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

A kind of Beamforming Method and system based on definite and uncertain collection constraint Download PDF

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CN105929369B
CN105929369B CN201610239510.3A CN201610239510A CN105929369B CN 105929369 B CN105929369 B CN 105929369B CN 201610239510 A CN201610239510 A CN 201610239510A CN 105929369 B CN105929369 B CN 105929369B
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CN105929369A (en
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廖斌
郭重涛
黄磊
杨丽鲜
黄辉平
王佳佳
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Shenzhen University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
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Abstract

Included the invention discloses a kind of based on the Beamforming Method and system, method that determine and uncertain collection constrains：The signal that array element receives in aerial array is obtained, summation is weighted to the signal that array element receives and obtains the output of Wave beam forming；Array covariance matrix is carried out according to several samples of beam forming process, is modeled according to the object function of the array covariance matrix after estimation, obtains the uncertain collection constraints after modeling；Uncertain collection constraint is changed into according to Berne type inequality and determines collection constraint；Object function is converted by Semidefinite Programming function according to semidefinite relaxation algorithm；Semidefinite Programming function is solved according to convex optimized algorithm, the Wave beam forming after output solution.The probabilistic constraints of random process are converted into certainty constraints by the present invention, convert the optimization that object function is carried out after Semidefinite Programming, overcome existing method array covariance matrix there are error it is serious the defects of, improve the robustness of system.

Description

A kind of Beamforming Method and system based on definite and uncertain collection constraint
Technical field
The present invention relates to the Wave beam forming field in array signal processing more particularly to one kind based on definite and uncertain collection The Beamforming Method and system of constraint.
Background technology
Array signal processing system is widely used in the fields such as sonar, space flight and radar, it includes two core technologies： Parameter Estimation and Wave beam forming.Adaptive beamformer technology from knowwhy moves towards engineer application and still faces many actually to ask Topic.Wherein, robust performance of the adaptive beam-forming algorithm under error condition directly influences actual application effect.Together When, it is uncertain present in actual environment also to bring serious influence to Adaptive beamformer technology, such as sensor position Put disturbance, local scattering, mutual coupling, steering vector mismatch etc..It is asked for the robust adaptive beamforming under uncertain collection constraint Topic, steering vector is reduced to triangle inequality by traditional method, then using rayleigh distributed, cut Bill and avenge husband's inequality and solve Algorithm, which is converted into Second-order cone programming problem.However there is no consider array of samples covariance square for traditional method The influence of battle array error, so its performance is restricted.Existing Beamforming Method is influenced by error, poor robustness, reduces battle array The measurement accuracy of column signal processing system.
Therefore, the prior art has yet to be improved and developed.
The content of the invention
In view of the deficiencies in the prior art, a kind of based on the wave beam determined and uncertain collection constrains present invention aims at providing Forming method and system, it is intended to solve Beamforming Method in the prior art and not account for array of samples covariance matrix error It influences, so its performance is restricted, system robustness is poor, the defects of reducing the measurement accuracy of array signal processing system.
Technical scheme is as follows：
A kind of Beamforming Method based on definite and uncertain collection constraint, wherein, 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 ripple The output that beam is formed；
B, array covariance matrix is carried out according to several samples of beam forming process, according to the array after estimation The object function of covariance 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 shape of parameter after output solution Into.
The Beamforming Method based on definite and uncertain collection constraint, wherein, 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 received, and s (k) is source signal, and a is steering vector, and i (k) is interference components, N (k) is 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)=wHx(k)
Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent the transposition of w.
The Beamforming Method based on definite and uncertain collection constraint, wherein, the step B is specifically included：
B1, array covariance matrix is carried out according to several samples of beam forming process, according to the battle array after estimation The object function of row covariance matrix is modeled, and object function isIt is uncertain intensive after modeling Beam condition is | | Δ | |≤γ,
WhereinExpression takes the maximum of () using Δ as independent variable,Expression takes () by independent variable of w Minimum value, Pr { } represent { } probability, array of samples covariance matrixΔ is covariance Matrix error,For a it is expected steering vector, δ is steering vector error.
The Beamforming Method based on definite and uncertain collection constraint, wherein, the step C is specifically included：
C1, uncertain collection is constrainedIt is converted into and determines collection constraint,
Wherein tr () represents the mark of (), | | | |FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u clothes It is the normal distribution that 0 covariance matrix is unit matrix I from mean vector, B=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is fixed Matrix, bw=BHwwHa∈CM。
The Beamforming Method based on definite and uncertain collection constraint, wherein, the step D is specifically included：
D1, orderObject function isConstraints isDue to
Utilize semidefinite decoding, it is assumed thatSo
Object function is converted into semidefinite decoding function after relaxation：Wherein constrain Condition isW >=0, wherein W=wwH, it is semidefinite order one Matrix, μ are the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||F 2。
A kind of Beam Forming System based on definite and uncertain collection constraint, wherein, system includes：
Signal acquisition module, for obtaining the signal that array element in aerial array receives, to the signal that array element receives into Row weighted sum obtains the output of Wave beam forming；
Object function modeling module is estimated for carrying out array covariance matrix according to several samples of beam forming process Meter, 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, after output solves The corresponding Wave beam forming of parameter.
The Beam Forming System based on definite and uncertain collection constraint, wherein, the signal acquisition module is specific Including：
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 received, and s (k) is source signal, and a is steering vector, and i (k) is interference components, N (k) is 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)=wHx(k)
Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent the transposition of w.
The Beam Forming System based on definite and uncertain collection constraint, wherein, the object function establishes module It specifically includes：
Object function and uncertain collection constraints acquiring unit, for several samples according to beam forming process into Row array 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,Expression takes () by independent variable of w Minimum value, Pr { } represent { } probability, array of samples covariance matrixΔ is covariance Matrix error,For a it is expected steering vector, δ is steering vector error.
The Beam Forming System based on definite and uncertain collection constraint, wherein, the constraint conversion module is specific Including：
Uncertain collection constraint is to definite collection constraint conversion unit, for uncertain collection to be constrained It is converted into and determines collection constraint,
Wherein tr () represents the mark of (), | | | |FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u clothes It is the normal distribution that 0 covariance matrix is unit matrix I from mean vector, B=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is fixed Matrix, bw=BHwwHa∈CM。
The Beam Forming System based on definite and uncertain collection constraint, wherein, the object function conversion module It specifically includes：
Semidefinite relaxation computing unit, for makingObject function isConstraints ForDue to
Utilize semidefinite decoding, it is assumed thatSo
Object function is converted into semidefinite decoding function after relaxation：Wherein constrain Condition isW >=0, wherein W=wwH, it is semidefinite order one Matrix, μ are the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||F 2。
Passed through the present invention provides a kind of based on the Beamforming Method and system, the present invention that determine and uncertain collection constrains The probabilistic constraints of random process are converted into certainty constraints by Berne type inequality, then will be true using semidefinite decoding Determine constraints conversion Semidefinite Programming, so as to carry out the optimization of object function, compensate for and only overcome in existing method Steering vector mismatch, without considering that sample array covariance matrix there are the serious defect problem of error, improves system Robustness improves the measurement accuracy of array signal processing system.
Description of the drawings
Fig. 1 is a kind of stream of preferred embodiment based on the Beamforming Method for determining and not knowing collection constraint of the present invention Cheng Tu.
Fig. 2 is a kind of concrete application embodiment based on the Beamforming Method for determining and not knowing collection constraint of the present invention Middle sample number is fixed as 100, exports Signal to Interference plus Noise Ratio-input signal-to-noise ratio graph.
Fig. 3 is a kind of concrete application embodiment based on the Beamforming Method for determining and not knowing collection constraint of the present invention SNR be fixed as 10dB, export Signal to Interference plus Noise Ratio-sample number curve figure.
Fig. 4 is a kind of work(of preferred embodiment based on the Beam Forming System for determining and not knowing collection constraint of the present invention It can functional block diagram.
Specific embodiment
To make the purpose of the present invention, technical solution and effect clearer, clear and definite, below to the present invention further specifically It is bright.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not intended to limit the present invention.
The present invention also provides a kind of preferred embodiments based on the Beamforming Method for determining and not knowing collection constraint Flow chart, as shown in Figure 1, wherein, method includes：
Step S100, the signal that array element receives in aerial array is obtained, the signal that array element receives is weighted and is asked With obtain the output of Wave beam forming；
Step S200, array covariance matrix is carried out according to several samples of beam forming process, according to estimation The object function of array covariance matrix afterwards is modeled, and obtains the uncertain collection constraints after modeling；
Step S300, uncertain collection constraint is changed into according to Berne type inequality and determines collection constraint；
Step S400, object function is converted by Semidefinite Programming function according to semidefinite relaxation algorithm；
Step S500, Semidefinite Programming function is solved according to convex optimized algorithm, the parameter after output solves is corresponding Wave beam forming.
When it is implemented, step S100 is specifically included：
Step S101, the signal that array element receives in aerial array is obtained, array element is set and receives signal as x (k),
X (k)=s (k) a+i (k)+n (k)
Wherein x (k) is all signals received, and s (k) is source signal, and a is steering vector, and i (k) is interference components, N (k) is noise component(s)；
Step S102, docking collection of letters x (k) is weighted summation and obtains the output y (k) of Wave beam forming,
Y (k)=wHx(k)
Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent the transposition of w.
Further, step S200 is specifically included：
Step S201, array covariance matrix is carried out according to several samples of beam forming process, according to estimation The object function of array covariance matrix afterwards is modeled, and object function isIt is not true after modeling Surely collection constraints is | | Δ | |≤γ,
WhereinExpression takes the maximum of () using Δ as independent variable,Expression takes () by independent variable of w Minimum value, Pr { } represent { } probability, array of samples covariance matrixΔ is covariance Matrix error,For a it is expected steering vector, δ is steering vector error.The general values of γ are 2, p be close to 1 it is normal Amount.
When it is implemented, step S300 is specifically included：
Step S301, uncertain collection is constrainedIt is converted into and determines collection constraint,
Wherein tr () represents the mark of (), | | | |FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u clothes It is the normal distribution that 0 covariance matrix is unit matrix I from mean vector, B=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is fixed Matrix, bw=BHwwHa∈CM。
When it is implemented, constraints in step S301It is equivalent toAnd if lemma T=zTAz+bTZ, wherein A ∈ RM ×M, b ∈ RM, z ∈ N (0, I), then Wherein
s-(X)=max { λmax(-X),0},λmax(X) be vectorial X maximum characteristic value, the mark of tr () representing matrix obtains Go out.
In further embodiment, step S400 is specifically included：
Step S401, makeObject function isConstraints isDue to
Utilize semidefinite decoding, it is assumed thatSo
Object function is converted into semidefinite decoding function after relaxation：Wherein constrain Condition isW >=0, wherein W=wwH, it is semidefinite order one Matrix, μ are the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||F 2。
By the order of the relaxed W of step S400 it is 1 in the step S500, whereinAnd then obtain mesh Scalar functions are converted intoConstraints is W≥0.Object function is convex function, and constraints is also convex function, therefore effective convex optimization method such as CVX can be utilized to solve Object function realizes maximum Signal to Interference plus Noise Ratio output.It is preferred that it is solved using the cvx bags in MATLAB.
The present invention also provides a kind of simulation results of concrete application example, and wherein simulated environment is as shown in table 1, analogous diagram Variable parameter is as shown in table 2.
Table 1
Table 2
The result difference emulated based on the simulation parameter that table 2 is set is as shown in Fig. 2, the emulation ginseng set based on table 3 Shown in the result figure 3 that number is emulated, simulation result shows in the case where sampling is there are error, the parameter pair after output solution The Wave beam forming answered is far longer than other methods, such as SMI (i.e. sample matrix inversion) methods and traditional Robust Minimum Variance Beamforming Method, the present invention improve the robustness of system, realize effective inhibition to interference signal.
The present invention also provides a kind of preferred embodiments based on the Beam Forming System for determining and not knowing collection constraint Functional schematic block diagram, as shown in figure 4, wherein, system includes：
Signal acquisition module 100, for obtaining the signal that array element in aerial array receives, the signal that is received to array element It is weighted summation and obtains the output of Wave beam forming；Specifically as above described in embodiment of the method.
Object function modeling module 200, for carrying out array covariance square according to several samples of beam forming process Battle array estimation, is modeled according to the object function of the array covariance matrix after estimation, obtains the uncertain collection constraint after modeling Condition；Specifically as above described in embodiment of the method.
Conversion module 300 is constrained, collection constraint is determined for changing into uncertain collection constraint according to Berne type inequality；Tool Body is as above described in embodiment of the method.
Object function conversion module 400, for object function to be converted into Semidefinite Programming function according to semidefinite relaxation algorithm； Specifically as above described in embodiment of the method.
Calculating and output module 500, for being solved according to convex optimized algorithm to Semidefinite Programming function, after output solves The corresponding Wave beam forming of parameter；Specifically as above described in embodiment of the method.
Further, the 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 received, and s (k) is source signal, and a is steering vector, and i (k) is interference components, N (k) is noise component(s)；Specifically as above described in embodiment of the method.
Computing unit is weighted summation for docking collection of letters x (k) and obtains the output y (k) of Wave beam forming,
Y (k)=wHx(k)
Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent the transposition of w；Specifically as above described in embodiment of the method.
Specifically, the object function is established module and is specifically included：
Object function and uncertain collection constraints acquiring unit, for several samples according to beam forming process into Row array 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,Specifically As above described in embodiment of the method.
WhereinExpression takes the maximum of () using Δ as independent variable,Expression takes () by independent variable of w Minimum value, Pr { } represent { } probability, array of samples covariance matrixΔ is covariance Matrix error,For a it is expected steering vector, δ is steering vector error；Specifically as above described in embodiment of the method.
Further, the constraint conversion module specifically includes：
Uncertain collection constraint is to definite collection constraint conversion unit, for uncertain collection to be constrained It is converted into and determines collection constraint,
Wherein tr () represents the mark of (), | | | |FRepresent Frobenius norms, 2 ρ=- ln (1-p), δ=Bu, u clothes It is the normal distribution that 0 covariance matrix is unit matrix I from mean vector, B=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is fixed Matrix, bw=BHwwHa∈CM；Specifically as above described in embodiment of the method.
Specifically, the object function conversion module specifically includes：
Semidefinite relaxation computing unit, for makingObject function isConstraints ForDue to
Utilize semidefinite decoding, it is assumed thatSo
Specifically as above method is real It applies described in example.
Object function is converted into semidefinite decoding function after relaxation：Wherein constrain Condition isW >=0, wherein W=wwH, it is semidefinite order one Matrix, μ are the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||F 2；Specifically such as Described in upper embodiment of the method.
In conclusion the present invention provides a kind of based on determining and the Beamforming Method and system of uncertain collection constraint, Method includes：The signal that array element receives in aerial array is obtained, summation is weighted to the signal that array element receives and obtains ripple The output that beam is formed；Array covariance matrix is carried out according to several samples of beam forming process, after estimation The object function of array covariance matrix is modeled, and obtains the uncertain collection constraints after modeling；It is differed according to Berne type Uncertain collection constraint is changed into and determines collection constraint by formula；Object function is converted by Semidefinite Programming letter according to semidefinite relaxation algorithm Number；Semidefinite Programming function is solved according to convex optimized algorithm, the corresponding Wave beam forming of parameter after output solution.The present invention The probabilistic constraints of random process are converted by certainty constraints by Berne type inequality, then utilize semidefinite decoding Definite constraints is converted into Semidefinite Programming, so as to carry out the optimization of object function, is compensated in existing method only gram Steering vector mismatch has been taken, without considering serious defect problem of the sample array covariance matrix there are error, has improved and is The robustness of system improves the measurement accuracy of array signal processing system.
It should be appreciated that the application of the present invention is not limited to the above, it for those of ordinary skills, can To be improved or converted according to the above description, all these modifications and variations should all belong to the guarantor of appended claims of the present invention Protect scope.

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)=wHx(k)
    Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent 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=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is set matrix, bw=BHwwHa∈CM。
  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=wwH, it is one matrix of semidefinite order, μ is the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||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)=wHx(k)
    Wherein w ∈ CMFor the weight vector of Wave beam forming, wHRepresent 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=Cδ 1/2∈CM×M, Bw=BHwwHB∈CM×M, it is set matrix, bw=BHwwHa∈CM。
  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=wwH, it is one matrix of semidefinite order, μ is the slack variable introduced, and vec () is the matrix-vector factor, | | vec (Bw) | |=| | Bw||F 2。
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