Robust adaptive beamforming method for small sample size of array radar and communication systems

Abstract

The application discloses a small sample robust adaptive beamforming method of array radar and communication system and belongs to the field of array signal processing. In view of the problem that the performance of an array with amplitude and phase errors obviously decreases under limited training samples, a spatial domain sparse model of interference power is established first, the amplitude and phase errors existing in the array channel and the power of interference signals are estimated, and the covariance matrix of the received signals is reconstructed; then, the above problem is solved by an alternating optimization algorithm; afterwards, the estimated amplitude and phase errors and the interference power are used to further obtain the estimated value of the covariance matrix of the received signals. Finally, the weight value of adaptive beamforming is calculated under the minimum variance distortionless response criterion, and the robust adaptive beamforming is realized. The method effectively improves the robustness of the adaptive beamforming of a non-ideal linear array under small samples.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing, specifically a robust adaptive beamforming method for array radar and communication systems under small sample conditions. Background Technology

[0002] Adaptive beamforming has become a key technology in many fields such as radar and communications due to its superior ability to enhance desired signals and suppress directional interference. Although Capon beamformers can achieve SINR optimization under ideal prior information, practical applications often face non-ideal physical factors such as limited snapshot number and amplitude and phase errors in array channels. These factors lead to deviations in steering vector mismatch and covariance matrix estimation. Traditional methods based on the sampling covariance matrix are prone to self-cancellation of the desired signal, resulting in severe performance degradation.

[0003] Currently, a representative method in existing technologies is to achieve efficient estimation of the interference plus noise covariance matrix (INCM) based on sparse reconstruction theory, thereby realizing robust adaptive beamforming. This type of method utilizes the sparse characteristics of the spatial distribution of interference signals, accurately estimating the steering vector and power of the interference components through sparse reconstruction algorithms, and then estimating the INCM, significantly improving the robustness of adaptive beamforming under small sample conditions. However, existing INCM reconstruction methods still have some technical shortcomings. Specifically, these methods are generally based on the assumption of an ideal array manifold, i.e., the pre-defined array amplitude and phase responses are precisely known. However, in practical engineering applications, there are inevitably non-ideal factors such as amplitude and phase inconsistencies, positional perturbations, and mutual coupling effects between array element channels, leading to steering vector model mismatch. When the aforementioned array amplitude and phase errors exist, the overcomplete dictionary on which sparse reconstruction relies will be distorted, the true azimuth of the interference source will deviate from the pre-defined discretized grid, causing a sharp decrease in the accuracy of the reconstructed INCM, a significant deterioration in the output signal-to-interference-plus-noise ratio performance of the beamformer, and difficulty in guaranteeing its robustness. Summary of the Invention

[0004] The purpose of this invention is to propose a robust adaptive beamforming method for array radar and communication systems under small sample conditions.

[0005] The technical solution for achieving the objective of this invention is as follows:

[0006] A robust adaptive beamforming method for array radar and communication systems under small sample conditions includes the following steps:

[0007] Step 1: Discretize the spatial domain into a grid and construct a sparse regularized optimization problem model for signals with amplitude and phase errors under small sample sizes.

[0008] Step 2: The sparse regular optimization problem is transformed into a constrained optimization problem that is easy to solve using the augmented Lagrangian technique;

[0009] Step 3: Iteratively solve using the alternating direction multiplier method to obtain estimates of the interference power vector and the equivalent amplitude and phase error vector, and finally estimate the interference plus noise covariance matrix of the received signal based on these estimates.

[0010] Step 4: Calculate the formation weights for robust adaptive beamforming by combining the minimum variance distortionless response criterion with the above-mentioned interference plus noise covariance matrix estimate.

[0011] Furthermore, the specific method for constructing the corresponding sparse regularized optimization problem model in step 1 is as follows:

[0012]

[0013] in, For the interference power vector, , The number of grids used to perform a uniform discretization operation over the spatial angular range. In The initial value range of each element is... , For the interference power vector The diagonal matrix formed ; For regularization parameters, and They represent The F norm, with the superscript 2 indicating the square calculation; and the superscript H indicating the conjugate transpose; For the equivalent amplitude and phase error vector The diagonal matrix formed , M is the number of antenna array elements. The initial value of each of the M elements in the array is 1. , To receive echo signals for the array, L represents the number of echo signal samples sampled in the time domain;

[0014] Spatial guidance vector dictionary Specifically:

[0015] ;

[0016] ;

[0017] j represents the imaginary unit. , ... These are the distances from the 2nd, 3rd, ..., Mth antenna elements to the first antenna element, respectively. After uniformly discretizing the spatial angular range, the first... Azimuth angle at discrete points , The wavelength of the echo signal.

[0018] Furthermore, the specific method for step 2 is as follows:

[0019] ;

[0020] For Lagrange multiplier matrices, The initial values ​​of each element in the array are all , As the first auxiliary variable, , As the first penalty factor, As the second penalty factor; As the second auxiliary variable, , , The initial value of each of the M elements in the array is 1. For the reason The diagonal matrix formed; and They are respectively and The first in One element, The upper bound of the maximum error express Norm.

[0021] Furthermore, in step 3, the specific method for obtaining the estimated values ​​of the interference power vector and the equivalent amplitude and phase error vector through iterative solution using the alternating direction multiplier method is as follows:

[0022] Step 301: Set the variable k, which indicates the number of iterations, and initialize k=1;

[0023] Step 302, Update :

[0024] ;

[0025] Step 303, update the interference power vector :

[0026] ;

[0027] in, It is a neighboring parameter, subscript Represents the first in the vector One element, , The superscript * indicates conjugate operation. Represents the Khatri-Rao product; This indicates finding the absolute value. , For vectorization operators;

[0028] Step 304, Build Update Optimization issues:

[0029] ;

[0030] Using convex optimization algorithm Solve the optimization problem to obtain The value;

[0031] Step 305, Build Update Optimization issues:

[0032]

[0033] Using convex optimization algorithm Solve the optimization problem to obtain The value;

[0034] Step 306, update the Lagrange multiplier matrix. :

[0035]

[0036] Step 307, if Then the iteration ends, and the solution of the last iteration is output. and As estimates of the interference power vector and the equivalent amplitude and phase error vector and Otherwise, let k = k + 1 and return to step 302. This is the preset termination precision.

[0037] Furthermore, the specific method for estimating the interference plus noise covariance matrix of the received signal in step 3 is as follows:

[0038] Step 308, based on the estimated value of the interference power vector The positions of all non-zero elements in Record its corresponding number Azimuth angle at discrete points Further construct the interference angle vector n is The number of all non-zero elements in the array. They are respectively the corresponding ;

[0039] Step 309: Calculate the array manifold matrix of the interference. ,right By taking the reciprocal of each element, we obtain an estimate of the array amplitude and phase error vector. Calculate the interference envelope The estimated value:

[0040]

[0041] The estimated value of the array amplitude and phase error vector The diagonal matrix formed;

[0042] The average power estimate of the interference was further calculated. :

[0043]

[0044] Step 310: Reconstruct the interference covariance matrix :

[0045] ;

[0046] For Hadamard product;

[0047] Step 311, the noise power is estimated as follows:

[0048]

[0049] in, Represents the trace operation of a matrix. ;

[0050] Therefore, the estimated value of the interference plus noise covariance matrix is:

[0051] ;

[0052] Let represent the M-dimensional identity matrix.

[0053] Furthermore, step 4 is specifically implemented as follows:

[0054]

[0055] The angle of arrival of the desired signal. For robust adaptive beamforming weights.

[0056] Due to the adoption of the above technical solution, the beneficial effects of this invention compared with the prior art are as follows:

[0057] This invention not only achieves high-precision joint estimation of array amplitude and phase error and interference power, but also effectively overcomes the performance degradation caused by small samples in practical engineering. Compared with the traditional robust adaptive beamforming strategy, this method can provide superior output signal-to-interference-plus-noise ratio (SINR) performance. Attached Figure Description

[0058] Figure 1 This is a simulation example of the present invention, showing the output SINR as a function of the input interference-to-noise ratio (INR) when the signal-to-noise ratio (SNR) is 10dB.

[0059] Figure 2 This is a simulation example of the present invention, showing the curve of output SINR changing with input INR when SNR=30dB.

[0060] Figure 3 This is a simulation of an example of the present invention, showing the change curve of SINR gain with interference noise ratio SNR when INR=30dB. Detailed Implementation

[0061] The present invention, namely a robust adaptive beamforming method for array radar and communication systems under small sample conditions, is further described below with reference to the accompanying drawings and embodiments. First, the method constructs a sparse model of interference power including array channel amplitude and phase errors, and establishes an optimization problem based on this sparse signal model. Then, the alternating direction multiplier method (ADMM) is used to reduce the order of the objective function of this problem, achieving iterative solutions for the amplitude and phase error vector and the interference power vector. Finally, based on the above estimates, the interference plus noise covariance matrix (INCM) is estimated, and the optimal beam weights are calculated using maximizing the signal-to-interference-plus-noise ratio (SINR) as the criterion, thereby achieving robust beamforming under non-ideal array conditions.

[0062] The specific implementation steps of this invention are as follows:

[0063] Step 1: Based on the far-field narrowband assumption and the linear array structure, generate a theoretical steering vector and establish an echo signal model when the array has amplitude and phase errors.

[0064] Assume the receiving array consists of M uncoupled antennas arranged in a linear array. , ... These represent the distances from the 2nd, 3rd, ..., Mth antenna elements to the first antenna element. Based on the linear array structure, construct the nominal steering vector. for:

[0065] (1)

[0066] in, The angle of arrival of the echo. The wavelength of the echo signal.

[0067] Under the condition of array channel amplitude and phase error, the echo signal is obtained by time-domain sampling. For each sample, the echo signal is represented as:

[0068] (2)

[0069] in, For Hadamard product, , and These represent the desired signal, interference, and noise, respectively. and Representing the desired signal and the first The complex amplitude vector of the interference, Indicates the number of interferences; The angle of arrival of the desired signal. The direction of arrival of the interfering signal. ; The amplitude and phase errors exist between array channels, where each element can be specifically represented as: , This represents the amplitude error on the antenna. This indicates the phase error on the upper antenna; To follow a pattern with a mean of 0, the covariance matrix is: The complex Gaussian distributed additive complex Gaussian white noise matrix. The variance of the noise is represented. Indicated An identity matrix of dimension 1.

[0070] Step 2: Discretize the angular range of the spatial domain into a grid, and construct a spatial steering vector dictionary using the steering vector of the receiving array. Construct a steering vector dictionary in the spatial domain based on the steering vector of the receiving array. Specifically:

[0071] (3)

[0072] in, After uniformly discretizing the spatial angular range, the first... Azimuth angle at discrete points , The number of grids used for the discretization operation;

[0073] Taking advantage of the spatial sparsity of interference signals, the echo signal is sparsely represented in the echo scenario where there is no desired signal as follows:

[0074] (4)

[0075] In the formula, Therefore A diagonal matrix with diagonal elements. This is the interference complex amplitude matrix. Indicates the direction of arrival. The complex amplitude vector of the signal at that point, if Equal to the direction of the interference signal wave hour, ;otherwise for A zero-dimensional vector.

[0076] Step 3: Based on the sparse signal model, multiply both sides of the sparse signal model by the inverse matrix of the amplitude and phase errors. Thus, the echo signal model can be equivalently represented as:

[0077] (5)

[0078] Performing cross-correlation processing on the above signals yields:

[0079] (6)

[0080] in, , By vectorizing both sides of the above equation, we obtain...

[0081] (7)

[0082] in, For vectorization operators, interference power vector , , This indicates the conjugate operation. Represents the Khatri-Rao product;

[0083] Based on the spatial sparsity of the interference, construct the inverse matrix of amplitude and phase error. and interference power vector Problem model:

[0084] (8)

[0085] in, , and They represent , and F norm, This is the regularization parameter.

[0086] Step 4: In order to reduce the order of the complex objective function and reconstruct the interference plus noise covariance matrix, this invention makes appropriate transformations to problem (8) so that ADMM can be used to solve it.

[0087] Step 4.1: Based on the ADMM framework, perform variable separation on the problem and introduce auxiliary variables to reduce the order of the objective function. Define auxiliary variables. , will Rewritten as:

[0088] (9)

[0089] Step 4.2: Introduce Lagrange multipliers Problem (9) is transformed into solving the augmented Lagrange function. Unconstrained optimization problem:

[0090] (10)

[0091] in, For Lagrange multiplier matrices, As a penalty factor, .

[0092] Step 4.3: Introduce auxiliary variables The fourth objective function The objective function is reduced to a quadratic function, thus rewriting problem (10) as follows:

[0093] (11)

[0094] Using penalty functions to constrain equality The relaxation process transforms problem (11) into the following optimization problem:

[0095] (12)

[0096] in, As a penalty factor, .

[0097] Step 4.4: Based on prior knowledge of amplitude error and phase error, determine the amplitude and phase error vectors. Maximum upper bound of error of elements Problem (12) can be transformed into the following problem:

[0098] (13)

[0099] Step 5: Use ADMM to iteratively solve problem (13). The specific method is as follows:

[0100] Step 5.1, Fix , , and ,in Indicates the first Update the variables after the next iteration. The optimization problem is expressed as:

[0101] (14)

[0102] Variables are obtained based on the first-order optimality condition. The update formula is:

[0103] (15)

[0104] Step 5.2, Fixed , , and Update variables The optimization problem is expressed as:

[0105] (16)

[0106] make Problem (16) can be expressed as:

[0107] (17)

[0108] Variables are obtained based on the iterative soft thresholding algorithm. The update formula is:

[0109] (18)

[0110] in, , It is a neighbor parameter. and Representing vectors respectively sum vector The first in One element;

[0111] Step 5.3, Fix , , and Update variables The optimization problem is expressed as:

[0112] (19)

[0113] This problem is a convex quadratic constrained quadratic programming problem. We can directly use existing convex optimization algorithms, such as the second-order cone programming method, to solve it and obtain the variables. The updated value.

[0114] Step 5.4, Fix , , and Update variables The optimization problem is expressed as:

[0115] (20)

[0116] This problem is also a convex quadratic constrained quadratic programming problem. We can directly use existing convex optimization algorithms, such as the second-order cone programming method, to solve for the variables. The updated value.

[0117] Among them, update , , After that, you can get the corresponding , , ;

[0118] Step 5.5, Lagrange dual variables The update formula is:

[0119] (twenty one)

[0120] The above variables are iteratively updated until the algorithm terminates, at which point the final iteration solution is output. and As estimates of the interference power vector and the equivalent amplitude and phase error vector and The initial settings for the algorithm are: and The initial values ​​are randomly generated, and the initial values ​​are all , and The initial value is A column vector of all 1s. The algorithm's iteration termination condition is... ,in For example, the preset termination precision .

[0121] Step 6: Based on the estimated value and The estimated value of the interference plus noise covariance matrix (INCM) is determined by the following method:

[0122] Estimated value based on interference power vector The positions of all non-zero elements in Record its corresponding number Azimuth angle at discrete points Further construct the interference angle vector n is The number of all non-zero elements in the array. They are respectively the corresponding ;

[0123] Calculate the array manifold matrix of the interference ,right By taking the reciprocal of each element, we obtain an estimate of the array amplitude and phase error vector. The interference envelope is calculated using least squares. The estimated value:

[0124] (twenty two)

[0125] The estimated value of the array amplitude and phase error vector The diagonal matrix formed;

[0126] Therefore, the average power estimate of the interference can be expressed as:

[0127] (twenty three) in, Indicates interference envelope middle The value of the element at that position;

[0128] Reconstruct the estimated value of the disturbance covariance matrix. :

[0129] (twenty four)

[0130] According to the received signal and The noise power is estimated to be:

[0131] (25)

[0132] in, Represents the trace operation of a matrix. Therefore, the estimated value of the covariance matrix of the interference plus noise is:

[0133] (26)

[0134] Step 7: Based on the estimated value of INCM And array amplitude and phase error estimates The robust adaptive beamforming weights are determined under the minimum variance distortion-free response criterion. The specific method is as follows:

[0135] Assume the output of the receiving beamformer is ,in This is the spatial beamforming weight vector. According to the minimum variance distortion-free response criterion, when the steering vector of the desired signal... The optimal weights for an adaptive beamformer when there are no errors:

[0136] (27)

[0137] in, It is the ideal INCM.

[0138] Using array amplitude and phase error estimates Steering vector for the desired signal The correction is made, and the corrected actual guide vector is: And using the estimated value of INCM Alternative The robust adaptive beamforming weights are obtained as follows:

[0139] (28)

[0140] Example

[0141] Matlab simulations are used to further illustrate the robust adaptive beamforming method for array radar communication systems under small sample conditions.

[0142] 1. Array parameter settings

[0143] This example uses a uniform linear array, and we set the number of array elements... ,wavelength For the range of airspace angles Discretize the grid uniformly at 1°. ,Right now , For the first The azimuth angles at discrete points. Therefore, the array's steering vector dictionary is: , No. The guide vectors for each discrete azimuth direction are: Assuming the maximum amplitude error is 10% and the maximum phase error is 5°, a uniformly distributed amplitude and phase error matrix can be constructed. , of which The amplitude and phase error of the number of array elements is expressed as: ,in , , Indicates in It is evenly distributed within the range.

[0144] 2. Signal Reception Settings

[0145] The complex amplitude matrix representing the interference signal, and the received noise signal. The array receives a signal that follows a complex Gaussian distribution. .

[0146] 3. Simulation testing of method performance

[0147] The robust adaptive beamforming algorithm based on covariance matrix reconstruction proposed in this invention is compared with two typical algorithms: diagonal loading and single-shot signal fitting. The theoretical upper limit of SINR performance is given when amplitude and phase errors and interference power are assumed to be known, thus verifying the performance of the proposed algorithm. The effects of input interference ratio (INR) and input signal-to-noise ratio (SNR) on output SINR are analyzed separately.

[0148] Two interference angles are randomly generated within the beam sidelobe, and the desired signal direction is... .

[0149] Figure 1 With an SNR of 10dB and a snapshot count of L=28, the amplitude and phase errors are respectively... , The graph shows the change in output SINR as a function of input INR.

[0150] Figure 2 With an SNR of 30dB, a snapshot count of L=28, and amplitude and phase errors of [missing information], [missing information]. , The graph shows the change in SINR gain as a function of input SNR.

[0151] Figure 3 With INR=30dB, shot count L=28, and amplitude and phase errors respectively... , The graph shows the change in output SINR as a function of input SNR.

[0152] Figure 1 The curves showing the output SINR versus input INR for three algorithms are presented. It can be seen that when INR is small, the SINR of the algorithm presented in this invention is higher than that of the two comparative algorithms, diagonal loading and single-shot signal fitting. However, as INR increases, the output SINR of both the proposed method and the single-shot signal fitting method decreases. This is because in the joint sparse reconstruction problem of amplitude and phase errors and interference signals, the estimation accuracy of amplitude and phase errors saturates with increasing INR, resulting in limited interference suppression performance. The diagonal loading method does not perform amplitude and phase error estimation, and its output SINR is not affected by INR, but its performance is limited. Figure 1 As can be seen, when the INR is no greater than 30dB, the output SINR of the method of this invention is higher than that of the two typical comparison algorithms.

[0153] Figure 2 The curves showing the SINR gain versus input INR for the three algorithms are given. It can be seen that... Figure 2 The results and Figure 1 It is a one-to-one correspondence. When the INR is small, the SINR gain of the method of this invention is the largest. When the INR is less than or equal to 15dB, the SINR gain of the method of this invention is very close to the theoretical upper limit, significantly better than the other two comparison methods. As the INR increases, the performance difference between the three methods gradually decreases. When the INR is greater than or equal to 35dB, the diagonal loading method outperforms the method of this invention. Figure 2 As can be seen, when the INR is no greater than 30dB, the SINR gain of the method of this invention is better than that of the two comparison algorithms.

[0154] Figure 3 Curves showing the output SINR as a function of input SNR for the three methods are presented. It can be seen that, with an INR fixed at 30 dB, the output SINR of the method of this invention is higher than that of the two comparative methods as the SNR increases.

[0155] By observing the above experimental results, it can be found that when the array has amplitude and phase errors, and the signal-to-noise ratio (SNR) is fixed, the algorithm of the present invention can output a higher SINR than the traditional algorithm when the interference noise ratio (INR) is low, and the gain for SINR is also higher. When the INR is fixed, the output SINR of the algorithm of the present invention increases with the increase of SNR, and the output SINR is better than the two comparison algorithms.

[0156] Through the performance simulation of adaptive beamforming under the above small sample and the performance comparison with other methods, it can be proved that the algorithm of the present invention is effective in the case of amplitude and phase error of the array, has the ability to provide higher SINR, and has better robustness than similar algorithms.

Claims

1. A robust adaptive beamforming method for array radar and communication systems under small sample conditions, characterized in that, Includes the following steps: Step 1: Discretize the spatial domain into a grid and construct a sparse regularized optimization problem model for signals with amplitude and phase errors under small sample sizes. Step 2: The sparse regular optimization problem is transformed into a constrained optimization problem that is easy to solve using the augmented Lagrangian technique; Step 3: Iteratively solve using the alternating direction multiplier method to obtain estimated values ​​of the interference power vector and the equivalent amplitude and phase error vector, and finally estimate the interference plus noise covariance matrix of the received signal based on these values. Step 4: Calculate the formation weights for robust adaptive beamforming by combining the minimum variance distortionless response criterion with the above-mentioned interference plus noise covariance matrix estimate.

2. The robust adaptive beamforming method for array radar and communication systems under small sample conditions according to claim 1, characterized in that, The specific method for constructing the corresponding sparse regularized optimization problem model in step 1 is as follows: ； in, For the interference power vector, , The number of grids used to perform a uniform discretization operation over the spatial angular range. In The initial value range of each element is... , For the interference power vector The diagonal matrix formed ; For regularization parameters, and They represent The F norm, with the superscript 2 indicating the square calculation; and the superscript H indicating the conjugate transpose; For the equivalent amplitude and phase error vector The diagonal matrix formed , M is the number of antenna array elements. The initial value of each of the M elements in the array is 1. , To receive echo signals for the array, L represents the number of echo signal samples sampled in the time domain; Spatial guidance vector dictionary Specifically: ； ； j represents the imaginary unit. , ... These are the distances from the 2nd, 3rd, ..., Mth antenna elements to the first antenna element, respectively. After uniformly discretizing the spatial angular range, the first... Azimuth angle at discrete points , The wavelength of the echo signal.

3. The robust adaptive beamforming method for array radar and communication systems under small sample conditions according to claim 2, characterized in that, The specific method for step 2 is as follows: ； For Lagrange multiplier matrices, The initial values ​​of each element in the array are all , As the first auxiliary variable, , As the first penalty factor, As the second penalty factor; As the second auxiliary variable, , , The initial value of each of the M elements in the array is 1. For the reason The diagonal matrix formed; and They are respectively and The first in One element, The upper bound of the maximum error express Norm.

4. The robust adaptive beamforming method for array radar and communication systems under small sample conditions according to claim 3, characterized in that, In step 3, the method of iteratively solving using alternating direction multipliers to obtain estimates of the amplitude and phase error vectors and interference power vectors is as follows: Step 301: Set the variable k, which indicates the number of iterations, and initialize k=1; Step 302, Update : ； Step 303, update the interference power vector : ； in, It is a neighboring parameter, subscript Represents the first in the vector One element, , The superscript * indicates conjugate operation. Represents the Khatri-Rao product; This indicates finding the absolute value. , For vectorization operators; Step 304, Build Update Optimization issues: ； Using convex optimization algorithm Solve the optimization problem to obtain The value; Step 305, Build Update Optimization issues: ； Using convex optimization algorithm Solve the optimization problem to obtain The value; Step 306, update the Lagrange multiplier matrix. : ； Step 307, if Then the iteration ends, and the solution of the last iteration is output. and As estimates of the interference power vector and the equivalent amplitude and phase error vector and Otherwise, let k = k + 1 and return to step 302. This is the preset termination precision.

5. The robust adaptive beamforming method for array radar and communication systems under small sample conditions according to claim 4, characterized in that, The specific method for estimating the interference plus noise covariance matrix of the received signal in step 3 is as follows: Step 308, based on the estimated value of the interference power vector The positions of all non-zero elements in Record its corresponding number Azimuth angle at discrete points Further construct the interference angle vector n is The number of all non-zero elements in the array. They are respectively the corresponding ; Step 309: Calculate the array manifold matrix of the interference. ,right By taking the reciprocal of each element, we obtain an estimate of the array amplitude and phase error vector. Calculate the interference envelope The estimated value: ； The estimated value of the array amplitude and phase error vector The diagonal matrix formed; The average power estimate of the interference was further calculated. : ； Step 310: Reconstruct the interference covariance matrix : ； For Hadamard product; Step 311, estimate the noise power for: ； in, Represents the trace operation of a matrix. ; Therefore, the interference plus noise covariance matrix The estimated value is: ； Let represent the M-dimensional identity matrix.

6. The robust adaptive beamforming method for array radar and communication systems under small sample conditions according to claim 5, characterized in that, The specific method for step 4 is as follows: ； The angle of arrival of the desired signal. For robust adaptive beamforming weights.

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