CN105699948B - Based on support vector machines and improve the Beamforming Method and system of mean square error performance - Google Patents

Abstract

The invention discloses a kind of based on support vector machines and improves the Beamforming Method and system of mean square error performance, it is above-mentioned based on support vector machines and to improve the Beamforming Method of mean square error performance and include the following steps: the regulation coefficient for being minimised as target and calculating robustness Beam-former weight with MSE, updated weight vector is obtained by adjusting coefficient；When calculating regulation coefficient, the actual value of desired signal is replaced using the estimated value of desired signal；It is minimised as target with array output power, using ε-insensitive loss function, converts SVR model for Wave beam forming problem；The SVR model is solved using IRWLS method and exports result.The present invention is minimised as target with MSE and is adjusted again to the Beam-former weight based on SVM, can improve the MSE performance of system in the case where not changing system SINR performance.

Description

Based on support vector machines and improve the Beamforming Method and system of mean square error performance

Technical field

It is especially a kind of based on support vector machines and to improve the wave of mean square error performance the invention belongs to Beamforming Method Beam forming method.

Background technique

Adaptive beamformer can automated to respond to the signal transmission environment of variation, press down while receiving desired signal Interference signal processed and noise have stronger flexibility and interference rejection capability compared with the conventional beamformer that non-data relies on.

However applicant's research shows that: traditional minimum variance is undistorted response (MVDR, Minimum Variance Distortionless Response) and Capon Beam-former available signal snap is few, direction of arrival (DOA, Direction of Arrival), estimation distortion or interference non-stationary variation etc. in actual working environments performance decline it is serious.

Signal to Interference plus Noise Ratio (SINR, Signal-to-Interference plus Noise Ratio) maximization is to work as prewave Beam formed in common measurement index and design criteria, but the criterion cannot be guaranteed the low distortion of signal, therefore can not help In the accurate estimation to signal.Least mean-square error (MMSE, Minimum Mean Squared Error) Wave beam forming is then straight It connects and target is minimised as with mean square error (MSE, Mean Squared Error), it is ensured that lesser distortion, but because of signal It is unknown and cannot be directly realized by.

Summary of the invention

Goal of the invention: provide it is a kind of based on support vector machines and improve the Beamforming Method of mean square error performance, with solution The above problem certainly of the existing technology.Further objective is that providing a kind of system for realizing the above method.

Technical solution: a kind of based on support vector machines and to improve the Beamforming Method of mean square error performance including as follows Step: step 1, the regulation coefficient for being minimised as target and calculating robustness Beam-former weight with MSE, by adjusting coefficient Obtain updated weight vector；When calculating regulation coefficient, the actual value of desired signal is replaced using the estimated value of desired signal；

Step 2 is minimised as target with array output power, using ε-insensitive loss function, by Wave beam forming problem It is converted into SVR model；

Step 3 solves the SVR model using IRWLS method and exports result.

It is a kind of based on support vector machines and to improve the Beam Forming System of mean square error performance, comprising:

Weight vector computing device, for the adjustment for being minimised as target and calculating robustness Beam-former weight of MSE Coefficient obtains updated weight vector by adjusting coefficient；When calculating regulation coefficient, replaced using the estimated value of desired signal The actual value of desired signal；

Model building device, for being minimised as target with array output power, using ε-insensitive loss function, by wave beam shape It is problematic to be converted into SVR model；

SVR model solution device, for solving the SVR model using IRWLS method and exporting result.

The Beam-former weight based on SVM is adjusted again the utility model has the advantages that the present invention is minimised as target with MSE It is whole, the MSE performance of system can be improved in the case where not changing system SINR performance.

Detailed description of the invention

Fig. 1 and Fig. 2 is simulation result schematic diagram of the invention.

Specific embodiment

The following specifically describes technical background of the invention and technical principles, first briefly describe main contents of the invention, with Research Thinking and beneficial effect of the invention are elaborated by a specific implementation case afterwards.

The present invention is based on support vector machines and the Beamforming Method that improves mean square error performance mainly includes the following steps:

Step 1, the regulation coefficient for being minimised as target and calculating robustness Beam-former weight with MSE, by adjusting Coefficient obtains updated weight vector；When calculating regulation coefficient, the reality of desired signal is replaced using the estimated value of desired signal Actual value.

In the step 1, the regulation coefficient are as follows:

Updated wave weight vector are as follows:

The estimated value of desired signal

It is described in formula | s | for the amplitude for hoping signal, the W is Beam-former weight vector, and footmark H indicates conjugate transposition, X (k) is the array data at k moment.

Step 2 is minimised as target with array output power, using ε-insensitive loss function, by Wave beam forming problem It is converted into SVR model；Cost function in the step 2, in SVR model are as follows:

In formula,Re () and Im () It respectively indicates and seeks real and imaginary parts；δ_RFor error matrix parameter, meaning is practical covariance matrix and sample covariance matrix Error matrix the Frobenius norm upper bound；L_ε(θ_i,d_i,f(θ_i)) it is ε-insensitive loss function, L_ε(θ_i, d_i, f (θ_i)) =| d_i-f(θ_i)|_ε=max 0, | d_i-f(θ_i)|-ε}；It is thereinIt indicates are as follows:

Wherein,C >=0 is punishment Coefficient, ε indicate error margin.

Step 3 solves the SVR model using IRWLS method and exports result.

Solve the specific steps of SVR model are as follows:

Step 31 passes through first order Taylor series expansion ε-insensitive loss function L_ε(θ_i,d_i,f(θ_i)), obtain kth time repeatedly GenerationWithI.e.

Step 32 reconstructs objective function using Two-order approximation, i.e.,

Wherein,It indicates to be directed to weightGradient operator, obtain least square cost function:

In formula, b_iIndicate all and weightUnrelated item, f_iFrom penalty term, then

Step 33 is solved using second iteration algorithm, is obtained:

Wherein,D_fIt is with f_iFor diagonal element, other complete zero Diagonal matrix,Penalty coefficient C is 1, error margin 0.001.

Based on the above method, constructs and a kind of based on support vector machines and improve mean square error for realizing the above method The Beam Forming System of energy, specifically includes that

Weight vector computing device, for the adjustment for being minimised as target and calculating robustness Beam-former weight of MSE Coefficient obtains updated weight vector by adjusting coefficient；When calculating regulation coefficient, replaced using the estimated value of desired signal The actual value of desired signal；

In the weight vector computing device,

The regulation coefficient are as follows:

Updated wave weight vector are as follows:

The estimated value of desired signal

It is described in formula | s | for the amplitude for hoping signal, the W is Beam-former weight vector, and footmark H indicates conjugate transposition, X (k) is the array data at k moment；

Model building device, for being minimised as target with array output power, using ε-insensitive loss function, by wave beam shape It is problematic to be converted into SVR model；Cost function in the model building device, in SVR model are as follows:

In formula,Re () and Im () It respectively indicates and seeks real and imaginary parts；δ_RFor error matrix parameter, meaning is practical covariance matrix and sample covariance matrix Error matrix the Frobenius norm upper bound；L_ε(θ_i,d_i,f(θ_i)) it is ε-insensitive loss function, L_ε(θ_i, d_i, f (θ_i)) =| d_i-f(θ_i)|_ε=max 0, | d_i-f(θ_i)|-ε}；It is thereinIt indicates are as follows:

Wherein,C >=0 is punishment Coefficient, ε indicate error margin, and penalty coefficient C is 1, and the error margin is 0.001.

SVR model solution device, for solving the SVR model using IRWLS method and exporting result.

The SVR model solution device comprises the following modules,

First module, for passing through first order Taylor series expansion ε-insensitive loss function L_ε(θ_i,d_i,f(θ_i)), it obtains Kth time iterationWithI.e.

Second module, for reconstructing objective function using Two-order approximation, i.e.,

Wherein,It indicates to be directed to weightGradient operator, obtain least square cost function:

In formula, b_iIndicate all and weightUnrelated item, f_iFrom penalty term, then

Third module, for obtaining using the solution of second iteration algorithm:

Wherein,D_fIt is with f_iFor diagonal element, other complete zero Diagonal matrix,

Case study on implementation

Construct narrowband model

If the one group far-field signal s coplanar with array_i(t), i=0 ... K is with different incidence angle θs_i, i=0 ... K arrives Up to an antenna array, bay number is M, and array element is non-directional and with the one-dimensional homogenous linear distribution of half-wave long spacing, due to each battle array First present position is different, and homogeneous plane wave has different time delays in the response of each array element output end, then m-th array element Output is:

In formula (1): to correspond to incidence angle θ₀S₀It (t) is desired signal, τ_m(θ₀) it is that desired signal reaches m-th of array element Relative to the time delay of reference array element, s_i(t) θ is come from_i, the interference signal in the direction i=1 ... K, τ_m(θ_i) it is corresponding interference letter Number reach time delay of m-th of the array element relative to reference array element, n_m(t) be m-th of array element additive noise.

Indicate that signal reaches the phase relation between each array element signals with signal steering vector.Enable [x_m,y_m] indicate m-th The position of array element, signal and array are in same plane, using y-axis as the zero direction of direction of arrival, then the amplitude and phase of m-th of array element Position response are as follows:

Wherein, g_mIndicate the gain of m-th of array element, λ indicates the wavelength of narrow band signal, θ_iIndicate that the wave of i-th of signal reaches Angle, θ ∈ (- pi/2, pi/2).

The array data at k moment is represented by

Wherein N_iFor the quantity of interference signal, s (k) and i_jIt (k) is signal and interference, n (k) is noise component(s), θ_sAnd θ_j, j =1 ..., N_iRespectively direction of arrival DOA, a (θ of signal and interference signal_s) and a (θ_j) it is corresponding steering vector:

a(θ_i)=[a₁(θ_i), a₂(θ_i) ..., a_M(θ_i)]^Τ。 (4)

Array signal model can also be written as vector form:

X (k)=A (Θ) s (k)+e (k) (5)

Wherein,

S (k)=[s_s(k), s₁(k) ..., s_p(k)]^Τ

E (k)=[s₁(k), s₂(k) ..., s_M(k)]^Τ

A (Θ)=[a (θ_s), a (θ₁) ..., a (θ_p)] (6)

Respectively indicate signal phasor, noise matrix and array guiding matrix.

The output of narrow-band beam shaper is

Y (k)=W^HX(k) (7)

Wherein W=[w₁,...,w_M]^TFor Beam-former weight vector, M is array number, ()^T()^HRespectively indicate transposition and Conjugate transposition.

Under the conditions of incoherent, array inputs correlation matrix are as follows:

Wherein E [] indicates expectation computing,WithJ=1 ..., N_iRespectively irrelevant incoming signal s (k) and i_j(k) power, Q are noise covariance matrix.

Existing a variety of robustness Wave beam forming schemes are based on linear constraint minimal variance (Linearly Constrained Minimum Variance, LCMV) method, because of this method more convenient various errors considered in actual environment or uncertain Factor.

LCMV Beam-former: under the Linear Constraints to desired DOA, minimize array output ENERGY E [| y (k)|²]=W^HR_XW, i.e.,

Wherein, complex constant g is the array response of desired DOA.

Under actual conditions, accurate covariance matrix R_X=E { X (k) X^Η(k) } unavailable, and common sampling covariance MatrixTo replace

Here N is observation number of snapshots.

As N → ∞, then array output energy can be obtained

Wherein σ_n ²For noise power.

As can be seen from the above equation, first item is desired output signal energy, and Section 2 is that interference signal exports the sum of energy, Section 3 is noise item.At this time if in W^Ha(θ_sArray is minimized under the Linear Constraints of)=g and exports energy, then wave beam shape Desired signal will only be extracted by growing up to be a useful person, and inhibit all other interference and noise signal.

The optimization problem can be solved with Lagrange algorithm, obtained LCMV Wave beam forming best weight value is

MVDR Wave beam forming best weight value when g=1 is

Wave beam forming based on SINR and MMSE criterion

1. being based on SINR criterion

SINR maximizes Wave beam forming and requires in the corresponding output signal of optimal weight vector W, and useful signal is made an uproar with interference The power ratio of acoustical signal is maximum, i.e.,

Wherein R_s=E { ss^ΗIt is desired signal covariance matrix, R_i=E { (i+n) (i+n)^ΗIt is interference noise covariance Matrix.

To obtain enabling the maximum weight vector of SINR, to (14) formula about W derivation, and it is enabled to be equal to zero, obtained

(15) solution of formula is exactly the optimal weight vector under SINR criterion.This is a joint eigenvalue problem, expression formulaValue value range between matrix R_i ^-1R_sBetween the maximum and minimal eigenvalue of W, and the expression formula is just (14) The expression formula of SINR represented by formula, therefore optimization problem is converted to following eigenvalue problem:

R_i ^-1R_sW=λ_maxW (16)

Optimal weight vector is feature vector corresponding to maximum eigenvalue.

Because of R_i=E { (i+n) (i+n)^ΗAnd R_s=E { ss^ΗBe not directly available, with signal covariance matrix R_XTo replace For R_i=E { (i+n) (i+n)^Η, then (14) formula can be converted to following form:

This optimization problem can easily be described as under the Linear Constraints to desired DOA, minimize the defeated of array Out ENERGY E [| y (k) |²]=W^HR_XW, i.e.,

2.MMSE Wave beam forming

By (14) formula it is found that for different zoom factor a, SINR (W)=SINR (aW), but this scaling will affect Other performance especially influences the estimation of signal waveform very big.

The basic thought of MMSE Beam-former is: directly minimizing MSE without requiring unbiased output.

Assuming that s (k)=s is that determining signal (for the sake of simplifying, omits k), the estimated value of s and itBetween MSE are as follows:

WhereinIt isCovariance matrix,It is its biasing, R_i=E { (i +n)(i+n)^ΗUnavailable under physical condition, commonly use sampling covariance matrixInstead of being indicated with R.Then (19) formula is converted Are as follows:

It minimizes above formula and obtains MMSE Beam-former:

W (s)=| s |²(R+|s|²aa^H)^-1A=β (s) W_MVDR, (21)

WhereinAnd 0 < β (s) < 1, with | s |²Monotonic increase.Therefore, to all | s |, W (s) < W_MVDR。

W (s) is substituted into (20) formula, the smallest MSE is obtained, uses MSE_OPTIt indicates

W_MVDR(20) formula of substitution, obtains the MSE of MVDR Beam-former, uses MSE_MVDRIt indicates

Clearly for all | s | > 0, MSE_OPT< MSE_MVDR.Therefore MMSE method can obtain smaller MSE.

Under physical condition, because signal s is unknown, MMSE Beam-former can not be realized directly, at this time can be by right The estimation of signal obtains preferable MSE performance.

3. improving the beamforming algorithm of MSE performance

With the regulation coefficient ρ for being minimised as target and calculating robustness Beam-former weight of MSE, that is, pass through

It obtains

Then by adjusting the updated weight of coefficient ρ are as follows:

The amplitude of desired signal is not aware that under physical condition | s |, the estimated value of desired signal can be usedTo replace s. The estimated value can be obtained by following formula:

Training sample set is T={ (θ₁,d₁),…(θ_P,d_P), θ_i∈ [- 90 °, 90 °], i=1 ... P is that wave reaches orientation Angle, beam pattern target response indicate are as follows:

It is minimised as target with array output power, using ε-insensitive loss function, Wave beam forming problem is modeled as SVR, cost function indicate are as follows:

Wherein,WithIt is respectively defined as:

Here, Re () and Im () are respectively indicated and are sought real and imaginary parts.δ_RFor error matrix parameter, meaning is practical The Frobenius norm upper bound of the error matrix of covariance matrix and sample covariance matrix.L_ε(θ_i,d_i,f(θ_i)) be ε-no Sensitive loss function, is selected as L herein_ε(θ_i, d_i, f (θ_i))=| d_i-f(θ_i)|_ε=max 0, | d_i-f(θ_i) |-ε },It is thereinIt indicates are as follows:

Here,C≥ 0 is penalty coefficient, and ε indicates error margin.

Power least square (Iteratively Reweighted Least Square, IRWLS) method is assigned using iteration to solve The above SVR problem.Pass through first order Taylor series expansion ε-insensitive loss function L_ε(θ_i,d_i,f(θ_i)), obtain kth time iteration 'sWithThen:

Objective function is reconstructed using Two-order approximation, i.e.,WhereinTable Show for weightGradient operator, obtain least square cost function:

In formula, b_iIndicate all and weightUnrelated item, f_iFrom penalty term, then

Second iteration algorithm can be used at this time easily to be solved:

WhereinD_fIt is with f_iFor diagonal element, other complete zero Diagonal matrix,

For the improvement situation for assessing MSE, this section has carried out comparison by Computer Simulation.Under the conditions of 30 snap, The Wave beam forming problem (hereinafter referred to as SVM_LS) of SVR of being modeled as is solved using IRWLS method, and will improve before MSE performance and change Into the SVM_LS method and traditional diagonal load MVDR (Diagonal Loading MVDR, MVDR_D) method after MSE performance It is compared.Simulated conditions setting are as follows:

One array number M=10, array element spacing are the one-dimensional even linear array of half-wavelength.All signal waveforms are orthorhombic phase Move keying (Quadrature Phase Shift Keying, QPSK) modulation, and independent same distribution.Space white Gaussian noise is Unit variance Q=I, dry ratio of making an uproarEach point is all made of being averaged of independently emulating for 500 times in emulation.Beam direction is taken out Sample parameter setting is P₁=20, P₂=40, P=P₁+P₂, Δ=2 °, punishment parameter C and error margin ε are set to 1 He 0.001.The error matrix parameter δ of introduction_RShould be related with signal sequence length and interference non-stationary degree, here only in accordance with letter Number sequence length is set as δ_R=| | R_X||/(10ln(M)+20).The diagonal loading factor of MVDR_D is set as 50.

MSE performance is characterized using the square root of normalized mean squared error (Normalized MSE, NMSE), calculation method Are as follows:

As can be seen from Figure 1, under accurate in desired signal estimation and interference smooth conditions, weight is adjusted coefficient more Newly, though improving mean square error performance not in whole SNR ranges, it is horizontal to maintain lower MSE.

As can be seen from Figure 2, estimate there are error and under interfering nonstationary condition in desired signal, the mean square error of SVM_LS Performance improvement is obvious, lower level is all maintained in whole SNR ranges, and improved SVM_LS method has entirely Office is better than the mean square error performance of MVDR_D.

In short, in view of these situations, it is real that the present invention is based on support vector machines (Support Vector Machine, SVM) Existing robustness Adaptive beamformer.It in cost function by introducing the evaluated error of signal covariance matrix and volume being added The outer constraint of DOA evaluated error and Sidelobe Suppression constraint, is configured to support vector regression (Support for Wave beam forming problem Vector Regression, SVR) problem.Using the inner link of two kinds of criterion, a kind of adjustment is generated using easy method The coefficient of weight, to improve the MSE performance of the beamforming algorithm based on SVM.

The preferred embodiment of the present invention has been described above in detail, still, during present invention is not limited to the embodiments described above Detail a variety of equivalents can be carried out to technical solution of the present invention within the scope of the technical concept of the present invention, this A little equivalents all belong to the scope of protection of the present invention.

Claims (1)

1. Based on support vector machines and the Beamforming Method of mean square error performance is improved 1. a kind of, which is characterized in that including as follows Step:

   Step 1, the regulation coefficient for being minimised as target and calculating robustness Beam-former weight with MSE, by adjusting coefficient Obtain updated weight vector；When calculating regulation coefficient, the actual value of desired signal is replaced using the estimated value of desired signal；

   Step 2 is minimised as target with array output power, and using ε-insensitive loss function, Wave beam forming problem is converted For SVR model；

   Step 3 solves the SVR model using IRWLS method and exports result；

   Construct narrowband model

   If the one group far-field signal s coplanar with array_i(t), i=0 ... K is with different incidence angle θs_i, i=0 ... K arrival one Antenna array, bay number are M, and array element is non-directional and with the one-dimensional homogenous linear distribution of half-wave long spacing, by each array element institute It is different to locate position, homogeneous plane wave has different time delays in the response of each array element output end, then the output of m-th of array element It is:

   In formula (1): to correspond to incidence angle θ₀S₀It (t) is desired signal, τ_m(θ₀) be desired signal reach m-th of array element relative to The time delay of reference array element, s_i(t) θ is come from_i, the interference signal in the direction i=1 ... K, τ_m(θ_i) it is that corresponding interference signal reaches Time delay of m-th of the array element relative to reference array element, n_m(t) be m-th of array element additive noise；

   Indicate that signal reaches the phase relation between each array element signals with signal steering vector；Enable [x_m,y_m] indicate m-th of array element Position, signal and array are in same plane, and using y-axis as the zero direction of direction of arrival, then the amplitude and phase of m-th array element are rung It answers are as follows:

   Wherein, g_mIndicate the gain of m-th of array element, λ indicates the wavelength of narrow band signal, θ_iIndicate the direction of arrival of i-th of signal, θ ∈ (-π/2,π/2)；

   The array data at k moment is expressed as

   Wherein N_iFor the quantity of interference signal, s (k) and i_jIt (k) is signal and interference, n (k) is noise component(s), θ_sAnd θ_j, j= 1,...,N_iRespectively direction of arrival DOA, a (θ of signal and interference signal_s) and a (θ_j) it is corresponding steering vector:

   a(θ_i)=[a₁(θ_i), a₂(θ_i) ..., a_M(θ_i)]^T (4)

   Array signal model is written as vector form:

   X (k)=A (Θ) s (k)+e (k) (5)

   Wherein,

   S (k)=[s_s(k), s₁(k) ..., s_p(k)]^T

   E (k)=[s₁(k), s₂(k) ..., s_M(k)]^T

   A (Θ)=[a (θ_s), a (θ₁) ..., a (θ_p)] (6)

   Respectively indicate signal phasor, noise matrix and array guiding matrix；

   The output of narrow-band beam shaper is

   Y (k)=W^HX(k) (7)

   Wherein W=[w₁,...,w_M]^TFor Beam-former weight vector, M is array number,WithRespectively indicate transposition and conjugation Transposition；

   Under the conditions of incoherent, array inputs correlation matrix are as follows:

   Wherein E [] indicates expectation computing,WithRespectively irrelevant incoming signal s (k) and i_j(k) Power, Q is noise covariance matrix；

   LCMV Beam-former: under the Linear Constraints to desired DOA, minimize array output ENERGY E [| y (k) |²] =W^HR_XW, i.e.,

   Wherein, complex constant g is the array response of desired DOA；

   Under actual conditions, accurate covariance matrix R_X=E { X (k) X^H(k) } it cannot obtain, and common sampling covariance matrixTo replace

   Here N is observation number of snapshots；

   As N → ∞, then array output energy is obtained

   Wherein σ_n ²For noise power；

   As seen from the above equation, first item is desired output signal energy, and Section 2 is that interference signal exports the sum of energy, and Section 3 is Noise item；At this time if in W^Ha(θ_sArray is minimized under the Linear Constraints of)=g and exports energy, then Beam-former will Desired signal is only extracted, and inhibits all other interference and noise signal；

   Lagrange algorithm solving optimization problem is used, obtained LCMV Wave beam forming best weight value is

   MVDR Wave beam forming best weight value when g=1 is

   Wave beam forming based on SINR and MMSE criterion

   1. being based on SINR criterion

   SINR maximizes Wave beam forming and requires in the corresponding output signal of optimal weight vector W, and useful signal and interference noise are believed Number power ratio it is maximum, i.e.,

   Wherein R_s=E { ss^HIt is desired signal covariance matrix, R_i=E { (i+n) (i+n)^HIt is interference noise covariance matrix；

   To obtain enabling the maximum weight vector of SINR, to (14) formula about W derivation, and it is enabled to be equal to zero, obtained

   (15) solution of formula is exactly the optimal weight vector under SINR criterion；This is a joint eigenvalue problem, expression formula Value value range between matrix R_i ^-1R_sBetween the maximum and minimal eigenvalue of W, and expression formula is just represented by (14) formula The expression formula of SINR, therefore optimization problem is converted to following eigenvalue problem:

   R_i ^-1R_sW=λ_maxW (16)

   Optimal weight vector is feature vector corresponding to maximum eigenvalue；

   Because of R_i=E { (i+n) (i+n)^HAnd R_s=E { ss^HBe not directly available, with signal covariance matrix R_XTo substitute R_i= E{(i+n)(i+n)^H, then (14) formula is converted to following form:

   This optimization problem is described as under the Linear Constraints to desired DOA, minimize array output ENERGY E [| y (k) |²]=W^HR_XW, i.e.,

   2.MMSE Wave beam forming

   Known by (14) formula, for different zoom factor a, SINR (W)=SINR (aW), but this scaling will affect other property Can, especially the estimation of signal waveform is influenced very big；

   The basic thought of MMSE Beam-former is: directly minimizing MSE without requiring unbiased output；

   Assuming that s (k)=s is determining signal, wherein for simplicity, omitting k, the estimated value of s and itBetween MSE are as follows:

   WhereinIt isCovariance matrix,It is its biasing, R_i=E { (i+n) (i+n)^HCannot be obtained under physical condition, commonly use sampling covariance matrixInstead of being indicated with R；Then (19) formula is converted are as follows:

   It minimizes above formula and obtains MMSE Beam-former:

   W (s)=| s |²(R+|s|² aa^H)^-1A=β (s) W_MVDR, (21)

   WhereinAnd 0 < β (s) < 1, with | s |²Monotonic increase；Therefore, to all | s |, W (s) < W_MVDR；

   W (s) is substituted into (20) formula, the smallest MSE is obtained, uses MSE_OPTIt indicates

   W_MVDR(20) formula of substitution, obtains the MSE of MVDR Beam-former, uses MSE_MVDRIt indicates

   Clearly for all | s | > 0, MSE_OPT< MSE_MVDR；Therefore MMSE method can obtain smaller MSE；

   Under physical condition, because signal s is unknown, MMSE Beam-former can not be realized directly, at this time by estimating to signal Meter obtains preferable MSE performance；

   3. improving the beamforming algorithm of MSE performance

   With the regulation coefficient ρ for being minimised as target and calculating robustness Beam-former weight of MSE, that is, pass through

   It obtains

   Then by adjusting the updated weight of coefficient ρ are as follows:

   The amplitude of desired signal is not aware that under physical condition | s |, using the estimated value of desired signalTo replace s；Estimated value by Following formula obtains:

   Training sample set is T={ (θ₁,d₁),…(θ_P,d_P), θ_i∈ [- 90 °, 90 °], i=1 ... P is that wave reaches azimuth, wave Beam figure target response indicates are as follows:

   It is minimised as target with array output power, using ε-insensitive loss function, Wave beam forming problem is modeled as SVR, Cost function indicates are as follows:

   Wherein,WithIt is respectively defined as:

   Here, Re () and Im () are respectively indicated and are sought real and imaginary parts；δ_RFor error matrix parameter, meaning is practical association side The Frobenius norm upper bound of the error matrix of poor matrix and sample covariance matrix；L_ε(θ_i,d_i,f(θ_i)) it is that ε-is insensitive Loss function is selected as L herein_ε(θ_i, d_i, f (θ_i))=| d_i-f(θ_i)|_ε=max 0, | d_i-f(θ_i) |-ε },It is thereinIt indicates are as follows:

   Here,C >=0 is to punish Penalty factor, ε indicate error margin；

   Power least square (Iteratively Reweighted Least Square, IRWLS) method solution or more is assigned using iteration SVR problem；Pass through first order Taylor series expansion ε-insensitive loss function L_ε(θ_i,d_i,f(θ_i)), obtain kth time iteration WithThen:

   Objective function is reconstructed using Two-order approximation, i.e.,WhereinIndicate needle To weightGradient operator, obtain least square cost function:

   In formula, b_iIndicate all and weightUnrelated item, f_iFrom penalty term, then

   It is solved at this time using second iteration algorithm:

   WhereinD_fIt is with f_iFor diagonal element, other complete zero it is diagonal Matrix,