CN105929369B - A kind of Beamforming Method and system based on definite and uncertain collection constraint - Google Patents

Abstract

The invention discloses a beamforming method and system based on definite and uncertain set constraints. The method includes: acquiring signals received by array elements in an antenna array, performing weighted summation on the signals received by the array elements to obtain beamforming output; According to several samples of the beamforming process, the array covariance matrix is estimated, and the objective function of the estimated array covariance matrix is modeled to obtain the modeled uncertain set constraints; according to the Berne-type inequality, the uncertain set The constraints are converted into definite set constraints; the objective function is converted into a semi-definite programming function according to the semi-definite relaxation algorithm; the semi-definite programming function is solved according to the convex optimization algorithm, and the beamforming after the solution is output. The present invention transforms the probability constraint condition of the random process into the deterministic constraint condition, optimizes the objective function after transforming the semidefinite programming problem, overcomes the serious defect that the array covariance matrix has errors in the existing method, and improves the robustness of the system sex.

Description

A Beamforming Method and System Based on Definite and Uncertain Set Constraints

technical field

The invention relates to the field of beamforming in array signal processing, in particular to a beamforming method and system based on certain and uncertain set constraints.

Background technique

The array signal processing system is widely used in sonar, spaceflight and radar and other fields, and it includes two core technologies: parameter estimation and beamforming. Adaptive beamforming technology still faces many practical problems from theoretical knowledge to engineering application. Among them, the robust performance of the adaptive beamforming algorithm in the case of errors directly affects the actual application effect. At the same time, the uncertainties in the actual environment also have a serious impact on the adaptive beamforming technology, such as sensor position disturbance, local scattering, mutual coupling, and steering vector mismatch. For the problem of robust adaptive beamforming under uncertain set constraints, the traditional method simplifies the steering vector into a triangle inequality, and then converts the problem into a second-order cone programming problem by using the Rayleigh distribution and the Chebyshev inequality algorithm . However, the traditional method does not consider the influence of the sample array covariance matrix error, so its performance is limited. The existing beamforming methods are affected by errors, have poor robustness, and reduce the measurement accuracy of the array signal processing system.

Therefore, the prior art still needs to be improved and developed.

Contents of the invention

In view of the deficiencies in the prior art, the purpose of the present invention is to provide a beamforming method and system based on certain and uncertain set constraints, aiming at solving the problem that the beamforming method in the prior art does not consider the influence of the sample array covariance matrix error, so Its performance is limited, the system robustness is poor, and the measurement accuracy of the array signal processing system is reduced.

Technical scheme of the present invention is as follows:

A beamforming method based on certain and uncertain set constraints, wherein the method comprises:

A. Obtain the signals received by the array elements in the antenna array, and perform weighted summation on the signals received by the array elements to obtain the beamforming output;

B. Estimate the array covariance matrix according to several samples of the beamforming process, perform modeling according to the objective function of the estimated array covariance matrix, and obtain the uncertain set constraints after modeling;

C. Transform uncertain set constraints into definite set constraints according to Berne-type inequality;

D. Transform the objective function into a semidefinite programming function according to the semidefinite relaxation algorithm;

E. Solve the semidefinite programming function according to the convex optimization algorithm, and output the beamforming corresponding to the solved parameters.

In the beamforming method based on definite and uncertain set constraints, step A specifically includes:

A1. Obtain the signal received by the array element in the antenna array, set the signal received by the array element as x(k),

x(k)=s(k)a+i(k)+n(k)

Where x(k) is all received signals, s(k) is the source signal, a is the steering vector, i(k) is the interference component, and n(k) is the noise component;

A2. Perform weighted summation on the received signal x(k) to obtain the output y(k) of the beamforming,

y(k)=w ^H x(k)

where w ∈ C ^M is the weight vector for beamforming, and w ^H represents the transpose of w.

In the beamforming method based on determination and uncertainty set constraints, the step B specifically includes:

B1. Estimate the array covariance matrix according to several samples in the beamforming process, and perform modeling according to the objective function of the estimated array covariance matrix. The objective function is The uncertain set constraints after modeling are ||Δ||≤γ,

in Indicates that taking Δ as the independent variable takes the maximum value of ( ), Indicates that w is the independent variable to take the minimum value of (·), Pr{·} represents the probability of {·}, the sample array covariance matrix Δ is the covariance matrix error, a is the expected steering vector, and δ is the steering vector error.

In the beamforming method based on determination and uncertainty set constraints, the step C specifically includes:

C1, the uncertain set constraints Converted to deterministic set constraints,

Wherein tr(·) represents the trace of (·), || || _F represents the Frobenius norm, 2ρ=-ln(1-p), δ=Bu, u obeys the mean value vector and is 0 and the covariance matrix is the identity matrix I Normal distribution, B=C _δ ^1/2 ∈C ^M×M , B _w ＝B ^H ww ^H B ∈ C ^M×M , is a fixed matrix, b _w ＝B ^H ww ^{H a} ∈ C ^M .

In the beamforming method based on determination and uncertainty set constraints, the step D specifically includes:

D1, order The objective function is The constraints are because

Using semidefinite relaxation, suppose So

After relaxation, the objective function is transformed into a semidefinite relaxation function: where the constraints are W≥0, where W=ww ^H is a matrix of semidefinite rank, μ is a slack variable introduced, vec(.) is a matrix vectorization factor, ||vec(B _w )||=||B _w | ^| _F2 .

A beamforming system based on certain and uncertain set constraints, wherein the system includes:

The signal acquisition module is used to acquire the signals received by the array elements in the antenna array, and perform weighted summation on the signals received by the array elements to obtain the beamforming output;

The objective function modeling module is used to estimate the array covariance matrix according to several samples of the beamforming process, perform modeling according to the objective function of the estimated array covariance matrix, and obtain the uncertain set constraints after modeling;

A constraint transformation module, used to transform the uncertain set constraints into certain set constraints according to Berne-type inequality;

An objective function conversion module, which is used to convert the objective function into a semidefinite programming function according to the semidefinite relaxation algorithm;

The calculation and output module is used to solve the semidefinite programming function according to the convex optimization algorithm, and output the beamforming corresponding to the solved parameters.

The beamforming system based on definite and uncertain set constraints, wherein the signal acquisition module specifically includes:

The signal obtaining unit is used to obtain the signal received by the array element in the antenna array, and the signal received by the array element is set as x(k),

x(k)=s(k)a+i(k)+n(k)

Where x(k) is all received signals, s(k) is the source signal, a is the steering vector, i(k) is the interference component, and n(k) is the noise component;

A computing unit, configured to weight and sum the received signals x(k) to obtain a beamformed output y(k),

y(k)=w ^H x(k)

where w ∈ C ^M is the weight vector for beamforming, and w ^H represents the transpose of w.

In the beamforming system based on definite and uncertain set constraints, the objective function establishment module specifically includes:

The objective function and uncertainty set constraint acquisition unit is used to estimate the array covariance matrix according to several samples in the beamforming process, and perform modeling according to the objective function of the estimated array covariance matrix. The objective function is The uncertain set constraints after modeling are ||Δ||≤γ,

in Indicates that taking Δ as the independent variable takes the maximum value of ( ), Indicates that w is the independent variable to take the minimum value of (·), Pr{·} represents the probability of {·}, the sample array covariance matrix Δ is the covariance matrix error, a is the expected steering vector, and δ is the steering vector error.

The beamforming system based on definite and uncertain set constraints, wherein the constraint conversion module specifically includes:

Uncertain set constraints to definite set constraints conversion unit, used to convert uncertain set constraints Converted to deterministic set constraints,

Wherein tr( ) represents the trace of ( ), || || _F represents the Frobenius norm, 2ρ=-ln(1-p), δ=Bu, u obeys the mean vector and is 0 covariance matrix is the identity matrix I Normal distribution, B=C _δ ^1/2 ∈C ^M×M , B _w ＝B ^H ww ^H B ∈ C ^M×M , is a fixed matrix, b _w ＝B ^H ww ^{H a} ∈ C ^M .

In the beamforming system based on definite and uncertain set constraints, the objective function conversion module specifically includes:

The semidefinite relaxation calculation unit is used to make The objective function is The constraints are because

Using semidefinite relaxation, suppose So

After relaxation, the objective function is transformed into a semidefinite relaxation function: where the constraints are W≥0, where W=ww ^H is a matrix of semidefinite rank, μ is a slack variable introduced, vec(.) is a matrix vectorization factor, ||vec(B _w )||=||B _w | ^| _F2 .

The present invention provides a beamforming method and system based on deterministic and uncertain set constraints. The present invention transforms the probability constraints of random processes into deterministic constraints through Berne-type inequalities, and then uses semidefinite relaxation to determine the constraints Transform the semi-definite programming problem, so as to optimize the objective function, make up for the serious defect that the existing method only overcomes the mismatch of the steering vector, but does not consider the error of the covariance matrix of the sample array, and improves the robustness of the system The performance improves the measurement accuracy of the array signal processing system.

Description of drawings

FIG. 1 is a flow chart of a preferred embodiment of a beamforming method based on certain and uncertain set constraints in the present invention.

Fig. 2 is a specific application example of a beamforming method based on definite and uncertain set constraints in the present invention, in which the number of sampling samples is fixed at 100, and the output SINR-input SNR curve.

Fig. 3 is a specific application example of a beamforming method based on deterministic and uncertain set constraints of the present invention, the SNR is fixed at 10dB, and the output signal-to-interference-noise ratio-sample number curve.

FIG. 4 is a functional block diagram of a preferred embodiment of a beamforming system based on certain and uncertain set constraints in the present invention.

Detailed ways

In order to make the object, technical solution and effect of the present invention more clear and definite, the present invention will be further described in detail below. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.

The present invention also provides a flowchart of a preferred embodiment of a beamforming method based on definite and uncertain set constraints, as shown in FIG. 1 , wherein the method includes:

Step S100, acquiring the signals received by the elements in the antenna array, performing weighted summation on the signals received by the elements to obtain the output of beamforming;

Step S200, perform array covariance matrix estimation according to several samples in the beamforming process, perform modeling according to the objective function of the estimated array covariance matrix, and obtain modeled uncertainty set constraints;

Step S300, converting the uncertain set constraints into definite set constraints according to Berne-type inequality;

Step S400, converting the objective function into a semidefinite programming function according to the semidefinite relaxation algorithm;

Step S500 , solving the semidefinite programming function according to the convex optimization algorithm, and outputting the beamforming corresponding to the solved parameters.

During specific implementation, step S100 specifically includes:

Step S101, obtain the signal received by the array element in the antenna array, set the signal received by the array element as x(k),

x(k)=s(k)a+i(k)+n(k)

Where x(k) is all received signals, s(k) is the source signal, a is the steering vector, i(k) is the interference component, and n(k) is the noise component;

Step S102, performing weighted summation on the received signal x(k) to obtain the beamforming output y(k),

y(k)=w ^H x(k)

where w ∈ C ^M is the weight vector for beamforming, and w ^H represents the transpose of w.

Further, step S200 specifically includes:

Step S201, perform array covariance matrix estimation according to several samples in the beamforming process, and perform modeling according to the objective function of the estimated array covariance matrix, and the objective function is The uncertain set constraints after modeling are ||Δ||≤γ,

in Indicates that taking Δ as the independent variable takes the maximum value of ( ), Indicates that w is the independent variable to take the minimum value of (·), Pr{·} represents the probability of {·}, the sample array covariance matrix Δ is the covariance matrix error, a is the expected steering vector, and δ is the steering vector error. γ generally takes a value of 2, and p is a constant close to 1.

During specific implementation, step S300 specifically includes:

Step S301, constraining the uncertain set Converted to deterministic set constraints,

Wherein tr( ) represents the trace of ( ), || || _F represents the Frobenius norm, 2ρ=-ln(1-p), δ=Bu, u obeys the mean vector and is 0 covariance matrix is the identity matrix I Normal distribution, B=C _δ ^1/2 ∈C ^M×M , B _w ＝B ^H ww ^H B ∈ C ^M×M , is a fixed matrix, b _w ＝B ^H ww ^{H a} ∈ C ^M .

During specific implementation, the constraints in step S301 Equivalent to And Lemma If T=z ^T Az+b ^T z, where A∈R ^{M ×M} , b∈R ^M , z∈N(0,I), then in

s ^- (X)=max{λ _max (-X),0}, λ _max (X) is the largest eigenvalue of the vector X, and tr(.) represents the trace of the matrix.

In a further embodiment, step S400 specifically includes:

Step S401, command The objective function is The constraints are because

Using semidefinite relaxation, suppose So

After relaxation, the objective function is transformed into a semidefinite relaxation function: where the constraints are W≥0, where W=ww ^H is a matrix of semidefinite rank, μ is a slack variable introduced, vec(.) is a matrix vectorization factor, ||vec(B _w )||=||B _w | ^| _F2 .

In the step S500, the rank of the relaxed W through the step S400 is 1, wherein Then the objective function can be transformed into The constraints are W≥0. The objective function is a convex function, and the constraints are also convex functions, so an effective convex optimization method such as CVX can be used to solve the objective function and achieve the maximum signal-to-interference-noise ratio output. Preferably, the cvx package in MATLAB can be used to solve the problem.

The present invention also provides a simulation result of a specific application example, wherein the simulation environment is shown in Table 1, and the variable parameters of the simulation graph are shown in Table 2.

Table 1

Table 2

The simulation results based on the simulation parameters set in Table 2 are shown in Figure 2, and the simulation results based on the simulation parameters set in Table 3 are shown in Figure 3. The simulation results show that in the case of sampling errors, the output parameters after solving The corresponding beamforming is much larger than other methods, such as the SMI (i.e. sample matrix inversion) method and the traditional robust minimum variance beamforming method. The present invention improves the robustness of the system and realizes effective suppression of interference signals .

The present invention also provides a functional block diagram of a preferred embodiment of a beamforming system based on certain and uncertain set constraints, as shown in FIG. 4 , wherein the system includes:

The signal acquisition module 100 is configured to acquire the signals received by the elements in the antenna array, and perform weighted summation on the signals received by the elements to obtain the beamformed output; the details are as described in the above method embodiments.

The objective function modeling module 200 is used to estimate the array covariance matrix according to several samples of the beamforming process, perform modeling according to the objective function of the estimated array covariance matrix, and obtain the uncertain set constraints after modeling; The details are as described in the above method embodiments.

The constraint conversion module 300 is configured to convert the uncertain set constraints into definite set constraints according to Berne-type inequality; the details are as described in the above method embodiments.

The objective function conversion module 400 is configured to convert the objective function into a semidefinite programming function according to the semidefinite relaxation algorithm; details are as described in the above method embodiments.

The calculation and output module 500 is configured to solve the semidefinite programming function according to the convex optimization algorithm, and output the beamforming corresponding to the solved parameters; the details are as described in the above method embodiment.

Further, the signal acquisition module specifically includes:

The signal obtaining unit is used to obtain the signal received by the array element in the antenna array, and the signal received by the array element is set as x(k),

x(k)=s(k)a+i(k)+n(k)

Where x(k) is all received signals, s(k) is a source signal, a is a steering vector, i(k) is an interference component, and n(k) is a noise component; the details are as described in the above method embodiment.

A computing unit, configured to weight and sum the received signals x(k) to obtain a beamformed output y(k),

y(k)=w ^H x(k)

Where ^w∈CM is the weight vector of beamforming, and w ^H represents the transposition of w; the details are as described in the above method embodiment.

Specifically, the objective function building module specifically includes:

The objective function and uncertainty set constraint acquisition unit is used to estimate the array covariance matrix according to several samples in the beamforming process, and perform modeling according to the objective function of the estimated array covariance matrix. The objective function is The uncertain set constraints after modeling are ||Δ||≤γ, The details are as described in the above method embodiments.

in Indicates that taking Δ as the independent variable takes the maximum value of ( ), Indicates that w is the independent variable to take the minimum value of (·), Pr{·} represents the probability of {·}, the sample array covariance matrix Δ is the covariance matrix error, a is the expected steering vector, and δ is the steering vector error; details are as described in the above method embodiment.

Further, the constraint conversion module specifically includes:

Uncertain set constraints to definite set constraints conversion unit, used to convert uncertain set constraints Converted to deterministic set constraints,

Wherein tr( ) represents the trace of ( ), || || _F represents the Frobenius norm, 2ρ=-ln(1-p), δ=Bu, u obeys the mean vector and is 0 covariance matrix is the identity matrix I Normal distribution, B=C _δ ^1/2 ∈C ^M×M , B _w ＝B ^H ww ^H B ∈ C ^M×M , is a definite matrix, b _w ＝B ^H ww ^H a ∈ C ^M ; the specific method is as above described in the examples.

Specifically, the objective function conversion module specifically includes:

The semidefinite relaxation calculation unit is used to make The objective function is The constraints are because

Using semidefinite relaxation, suppose So

The details are as described in the above method embodiments.

After relaxation, the objective function is transformed into a semidefinite relaxation function: where the constraints are W≥0, where W=ww ^H is a matrix of semidefinite rank, μ is a slack variable introduced, vec(.) is a matrix vectorization factor, ||vec(B _w )||=||B _w | | _F ² ; specifically as described in the above method embodiment.

To sum up, the present invention provides a beamforming method and system based on certain and uncertain set constraints. The method includes: obtaining the signals received by the elements in the antenna array, and performing weighted summation on the signals received by the elements to obtain The output of beamforming; the array covariance matrix is estimated according to several samples in the beamforming process, and the modeling is carried out according to the objective function of the estimated array covariance matrix, and the uncertain set constraints after modeling are obtained; according to the Berne type Inequalities convert uncertain set constraints into definite set constraints; transform the objective function into a semidefinite programming function according to the semidefinite relaxation algorithm; solve the semidefinite programming function according to the convex optimization algorithm, and output the beamforming corresponding to the solved parameters. The present invention converts the probability constraints of the stochastic process into deterministic constraints through Berne-type inequalities, and then uses semidefinite relaxation to convert the definite constraints into semidefinite programming problems, thereby optimizing the objective function and making up for the existing methods. It only overcomes the mismatch of the steering vector, but does not consider the serious defect of the error in the covariance matrix of the sample array, improves the robustness of the system, and improves the measurement accuracy of the array signal processing system.

It should be understood that the application of the present invention is not limited to the above examples, and those skilled in the art can make improvements or transformations according to the above descriptions, and all these improvements and transformations should belong to the protection scope of the appended claims of the present invention.

Claims (10)

1. It is 1. a kind of based on the Beamforming Method determined and uncertain collection constrains, which is characterized in that method includes：

   A, the signal that array element receives in aerial array is obtained, summation is weighted to the signal that array element receives and obtains wave beam shape Into output；

   B, array covariance matrix is carried out according to several samples of beam forming process, according to the array association side after estimation The object function of poor matrix is modeled, and obtains the uncertain collection constraints after modeling；

   C, uncertain collection constraint is changed into according to Berne type inequality and determines collection constraint；

   D, object function is converted by Semidefinite Programming function according to semidefinite relaxation algorithm；

   E, Semidefinite Programming function is solved according to convex optimized algorithm, the corresponding Wave beam forming of parameter after output solution.
2. It is 2. according to claim 1 based on the Beamforming Method determined and uncertain collection constrains, which is characterized in that step A is specifically included：

   A1, the signal that array element receives in aerial array is obtained, sets array element and receive signal as x (k),

   X (k)=s (k) a+i (k)+n (k)

   Wherein x (k) is all signals for receiving, and s (k) is source signal, and a is steering vector, and i (k) is interference components, n (k) For noise component(s)；

   A2, docking collection of letters x (k) are weighted summation and obtain the output y (k) of Wave beam forming,

   Y (k)=w^Hx(k)

   Wherein w ∈ C^MFor the weight vector of Wave beam forming, w^HRepresent the transposition of w.
3. It is 3. according to claim 2 based on the Beamforming Method determined and uncertain collection constrains, which is characterized in that described Step B is specifically included：

   B1, array covariance matrix is carried out according to several samples of beam forming process, is assisted according to the array after estimation The object function of variance matrix is modeled, and object function isUncertain collection constraint after modeling Condition is | | Δ | |≤γ,

   WhereinExpression takes the maximum of () using Δ as independent variable,It represents to take () most by independent variable of w Small value, Pr { } represent the probability of { }, array of samples covariance matrixΔ is covariance matrix Error,A is it is expected steering vector, and δ is steering vector error, and K is sample number.
4. It is 4. according to claim 3 based on the Beamforming Method determined and uncertain collection constrains, which is characterized in that described Step C is specifically included：

   C1, uncertain collection is constrainedIt is converted into and determines collection constraint,

   <mrow> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>&amp;GreaterEqual;</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>a</mi> <mi>H</mi> </msup> <msup> <mi>ww</mi> <mi>H</mi> </msup> <mi>a</mi> </mrow>

   Wherein tr () represents the mark of (), | | | |_FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u obey average The normal distribution that vector is unit matrix I for 0 covariance matrix, B=C_δ ^1/2∈C^M×M, B_w=B^Hww^HB∈C^M×M, it is set matrix, b_w=B^Hww^Ha∈C^M。
5. It is 5. according to claim 4 based on the Beamforming Method determined and uncertain collection constrains, which is characterized in that described Step D is specifically included：

   D1, orderObject function isConstraints isDue to

   <mrow> <msup> <mi>w</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>&amp;gamma;</mi> <mi>I</mi> <mo>)</mo> </mrow> <mi>w</mi> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mo>(</mo> <mrow> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>&amp;gamma;</mi> <mi>I</mi> </mrow> <mo>)</mo> <mi>w</mi> <mo>)</mo> </mrow> </mrow>

   <mrow> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <mo>|</mo> <mo>|</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>v</mi> <mi>e</mi> <msub> <mi>c</mi> <mi>w</mi> </msub> <mo>(</mo> <msub> <mtable> <mtr> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> <mi>w</mi> </msub> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>b</mi> <mi>w</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>|</mo> <mo>|</mo> </mrow>

   Utilize semidefinite decoding, it is assumed thatSo

   <mrow> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>&amp;GreaterEqual;</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <mi>&amp;mu;</mi> <mo>&amp;GreaterEqual;</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>a</mi> <mi>H</mi> </msup> <mi>w</mi> <mi>a</mi> <mo>;</mo> </mrow>

   Object function is converted into semidefinite decoding function after relaxation：Wherein constraints ForW >=0, wherein W=ww^H, it is one matrix of semidefinite order, μ is the slack variable introduced, and vec () is the matrix-vector factor, | | vec (B_w) | |=| | B_w||_F ²。
6. It is 6. a kind of based on the Beam Forming System determined and uncertain collection constrains, which is characterized in that system includes：

   Signal acquisition module for obtaining the signal that array element in aerial array receives, adds the signal that array element receives Power summation obtains the output of Wave beam forming；

   Object function modeling module, for carrying out array covariance matrix according to several samples of beam forming process, It is modeled according to the object function of the array covariance matrix after estimation, obtains the uncertain collection constraints after modeling；

   Conversion module is constrained, collection constraint is determined for changing into uncertain collection constraint according to Berne type inequality；

   Object function conversion module, for object function to be converted into Semidefinite Programming function according to semidefinite relaxation algorithm；

   Calculating and output module, for being solved according to convex optimized algorithm to Semidefinite Programming function, the parameter after output solution Corresponding Wave beam forming.
7. It is 7. according to claim 6 based on the Beam Forming System determined and uncertain collection constrains, which is characterized in that described Signal acquisition module specifically includes：

   Signal acquiring unit for obtaining the signal that array element in aerial array receives, sets array element and receives signal as x (k),

   X (k)=s (k) a+i (k)+n (k)

   Wherein x (k) is all signals for receiving, and s (k) is source signal, and a is steering vector, and i (k) is interference components, n (k) For noise component(s)；

   Computing unit is weighted summation for docking collection of letters x (k) and obtains the output y (k) of Wave beam forming,

   Y (k)=w^Hx(k)

   Wherein w ∈ C^MFor the weight vector of Wave beam forming, w^HRepresent the transposition of w.
8. It is 8. according to claim 7 based on the Beam Forming System determined and uncertain collection constrains, which is characterized in that described Object function is established module and is specifically included：

   Object function and uncertain collection constraints acquiring unit, for carrying out battle array according to several samples of beam forming process Row covariance matrix is modeled according to the object function of the array covariance matrix after estimation, and object function isUncertain collection constraints after modeling is | | Δ | |≤γ,

   WhereinExpression takes the maximum of () using Δ as independent variable,It represents to take () most by independent variable of w Small value, Pr { } represent the probability of { }, array of samples covariance matrixΔ is covariance matrix Error,A is it is expected steering vector, and δ is steering vector error, and K is sample number.
9. It is 9. according to claim 8 based on the Beam Forming System determined and uncertain collection constrains, which is characterized in that described Constraint conversion module specifically includes：

   Uncertain collection constraint is to definite collection constraint conversion unit, for uncertain collection to be constrainedIt is converted into Determine collection constraint,

   <mrow> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>&amp;GreaterEqual;</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>a</mi> <mi>H</mi> </msup> <msup> <mi>ww</mi> <mi>H</mi> </msup> <mi>a</mi> </mrow>

   Wherein tr () represents the mark of (), | | | |_FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u obey average The normal distribution that vector is unit matrix I for 0 covariance matrix, B=C_δ ^1/2∈C^M×M, B_w=B^Hww^HB∈C^M×M, it is set matrix, b_w=B^Hww^Ha∈C^M。
10. It is 10. according to claim 9 based on the Beam Forming System determined and uncertain collection constrains, which is characterized in that institute Object function conversion module is stated to specifically include：

    Semidefinite relaxation computing unit, for makingObject function isConstraints isDue to

    <mrow> <msup> <mi>w</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>&amp;gamma;</mi> <mi>I</mi> <mo>)</mo> </mrow> <mi>w</mi> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mo>(</mo> <mrow> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>&amp;gamma;</mi> <mi>I</mi> </mrow> <mo>)</mo> <mi>w</mi> <mo>)</mo> </mrow> </mrow>

    <mrow> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <mo>|</mo> <mo>|</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>v</mi> <mi>e</mi> <msub> <mi>c</mi> <mi>w</mi> </msub> <mo>(</mo> <msub> <mtable> <mtr> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> <mi>w</mi> </msub> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>b</mi> <mi>w</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>|</mo> <mo>|</mo> </mrow>

    Utilize semidefinite decoding, it is assumed thatSo

    <mrow> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <msub> <mo>|</mo> <mi>F</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mn>2</mn> <msub> <mi>b</mi> <mi>w</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mo>&amp;GreaterEqual;</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <mn>2</mn> <mi>&amp;rho;</mi> </mrow> </msqrt> <mi>&amp;mu;</mi> <mo>&amp;GreaterEqual;</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>a</mi> <mi>H</mi> </msup> <mi>w</mi> <mi>a</mi> <mo>;</mo> </mrow>

    Object function is converted into semidefinite decoding function after relaxation：Wherein constraints ForW >=0, wherein W=ww^H, it is one matrix of semidefinite order, μ is the slack variable introduced, and vec () is the matrix-vector factor, | | vec (B_w) | |=| | B_w||_F ²。