CN104408278A - A method for forming steady beam based on interfering noise covariance matrix estimation - Google Patents

Abstract

The present invention relates to the field of self-adaption beam forming during array signal processing, in particular, relates to a method for forming a steady beam based on an interfering noise covariance matrix estimation. The present invention includes using an antenna array to receive a distant field narrowband incident signal, using a limited snapshot number to estimate the covariance matrix for receiving data, then performing characteristic decomposition on the covariance matrix for receiving data, wherein prior information of the number of an incident signal source can be obtained via a method of estimating the number of the signal source; optimally obtaining a more accurate desired signal steering vector at a projection of a noise subspace according to an estimated desired signal steering vector, and performing estimation obtaining of the interfering noise covariance matrix according to interference signal steering vector prior information and an estimated noise power so as to perform beam forming. The present invention first performs the interfering noise covariance matrix estimation to get rid of the component of the desired signal in the training data, which can avoid the generation of signal cancellation.

Description

A kind of robust ada-ptive beamformer method based on interference noise covariance matrix

Technical field

The present invention relates to Adaptive beamformer field in Array Signal Processing, be specifically related to a kind of robust ada-ptive beamformer method based on interference noise covariance matrix.

Background technology

Since last century six the seventies, researchist starts one-dimensional signal process to extend to multidimensional processiug field gradually, opens this new important research field of Array Signal Processing.Array Signal Processing is that several or tens sensors are lined up sensor array according to certain array mode at space diverse location, utilizes it to carry out reception process to spacing wave, extracts the signal characteristic of useful signal, the information comprised in analytic signal.Compared with the one-dimensional signal process of single-sensor Received signal strength, sensor array has the advantage such as wave beam control flexibly, higher processing gain, extremely strong AF panel and higher spatial resolution.Thus decades, array signal obtained development at full speed recently, current Array Signal Processing has become a very important branch of signal transacting field, is widely used in multiple national economy and the Military Application fields such as radar, sonar, communication, seismic survey, dynamo-electric measurement, radio astronomy and medical diagnosis.

Capon (Minimum Variance Distortionless Respons, MVDR) beamforming algorithm makes that wanted signal is undistorted to be passed through, and has better inhibiting adaptive array signal treatment technology for Noise and Interference simultaneously.In actual applications, usually adopt method that array received signal covariance is inverted to construct MVDR, the method is better performances when Received signal strength does not comprise wanted signal.But in actual applications, wanted signal is often included in array received signal, and simultaneously communication process medium wave deformation distortion, sensor position uncertainties, signal wave reach angle (Direction of Arrive) evaluated error, array element inconsistency equal error will cause wanted signal steering vector to occur error.Capon beamforming algorithm is very responsive for steering vector error, and now wanted signal will be treated as interference and suppresses, and produces wanted signal and disappears mutually, cause the sharply decline of performance.Therefore sane Beamforming Method is one of study hotspot of scholar.

Summary of the invention

The object of the present invention is to provide one to avoid signal cancellation, a kind of robust ada-ptive beamformer method based on interference noise covariance matrix of more sane wave beam is provided.

The object of the present invention is achieved like this:

(1) arrowband, antenna array receiver far field incoming signal is utilized, get limited fast umber of beats to carry out estimating reception data covariance matrix, then carry out feature decomposition to reception data covariance matrix, the prior imformation according to incident signal source number obtains noise subspace U _nand estimating noise power the prior imformation of incident signal source number can be obtained by source number estimation method, the desired homogeneous uniform line-array that sensor array is made up of N number of array element, far field space target wanted signal and P undesired signal, signal and interference, uncorrelated mutually between interference and interference, each channel noise is mutual independently zero mean Gaussian white noise, with signal, disturb all uncorrelated, Array Model, N × 1 tie up array received signal x (t) be:

$\begin{array}{c}x\left(t\right)=\underset{m=0}{\overset{P}{\mathrm{\Σ}}}a\left({\mathrm{\θ}}_m\right)s_m\left(t\right)+n\left(t\right)\\ =a\left({\mathrm{\θ}}_0\right)s_0\left(t\right)+A_I{S}_I\left(t\right)+n\left(t\right)\end{array};$

θ _m, m=1,2 ..., P is that undesired signal ripple reaches angular direction, θ ₀for expecting that signal wave reaches angular direction, a (θ _m), m=0,1,2 ..., P represents the steering vector of wanted signal, undesired signal, a (θ ₀) for expecting signal guide vector, s ₀t () is for expecting complex envelope, A _i=[a (θ ₁), a (θ ₂) ..., a (θ _p)] for undesired signal steering vector composition array manifold, S _i(t)=[s ₁(t), s ₂(t) ..., s _p(t)] ^tfor undesired signal complex envelope, n (t)=[n ₁(t), n ₂(t) ..., n _n(t)] ^tit is white Gaussian noise.() ^trepresent transpose operation, the covariance matrix R of array received signal is:

$\begin{array}{c}R=R_s+R_{i+n}={\mathrm{\σ}}_0^{2}a\left({\mathrm{\θ}}_0\right)a^H\left({\mathrm{\θ}}_0\right)+\\ \underset{k=1}{\overset{P}{\mathrm{\Σ}}}{\mathrm{\σ}}_k^{2}a\left({\mathrm{\θ}}_k\right)a^H\left({\mathrm{\θ}}_k\right)+{\mathrm{\σ}}_n^{2}I\end{array};$

R _sand R _i+nbe respectively wanted signal covariance matrix and interference noise covariance matrix, represent the power of wanted signal, a P undesired signal and space white noise signal respectively, I representation unit matrix, () ^hrepresent complex-conjugate transpose computing, the snap data estimation Received signal strength data covariance matrix by limited:

$\hat{R}=\frac{1}{L}\underset{k=1}{\overset{L}{\mathrm{\Σ}}}{x}_k\left(t\right)x_k^H\left(t\right);$

L represents fast umber of beats, x _kt () represents a kth snap;

Number of sources P+1<N is right carry out feature decomposition, known according to incident signal source number:

$\hat{R}=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i{u}_i{u}_i^H=U_s{\mathrm{\&lambda;}}_s{U}_s^H+U_n{\mathrm{\&lambda;}}_n{U}_n^H;$

Wherein n number of eigenwert, u _i, i=1,2 ..., N is its characteristic of correspondence vector, λ _s=diag{ λ ₁, λ ₂..., λ _p+1and λ _n=diag{ λ _p+2, λ _p+3..., λ _nrepresent larger eigenwert, less eigenvalue cluster diagonally matrix respectively, U _sand U _nthen represent signal disturbing subspace and noise subspace respectively;

estimation of Mean can be got obtain by the little eigenwert that Received signal strength covariance matrix feature decomposition is corresponding, namely

${\mathrm{\&sigma;}}_n^2={\widehat{\mathrm{\&sigma;}}}_n^2=\frac{1}{N-P-1}\underset{i=P+2}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i;$

(2) by wanted signal steering vector error constraints in spherical uncertain concentrate, according to estimate wanted signal steering vector at noise subspace U _nprojection optimum obtain wanted signal steering vector more accurately true wanted signal steering vector and U _sthe same subspace of generate, due to U _swith U _northogonal, then wanted signal steering vector is orthogonal with noise subspace:

$u_i^{H}a\left({\mathrm{\&theta;}}_0\right)=0,i=P+2,P+3,...,N,$

${\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=a^H\left({\mathrm{\&theta;}}_0\right)U_n{U}_n^{H}a\left({\mathrm{\&theta;}}_0\right)=0,$

Wanted signal steering vector a (θ ₀) there is evaluated error, the wanted signal steering vector of estimation surplus is left in noise subspace:

${\left|\right|U_n^H\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\&GreaterEqual;{\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0,$

When with a (θ ₀) equal time equal sign set up

$\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^2={\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0,$

Wanted signal steering vector error constraints uncertainly to be concentrated in spherical, can based on the sane wave beam of spherical uncertain collection:

$\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^{2}s.t.{\left|\right|a-\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\&le;\mathrm{\&epsiv;};$

ε is the border of spherical uncertain collection, carries out solving obtaining optimum steering vector

(3) noise power of foundation undesired signal steering vector prior imformation and estimation carry out interference noise covariance matrix to obtain undesired signal steering vector prior imformation is obtained by Power estimation method and Array Model,

The power spectrum of Capon Beam-former is:

$p=\frac{1}{a^H\left(\mathrm{\&theta;}\right)R^{-1}a\left(\mathrm{\&theta;}\right)};$

Noise is independently white Gaussian noise, and undesired signal steering vector prior imformation is known, then interference noise covariance matrix for:

${\hat{R}}_{i+n}=\underset{k=1}{\overset{P}{\mathrm{\&Sigma;}}}\frac{a\left({\mathrm{\&theta;}}_k\right)a^H\left({\mathrm{\&theta;}}_k\right)}{a^H\left({\mathrm{\&theta;}}_k\right)R^{-1}a\left({\mathrm{\&theta;}}_k\right)}+{\widehat{\mathrm{\&sigma;}}}_n^{2}I;$

(4) optimum weight vector w is calculated _opt, carry out Wave beam forming,

Optimum weight vector:

$w_{\mathrm{opt}}=\frac{{\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)}{{\stackrel{\&OverBar;}{a}}^H\left({\mathrm{\&theta;}}_0\right){\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)};$

Then aerial array output and Wave beam forming are:

$y=w_{\mathrm{opt}}^{H}x\left(t\right).$

Beneficial effect of the present invention is:

(1) first the present invention has carried out interference noise covariance matrix, eliminates the wanted signal composition in training data, can avoid producing signal cancellation.

(2) the present invention is based on wanted signal steering vector at the optimum acquisition of noise subspace projection steering vector more accurately, therefore for wanted signal steering vector mismatch, there is very strong robustness, and array pattern can point to correct wanted signal come to.

(3) the present invention has higher array output Signal to Interference plus Noise Ratio, and array output performance declines not obvious when higher input signal-to-noise ratio.

Accompanying drawing explanation

Fig. 1 even linear array model;

Fig. 2 algorithm flow chart of the present invention;

Fig. 3 algorithm effect figure of the present invention.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described further.

In order to avoid signal cancellation, the present invention proposes a kind of robust adaptive beamforming method based on interference noise covariance matrix.The method is the mutual independent signal in arrowband, far field for incoming signal, and noise is the situation of white Gaussian noise, and need the prior imformations such as the steering vector knowing incident signal source number and incoming signal, concrete step is as follows:

Step 1: utilize arrowband, antenna array receiver far field incoming signal, get limited fast umber of beats to carry out estimating reception data covariance matrix, then carry out feature decomposition to reception data covariance matrix, the prior imformation according to incident signal source number obtains noise subspace U _nand estimating noise power the prior imformation of incident signal source number can be obtained by source number estimation method.

Step 2: by wanted signal steering vector error constraints in spherical uncertain concentrated, according to the wanted signal steering vector estimated at noise subspace U _nprojection optimum obtain wanted signal steering vector more accurately

Step 3: the noise power of foundation undesired signal steering vector prior imformation and estimation carry out interference noise covariance matrix to obtain undesired signal steering vector prior imformation can be obtained by Power estimation method and Array Model.

Step 4: calculate optimum weight vector w according to MVDR Beamforming Method _opt, then carry out Wave beam forming.

The method that the present invention describes is a kind of robust ada-ptive beamformer method based on interference noise covariance matrix, compared with conventional beamformer method, has robustness for wanted signal steering vector mismatch, and can avoid the generation of signal cancellation phenomenon.

Concrete steps are as follows:

Step 1: utilize arrowband, antenna array receiver far field incoming signal, get limited fast umber of beats to carry out estimating reception data covariance matrix, then carry out feature decomposition to reception data covariance matrix, the prior imformation according to incident signal source number obtains noise subspace U _nand estimating noise power the prior imformation of incident signal source number can be obtained by source number estimation method.

Considering the desired homogeneous uniform line-array that is made up of N number of array element of sensor array, there is a target wanted signal and P undesired signal in far field space, signal and interference, uncorrelated mutually between interference and interference.Each channel noise is mutual independently zero mean Gaussian white noise, with signal, disturb all uncorrelated.Array Model as shown in Figure 1.Array received signal x (t) that N × 1 is tieed up can be expressed as:

$\begin{array}{c}x\left(t\right)=\underset{m=0}{\overset{P}{\mathrm{\&Sigma;}}}a\left({\mathrm{\&theta;}}_m\right)s_m\left(t\right)+n\left(t\right)\\ =a\left({\mathrm{\&theta;}}_0\right)s_0\left(t\right)+A_I{S}_I\left(t\right)+n\left(t\right)\end{array}---\left(1\right)$

In formula: θ _m, m=1,2 ..., P is that undesired signal ripple reaches angular direction, θ ₀for expecting that signal wave reaches angular direction, a (θ _m), m=0,1,2 ..., P represents the steering vector of wanted signal, undesired signal, a (θ ₀) for expecting signal guide vector, s ₀t () wraps again for wanted signal

Network, A _i=[a (θ ₁), a (θ ₂) ..., a (θ _p)] for undesired signal steering vector composition array manifold, S _i(t)=[s ₁(t), s ₂(t) ..., s _p(t)] ^tfor undesired signal complex envelope, n (t)=[n ₁(t), n ₂(t) ..., n _n(t)] ^tit is white Gaussian noise.() ^trepresent transpose operation.So the covariance matrix R of array received signal is:

$\begin{array}{c}R=R_s+R_{i+n}={\mathrm{\&sigma;}}_0^{2}a\left({\mathrm{\&theta;}}_0\right)a^H\left({\mathrm{\&theta;}}_0\right)+\\ \underset{k=1}{\overset{P}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_k^{2}a\left({\mathrm{\&theta;}}_k\right)a^H\left({\mathrm{\&theta;}}_k\right)+{\mathrm{\&sigma;}}_n^{2}I\end{array}---\left(2\right)$

In formula: R _sand R _i+nbe respectively wanted signal covariance matrix and interference noise covariance matrix. represent the power of wanted signal, a P undesired signal and space white noise signal respectively.I representation unit matrix, () ^hrepresent complex-conjugate transpose computing.In actual applications, often through limited snap data estimation Received signal strength data covariance matrix, that is:

$\hat{R}=\frac{1}{L}\underset{k=1}{\overset{L}{\mathrm{\&Sigma;}}}{x}_k\left(t\right)x_k^H\left(t\right)---\left(3\right)$

In formula: L represents fast umber of beats, x _kt () represents a kth snap.

Suppose number of sources P+1<N, right carry out feature decomposition, known according to incident signal source number:

$\hat{R}=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i{u}_i{u}_i^H=U_s{\mathrm{\&lambda;}}_s{U}_s^H+U_n{\mathrm{\&lambda;}}_n{U}_n^H---\left(4\right)$

Wherein n number of eigenwert, u _i, i=1,2 ..., N is its characteristic of correspondence vector.λ _s=diag{ λ ₁, λ ₂..., λ _p+1and λ _n=diag{ λ _p+2, λ _p+3..., λ _nrepresent larger eigenwert, the paired angular moment of less eigenvalue cluster respectively

Battle array.U _sand U _nthen represent signal disturbing subspace and noise subspace respectively.

estimation of Mean can be got obtain by the little eigenwert that Received signal strength covariance matrix feature decomposition is corresponding, namely

${\mathrm{\&sigma;}}_n^2={\widehat{\mathrm{\&sigma;}}}_n^2=\frac{1}{N-P-1}\underset{i=P+2}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i---\left(5\right)$

Step 2: by wanted signal steering vector error constraints in spherical uncertain concentrated, according to the wanted signal steering vector estimated at noise subspace U _nprojection optimum obtain wanted signal steering vector more accurately

From subspace theory, true wanted signal steering vector and U _sthe same subspace of generate, again due to U _swith U _northogonal, then wanted signal steering vector is orthogonal with noise subspace.That is:

$u_i^{H}a\left({\mathrm{\&theta;}}_0\right)=0,i=P+2,P+3,...,N---\left(6\right)$

It can thus be appreciated that:

${\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=a^H\left({\mathrm{\&theta;}}_0\right)U_n{U}_n^{H}a\left({\mathrm{\&theta;}}_0\right)=0---\left(7\right)$

In actual applications, real wanted signal steering vector a (θ ₀) usually there is evaluated error, the wanted signal steering vector therefore estimated in noise subspace, leave surplus, therefore have inequality below to set up:

${\left|\right|U_n^H\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\&GreaterEqual;{\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0---\left(8\right)$

Obviously, when with a (θ ₀) equal time equal sign set up.Namely

$\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^2={\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0---\left(9\right)$

By wanted signal steering vector error constraints in spherical uncertain concentrated, thus can be as follows based on the robust ada-ptive beamformer algorithm of spherical uncertain collection:

$\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^{2}s.t.{\left|\right|a-\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\&le;\mathrm{\&epsiv;}---\left(10\right)$

ε is the border of spherical uncertain collection, and above formula is the convex optimization problem of second order, and CVX kit can be utilized to carry out solving the steering vector obtaining optimum

Step 3: the noise power of foundation undesired signal steering vector prior imformation and estimation carry out interference noise covariance matrix to obtain undesired signal steering vector prior imformation can be obtained by Power estimation method and Array Model.

The power spectrum of Capon Beam-former can be expressed from the next:

$p=\frac{1}{a^H\left(\mathrm{\&theta;}\right)R^{-1}a\left(\mathrm{\&theta;}\right)}---\left(11\right)$

Because undesired signal number is usually limited in spatial domain, noise is assumed to independently white Gaussian noise, and undesired signal steering vector prior imformation is known, then interference noise covariance matrix can be expressed as:

${\hat{R}}_{i+n}=\underset{k=1}{\overset{P}{\mathrm{\&Sigma;}}}\frac{a\left({\mathrm{\&theta;}}_k\right)a^H\left({\mathrm{\&theta;}}_k\right)}{a^H\left({\mathrm{\&theta;}}_k\right)R^{-1}a\left({\mathrm{\&theta;}}_k\right)}+{\widehat{\mathrm{\&sigma;}}}_n^{2}I---\left(12\right)$

Step 4: calculate optimum weight vector w according to MVDR Beamforming Method _opt, and carry out Wave beam forming.

Optimum weight vector can be tried to achieve by following formula:

$w_{\mathrm{opt}}=\frac{{\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)}{{\stackrel{\&OverBar;}{a}}^H\left({\mathrm{\&theta;}}_0\right){\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)}---\left(13\right)$

Then array output and Wave beam forming are:

$y=w_{\mathrm{opt}}^{H}x\left(t\right)---\left(14\right).$

Claims (1)

1., based on a robust ada-ptive beamformer method for interference noise covariance matrix, it is characterized in that, comprise the following steps:

(1) arrowband, antenna array receiver far field incoming signal is utilized, get limited fast umber of beats to carry out estimating reception data covariance matrix, then carry out feature decomposition to reception data covariance matrix, the prior imformation according to incident signal source number obtains noise subspace U _nand estimating noise power the prior imformation of incident signal source number can be obtained by source number estimation method, the desired homogeneous uniform line-array that sensor array is made up of N number of array element, far field space target wanted signal and P undesired signal, signal and interference, uncorrelated mutually between interference and interference, each channel noise is mutual independently zero mean Gaussian white noise, with signal, disturb all uncorrelated, Array Model, N × 1 tie up array received signal x (t) be:

$\begin{array}{c}x\left(t\right)=\underset{m=0}{\overset{P}{\mathrm{\&Sigma;}}}a\left({\mathrm{\&theta;}}_m\right)s_m\left(t\right)+n\left(t\right)\\ =a\left({\mathrm{\&theta;}}_0\right)s_0\left(t\right)+A_I{S}_I\left(t\right)+n\left(t\right);\end{array}$

θ _m, m=1,2 ..., P is that undesired signal ripple reaches angular direction, θ ₀for expecting that signal wave reaches angular direction, a (θ _m), m=0,1,2 ..., P represents the steering vector of wanted signal, undesired signal, a (θ ₀) for expecting signal guide vector, s ₀t () is for expecting complex envelope, A _i=[a (θ ₁), a (θ ₂) ..., a (θ _p)] for undesired signal steering vector composition array manifold, S _i(t)=[s ₁(t), s ₂(t) ..., s _p(t)] ^tfor undesired signal complex envelope, n (t)=[n ₁(t), n ₂(t) ..., n _n(t)] ^tit is white Gaussian noise.() ^trepresent transpose operation, the covariance matrix R of array received signal is:

$\begin{array}{c}R=R_s+R_{i+n}={\mathrm{\&sigma;}}_0^{2}a\left({\mathrm{\&theta;}}_0\right)a^H\left({\mathrm{\&theta;}}_0\right)+\\ \underset{k=1}{\overset{P}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_k^{2}a\left({\mathrm{\&theta;}}_k\right)a^H\left({\mathrm{\&theta;}}_k\right)+{\mathrm{\&sigma;}}_n^{2}I;\end{array}$

R _sand R _i+nbe respectively wanted signal covariance matrix and interference noise covariance matrix, represent the power of wanted signal, a P undesired signal and space white noise signal respectively, I representation unit matrix, () ^hrepresent complex-conjugate transpose computing, the snap data estimation Received signal strength data covariance matrix by limited:

$\hat{R}=\frac{1}{L}\underset{k=1}{\overset{L}{\mathrm{\&Sigma;}}}{x}_k\left(t\right)x_k^H\left(t\right);$

L represents fast umber of beats, x _kt () represents a kth snap;

Number of sources P+1<N is right carry out feature decomposition, known according to incident signal source number:

$\hat{R}=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i{u}_i{u}_i^H=U_s{\mathrm{\&lambda;}}_s{U}_s^H+U_n{\mathrm{\&lambda;}}_n{U}_n^H;$

Wherein ${\mathrm{\&lambda;}}_1\&GreaterEqual;{\mathrm{\&lambda;}}_2\&GreaterEqual;\&CenterDot;\&CenterDot;\&CenterDot;\&GreaterEqual;{\mathrm{\&lambda;}}_P\&GreaterEqual;{\mathrm{\&lambda;}}_{P+1}\&GreaterEqual;{\mathrm{\&lambda;}}_{P+2}=\&CenterDot;\&CenterDot;\&CenterDot;={\mathrm{\&lambda;}}_N={\mathrm{\&sigma;}}_n^2$ N number of eigenwert, u _i, i=1,2 ..., N is its characteristic of correspondence vector, λ _s=diag{ λ ₁, λ ₂..., λ _p+1and λ _n=diag{ λ _p+2, λ _p+3..., λ _nrepresent larger eigenwert, less eigenvalue cluster diagonally matrix respectively, U _sand U _nthen represent signal disturbing subspace and noise subspace respectively;

estimation of Mean can be got obtain by the little eigenwert that Received signal strength covariance matrix feature decomposition is corresponding, namely

${\mathrm{\&sigma;}}_n^2={\widehat{\mathrm{\&sigma;}}}_n^2=\frac{1}{N-P-1}\underset{i=P+2}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i;$

(2) by wanted signal steering vector error constraints in spherical uncertain concentrate, according to estimate wanted signal steering vector at noise subspace U _nprojection optimum obtain wanted signal steering vector more accurately true wanted signal steering vector and U _sthe same subspace of generate, due to U _swith U _northogonal, then wanted signal steering vector is orthogonal with noise subspace:

$\begin{array}{cc}{u}_i^{H}a\left({\mathrm{\&theta;}}_0\right)=0& i=P+2,P+3,\&CenterDot;\&CenterDot;\&CenterDot;,N,\end{array}$

${\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=a^H\left({\mathrm{\&theta;}}_0\right)U_n{U}_n^{H}a\left({\mathrm{\&theta;}}_0\right)=0,$

Wanted signal steering vector a (θ ₀) there is evaluated error, the wanted signal steering vector of estimation surplus is left in noise subspace:

${\left|\right|U_n^H\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\&GreaterEqual;{\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0,$

When with a (θ ₀) equal time equal sign set up

$\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^2={\left|\right|U_n^{H}a\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2=0,$

Wanted signal steering vector error constraints uncertainly to be concentrated in spherical, can based on the sane wave beam of spherical uncertain collection:

$\begin{array}{ccc}\underset{a}{\min }{\left|\right|U_n^{H}a\left|\right|}_2^2& s.t.& {\left|\right|a-\hat{a}\left({\mathrm{\&theta;}}_0\right)\left|\right|}_2^2\end{array},\&le;\mathrm{\&epsiv;};$

ε is the border of spherical uncertain collection, carries out solving obtaining optimum steering vector

(3) noise power of foundation undesired signal steering vector prior imformation and estimation carry out interference noise covariance matrix to obtain undesired signal steering vector prior imformation is obtained by Power estimation method and Array Model,

The power spectrum of Capon Beam-former is:

$p=\frac{1}{a^H\left(\mathrm{\&theta;}\right)R^{-1}a\left(\mathrm{\&theta;}\right)};$

Noise is independently white Gaussian noise, and undesired signal steering vector prior imformation is known, then interference noise covariance matrix for:

${\hat{R}}_{i+n}=\underset{k=1}{\overset{P}{\mathrm{\&Sigma;}}}\frac{a\left({\mathrm{\&theta;}}_k\right)a^H\left({\mathrm{\&theta;}}_k\right)}{a^H\left({\mathrm{\&theta;}}_k\right)R^{-1}a\left({\mathrm{\&theta;}}_k\right)}+{\hat{R}}_n^{2}I;$

(4) optimum weight vector w is calculated _opt, carry out Wave beam forming,

Optimum weight vector:

$w_{\mathrm{opt}}=\frac{{\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)}{{\stackrel{\&OverBar;}{a}}^H\left({\mathrm{\&theta;}}_0\right){\hat{R}}_{i+n}^{-1}\stackrel{\&OverBar;}{a}\left({\mathrm{\&theta;}}_0\right)};$

Then aerial array output and Wave beam forming are:

$y=w_{\mathrm{opt}}^{H}x\left(t\right).$