) means taking the real part,

[0114] Let the gradient be equal to 0 and the iterative formula for the wave arrival direction and amplitude and phase error is:

[0115]

[0116] in,(·) (q) Represents the value after q iterations.

[0117] S32, based on the iterative formula, iterate using the initial value of the wave direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, when the cost function difference J(α (q) )-J(α (q-1) ) is less than the preset accuracy p, the iteration is stopped to obtain the array element amplitude error estimation value, array element phase error estimation value and wave direction of arrival estimation value:

[0118]

[0119] This embodiment studies the problem of estimating the direction of arrival when there are amplitude and phase errors in the array, and proposes a method for jointly estimating the amplitude and phase errors and the direction of arrival using the Gauss-Newton method. First, a likelihood function is constructed based on the array observation model to obtain the objective function of the optimization problem, thereby transforming the parameter estimation problem into an optimization problem. Then, the MUSIC algorithm is used to obtain the initial value of the direction of arrival, and the Gauss-Newton method is used to iteratively solve the amplitude error of the array element, the phase error of the array element, and the direction of arrival, thereby reducing the influence of the array amplitude and phase errors on the accuracy of the direction of arrival estimation and improving the performance of the direction of arrival estimation. The use of the Gauss-Newton method greatly reduces the computational complexity and has good convergence.

[0120] The method and beneficial effects of this embodiment are further described below through simulation.

[0121] Consider a uniform linear array with 9 elements, where the element spacing d is half the wavelength, i.e. d = λ / 2, the number of targets is 1, and the direction of arrival is -10°. The number of snapshots is set to 200. The amplitude error follows a uniform distribution in the interval [0,1], and the phase error follows a uniform distribution in the interval [-5°,5°].

[0122] See also Figure 2 , Figure 2 The figure is a comparison of the results of using the MUSIC algorithm to estimate the direction of arrival without compensating the array error when the signal-to-noise ratio is 20 dB, and using the method of this embodiment to jointly estimate the array error and the direction of arrival. Figure 2 It can be seen that when the array received data is directly used for DOA estimation using the MUSIC algorithm, the estimation result deviates from the target true angle; when the array received data is used for DOA estimation using the method of this embodiment, the estimation result is closer to the target true angle than when the MUSIC algorithm is directly used.

[0123] See also Figure 3 , Figure 3 The figure is a performance result diagram of single target direction of arrival estimation under different signal-to-noise ratios according to an embodiment of the present invention, wherein the signal-to-noise ratio ranges from 0 dB to 40 dB. Figure 3 The comparison results of the root mean square error and CRLB obtained by performing 1000 Monte Carlo simulation experiments in two cases, namely, directly using the MUSIC algorithm and using the method of this embodiment for error estimation and compensation, are given, where CRLB represents the Cramer-Rao lower bound of the corresponding direction of arrival estimation, which serves as a reference standard for estimation performance. The root mean square error calculation formula is:

[0124]

[0125] In the formula, θ n represents the estimated position of the nth experiment, and N represents the total number of experiments.

[0126] from Figure 3 It can be seen that under the same conditions, compared with directly using the MUSIC algorithm to estimate the direction of arrival, the method of this embodiment has significantly improved effect and can achieve the Cramer-Rao lower bound.

[0127] Figure 4 This is a diagram showing the performance results of estimating the direction of arrival of a new target at different signal-to-noise ratios after the array error is compensated according to an embodiment of the present invention. Figure 4The following table shows the comparison results of the root mean square error (RMS) and CRLB obtained by performing 1000 Monte Carlo simulation experiments under different signal-to-noise ratios (the signal-to-noise ratio range is 0dB to 40dB) and using the MUSIC algorithm to estimate the direction of arrival for a new angle of 15°. Figure 4 It can be seen that for estimating the new angle, under the same conditions, compared with directly using the MUSIC algorithm for DOA estimation, the estimation effect is significantly improved by using the MUSIC algorithm after compensating for the array error, and when the signal-to-noise ratio is high, the Cramer-Rao lower bound can be achieved.

[0128] This embodiment verifies through simulation that this method has good estimation accuracy when there are amplitude and phase errors in the array, and can achieve the corresponding Cramer-Rao lower bound.

[0129] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, several simple deductions or substitutions can be made without departing from the concept of the present invention, which should be regarded as falling within the protection scope of the present invention.

Claims

1. A method for jointly estimating amplitude and phase error and direction of arrival based on Gauss-Newton method, characterized in that: Includes steps: Constructing a likelihood function of array element amplitude error, array element phase error and wave direction of arrival according to the array observation model, and calculating a cost function according to the likelihood function, transforming the parameter estimation problem of the likelihood function into a parameter optimization problem, and obtaining an objective function; The MUSIC algorithm is used to calculate the initial value of the direction of arrival in the objective function, and the initial value of the array element amplitude error and the initial value of the array element phase error are set; According to the initial value of the direction of arrival, the initial value of the array element amplitude error and the initial value of the array element phase error, the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error and the direction of arrival to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error and a estimated value of the direction of arrival.

2. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 1, characterized in that: According to the array observation model, a likelihood function of array element amplitude error, array element phase error and wave direction of arrival is constructed, and a cost function is calculated according to the likelihood function, and the parameter estimation problem of the likelihood function is transformed into a parameter optimization problem to obtain an objective function, including: Constructing a likelihood function of array element amplitude error, array element phase error and wave direction of arrival based on the array received signal of the uniform linear array in the array observation model; Taking the negative logarithm of the likelihood function and removing the constant term to obtain a cost function regarding array element amplitude error, array element phase error and direction of arrival; Based on the cost function, the parameter estimation problem of the likelihood function is transformed into a parameter optimization problem to obtain an objective function.

3. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 2, characterized in that: The array receives the signal: x(t)=ΓA(θ)s(t)+v(t),t=1,2,…,N; Where x(t) represents the array receiving signal at time t, A(θ) = [a(θ1), a(θ2), …, a(θ K )] represents the array manifold, j represents the imaginary unit, d represents the array element spacing, λ represents the wavelength, θ k represents the angle of the kth source, θ=[θ1,θ2,…,θ K ] T represents the direction of arrival, K represents the number of signal sources, M represents the number of array elements, s(t)=[s1(t),s2(t),…,s K (t)] T , represents the kth signal at time t, a tk represents the amplitude of the kth incoming signal at time t, φ tk represents the phase of the kth incoming signal at time t, v(t) = [v1(t), v2(t), …, v M (t)] T represents noise, v(t)~CN(0,Q v ) means the mean is zero and the covariance matrix is ​​Q v The complex circularly symmetric Gaussian white noise is N, N represents the number of snapshots, Γ represents the array amplitude and phase error matrix; taking the amplitude and phase error of the last array element as a reference, the array amplitude and phase error matrix is ​​expressed as a diagonal matrix: ρ m It represents the ratio of the amplitude error of the mth array element to the Mth array element, It represents the difference between the phase error of the mth array element and the Mth array element.

4. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 3, characterized in that: The likelihood function is: in, ρ=[ρ1,ρ2,…,ρ M-1 ] T represents the array element amplitude error, represents the array element phase error, I N represents the N-order identity matrix, represents the Kronecker product, and The cost function is: in, The objective function is: in, They represent the estimated value of direction of arrival, the estimated value of array element amplitude error, and the estimated value of array element phase error respectively.

5. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 1, characterized in that: The MUSIC algorithm is used to calculate the initial value of the direction of arrival in the objective function, including: The covariance matrix is ​​obtained by calculating the autocorrelation of the array element receiving signal matrix, and the covariance is subjected to singular value decomposition, and the covariance matrix after the singular value decomposition is divided into the form of the sum of two matrices according to the number of information sources, and the eigenvectors corresponding to MK eigenvalues ​​are obtained, where M represents the number of array elements and K represents the number of information sources; Constructing a space spectrum function using the eigenvectors corresponding to the MK eigenvalues; According to the spatial spectrum function, the angles corresponding to the first K larger peaks of the spatial spectrum are used as initial values ​​of the direction of arrival.

6. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 5, characterized in that: The covariance matrix divided into the form of the sum of two matrices is: R XX =U S Σ S IN S H +U V Σ V IN V H ; Among them, Σ S is the diagonal matrix consisting of the first K larger eigenvalues ​​of Σ, U S is the eigenvector corresponding to the K largest eigenvalues, Σ V is the diagonal matrix consisting of the last MK smaller eigenvalues ​​of Σ, U V is the eigenvector corresponding to the MK smaller eigenvalues, Σ is the diagonal matrix formed by the M eigenvalues ​​obtained by performing singular value decomposition on the covariance matrix; The spatial spectrum function is: Among them, P MUSIC (θ) represents the spatial spectrum, j represents the imaginary unit, d represents the array element spacing, λ represents the wavelength, and M represents the number of array elements; The initial value of the direction of arrival is: Wherein, K represents the number of information sources.

7. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 1, characterized in that: Set the initial value of the array element amplitude error and the initial value of the array element phase error, including: Set the initial value of the array element amplitude error to: Wherein, M represents the number of array elements; Set the initial value of the array element phase error to:

8. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 1, characterized in that: According to the initial value of the direction of arrival, the initial value of the array element amplitude error, and the initial value of the array element phase error, the Gauss-Newton method is used to jointly estimate the array element amplitude error, the array element phase error, and the direction of arrival to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error, and an estimated value of the direction of arrival, including: The Gauss-Newton method is used to obtain the iterative formulas for the direction of arrival and amplitude and phase errors; Based on the iterative formula, iteration is performed using the initial value of the direction of arrival, the initial value of the array element amplitude error, and the initial value of the array element phase error. When the difference between the cost functions of two adjacent iterations is less than a preset accuracy, the iteration is stopped to obtain an estimated value of the array element amplitude error, an estimated value of the array element phase error, and an estimated value of the direction of arrival.

9. The method for jointly estimating amplitude-phase error and direction of arrival based on Gauss-Newton method according to claim 8, characterized in that: The iteration formula is: in,(·) (q) represents the value after q iterations, Re(·) means taking the real part, I N represents the N-order identity matrix, represents the Kronecker product, Q v represents the covariance matrix of complex-valued circularly symmetric Gaussian white noise, Jac(α) is The Jacobian matrix of α is: in, The array element amplitude error estimation value, array element phase error estimation value and wave direction of arrival estimation value are: in, They represent the estimated value of direction of arrival, the estimated value of array element amplitude error, and the estimated value of array element phase error respectively.

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