CN114818784B - Improved robust beam forming method combining covariance matrix and ADMM algorithm - Google Patents

Abstract

The invention belongs to the technical field of array signal processing, and particularly relates to an improved robust beam forming method of a combined covariance matrix and an ADMM algorithm, which constructs an improved robust beam forming guide vector optimization model, and improves the guide vector optimization model on the basis of covariance matrix reconstruction; firstly, utilizing SDP relaxation, secondly, utilizing ADMM algorithm to solve the model, obtaining closed optimal solution in each iteration process, calculating optimal weighting vector, and carrying out multi-beam weighted summation on the optimal weighting vector and array received data vector to form output robust beam. The invention better meets the robustness requirement, effectively improves the anti-interference motion capability and the steering vector mismatch capability of the traditional beam forming device, simultaneously effectively improves the condition of outputting SINR under the condition of low input SNR, and effectively prevents the self-cancellation phenomenon.

Description

Improved robust beam forming method combining covariance matrix and ADMM algorithm

Technical Field

The invention relates to the technical field of array signal processing, in particular to an improved robust beam forming method combining a covariance matrix and an ADMM algorithm.

Background

Robust adaptive beamforming has been recognized as a fundamental problem in array signal processing and has attracted extensive research and attention. This is because conventional adaptive beamforming techniques are less immune to minor or modest differences between the hypothesized and actual signal steering vectors, as well as antenna calibration errors and other mismatches.

Adaptive beamforming has been widely used in the fields of sonar, radar, bioscience, speech signal processing, medical engineering, etc., and is one of the research hotspots in the field of array signal processing. The adaptive beamforming algorithm is very sensitive to errors in steering vector mismatch, and even small steering vector errors, such as directional errors, array disturbances, moving targets, etc., can lead to dramatic degradation in algorithm performance. In addition, when the training data contains the desired signal, the desired signal may be regarded as an interference signal, resulting in a self-cancellation phenomenon. Conventional beamforming algorithms form very narrow nulls at the interference, which, if an array disturbance occurs, necessarily causes the interference to deviate from the null position and even the algorithm to fail entirely. There is therefore a need to investigate the robustness of enhancement algorithms to overcome the above-mentioned problems.

In order to effectively improve the algorithm performance of the traditional beam former under disturbance and steering vector mismatch at the interference position, an improved robust beam forming method combining covariance matrix reconstruction and ADMM algorithm is provided.

Robustness of enhancement algorithms can be broadly divided into two categories: one class is covariance matrix-based algorithms: diagonal Loading (DL) algorithms, feature space algorithms, and covariance matrix reconstruction algorithms (Interference Plus Noise, IPN). The DL algorithm adds a loading factor to the diagonal of the covariance matrix, so as to suppress noise in the weight vector, but the selection of the optimal loading factor is difficult to determine. The feature space algorithm is to solve the feature value of the covariance matrix and divide the feature value, the guide vector corresponding to the large feature value is a subspace of the expected signal and the interference signal, and the guide vector corresponding to the small feature value is a noise subspace. And projecting the expected signal with the error to a subspace of the expected signal plus the interference signal, thereby eliminating the error.

Thus, in order to meet the robustness requirements, a number of robust adaptive beamforming techniques have been established.

Disclosure of Invention

The technical problem to be solved by the invention is to provide an improved robust beam forming method combining covariance matrix and ADMM algorithm aiming at the defects in the prior art.

The technical scheme adopted for solving the technical problems is as follows:

the invention provides an improved robust beam forming method combining covariance matrix and ADMM algorithm, which comprises the following steps:

step 1, a receiving end is set to be a uniform linear array formed by N array elements, and the output of a beam forming device at the moment k is as follows: y (k) =w ^H x (k); where w is a weight vector of Nx1, (. Cndot.) for the display of the display, the display is a display of the display ^H Representing a conjugate transpose, x (k) being the array receive data vector;

step 2, constructing a guide vector optimization model of robust beam forming:

N(1-η ₁ )≤||a|| ² ≤N(1+η ₂ )

||a-a ₀ || ² ≤ε

the covariance matrix is: Δ ₀ expressed as:

wherein d (θ) is defined as the relative steering vector in the θ direction, θ= [ θ ] _min ,θ _max ]Indicating that the desired signal is stationaryWithin the interval defined, it is assumed that the mismatch interval is smaller than Θ and angularly separated from the interfering signal,representing the complement of Θ; k is the snapshot number; a is an array steering vector, a ₀ ＝d(θ ₀ ),θ ₀ ＝(θ _min +θ _max ) And/2 is defined as interval Θ= [ θ ] _min ,θ _max ]Is a median of (2); η (eta) ₁ 、η ₂ And epsilon are defined parameters, which are allowed error norm limits;

step 3, solving a model: on the basis of covariance matrix reconstruction, improving a guide vector optimization model; firstly, utilizing SDP relaxation, secondly, utilizing ADMM algorithm to solve the model, and obtaining a closed optimal solution in each iteration process, thereby obtaining an optimal guide vector a ^* ；

Step 4, calculating an optimal weighting vector:and carrying out multi-beam weighted summation on the data vector x (k) and the array received data vector x (k) to form an output robust beam.

Further, in the method, the covariance matrix of the interference plus noise is:

wherein ,a spatial power spectrum which is a Capon algorithm;

to make the interference area rangeWherein Δδ is the required null range, and the reconstructed interference covariance matrix is:

the reconstructed interference plus noise covariance matrix is then as follows:

wherein ,is the noise power; select->For->The minimum eigenvalue corresponding to the eigenvalue, a (theta), represents the steering vector, theta _i Indicating the direction of the interference signal, delta _i Representing the defined null width, P represents the total number of divided interference interval precision.

Further, the method for loosening SDP in the method is as follows:

the global optimal solution of the steering vector optimization model is given by the optimal solution of the corresponding SDP relaxation problem:

N(1-η ₁ )≤trX≤N(1+η ₂ )

the rewriting is as follows:

according to the interior point method, expressed as:

mintr(c ^T Y)

s.t.tr(QY)≤P

wherein ,P ₁ ＝Δ ₀ ，P ₂ ＝N(1+η ₂ )，P ₃ ＝N(η ₁ -1)，P ₄ ＝ε，

x is an N multiplied by N Hermitian matrix, y is the optimal solution of the constructed SDP relaxation model, and has no specific physical meaning, and I represents an identity matrix;

putting the objective of the problem and the constraint condition of the problem into a function by using the Lagrangian method to obtain the formula:

wherein m' represents the sequence number of Q, P involved in accumulation;

according to the thought of obstacle function, letThe formula:

wherein t is a parameter for adjusting approximation; respectively solving the first-order derivative and the second-order derivative of the above formula, adjusting the value of t to enable the value of t to be converged, and obtaining the optimal X ^* ；

Order theThe steering vector optimization model is then transformed into:

||a|| ² ＝b ₂

||a-a ₀ || ² ≤ε

due to, ||a ^* -a ₀ ^H a ₀ || ² ＝a ^*H (I-a ₀ a ₀ ^H )a ^* ≤||a-a ₀ || ²≤ε, wherein a^* Is also the optimal solution to the SDP relaxation problem, so the SDP relaxation problem is re-representedFor new optimization problems:

||a|| ² ＝b ₂

a ^H Ba≤ε

wherein b=i-a ₀ a ₀ ^H 。

Further, the method for solving by using the ADMM algorithm comprises the following steps:

introducing an auxiliary variable z, h, and substituting the auxiliary variable z, h into a new optimization problem to obtain:

h ^H a＝b ₂

h ^H Ba+z＝ε

h-a＝0

z≥0

solving the above equation by using the scaled form of ADMM, and introducing auxiliary variables s, u, m, v according to the ADMM frame, the extended Lagrangian function of the above equation is:

wherein ,ρ₁ ,ρ ₂ ,ρ ₃ ,ρ ₄ >0 is penalty coefficient;

the closed solution is obtained by using ADMM in a loop mode, and in the (n+1) th iteration process, the updating of { a, h, z, s, u, m, v } is respectively as follows:

u ⁿ⁺¹ ＝u ⁿ +(h ⁿ⁺¹ ) ^H a ⁿ⁺¹ -b ₂

m ⁿ⁺¹ ＝m ⁿ +(h ⁿ⁺¹ ) ^H Ba ⁿ⁺¹ +z ⁿ⁺¹ -ε

v ⁿ⁺¹ ＝v ⁿ +h ⁿ⁺¹ -a ⁿ⁺¹

the solving process is as follows:

1) Updating h:

for a given { a } ⁿ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of h is obtained by solving the following problem:

to obtain the minimum of the above equation, the first derivative with respect to h is solved for the above equation and the derivative is made zero, i.eAnd (3) solving to obtain:

h ⁿ⁺¹ ＝Γ ^-1 Φ

wherein Γ and Φ are defined as follows:

2) Updating a:

for a given { h ⁿ⁺¹ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of a is obtained by solving the following problem:

similarly, the four items in the above are respectively derived and theAnd (3) solving to obtain:

a ⁿ⁺¹ ＝Π ^-1 Ψ

wherein, pi and ψ are defined as follows:

3) Updating z:

for a given { a } ⁿ⁺¹ ,h ⁿ⁺¹ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of z is obtained by solving the following problem:

s.t z≥0

to the above-mentioned formula to make it be0, solve to z ⁿ⁺¹ ＝ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ Therefore:

z ⁿ⁺¹ ＝max{0,ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ }。

the invention has the beneficial effects that: the improved robust beam forming method of the combined covariance matrix and ADMM algorithm of the invention improves performance by introducing more practical constraints to the power maximization problem. In order to find the optimal guiding vector, bilateral norm disturbance constraint and secondary similarity constraint are added, so that the arrival direction of the SOI is ensured to be far away from the DOA area of all linear combinations of the disturbance guiding vectors, and the DOA of the optimal guiding vector is ensured to be positioned in the angular sector area of the SOI. The result shows that the problem of maximizing the output power of the array is a non-convex quadratic constraint quadratic programming problem with non-homogeneous constraint, the method proves that the problem is still solvable, and the optimal steering vector is found by utilizing ADMM, so that the formation of a steady beam is realized; the invention better meets the robustness requirement, effectively improves the anti-interference motion capability and the steering vector mismatch capability of the traditional beam forming device, simultaneously effectively improves the condition of outputting SINR under the condition of low input SNR, and effectively prevents the self-cancellation phenomenon.

Drawings

The invention will be further described with reference to the accompanying drawings and examples, in which:

FIG. 1 is a normalized beam pattern comparison for different algorithms for different mismatch angles; (a) Estimating an angle(b) Estimate angle->

Fig. 2 is a graph of output SINR as a function of input SNR; (a) Estimating an angle(b) Estimate angle->

Fig. 3 is a graph showing the variation of the output SINR with the number of bursts;

fig. 4 is a graph of variation in output SINR at different mismatch angles;

fig. 5 is a flow chart of the method of the present invention.

Detailed Description

The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.

1. Constructing a signal model;

without loss of generality, the embodiment of the invention considers that the receiving end is a Uniform Linear Array (ULA) formed by N array elements, the array element spacing is d, and then the output of the narrow-band beam forming device at the moment k is

y(k)＝w ^H x(k) (1)

Where w is a weight vector of Nx1, (. Cndot.) for the display of the display, the display is a display of the display ^H Representing the conjugate transpose, x (k) is the array receive data vector.

x(k)＝s(k)+i(k)+n(k) (2)

Wherein s (k), i (k), and n (k) correspond to the desired signal, the interference signal, and the noise signal, respectively.

From the minimum variance distortion-free response beamformer, the optimal steering vector value a can be found ^* And according to the actual covariance matrix

Wherein K is the snapshot number.

And then obtaining a weighted vector as a weight vector according to the SMI algorithm

In order to improve the array output SINR and output power of the robust adaptive beamformer, an improved method is proposed, a compromise between sensitivity and robustness is to obtain more a priori information, and a new problem of optimal estimation of SOI direction vectors is presented.

Consider the following general steering vector estimation problem:

where Λ is the uncertainty set of steering vectors. The most common Λ includes the norm constraint a ² ＝N。

Consider the following two sets of conditional defined uncertainties:

Λ＝{a|N(1-η ₁ )≤||a|| ² ≤N(1+η ₂ ),||a-a ₀ || ² ≤ε} (6)

wherein ,a₀ ＝d(θ ₀ ),θ ₀ ＝(θ _min +θ _max ) And/2 is the interval Θ= [ θ ] _min ,θ _max ]Is a median value of (c). η (eta) ₁ 、η ₂ And epsilon are defined parameters, which are allowed error norm limits.

In the uncertainty set, a first bilateral norm constraint allows for a range of errors of the array steering vector a in consideration of gain disturbances of the steering vector caused by amplitude errors, phase errors, position errors, etc. caused by the antenna; the second constraint utilizes a ₀ And epsilon is defined as a generalized similarity constraint, meaning that the uncertainty set of steering vector a is a convex set. And setting the direction of the SOI steering vector instead of the set of directions in which all of the interfering steering vectors are linearly combined ensures that the DOA of the optimal steering vector is located within the angular sector of the SOI. Combining formula (5) and formula (6), the proposed model is:

in equation (7), the new constraint expands the viable set of problems more likely, thereby enabling a better estimation of the true steering vector. And, in equation (7), the only required a priori conditions include three allowable error norm limits, which is a non-convex quadratic constraint quadratic programming problem with three non-homogeneous constraints.

2. Solving a problem model by improved null widening robust beam forming;

and on the basis of covariance matrix reconstruction, improving a guide vector optimization model. Equation (7) is an NP-hard problem, first, using SDP relaxation, then using ADMM algorithm to solve the model, and obtaining a closed optimal solution in each iteration process, and then obtaining an optimal weighting vector according to equation (4).

The qqp problem, equation (7), is solvable, and its globally optimal solution can be given by the optimal solution of the corresponding SDP relaxation problem.

Further, the expression (8) can be rewritten as:

according to the interior point method, formula (9) can be expressed as:

wherein ,P ₁ ＝Δ ₀ ，P ₂ ＝N(1+η ₂ )，P ₃ ＝N(η ₁ -1)，P ₄ ＝ε/>the original problem can be expressed as a general form of linear programming. Make the following stepsPutting the objective of the problem and the constraint condition of the problem into a function by using a Lagrangian method to obtain the formula:

according to the thought of obstacle function, letObtained (12)

Wherein t is a parameter for adjusting approximation;

finally, respectively solving the first-order derivative and the second-order derivative of the formula (12), and adjusting the value of t to enable the formula (12) to converge so as to obtain the optimal X ^* 。

Order theThen the formula (7) is converted into

Due to, ||a ^* -a ₀ ^H a ₀ || ² ＝a ^*H (I-a ₀ a ₀ ^H )a ^* ≤||a-a ₀ || ²≤ε, wherein a^* Also the optimal solution of equation (8), so equation (8) can be re-expressed as a new optimization problem:

wherein b=i-a ₀ a ₀ ^H 。

For equation (14), the solution is performed using the ADMM algorithm. First, the auxiliary variables z, h are introduced, and substituted into formula (14) to obtain:

solving the formula (15) by using the scaled version of ADMM and introducing the auxiliary variables s, u, m, v according to the ADMM frame, the augmented Lagrangian function of the formula (15) is

wherein ,ρ₁ ,ρ ₂ ,ρ ₃ ,ρ ₄ >And 0 is a penalty coefficient.

In general, in ADMM, the update of the original variable is obtained by minimizing the augmented lagrangian multiplier, while the update of the lagrangian multiplier is obtained by the dual-rise method. Based on equation (16), a closed-loop solution can be obtained by using ADMM in the following loop manner. In the (n+1) th iteration, the updates of { a, h, z, s, u, m, v } are as follows, respectively

u ⁿ⁺¹ ＝u ⁿ +(h ⁿ⁺¹ ) ^H a ⁿ⁺¹ -b ₂ (21)

m ⁿ⁺¹ ＝m ⁿ +(h ⁿ⁺¹ ) ^H Ba ⁿ⁺¹ +z ⁿ⁺¹ -ε (22)

v ⁿ⁺¹ ＝v ⁿ +h ⁿ⁺¹ -a ⁿ⁺¹ (23)

The solutions of the formulas (17), (18) and (19) will be specifically considered below.

1. Updating h:

for a given { a } ⁿ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of h is obtained by solving the following problem

To obtain the minimum of equation (24), equation (24) is solved herein for a first derivative about h and the derivative is made zero, i.eSolving to obtain

h ⁿ⁺¹ ＝Γ ^-1 Φ (25)

Wherein Γ and Φ are defined as follows respectively

2. Updating a:

for a given { h ⁿ⁺¹ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of a can be obtained by solving the following problem

Similarly, the four terms in the formula (28) are respectively derived and the following is causedRelieve->

a ⁿ⁺¹ ＝Π ^-1 Ψ (29)

Wherein, omega and xi are defined as follows

3. Updating z:

for a given { a } ⁿ⁺¹ ,h ⁿ⁺¹ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of z can be obtained by solving the following problem

Deriving formula (32) and making it 0 to obtain z ⁿ⁺¹ ＝ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ Therefore, it is

z ⁿ⁺¹ ＝max{0,ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ } (33)

In summary, the embodiment of the invention provides a zero notch widening robust wave beam forming method based on covariance matrix reconstruction and ADMM.

3. Simulation experiment;

in order to verify the effectiveness of the proposed method, simulation experiments verify that the DL algorithm and the SMI algorithm are compared with the performance analysis of the method of the invention under the condition that interference, clutter and steering vector mismatch exists. In the experiment, the array element number is N=10, the input SNR=10 dB, and the preset null width of two interferences is delta respectively ₁＝8° and Δδ₂ =5°; assuming that the actual expected direction angle is θ ₀ =10°, the desired signal sampling region Θ isThe actual azimuth angle of the interference signal is theta _i1 ＝-40°、θ _i2 =70°, interference signal sampling region [ θ ] _i1 -8°,θ _i1 +8°]、[θ _i2 -5°,θ _i2 +5°]. The sampling points are 100, and all results are obtained by 100 independent Monte Carlo experiment statistics.

When the pointing error exists in the experiment 1, the normalized wave beam patterns of different algorithms are compared;

in this experiment, the current actual expected direction angle is θ ₀ =10°, assuming that the estimated desired direction angle is and />Inr=10 dB. Fig. 1 shows a beam pattern comparison at different mismatch angle conditions. As can be seen from fig. 1, the improved algorithm corrects the mismatched steering vectors when the mismatch angle is 8 ° and 5 °. In addition, the improved algorithm is lower in output level than the algorithm of the invention, whether at the nulls or the side lobes. In summary, the proposed algorithm also improves the robustness against systematic errors.

Outputting SINR analysis under different input SNR of experiment 2;

in this experiment, the experimental conditions were the same as in experiment 2. Fig. 2 comparatively analyzes the effect of different input SNRs on the output SINR. It can be seen that the problem of mismatching of the guiding vectors cannot be solved by the SMI algorithm and the DL algorithm, so that the performance is far away from the theoretical optimal value, the guiding vectors can be corrected by the improved algorithm, and the output SNIR performance is relatively good when the estimated angle is 8 degrees and 5 degrees. The improved algorithm ensures better SINR output under the condition of low SNR input, so that the array system is less influenced by SNR and is more stable. In summary, the improved algorithm can maintain stability under the conditions of angle mismatch and interference disturbance, the performance is superior to other algorithms, and the improved algorithm has better performance at low input SINR.

Outputting SINR analysis under different snapshot numbers in experiment 3;

the current actual desired direction angle is the estimated desired direction angle, the width of the two interference nulls is 8 ° and 5 °, respectively, and the input snr=30 dB. Fig. 3 shows the variation of the output SINR with the number of bursts. The SMI algorithm is close to convergence when the snapshot number is more than 30, and the DL algorithm is greatly influenced by the snapshot. The improved algorithm is obviously superior to other comparison algorithms, can output high SINR under the condition of low snapshot number, and is closest to the theoretical optimal value.

Outputting SINR analysis under different mismatch angles in experiment 4;

the current actual expected direction angle is that the input snapshot number N=100, the width of two interference nulls is 8 degrees and 5 degrees respectively, the input SNR=30 dB, the mismatch angle is changed from-8 degrees to 8 degrees, and 100 independent experiments are carried out on each snapshot number. As can be seen from fig. 4, when the mismatch angle is large, the improved algorithm can output SINR close to the optimal value, and the performance is significantly better than other comparison algorithms. In summary, the proposed algorithm can correct the steering vector and output a signal-to-noise ratio close to the optimal value.

It will be understood that modifications and variations will be apparent to those skilled in the art from the foregoing description, and it is intended that all such modifications and variations be included within the scope of the following claims.

Claims (2)

1. An improved method of robust beamforming for a joint covariance matrix and ADMM algorithm, comprising the steps of:

step 1, a receiving end is set to be a uniform linear array formed by N array elements, and the output of a beam forming device at the moment k is as follows: y (k) =w ^H x (k); where w is a weight vector of Nx1, (. Cndot.) for the display of the display, the display is a display of the display ^H Representing a conjugate transpose, x (k) being the array receive data vector;

step 2, constructing a guide vector optimization model of robust beam forming:

N(1-η ₁ )≤||a|| ² ≤N(1+η ₂ )

||a-a ₀ || ² ≤ε

the covariance matrix is: Δ ₀ expressed as:

wherein d (θ) is defined as the relative steering vector in the θ direction, θ= [ θ ] _min ,θ _max ]Meaning that the desired signal is within a defined interval, assuming that the mismatch interval is smaller than theta and angularly separated from the interfering signal,representing the complement of Θ; k is the snapshot number; a is an array steering vector, a ₀ ＝d(θ ₀ ),θ ₀ ＝(θ _min +θ _max ) And/2 is defined as interval Θ= [ θ ] _min ,θ _max ]Is a median of (2); η (eta) ₁ 、η ₂ And epsilon are defined parameters, which are allowed error norm limits;

step 3, solving a model: on the basis of covariance matrix reconstruction, improving a guide vector optimization model; firstly, utilizing SDP relaxation, secondly, utilizing ADMM algorithm to solve the model, and obtaining a closed optimal solution in each iteration process, thereby obtaining an optimal directorQuantity a ^* ；

Step 4, calculating an optimal weighting vector: a reconstructed covariance matrix; carrying out multi-beam weighted summation on the data vector x (k) and the array received data vector x (k) to form an output robust beam;

the method for reconstructing the covariance matrix comprises the following steps:

the covariance matrix of the interference plus noise is:

wherein ,a spatial power spectrum which is a Capon algorithm; a (θ) represents an array steering vector;

to make the interference area rangeWherein Δδ is the required null range, and the reconstructed interference covariance matrix is:

wherein ,θ_i Indicating the direction of the interference signal; delta _i Representing a defined null width; p represents the total number of precision dividing interference intervals;

the reconstructed interference plus noise covariance matrix is then as follows:

wherein ,is the noise power; select->For->A minimum feature value corresponding to the feature decomposition;

the SDP relaxation method in the method comprises the following steps:

the global optimal solution of the steering vector optimization model is given by the optimal solution of the corresponding SDP relaxation problem:

N(1-η ₁ )≤trX≤N(1+η ₂ )

the rewriting is as follows:

according to the interior point method, expressed as:

mintr(c ^T Y)

s.t.tr(QY)≤P

wherein ,P＝[P ₁ P ₂ P ₃ P ₄ ] ^T ，P ₁ ＝Δ ₀ ，P ₂ ＝N(1+η ₂ )，P ₃ ＝N(η ₁ -1)，P ₄ ＝ε；Q＝[Q ₁ Q ₂ Q ₃ Q ₄ ] ^T ，/>

x is an N multiplied by N Hermitian matrix, y is the optimal solution of the constructed SDP relaxation model, and has no specific physical meaning, and I represents an identity matrix;

putting the objective of the problem and the constraint condition of the problem into a function by using the Lagrangian method to obtain the formula:

wherein m' represents the sequence number of Q, P involved in accumulation;

based on the idea of barrier functionOrder-makingThe formula:

wherein t is a parameter for adjusting approximation; respectively solving the first-order derivative and the second-order derivative of the above formula, adjusting the value of t to enable the value of t to be converged, and obtaining the optimal X ^* ；

Order theThe steering vector optimization model is then transformed into:

||a|| ² ＝b ₂

||a-a ₀ || ² ≤ε

due to, ||a ^* -a ₀ ^H a ₀ || ² ＝a ^*H (I-a ₀ a ₀ ^H )a ^* ≤||a-a ₀ || ²≤ε, wherein a^* Is also the optimal solution to the SDP relaxation problem, so the SDP relaxation problem is re-expressed as a new optimization problem:

||a|| ² ＝b ₂

a ^H Ba≤ε

wherein b=i-a ₀ a ₀ ^H 。

2. The improved method of robust beamforming for a joint covariance matrix and ADMM algorithm according to claim 1, wherein the method of solving using the ADMM algorithm comprises:

introducing an auxiliary variable z, h, and substituting the auxiliary variable z, h into a new optimization problem to obtain:

h ^H a＝b ₂

h ^H Ba+z＝ε

h-a＝0

z≥0

solving the above equation by using the scaled form of ADMM, and introducing auxiliary variables s, u, m, v according to the ADMM frame, the extended Lagrangian function of the above equation is:

wherein ,ρ₁ ,ρ ₂ ,ρ ₃ ,ρ ₄ >0 is penalty coefficient;

the closed solution is obtained by using ADMM in a loop mode, and in the (n+1) th iteration process, the updating of { a, h, z, s, u, m, v } is respectively as follows:

u ⁿ⁺¹ ＝u ⁿ +(h ⁿ⁺¹ ) ^H a ⁿ⁺¹ -b ₂

m ⁿ⁺¹ ＝m ⁿ +(h ⁿ⁺¹ ) ^H Ba ⁿ⁺¹ +z ⁿ⁺¹ -ε

v ⁿ⁺¹ ＝v ⁿ +h ⁿ⁺¹ -a ⁿ⁺¹

the solving process is as follows:

1) Updating h:

for a given { a } ⁿ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of h is obtained by solving the following problem:

to obtain the minimum of the above equation, the first derivative with respect to h is solved for the above equation and the derivative is made zero, i.eAnd (3) solving to obtain:

h ⁿ⁺¹ ＝Γ ^-1 Φ

wherein Γ and Φ are defined as follows:

2) Updating a:

for a given { h ⁿ⁺¹ ,z ⁿ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of a is obtained by solving the following problem:

similarly, the four items in the above are respectively derived and theAnd (3) solving to obtain:

a ⁿ⁺¹ ＝Π ^-1 Ψ

wherein, pi and ψ are defined as follows:

3) Updating z:

for a given { a } ⁿ⁺¹ ,h ⁿ⁺¹ ,s ⁿ ,u ⁿ ,m ⁿ ,v ⁿ The update of z is obtained by solving the following problem:

s.t z≥0

deriving the above value to 0 and solving for z ⁿ⁺¹ ＝ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ Therefore:

z ⁿ⁺¹ ＝max{0,ε-m ⁿ -h ⁿ⁺¹ Ba ⁿ⁺¹ }。