CN105354171A - Improved eigenvector projection subspace estimation adaptive beam forming method - Google Patents

Abstract

The invention relates to an improved eigenvector projection subspace estimation adaptive beam forming method. The method comprises: an array antenna performing sampling on an input signal; constructing a KR signal covariance vector by Khatri-Rao; estimating a guide vector with an eigenvector projection subspace method; and obtaining an output signal of the array antenna. Compared with a conventional beam former, convergence can be obtained in a relatively small snapshot; relatively high convergence precision is achieved; and the technology can be suitable for systems with relatively high real-time requirements.

Description

Projection subspace estimation adaptive beam forming method for improving feature vector

Technical Field

The invention relates to a projection subspace estimation self-adaptive beam forming method for improving a feature vector.

Background

As a new emerging object, array signal processing has been widely applied to many fields such as radar, sonar, navigation, and communication. The array antenna adaptive beamforming has a strong interference suppression capability as an important aspect of array signal processing, and can effectively improve the signal to interference plus noise ratio (SINR) of an output signal, so that the array antenna adaptive beamforming has a high practical application value. Among them, the Minimum Variance Distortionless Response (MVDR) beamforming method is a classical algorithm. However, in practical applications, the sampling is snapshot-limited, so that the sampling covariance matrix has a large error. The prior information of the guide vector is inaccurate due to factors such as position error of the array elements, inconsistent phase response of the array elements and the like. How to make the algorithm have errors in the input guide vector prior information and obtain convergence under a smaller sampling snapshot, and the output signal has higher SINR, thus having important engineering application value.

Numerous robustness algorithms have been proposed by a large number of scholars, and typical algorithms include a Diagonal Loading (DL) [7] beamforming algorithm, an Eigen Subspace (ESB) beamforming algorithm, and the like. The adaptive beam-forming algorithm with large angle mismatch Robustness (RLM) proposed in document [10] linearly represents the desired signal steering vector by orthogonal subspace feature vectors. An adaptive beam-forming algorithm (EPS) for subspace estimation of feature vector projections proposed in the document modifiedprojectionaproformbustedapromethodbeamelse estimate the desired signal subspace by projecting onto an assumed steering vector, which reduces the error by projecting onto the desired signal subspace. The subspace method is also a common method for improving the performance of the beamforming algorithm to reduce the covariance matrix estimation error. The document (DOAestion of quasi-stationaryisnavia Khatri-raosubspace) proposes a Khati-Rao (KR) incoming wave angle estimation method, and by researching a secondary data structure of a quasi-stationary signal, a KR subspace can effectively eliminate a noise covariance matrix and simultaneously increase the aperture of an array element. Document [ Sparsecovariancefixing for directionFarrivalidations ] sparse representation of an array covariance matrix using a steering vector construction dictionary structure. However, the above algorithm does not effectively correct the estimation error of the steering vector when the estimation error of the sampling covariance matrix is reduced, so that the algorithm can be converged under a small snapshot, and the output signal has a high SINR.

In view of the above deficiency, a method for improving subspace estimation (MEPS) adaptive beamforming of feature vector projection is proposed herein. The KR signal subspace is first constructed to reduce the estimation error of the sampled covariance matrix. And secondly, constructing an interference and noise subspace by using the corrected covariance matrix eigenvector, and reducing the estimation error of the sampling covariance matrix by orthogonally projecting the assumed guide vector to the interference and noise subspace.

Disclosure of Invention

The invention aims to provide a projection subspace estimation adaptive beam forming method capable of converging to an improved feature vector with higher precision under fewer snapshots.

The purpose of the invention is realized as follows:

the method comprises the following steps:

(1) the array antenna samples an input signal:

the sampled data is represented by a sampled data covariance matrix as:

$R=\frac{1}{K}\underset{i=1}{\overset{K}{\Σ}}{x}^{H}x$

wherein, K is the sampling fast beat number, and x is the input signal;

(2) and constructing a KR signal covariance vector by using a Khatri-Rao product:

$d\left(\mathrm{\θ}\right)=a\left(\mathrm{\θ}\right)\⊗a\left(\mathrm{\θ}\right)$

vec (-) is the stacking of the matrix columns into the first column:

d(θ)＝vec(a(θ)a^H(θ))

wherein a (θ) is an input signal steering vector;

and (3) solving covariance vectors on two sides to obtain:

$\begin{array}{c}vec(R-{\mathrm{\σ}}_{noise}^{2}I)=\underset{i=1}{\overset{N}{\Σ}}{\mathrm{\σ}}_i^{2}vec\left(a\right({\mathrm{\θ}}_i\left)a^H\right({\mathrm{\θ}}_i\left)\right)\\ =\underset{i=1}{\overset{N}{\Σ}}{\mathrm{\σ}}_i^{2}d\left({\mathrm{\θ}}_i\right)\end{array}$

for a linear combination of the covariance vectors of the desired signal and the interfering signal, a matrix S is constructed:

$S=\underset{0}{\overset{\mathrm{\π}}{\∫}}d\left(\mathrm{\θ}\right)d^H\left(\mathrm{\θ}\right)d\mathrm{\θ}$

signal covariance vectorLinearly expressed by the eigenvector of the matrix S, the d (theta) subspace dimension is equal to the number of non-zero eigenvalues of the matrix S;

constructing a matrix Q by the eigenvectors corresponding to the non-zero eigenvalues of the matrix S:

Q＝[e₁,e₂,...,e_K]

wherein,the characteristic vectors corresponding to the non-zero eigenvalues of the matrix Q are obtained, and the columns of Q are mutually orthogonal to form a characteristic subspace of the signal covariance vector;

$vec\left(R_1\right)=Q{\left(Q^{H}Q\right)}^{-1}{Q}^{H}vec\left(\hat{R}\right)$

wherein, Q (Q)^HQ)^-1Q^HA projection matrix which is a signal covariance vector subspace;for sampling the covariance matrix, R₁To representProjecting to a signal covariance vector quantum space;

(3) estimating a guide vector by a feature vector projection subspace method:

if the interference and noise subspace projection matrix U is obtained_i+nThe estimation error of the desired steering vector is eliminated by projection into the orthogonal subspace:

$\tilde{a}=(I-U_{i+n}{U}_{i+n}^H)\hat{a}$

wherein,in order for the estimate of the steering vector to be desired,r obtained for desired guide vector correction value₁The eigenvalue decomposition is performed as:

$R_1=\underset{i=1}{\overset{N}{\Σ}}{\mathrm{\λ}}_i{e}_i{e}_i^H$

wherein λ is₁＞λ₂＞...＞λ_K+1＝...＝λ_N＝σ²Is a covariance matrix R₁Characteristic value of e_iThe characteristic vector corresponding to the characteristic value;

each feature vector e_iAll-directional guide vectorProjection, selecting feature vector corresponding to projection to construct interference plus noise signal subspace U_i+n；

Projection of feature vectors onto steering vectorsThe descending order is arranged to obtain: p is a radical of_[M]≥p_[M-1]≥…≥p_[1]；

Constructing an interference plus noise subspace using the feature vectors;

$(p_{\&lsqb;m\&rsqb;}+p_{\&lsqb;m-1\&rsqb;}\&GreaterEqual;...\&GreaterEqual;p_{\&lsqb;1\&rsqb;})/\underset{i=1}{\overset{M}{\&Sigma;}}\left(i\right)<\mathrm{\&rho;}$

as m increases, the value of m when p is first exceeded is the evaluated value, constructing the interference plus noise subspace:

U_i+n＝[e_[1],e_[2],…,e_[m-1]]

the interference plus noise subspace projection matrix is then:

$P_{i+n}=U_{i+n}{U}_{i+n}^H$

further, the desired subspace projection matrix is obtained as:

P_s＝I-P_i+n

eliminating the estimation error of the interference plus noise subspace part, and improving the estimation precision of the guide vector:

$\tilde{a}=P_s\hat{a}$

improved beamforming weight vector:

$w=\frac{R_1\tilde{a}}{{\tilde{a}}^H{R}_1^{-1}\tilde{a}}$

(4) obtaining an array antenna output signal:

y＝w^Hx

where y is the output signal.

The invention has the beneficial effects that:

compared with the traditional adaptive beam synthesizer, the invention has the following main advantages:

compared with the traditional beam synthesizer, the invention can obtain convergence under smaller fast shooting;

compared with the traditional beam synthesizer, the technology of the invention has higher convergence precision;

the technology of the invention can be suitable for systems with higher real-time requirements.

The MEPS beam forming method has the advantages that the MEPS beam forming method adopts a sampling covariance matrix with higher precision, and meanwhile, the assumed guide vector is projected to a signal subspace, so that the estimation error of the guide vector is reduced, and the MEPS can obtain a higher SINR output signal under a smaller snapshot.

Drawings

Fig. 1 is a 5-array element uniform linear array, and the eigenvalues of the matrix S are arranged from small to large in distribution.

FIG. 2 is a graph of convergence rate of the MEPS, RLM, EPS and MVDR algorithms without the desired signal angle mismatch.

FIG. 3 is a graph of convergence accuracy of the MEPS, RLM, EPS and MVDR algorithms without the desired signal angle mismatch.

FIG. 4 is a graph of the convergence rate of the MEPS, RLM, EPS and MVDR algorithms in the presence of an angle mismatch of the desired signal.

FIG. 5 is a graph of convergence accuracy of the MEPS, RLM, EPS and MVDR algorithms in the presence of an expected signal angle mismatch.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

Array antenna beamforming, i.e. receiving spatial signals by using multiple antennas. And filtering the array signals by applying an array signal processing algorithm, and further performing weighted summation to obtain an output signal with a larger signal-to-interference-and-noise ratio (SINR). After weighting, the beam pattern is displayed as a main lobe aligned with the expected signal and a null aligned with the interference signal. Because the algorithm has strong interference suppression capability, the beam pattern should have a low side lobe level. How to improve the convergence rate of the beam synthesis algorithm under the condition of small snapshot and obtain more stable SINR output is the key for obtaining engineering application of the adaptive beam synthesis algorithm. An MEPS adaptive beamforming algorithm is proposed, which first constructs KR signal subspace to eliminate the estimation error of the covariance matrix in the noise subspace. And secondly, constructing an expected signal subspace by using the corrected main characteristic vector of the covariance matrix, and assuming that a guide vector is projected to the signal subspace, reducing the estimation error of the guide vector, so that the MEPS can obtain a higher SINR output signal under a smaller snapshot.

A signal covariance matrix KR subspace is constructed based on a Khatri-Rao (KR) covariance matrix construction method to eliminate estimation errors of covariance matrices. Secondly, an expected signal subspace is constructed by using the corrected main characteristic vector of the covariance matrix, and a pilot vector is supposed to project to the signal subspace, so that the estimation error of the pilot vector is reduced, and the MEPS can obtain a higher SINR output signal under a smaller snapshot.

Firstly, an array antenna samples an input signal.

The sampled data is represented by a sampled data covariance matrix as:

$\begin{array}{c}R=\frac{1}{K}\underset{i=1}{\overset{K}{\&Sigma;}}{x}^{H}x\\ =\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&sigma;}}_i^{2}a\left({\mathrm{\&theta;}}_i\right)a^H\left({\mathrm{\&theta;}}_i\right)+{\mathrm{\&sigma;}}_{noise}^{2}I\end{array}---\left(1\right)$

wherein, K is the sampling fast beat number, and x is the input signal.

Step two, constructing a KR signal covariance vector by a Khatri-Rao product:

$d\left(\mathrm{\&theta;}\right)=a\left(\mathrm{\&theta;}\right)\&CircleTimes;a\left(\mathrm{\&theta;}\right)---\left(2\right)$

defining vec (·) as the stacking of the matrix columns into the first column, referred to herein as the "covariance vector," then equation (2) can be expressed as:

d(θ)＝vec(a(θ)a^H(θ))(3)

where a (θ) is the input signal steering amount.

And (3) obtaining a covariance vector on two sides of the formula (1):

$\begin{array}{c}vec(R-{\mathrm{\&sigma;}}_{noise}^{2}I)=\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&sigma;}}_i^{2}vec\left(a\right({\mathrm{\&theta;}}_i\left)a^H\right({\mathrm{\&theta;}}_i\left)\right)\\ =\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&sigma;}}_i^{2}d\left({\mathrm{\&theta;}}_i\right)\end{array}---\left(4\right)$

in a clear view of the above, it is known that,is a linear combination of the covariance vectors of the desired signal and the interfering signal, and, therefore, if a covariance vector subspace is constructed,must be in the sub-spaceIt can be demonstrated that if the received signals are uncorrelated with each other, the N × N received signal covariance matrix is the Toeplitz matrix and can be fully characterized by a 2N-1 dimensional subspace.

$S=\underset{0}{\overset{\mathrm{\&pi;}}{\&Integral;}}d\left(\mathrm{\&theta;}\right)d^H\left(\mathrm{\&theta;}\right)d\mathrm{\&theta;}---\left(5\right)$

Then the signal covariance vectorCan be linearly represented by the principal eigenvector of the matrix S. The d (θ) subspace dimension is equal to the number of non-zero eigenvalues of the matrix S.

And constructing a matrix Q by using the eigenvectors corresponding to the non-zero eigenvalues of the matrix S, as shown in formula (6):

Q＝[e₁,e₂,...,e_K](6)

wherein,the eigenvectors corresponding to the non-zero eigenvalues of the matrix Q. Therefore, the columns of Q are orthogonal to each other, constituting the feature subspace of the signal covariance vector.

Definition of R₁The following were used:

$vec\left(R_1\right)=Q{\left(Q^{H}Q\right)}^{-1}{Q}^{H}vec\left(\hat{R}\right)---\left(7\right)$

wherein, Q (Q)^HQ)^-1Q^HA projection matrix which is a signal covariance vector subspace;for sampling the covariance matrix, then R₁To representAnd projecting to the signal covariance vector quantum space.

Step three, estimating a guide vector by a feature vector projection subspace method

Similar to the covariance matrix error correction, if the interference and noise subspace projection matrix U is solved_i+nThen the estimation error of the desired steering vector can be eliminated by projecting to its orthogonal subspace, i.e.:

$\tilde{a}=(I-U_{i+n}{U}_{i+n}^H)\hat{a}---\left(8\right)$

wherein,in order for the estimate of the steering vector to be desired,the correction value is the desired steering vector. R which can be obtained by the formula (7) can be used herein₁The eigenvalue decomposition is performed as:

$R_1=\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&lambda;}}_i{e}_i{e}_i^H---\left(9\right)$

wherein λ is₁＞λ₂＞...＞λ_K+1＝...＝λ_N＝σ²Is a covariance matrix R₁Characteristic value of e_iAnd the feature vector is corresponding to the feature value.

Due to the assumption of a steering vectorWith the true steering vector a (theta)₀) The angle difference is not large, so each characterEigenvector e_iHomodromous hypothesis steering vectorProjection, selecting feature vector corresponding to smaller projection to construct interference plus noise signal subspace U_i+n。

Projection of feature vectors onto hypothetical steering vectorsThe descending order is arranged to obtain:

since larger projection-corresponding feature vectors can construct the desired signal subspace, smaller feature vectors can construct the interference-plus-noise subspace.

$(p_{\&lsqb;m\&rsqb;}+p_{\&lsqb;m-1\&rsqb;}\&GreaterEqual;...\&GreaterEqual;p_{\&lsqb;1\&rsqb;})/\underset{i=1}{\overset{M}{\&Sigma;}}p\left(i\right)<\mathrm{\&rho;}---\left(10\right)$

As m increases, the value of m when the right equation is first exceeded is evaluated, and the interference-plus-noise subspace is constructed as:

U_i+n＝[e_[1],e_[2],…,e_[m-1]](11)

the interference plus noise subspace projection matrix may be expressed as:

$P_{i+n}=U_{i+n}{U}_{i+n}^H---\left(12\right)$

further, the desired subspace projection matrix is obtained as:

P_s＝I-P_i+n(13)

similar to the covariance matrix error correction algorithm, the estimation error of the interference plus noise subspace part is eliminated by assuming that the guide vector projection to the expected signal subspace can be used for mapping the guide vector part in the signal subspace without distortion. The estimation accuracy of the steering vector can be improved. The specific calculation formula is as follows:

$\tilde{a}=P_s\hat{a}---\left(14\right)$

the improved beam forming weight vector calculation formula is as follows:

$w=\frac{R_1\tilde{a}}{{\tilde{a}}^H{R}_1^{-1}\tilde{a}}---\left(15\right)$

and step four, obtaining the output signal of the array antenna.

y＝w^Hx(16)

Where y is the output signal.

The effect of the embodiment is as follows:

the phase of the invention and the traditional beam forming method have the following advantages:

an MEPS adaptive beamforming algorithm is proposed to reduce the estimation error of the sampling covariance matrix while correcting the estimation error of the hypothesized steering vector. The algorithm firstly constructs a KR signal subspace to estimate an array covariance matrix, and the algorithm can be guaranteed to be converged under fewer snapshots; secondly, an interference plus noise subspace is constructed by using the estimated covariance matrix eigenvector, and estimation errors are reduced by projecting the guide vector to the orthogonal subspace of the guide vector. The algorithm can accurately estimate the signal subspace without the number of the information sources, and greatly improves the flexibility of engineering application. Simulation results show that the improved algorithm can be converged at less fast beats, obtain higher SINR output at a larger range of SNR input and cannot generate obvious signal cancellation at the higher SNR input.

Compared with the traditional algorithm, the embodiment can obtain accurate sampling signals under smaller fast shooting;

compared with the traditional algorithm, the technology of the embodiment can effectively correct the error of the expected signal guide vector;

and thirdly, compared with the traditional algorithm, the technology of the embodiment can be converged to higher precision under smaller fast shooting.

The following simulation tests were performed to verify:

FIG. 2 shows the convergence rate curves of MEPS, RLM, EPS and MVDR, respectively. Where the input signal-to-noise ratio (SNR) is 20dB, so the array optimal output SINR is 30 dB. In the figure, K is the number of sampling fast beats. In the case that the steering vector is precisely known, the estimation accuracy of the covariance matrix directly affects the convergence speed of the algorithm. Comparing the output SINR of the four algorithms with the fast sampling rate can draw the following conclusions: the MEPS can converge at fewer snapshots, so that the MEPS can effectively reduce the estimation error of the sampling covariance matrix. RLM, EPS and MVDR have a large error due to a small snapshot down-sampling covariance matrix, and the convergence rate of the algorithm is low.

Fig. 3 is a convergence accuracy curve of the MEPS, RLM, EPS, and MVDR algorithms, respectively. Comparing the output SINR versus input SNR for the three algorithms can be seen: with the change of the input SNR, the MEPS always keeps higher SINR output, but the performance of the three algorithms of RLM, EPS and MVDR is unstable. Namely: EPS has higher SINR output when SNR is less than 0dB, and the output SINR drops rapidly when SNR is more than 5 dB; the output SINR obtained by RLM and MVDR are both lower. In fact, when the input SNR increases, the effect of the covariance matrix estimation error on the output SINR increases, and RLM, EPS and MVDR will produce severe signal cancellation, so the output SINR is low. The MEPS algorithm constructs a KR signal covariance vector subspace and reduces the estimation error of a covariance matrix by means of projection, so that the estimation accuracy of the covariance matrix is improved, and the output SINR is high.

Fig. 4 is a curve of variation of output SINR with fast beat number of four algorithms of MEPS, RLM, EPS and ESB when the desired signal steering vector is mismatched. The desired signal angle is estimated to be 6 °. It can be seen from fig. 4 that when the guide vector mismatch exists, the convergence speed of the MEPS algorithm is slower than that of the MEPS algorithm without the mismatch, but since the sampling covariance matrix is projected in the covariance vector quantum space, the error is reduced, the MEPS algorithm still has a faster convergence speed compared with the other three algorithms, and the MEPS algorithm has stronger robustness to the guide vector mismatch. The covariance matrixes of the RLM, the EPS and the ESB have large errors, so that the convergence rate of the algorithm is reduced; the covariance matrix with large error causes large estimation error of the steering vector, and influences the output SINR.

FIG. 5 is a graph of output SINR versus input SNR for MEPS, RLM, EPS, and ESB. The desired signal angle is estimated to be 6 °. It can be seen that when the input SNR changes, the RLM, EPS, and ESB algorithms cannot maintain a high SINR output, while the MEPS algorithm always has a high SINR output. In fact, the covariance matrix adopted by the MEPS algorithm has high estimation precision, so that a more accurate expected guide vector can be effectively estimated. However, the covariance matrix covariance of the RLM, EPS and ESB algorithms has a large error, so that a more accurate expected guide vector cannot be estimated, and thus the output SINR is unstable. And when the input SNR is increased, the RLM, EPS and ESB algorithms have expected signal cancellation, so that the output SINR is reduced quickly.

In summary, the MEPS adaptive beamforming algorithm reduces the estimation error of the sampling covariance matrix and corrects the estimation error of the assumed steering vector. The algorithm can accurately estimate the signal subspace without the number of the information sources, and greatly improves the flexibility of engineering application. The improved algorithm herein is capable of achieving convergence with fewer snapshots and a higher SINR output at a larger range of SNR inputs without significant signal cancellation at the higher SNR inputs.

Claims (1)

1. A projection subspace estimation adaptive beam forming method for improving feature vectors is characterized by comprising the following steps:

(1) the array antenna samples an input signal:

the sampled data is represented by a sampled data covariance matrix as:

$R=\frac{1}{K}\underset{i=1}{\overset{K}{\&Sigma;}}{x}^{H}x$

wherein, K is the sampling fast beat number, and x is the input signal;

(2) and constructing a KR signal covariance vector by using a Khatri-Rao product:

$d\left(\mathrm{\&theta;}\right)=a\left(\mathrm{\&theta;}\right)\&CircleTimes;a\left(\mathrm{\&theta;}\right)$

vec (-) is the stacking of the matrix columns into the first column:

d(θ)＝vec(a(θ)a^H(θ))

wherein a (θ) is an input signal steering vector;

and (3) solving covariance vectors on two sides to obtain:

$\begin{array}{c}vec(R-{\mathrm{\&sigma;}}_{noise}^{2}I)=\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&sigma;}}_i^{2}vec\left(a\right({\mathrm{\&theta;}}_i\left)a^H\right({\mathrm{\&theta;}}_i\left)\right)\\ =\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&sigma;}}_i^{2}d\left({\mathrm{\&theta;}}_i\right)\end{array}$

for a linear combination of the covariance vectors of the desired signal and the interfering signal, a matrix S is constructed:

$S=\underset{0}{\overset{\mathrm{\&pi;}}{\&Integral;}}d\left(\mathrm{\&theta;}\right)d^H\left(\mathrm{\&theta;}\right)d\mathrm{\&theta;}$

signal covariance vectorLinearly expressed by the eigenvector of the matrix S, the d (theta) subspace dimension is equal to the number of non-zero eigenvalues of the matrix S;

constructing a matrix Q by the eigenvectors corresponding to the non-zero eigenvalues of the matrix S:

Q＝[e₁,e₂,...,e_K]

wherein,the characteristic vectors corresponding to the non-zero eigenvalues of the matrix Q are obtained, and the columns of Q are mutually orthogonal to form a characteristic subspace of the signal covariance vector;

$vec\left(R_1\right)=Q{\left(Q^{H}Q\right)}^{-1}{Q}^{H}vec\left(\hat{R}\right)$

wherein, Q (Q)^HQ)^-1Q^HA projection matrix which is a signal covariance vector subspace;for sampling the covariance matrix, R₁To representProjecting to a signal covariance vector quantum space;

(3) estimating a guide vector by a feature vector projection subspace method:

if the interference and noise subspace projection matrix U is obtained_i+nThe estimation error of the desired steering vector is eliminated by projection into the orthogonal subspace:

$\tilde{a}=(I-U_{i+n}{U}_{i+n}^H)\hat{a}$

wherein,in order for the estimate of the steering vector to be desired,r obtained for desired guide vector correction value₁The eigenvalue decomposition is performed as:

$R_1=\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&lambda;}}_i{e}_i{e}_i^H$

wherein λ is₁＞λ₂＞...＞λ_K+1＝...＝λ_N＝σ²Is a covariance matrix R₁Characteristic value of e_iThe characteristic vector corresponding to the characteristic value;

each feature vector e_iAll-directional guide vectorProjection, selecting feature vector corresponding to projection to construct interference plus noise signal subspace U_i+n；

Projection of feature vectors onto steering vectorsThe descending order is arranged to obtain: p is a radical of_[M]≥p_[M-1]≥…≥p_[1]；

Constructing an interference plus noise subspace using the feature vectors;

$(p_{\&lsqb;m\&rsqb;}+p_{\&lsqb;m-1\&rsqb;}\&GreaterEqual;...\&GreaterEqual;p_{\&lsqb;1\&rsqb;})/\underset{i=1}{\overset{M}{\&Sigma;}}p\left(i\right)<\mathrm{\&rho;}$

as m increases, the value of m when p is first exceeded is the evaluated value, constructing the interference plus noise subspace:

U_i+n＝[e_[1],e_[2],…,e_[m-1]]

the interference plus noise subspace projection matrix is then:

$P_{i+n}=U_{i+n}{U}_{i+n}^H$

further, the desired subspace projection matrix is obtained as:

P_s＝I-P_i+n

eliminating the estimation error of the interference plus noise subspace part, and improving the estimation precision of the guide vector:

$\tilde{a}=P_s\hat{a}$

improved beamforming weight vector:

$w=\frac{R_1\tilde{a}}{{\tilde{a}}^H{R}_1^{-1}\tilde{a}}$

(4) obtaining array antenna output signals

y＝w^Hx

Where y is the output signal.