Adaptive beamforming method based on joint reconstruction of amplitude and phase errors and covariance matrix

Abstract

The present invention discloses an adaptive beamforming method based on the joint reconstruction of amplitude-phase error and covariance matrix, comprising: establishing a sparse representation model of interference signal power when there are amplitude-phase errors in array channels, reconstructing the real interference plus noise covariance matrix into a joint estimation of amplitude-phase error and interference power, and equivalently expressing it as an improved basis pursuit denoising problem; using the alternating direction multiplier method and the penalty function method to perform variable separation and objective function reduction on the problem, thereby achieving iterative solution of amplitude-phase error and interference power; using the estimated amplitude-phase error and interference power to reconstruct the interference plus noise covariance matrix, and generating beamforming weights based on the maximization signal-to-interference-noise ratio criterion to achieve robust adaptive beamforming. The present invention can effectively achieve robust adaptive beamforming when there are amplitude-phase errors between array channels, and effectively solves the performance degradation problem caused by the limited number of snapshots in practical applications.

Description

Technical Field

[0001] The present invention belongs to the field of array signal processing, and in particular relates to an adaptive beamforming method based on joint reconstruction of amplitude and phase errors and covariance matrix (INCM). Background Art

[0002] Adaptive beamforming technology, by properly setting the weights of array antennas, can form a high-gain beam in the direction of the desired signal and a null in the direction of interference, effectively improving the signal-to-interference-plus-noise ratio (SINR) of the output signal. It can adapt to environmental changes by updating weights in real time and is currently widely used in many fields, including radar, communications, medical imaging, and integrated communications and perception. The Capon beamformer is theoretically optimal when the array structure and the covariance matrix of the received data are accurately known. However, in practical applications, factors such as amplitude and phase errors between array channels, a limited number of snapshots, and the complexity of interfering signals lead to discrepancies between the actual desired signal's steering vector and the interference plus noise covariance matrix (INCM) and the theoretical values, severely degrading the performance of the adaptive beamformer.

[0003] In recent years, methods based on interference-plus-noise covariance matrix (INCM) reconstruction have become a research hotspot. These methods utilize sparse reconstruction techniques, exploiting the spatial sparsity of interference signals, to effectively improve the performance of adaptive beamformers with a limited number of snapshots. However, when amplitude and phase errors exist between array channels, the output signal-to-interference-plus-noise ratio (SINR) performance of the INCM reconstruction method degrades dramatically. Furthermore, the INCM reconstruction method is computationally complex and slow. Summary of the Invention

[0004] The object of the present invention is to provide an adaptive beamforming method based on joint reconstruction of amplitude and phase errors and covariance matrix.

[0005] The technical solution adopted by the present invention is: an adaptive beamforming method based on joint reconstruction of amplitude and phase error and interference plus noise covariance matrix, characterized by the following specific implementation steps:

[0006] Step 1: Based on the uniform linear array structure, a nominal steering vector is constructed under the far-field narrowband assumption, and an echo signal model is established when the array has amplitude and phase errors.

[0007] Step 2: Discretize the spatial domain and establish a sparse signal model when the array has amplitude and phase errors, based on the sparse distribution of interference signals in the spatial domain.

[0008] Step 3: Based on the sparse signal model, a model for jointly solving the array amplitude and phase errors and interference power is constructed;

[0009] Step 4: Use the alternating direction multiplication method and penalty function method to perform variable separation and objective function reduction on the joint solution model of array amplitude and phase error and interference power, realize the iterative solution of amplitude and phase error and interference power, and reconstruct the interference plus noise covariance matrix using the solved amplitude and phase error and interference power;

[0010] Step 5: Based on the reconstructed interference plus noise covariance matrix, the beamforming weights are generated based on the maximization signal-to-interference-and-noise ratio criterion to achieve robust adaptive beamforming.

[0011] Compared with the existing technology, the present invention has the following significant advantages: it can accurately estimate the array amplitude and phase errors and the interference signal power, and can effectively solve the performance degradation problem caused by the limited number of snapshots in practical applications; at the same time, compared with the traditional robust beamforming algorithm, its output SINR performance is significantly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 This is a curve diagram of the output SINR changing with the input interference-noise ratio (INR) when the signal-to-noise ratio (SNR) is 25dB in a simulation of an example of the present invention.

[0013] Figure 2 This is a curve diagram of the change of output SINR with input SNR when INR=30dB in an example simulation of the present invention.

[0014] Figure 3 This is a curve diagram of the output SINR changing with the number of snapshots L when SNR=25dB and INR=30dB in an example simulation of the present invention.

[0015] Figure 4 This is a curve diagram of the output SINR changing with the input amplitude error when SNR=25dB and INR=30dB in a simulation of an example of the present invention.

[0016] Figure 5 This is a curve diagram of the output SINR changing with phase error when SNR=25dB and INR=30dB in a simulation of an example of the present invention. DETAILED DESCRIPTION

[0017] The present invention proposes an adaptive beamforming method based on joint reconstruction of amplitude-phase error and covariance matrix (INCM). First, a sparse representation model for interference signal power in the presence of array channel amplitude-phase error is established. The reconstruction of the true interference-plus-noise covariance matrix (INCM) is converted into a joint estimation of the amplitude-phase error and interference power, which is equivalently formulated as an improved basis pursuit denoising problem. The problem is then subjected to variable separation and objective function reduction using the alternating direction multiplier method (ADMM) and penalty function method, achieving an iterative solution for the amplitude-phase error and interference power. Finally, the estimated amplitude-phase error and interference power are used to reconstruct the interference-plus-noise covariance matrix (INCM), and beamforming weights are generated based on the criterion of maximizing the signal-to-interference-noise ratio, achieving robust adaptive beamforming.

[0018] The specific implementation steps of the present invention are as follows:

[0019] 1. Uniform linear array structure, constructing nominal steering vectors under the assumption of far-field narrowband, and establishing an echo signal model when the array has amplitude and phase errors.

[0020] Under far-field narrowband conditions, it is assumed that the receiving array contains M antennas arranged as a uniform linear array, and the spacing between the antennas is , is the wavelength, and there is no mutual coupling between the array elements;

[0021] Based on the uniform linear array structure, the nominal steering vector is constructed under the assumption of far-field narrowband. ( represents an M×1 dimensional complex matrix), the specific expression is:

[0022]

[0023] in, is the normalized spatial frequency, is the arrival angle of the echo;

[0024] Considering the amplitude and phase errors in the array channels, the echo signal is sampled in the time domain, L snapshots are taken, and the number of sampling points in the normalized spatial frequency range [-1, 1] is set to . Sampling to obtain L snapshot echo signals ( represents an M×L dimensional complex matrix) is expressed as:

[0025]

[0026] in It is Hadamard. 、 and represent the desired signal, interference, and noise respectively, represents an M×L dimensional complex matrix, and represent the complex amplitude vectors of the desired signal and the kth interference respectively ( ), is the number of interferences, represents an L×1 dimensional complex matrix; is the normalized spatial frequency of the desired signal, is the normalized spatial frequency of the interference signal; is the amplitude and phase error vector of the array channel, Represents an M×1 dimensional complex matrix, where each element is: , is the amplitude error on each antenna, is the phase error; is an additive complex Gaussian white noise matrix with a mean of 0 and a covariance matrix of The complex Gaussian distribution of That is, the echo signal model when there are amplitude and phase errors in the array.

[0027] 2. Discretize the spatial domain and, based on the sparse distribution of interference signals in the spatial domain, establish a sparse signal model when the array has amplitude and phase errors.

[0028] Construct a spatial steering vector dictionary based on the receiving array steering vectors ( represent dimensional complex matrix) is specifically:

[0029]

[0030] in, Indicates the Normalized spatial frequencies, , is the number of samples of normalized spatial frequency;

[0031] Since the interference is sparsely distributed in the spatial domain, when the echo signal does not contain the desired signal, the echo signal received by the array can be sparsely represented as:

[0032]

[0033] Where, is a vector of magnitude and phase error The constructed diagonal matrix, represent -dimensional complex matrix whose diagonal elements form the vector ,Right now , represents an operator that forms a column vector from the diagonal elements of a matrix, is the interference complex amplitude matrix, represent -dimensional complex matrix, This is the sparse signal model.

[0034] 3. Based on the sparse signal model, a model for jointly solving the array amplitude and phase error and interference power problem is constructed.

[0035] Multiply both sides of the sparse signal model by the inverse matrix of the amplitude and phase error , then can be rewritten as

[0036]

[0037] Since the interference and noise are uncorrelated, and the noise received by each array element is orthogonal, according to the formula Can get

[0038]

[0039] Remember the matrix ( represent -dimensional complex matrix), ( represent dimensional complex matrix), formula Can be rewritten as

[0040]

[0041] Pair Perform vectorization processing and get:

[0042]

[0043] in, represents the Khatri-Rao product, represents the vectorized operator, Vector-oriented dictionary The conjugate of , ;

[0044] Since the interference is sparsely distributed in the airspace, the present invention uses the interference power Sum amplitude and phase error The joint solution problem model is equivalently expressed as follows:

[0045]

[0046] in, and Respectively and the F-norm, is the regularization parameter.

[0047] 4. The alternating direction multiplier method (ADMM) and penalty function method are used to separate variables and reduce the objective function, so as to iteratively solve the amplitude and phase errors and interference power. The interference plus noise covariance matrix (INCM) is reconstructed using the estimated amplitude and phase errors and interference power.

[0048] Implementing problem solving based on ADMM framework By performing variable separation optimization and reducing the quartic objective function to a quadratic function by introducing auxiliary variables and utilizing a penalty function method, the present invention proposes a joint estimation method for amplitude and phase errors and interference power.

[0049] Defining auxiliary variables , will be can be rewritten as:

[0050]

[0051] According to the ADMM principle, the problem Can be transformed into The unconstrained optimization problem of the objective function is:

[0052]

[0053] in, is the Lagrange multiplier matrix, is the penalty factor, , represent -dimensional complex matrix;

[0054] In the question middle, It is a variable The present invention introduces an auxiliary variable , reducing the quartic objective function to a quadratic objective function. So, the problem Rewritten as:

[0055]

[0056] For the problem , the penalty function method is used to constrain the equality Expressed as a penalty function term in the objective function, it is transformed into the following problem:

[0057]

[0058] in, is the penalty factor, .

[0059] To avoid problems Falling into a false optimal solution , introducing an additional penalty term into its objective function. So, the problem Converted into the following form:

[0060]

[0061] in, and Both are penalty factors.

[0062] Using ADMM technology to solve the problem Perform iterative solution to iteratively solve the interference power and equivalent amplitude and phase errors , the estimated amplitude and phase errors are , and then the interference plus noise covariance matrix (INCM) is reconstructed using the solved amplitude and phase errors and interference power.

[0063] 5. The specific method of iteratively solving the final quadratic objective function using ADMM technology is:

[0064] Step 5.1: Fixation , , and ,in Indicates the Variables after iterations, update variables The optimization problem is expressed as:

[0065]

[0066] question It is an unconstrained convex quadratic programming problem. According to the first-order optimal condition, its closed-form optimal solution is:

[0067]

[0068] Step 5.2: Fixed , , and , update the variable The optimization problem is expressed as:

[0069]

[0070] make ,question It can be equivalently expressed as:

[0071]

[0072] Mode It is a basis pursuit denoising problem, which is solved by the iterative soft threshold (IST) algorithm. The iterative process is:

[0073]

[0074] Among them, the formula All operations in are performed element-wise. , is a proximity parameter.

[0075] Step 5.3: Fix , , and , update the variable The optimization problem is expressed as:

[0076]

[0077] question It is an unconstrained convex quadratic programming problem. According to the first-order optimal condition, its closed-form optimal solution is:

[0078]

[0079] in, as well as , and The matrices and The (m, i)th element of , .

[0080] Step 5.4: Fixing , , and , update the variable The optimization problem is expressed as:

[0081]

[0082] question It is an unconstrained convex quadratic programming problem. According to the first-order optimal condition, its closed-form optimal solution is:

[0083]

[0084] in, , as well as .

[0085] Step 5.5: Lagrange dual variables Update as follows:

[0086]

[0087] In summary, through iterative update 、 、 、 and , until the set number of iterations is reached, the interference power and equivalent amplitude and phase error obtained in the last iteration are the interference power finally solved and equivalent amplitude and phase errors .

[0088] 6. The specific method for determining the interference plus noise covariance matrix (INCM) is:

[0089] According to the estimated interference power and the estimated amplitude and phase errors , reconstruct the interference covariance matrix, specifically:

[0090]

[0091] The obtained sampled interference plus noise covariance matrix (INCM) is:

[0092]

[0093] use minus , obtain the sampled noise covariance matrix .

[0094] according to , we get an estimate of the noise power:

[0095]

[0096] in, represents the trace of the matrix.

[0097] The estimated value of the noise covariance matrix is ​​determined as:

[0098]

[0099] according to and Reconstruct the interference plus noise covariance matrix (INCM):

[0100]

[0101] 7. Based on the reconstructed interference plus noise covariance matrix (INCM), beamforming weights are generated based on the criterion of maximizing the signal-to-interference-noise ratio. The specific method for implementing robust adaptive beamforming is as follows:

[0102] Receive beamformer output ( represents a 1×L-dimensional complex matrix) composed of the received signal matrix ( represents an M×L dimensional complex matrix) after beamforming weight vector ( represents an M×1 dimensional complex matrix) spatial domain filtering, that is:

[0103]

[0104] According to the maximization SINR criterion, the optimal weight vector is obtained as:

[0105]

[0106] in, is the power of the desired signal, is the ideal interference plus noise covariance matrix (INCM), is the steering vector of the desired signal, 、 、 represent the desired signal, interference signal and noise respectively.

[0107] Mode It is equivalent to the minimum variance distortionless response (MVDR) beamformer, and its optimal beamforming weight is:

[0108]

[0109] When there are amplitude and phase errors in the array channels, Eq. The signal steering vector in is corrected to At this time, the corresponding optimal adaptive beamformer is:

[0110]

[0111] The estimated value and Substitution , and get the adaptive beam weights:

[0112]

[0113] The present invention accurately estimates array amplitude and phase errors and interference signal power, effectively addressing performance degradation caused by a limited number of snapshots in practical applications. Furthermore, compared with traditional robust beamforming algorithms, its output signal-to-interference-and-noise ratio (SIGINI) performance is significantly improved. The present invention is further illustrated by an example, which is simulated using Matlab software.

[0114] Example

[0115] 1. Array parameter settings

[0116] Number of array elements , array element spacing ,wavelength , the spatial domain is uniformly discretized into ,Right now , For the The nominal steering vector dictionary of the array is thus , No. The nominal steering vector for each discrete azimuth direction is ,in is the normalized spatial frequency. Construct an amplitude and phase error matrix that obeys uniform distribution , among which The amplitude and phase error of the number of array elements is expressed as ,in , .

[0117] 2. Receive signal settings

[0118] is the interference complex amplitude matrix, the received noise signal Satisfying the complex Gaussian distribution, the array received signal is expressed as , .

[0119] 3. Simulation test of the performance of this method

[0120] The adaptive beamforming method based on joint reconstruction of amplitude and phase errors and the covariance matrix was compared with four conventional algorithms: the interference-plus-noise covariance matrix (INCM) reconstruction (RAB) algorithm, single-snapshot signal fitting, single-snapshot signal fitting followed by multiple averaging, and sampled covariance matrix inversion (SMI) to verify the performance of the GP-INCM joint reconstruction algorithm. Furthermore, an optimal MVDR beamformer with precisely known amplitude and phase errors was presented as a benchmark for performance comparison. The impact of input interference-to-noise ratio (INR), input signal-to-noise ratio (SNR), number of snapshots (L), and amplitude and phase errors on the output SINR was analyzed.

[0121] Set the directions of the two interferences to be and , the direction of the desired signal is .

[0122] Figure 1 For SNR=25dB, the number of snapshots is L=28, and the amplitude and phase errors are , When , the output SINR changes with the input INR.

[0123] Figure 2 For INR=30dB, the number of snapshots is L=28, and the amplitude and phase errors are , When , the output SINR changes with the input SNR.

[0124] Figure 3 For SNR=25dB, INR=30dB, the amplitude and phase errors are , When , the output SINR changes with the number of snapshots L.

[0125] Figure 4 For SNR=25dB, INR=30dB, the number of snapshots is L=28, and the phase error When , the output SINR changes with the input amplitude error.

[0126] Figure 5 For SNR=25dB, INR=30dB, the number of snapshots is L=28, the amplitude and phase errors are When , the output SINR changes with the phase error.

[0127] As can be seen from the above, the adaptive beamforming method based on the joint reconstruction of amplitude and phase errors and the interference plus noise covariance matrix (INCM) outperforms the traditional beamforming algorithm in the case of low drying ratio, limited number of snapshots, and amplitude and phase errors in the array, outputs a higher SINR, and achieves robust adaptive beamforming.

[0128] The present invention simulates the performance of an adaptive beamforming method based on joint reconstruction of amplitude and phase errors and the interference plus noise covariance matrix (INCM) and compares it with other methods. The feasibility and superiority of the present invention can be demonstrated, and it can provide higher SINR performance and better robustness to changes in amplitude and phase errors.

Claims

1. An adaptive beamforming method based on joint reconstruction of amplitude and phase errors and covariance matrix, characterized in that: The specific implementation steps are as follows: Step 1: Based on the uniform linear array structure, a nominal steering vector is constructed under the far-field narrowband assumption, and an echo signal model is established when the array has amplitude and phase errors. Step 2: Discretize the spatial domain and establish a sparse signal model when the array has amplitude and phase errors, based on the sparse distribution of interference signals in the spatial domain. Step 3: Based on the sparse signal model, a model for jointly solving the array amplitude and phase errors and interference power is constructed; Step 4: Use the alternating direction multiplier method and penalty function method to separate variables and reduce the objective function of the joint solution model of array amplitude and phase error and interference power, realize the iterative solution of amplitude and phase error and interference power, and reconstruct the interference plus noise covariance matrix using the solved amplitude and phase error and interference power. The specific method is as follows: Defining auxiliary variables , the joint solution model of array amplitude and phase error and interference power is rewritten as: ， According to the ADMM principle, the rewritten array amplitude and phase error and interference power joint solution model is transformed into The unconstrained optimization problem of the objective function is: ,in, is the Lagrange multiplier matrix, is the penalty factor, , represent -dimensional complex matrix; Introducing auxiliary variables , the quartic objective function Reduced to a quadratic objective function, that is: ， The penalty function method is used to constrain the equality Expressed as a penalty function term in the objective function, the quadratic objective function is transformed into the following problem: ， in, is the penalty factor, ; Introducing the penalty term, the quadratic objective function is transformed into the following form: ， in, and Both are penalty factors; The ADMM technique is used to iteratively solve the final quadratic objective function to obtain the estimated interference power and equivalent amplitude and phase errors , the estimated amplitude and phase errors are , and then reconstruct the interference plus noise covariance matrix using the solved amplitude and phase errors and interference power; Step 5: Based on the reconstructed interference plus noise covariance matrix, the beamforming weights are generated based on the maximization signal-to-interference-and-noise ratio criterion to achieve robust adaptive beamforming.

2. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 1, characterized in that: Under the far-field narrowband assumption, the specific method for constructing the nominal steering vector based on the uniform linear array structure and establishing the echo signal model when the array has amplitude and phase errors is as follows: Under far-field narrowband conditions, the receiving array contains M antennas arranged as a uniform linear array, and the spacing between the antennas is , is the wavelength, and there is no mutual coupling between the array elements; Based on the uniform linear array structure, the nominal steering vector is constructed under the assumption of far-field narrowband. , Represents an M×1 dimensional complex matrix, and the specific expression is: ， in, is the normalized spatial frequency, is the arrival angle of the echo; Considering the amplitude and phase errors in the array channels, the echo signal is sampled in the time domain, L snapshots are taken, and the number of sampling points in the normalized spatial frequency range [-1, 1] is set to , sampling to obtain L snapshot echo signals Expressed as: ， in, For Hadamard, 、 and represent the desired signal, interference, and noise respectively, represents an M×L dimensional complex matrix, and denote the complex amplitude vectors of the desired signal and the kth interference, respectively. , is the number of interferences, represents an L×1 dimensional complex matrix; is the normalized spatial frequency of the desired signal, is the normalized spatial frequency of the interference signal; is the array amplitude and phase error, Represents an M×1 dimensional complex matrix, where each element is: , is the amplitude error on each antenna, is the phase error; is an additive complex Gaussian white noise matrix with a mean of 0 and a covariance matrix of The complex Gaussian distribution of That is, the echo signal model when there are amplitude and phase errors in the array.

3. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 1, characterized in that: Discretize the spatial domain. Based on the sparse distribution of interference signals in the spatial domain, the specific method to establish a sparse signal model when the array has amplitude and phase errors is as follows: Construct a spatial steering vector dictionary based on the receiving array steering vectors Specifically: ， in, Indicates the Normalized spatial frequencies, , is the number of samples of normalized spatial frequency, represent -dimensional complex matrix; When the echo signal does not contain the desired signal, the echo signal received by the array is sparsely represented as: ， Where, is a result of the array amplitude and phase error The constructed diagonal matrix, represent -dimensional complex matrix, is the interference complex amplitude matrix, represent -dimensional complex matrix, This is the sparse signal model.

4. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 3, characterized in that: Based on the sparse signal model, the specific method of constructing a model for jointly solving the array amplitude and phase error and interference power problem is as follows: Multiply both sides of the sparse signal model by the inverse matrix of the amplitude and phase error , then the sparse signal model can be rewritten as: ， According to the rewritten sparse signal model, we can get ， Remember the matrix , represent -dimensional complex matrix, , represent dimensional complex matrix, the sparse signal model is rewritten as: ， The rewritten sparse signal model is vectorized to obtain: ， in, represents the Khatri-Rao product, represents the vectorized operator, Vector-oriented dictionary The conjugate of , ; Constructing array amplitude and phase errors and interference power The joint problem solving model is as follows: ， in, and Respectively and the F-norm, is the regularization parameter.

5. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 1, characterized in that: The specific method of iteratively solving the final quadratic objective function using ADMM technology is: Step 6.1: Fixation , , and ,in Indicates the Variables after iterations, update variables The optimization problem is expressed as: ， According to the first-order optimal condition, its closed-form optimal solution is: ， Step 6.2: Fixing , , and , update the variable The optimization problem is expressed as: ， make , update the variable The optimization problem is equivalently expressed as: ， Iterative soft thresholding algorithm is used to update the variables The optimization problem is solved as follows: ， in, , is a proximity parameter; Step 6.3: Fixing , , and , update the variable The optimization problem is expressed as: ， According to the first-order optimal conditions, update the variables The closed-form optimal solution to the optimization problem is: ， in, as well as , and The matrices and The (m, i)th element of , ; Step 6.4: Fixing , , and , update the variable The optimization problem is expressed as: ， According to the first-order optimal conditions, update the variables The closed-form optimal solution to the optimization problem is: ， in, , as well as ; Step 6.5: Lagrange Dual Variables Update as follows: ， Return to step 6.1 until the set number of iterations is reached. The interference power and equivalent amplitude and phase error obtained in the last iteration are the estimated interference power. and equivalent amplitude and phase errors .

6. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 5, characterized in that: The specific method for determining the interference plus noise covariance matrix is: Based on the estimated interference power and the estimated amplitude and phase errors , reconstruct the interference covariance matrix, specifically: ， The obtained sampling interference plus noise covariance matrix is: ， use minus , obtain the sampled noise covariance matrix ; according to , we get an estimate of the noise power: ， in, represents the trace of the matrix; Determine the estimate of the noise covariance matrix as: ， according to and Reconstruct the interference plus noise covariance matrix: 。 7. The adaptive beamforming method based on joint reconstruction of amplitude and phase error and covariance matrix according to claim 6, characterized in that: Based on the reconstructed interference plus noise covariance matrix, the beamforming weights are generated based on the maximization signal-to-interference-noise ratio criterion. The specific method for implementing robust adaptive beamforming is as follows: Receive beamformer output By the received signal matrix After beamforming weight vector Spatial filtering is used to obtain represents a 1×L-dimensional complex matrix, represents an M×L dimensional complex matrix, Represents an M×1 dimensional complex matrix, that is: ， According to the criterion of maximizing the signal-to-interference-noise ratio, the optimal weight vector is obtained as: ， in, is the power of the desired signal, is the interference plus noise covariance matrix, is the steering vector of the desired signal, 、 、 represent the desired signal, interference signal and noise respectively; Maximizing the signal-to-interference-noise ratio is equivalent to the minimum variance distortionless response beamformer, and the optimal beamforming weights are determined as: ， When there are amplitude and phase errors in the array channel, the signal steering vector is corrected to , the corresponding optimal adaptive beamformer is: ， The estimated value and Substituting into the above formula, we can get the adaptive beam weight: 。

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