CN103412294A - Airborne radar space-time three-dimensional clutter suppression method based on double direct product decomposition - Google Patents

Abstract

The invention discloses an airborne radar space-time three-dimensional clutter suppression method based on double direct product decomposition. The problems that airborne radar space-time three-dimensional clutter suppression calculation complexity is high, the quantity of demanded samples is large, and array element errors obviously reduce detection performance are mainly solved. The implementation process of the method comprises the first step of carrying out primary direct product decomposition on the weight vector of a space-time three-dimensional filter, the second step of carrying out secondary direct product decomposition on a space-domain weight vector, the third step of obtaining the double direct product decomposition mode of the weight vector of the space-time three-dimensional filter through truncation processing, the fourth step of solving a time-domain weight vector set, a space-domain orientation dimension weight vector set and a space-domain pitching dimension weight vector set through a double double-iterative algorithm and the fifth step of restoring the space-time three-dimensional filter weight vector according to the solved time-domain weight vector set, space-domain orientation dimension weight vector set and space-domain pitching dimension weight vector set and carrying out clutter suppression. The method is low in computation complexity, the quantity of the demanded samples is small, good error robustness is achieved, and moving target detection performance can be improved.

Description

Three-dimensional clutter suppression method during based on the airborne radar space of dual the direct product decomposition

Technical field

The invention belongs to the Radar Signal Processing Technology field, while being specifically related to a kind of airborne radar space based on dual the direct product decomposition, three-dimensional dimensionality reduction self-adapting clutter inhibition method, can be used for improving the detection performance to moving target.

Background technology

Airborne radar is used to identify the moving target in strong clutter environment, and the clutter scattering unit is due to the direction difference, and how general corresponding clutter rate frequency be also different, coupled characteristic when this phenomenon is referred to as sky.Find simple clutter suppression method efficiently, be one of the research emphasis in airborne radar field always.Self-adaptive processing during three-dimensional space (3D-STAP) is self-adaptive processing on the two peacekeeping time domains of spatial domain, can take full advantage of pitching dimension degree of freedom, and the array element error is had to better robustness.Simultaneously, under Far Range and certain pulse repetition rate (PRF), pitching dimension adaptive beam can suppress the range ambiguity clutter, so 3D-STAP can alleviate the range ambiguity problem.Although 3D-STAP has some better performances, along with adding the pitching dimension to process, entirely tie up the adaptive processor dimension higher, computation complexity is larger, be difficult to real-time processing, and along with the increase of processor dimension, 3D processing needs how independent identically distributed sample.In order to reduce computation complexity and sample demand, guarantee simultaneously self-adaptation profit and loss minimum, the 3D-STAP dimensionality reduction technology is furtherd investigate.

At present, in airborne radar 3D-STAP, through method commonly used, spreading factor method (EFA) is arranged, the method is first carried out doppler filtering in time domain, then in spatial domain pitching peacekeeping azimuth dimension, combine self-adaptive processing, when having the array element error, clutter spectrum spreads in spatial domain, has more clutter to enter time domain Doppler passage, algorithm performance descends, and the method also needs that spatial domain is tieed up to processor entirely and inverts, and computation complexity is high, is unfavorable for real-time processing.In addition, verified, for guaranteeing the self-adaptation profit and loss, be less than 3dB, the required independent same distribution sample number of EFA method should be greater than spatial domain and entirely tie up more than 2 times of processor dimension, and, in the clutter environment of actual fast time variant, usually be difficult to obtain so many independent same distribution sample.Equally, existing some other airborne radar 3D-STAP method, also there will be similar problem.

Summary of the invention

The object of the invention is to the deficiency for above-mentioned prior art, propose a kind of airborne radar clutter suppression method based on dual the direct product decomposition, reduce computation complexity and sample demand, improve the error robustness, improve moving-target and detect performance.

The technical scheme that realizes the object of the invention may be summarized to be: at first, during by sky, the three-dimensional filter weight vector carries out a subdirect product decomposition, and while obtaining sky, the three-dimensional filter weight vector is about the direct product form of the composition of time domain weight vector group, spatial domain weight vector group; Secondly, spatial domain weight vector group is carried out to two subdirect product decompositions, obtain the direct product form of the composition of spatial domain weight vector group about spatial domain azimuth dimension weight vector group, spatial domain pitching right-safeguarding set of vectors; Three-dimensional filter weight vector when then, the multipair time domain weight vector group of intercepting, spatial domain azimuth dimension weight vector group and spatial domain pitching right-safeguarding set of vectors are approached sky; Then, utilize dual double iterative algorithm (BBIA), solve multipair time domain weight vector group, spatial domain azimuth dimension weight vector group and the spatial domain pitching right-safeguarding set of vectors of intercepting; Finally, three-dimensional filter weight vector while recovering clearancen by the multipair weight vector group of trying to achieve, be used for clutter reduction.

Suppose that airborne radar array is M * N rectangular surfaces battle array, M presentation surface battle array line number wherein, N presentation surface number of arrays, the umber of pulse in the relevant processing time (CPI) is K.

The specific implementation process is as follows:

During (1) by sky, three-dimensional filter weight vector w carries out a subdirect product decomposition, three-dimensional filter weight vector w=[w during by sky _1,1,1W _{M, 1,1}W _1,2,1W _{M, 2,1}W _{1, N, K}W _{M, N, K}] ^TBe arranged in the weight matrix W of a K * MN dimension ₁, wherein symbol T means transposition, utilizes the full rank decomposition method by weight matrix W ₁Be decomposed into

R wherein ₁Mean weight matrix W ₁Order, u _nMean spatial domain weight vector group, d _nMean time domain weight vector group, symbol H means conjugate transpose; By equation

The right and left column vector, obtain a subdirect product decomposition

Wherein symbol * means to get conjugation, symbol

Mean direct product;

(2) by spatial domain weight vector group u _nCarry out two subdirect product decompositions, by spatial domain weight vector u _nLine up M * N and tie up matrix U _n, utilize the full rank decomposition method by weight matrix U _nBe decomposed into

R wherein ₂Mean U _nOrder, θ _nMean spatial domain pitching right-safeguarding set of vectors,

Mean spatial domain azimuth dimension weight vector group; By equation The right and left column vector, obtain two subdirect product decompositions

(3) by the subdirect product decomposition of weight vector w in step (1)

And spatial domain weight vector group u in step (2) _nTwo subdirect product decompositions

Carry out truncation, the dual the direct product decomposition form of three-dimensional filter weight vector w while obtaining sky:

D<<r ₁, symbol<<mean much smaller than;

(4) respectively by d _n, And θ _n(n=1 ..., D) column vector, form column vector $p_d={[d_1^T,...,d_D^T]}^T,$ $p_{\mathrm{\&theta;}}={[{\mathrm{\&theta;}}_1^T,...,{\mathrm{\&theta;}}_D^T]}^T,$ And utilize dual double iterative algorithm, solve column vector p _d,

And p _θ

(5) according to the column vector p tried to achieve _d,

And p _θThree-dimensional filter weight vector w while recovering clearancen, be used for clutter reduction.

The present invention compared with prior art has following characteristics:

1, compared with prior art, the sample demand is low in the present invention, under Small Sample Size, can obtain good performance.Theoretical analysis proves, is less than 3dB for guaranteeing the self-adaptation profit and loss, and independent identically distributed sample number should be greater than 2 times of clutter covariance matrix dimensions.The spreading factor method (EFA) of take is example, when the method is combined self-adaptive processing in spatial domain, need to invert to the covariance matrix that dimension is 3MN, and the sample demand should be greater than 6MN.Three-dimensional filter weight vector when the present invention approaches sky with D (D≤3) group spatial domain and time domain weight vector, utilize dual double iterative algorithm to solve D group spatial domain and time domain weight vector, the covariance matrix dimension wherein related to is respectively DM, DN and DK, nonsingular for guaranteeing covariance matrix, the sample number that the inventive method needs only need be not less than max{DM, DN, DK}, max mean to get the maximal value of number in bracket.Generally, max{DM, DN, DK}<<6MN, the present invention greatly reduces the sample demand.

2, the present invention compared with prior art, has reduced computation complexity, is conducive to real-time processing.The typical spreading factor method (EFA) of take is example, and the covariance matrix of 3MN dimension is inverted, and computation complexity is O (27 (MN) ³).The covariance matrix dimension related in the present invention is respectively DM, DN and DK, and the complexity of inversion operation is respectively O ((DM) ³), O ((DN) ³) and O ((DK) ³), ectomesoderm double iterative algorithm 2 steps of the present invention can restrain, and internal layer double iterative algorithm 5 steps can restrain, and the total computation complexity of the present invention is O (10 (DM) ³+ 10 (DN) ³+ 2 (DK) ³).Due to D value very little (D<<3), common O (10 (DM) ³+ 10 (DN) ³+ 2 (DK) ³The O of)<<(27 (MN) ³), so the present invention greatly reduces computation complexity, is conducive to real-time processing.

3, the present invention compared with prior art, when having the array element error, has better moving-target Doppler frequency and detects performance, namely has better error robustness.The typical EFA algorithm of take is example, owing to there being the array element error, clutter spectrum spreads in spatial domain, when using Doppler filter to carry out filtering, there is more clutter to enter Doppler's main lobe, cause the formed notch width of space domain self-adapted processing and the degree of depth to be difficult to meet clutter and suppress requirement, make this algorithm performance descend.The present invention carries out double iterative associating self-adaptive processing in time domain dimension, spatial domain azimuth dimension and spatial domain pitching dimension, makes beam pattern form deep notch in spatial domain, has further improved clutter and has suppressed ability.

The accompanying drawing explanation

Fig. 1 is airborne radar Array Model figure

Fig. 2 is realization flow figure of the present invention

Fig. 3 is the internal layer iteration convergence curve of dual double iterative algorithm in the present invention

Fig. 4 is the external iteration convergence curve of dual double iterative algorithm in the present invention

Fig. 5 is that the present invention and EFA method output signal-to-noise ratio improve comparison diagram

When Fig. 6 is the present invention and EFA method processing emulated data, residual spur normalization output power comparison diagram

Fig. 7 is that the output signal-to-noise ratio of the present invention and EFA method improves the change curve with sample number

Embodiment

With reference to Fig. 2, specific implementation step of the present invention is as follows:

1. during by sky, three-dimensional filter weight vector w carries out a subdirect product decomposition, three-dimensional filter weight vector w=[w during by sky _1,1,1W _{M, 1,1}W _1,2,1W _{M, 2,1}W _{1, N, K}W _{M, N, K}] ^TBe arranged in the weight matrix W of a K * MN dimension ₁, utilize the full rank decomposition method by weight matrix W ₁Be decomposed into the product form of spatial domain weight vector group and time domain weight vector group, obtain

R wherein ₁For weight matrix W ₁Order, u _nMean spatial domain weight vector group, d _nMean time domain weight vector group; By weight matrix W ₁Column vectorization can recover w, by equation The right and left column vector, can obtain a subdirect product decomposition

2. by spatial domain weight vector group u _nCarry out two subdirect product decompositions, by spatial domain weight vector group u _nLine up M * N and tie up spatial domain weight matrix U _n, utilize the full rank decomposition method by spatial domain weight matrix U _nBe decomposed into the product form of spatial domain azimuth dimension weight vector group and spatial domain pitching right-safeguarding set of vectors, obtain

R wherein ₂For weight matrix U _nOrder, θ _nMean spatial domain pitching right-safeguarding set of vectors,

Mean spatial domain azimuth dimension weight vector group; By spatial domain weight matrix U _nColumn vectorization can recover spatial domain weight vector group u _n, by equation

The right and left is column vector all, can obtain two subdirect product decomposition forms

The full rank decomposition method of the present invention application is: for order, be the matrix Z of r, can be decomposed into matrix F that a row full rank and order are r and capable full rank and order is the product form of the matrix G of r, namely has

F=[f wherein ₁..., f _r], G=[g ₁..., g _r] ^H, column vector f _i(i=1 ..., r) uncorrelated each other, column vector g _i(i=1 ..., r) uncorrelated each other.

3. to the subdirect product decomposition of the weight vector w that obtains in step 1

With the spatial domain weight vector group u obtained in step 2 _nTwo subdirect product decompositions

Carry out truncation, first intercept D time domain weight vector group

With D spatial domain weight vector group u _nMean weight vector w, obtain

Intercept again 1 spatial domain pitching right-safeguarding set of vectors

With 1 spatial domain azimuth dimension weight vector group θ _nMean spatial domain weight vector group u _n, obtain

Carry it into

Obtain the dual the direct product decomposition form of weight vector w:

4. respectively by d _n,

And θ _n(n=1 ..., D) column vector, form column vector

$p_{\mathrm{\&theta;}}={[{\mathrm{\&theta;}}_1^T,...,{\mathrm{\&theta;}}_D^T]}^T,$ Then utilize dual double iterative algorithm to solve column vector p _d,

And p _θ.

Lower mask body is introduced solution procedure.

Suppose x=[x _1,1,1X _{M, 1,1}X _1,2,1X _{M, 2,1}X _{1, N, K}X _{M, N, K}] ^TThe three-dimensional sampled data vector that receives during for sky, target time domain steering vector is a _d, target spatial domain azimuth dimension steering vector is Target spatial domain pitching dimension steering vector is a _θ, the goal orientation vector is

1) fixing p _d, ask

p _θOptimum solution.Given initial value Set up following cost function:

Wherein $X=\mathrm{diag}(\mathrm{reshape}((d_1^H\&CircleTimes;I_{\mathrm{MN}})x,M,N),...,\mathrm{reshape}((d_D^H\&CircleTimes;I_{\mathrm{MN}})x,M,N)),$ Symbol E means to ask expectation, $\mathrm{reshape}((d_i^H\&CircleTimes;I_{\mathrm{MN}})x,M,N),i=1,...,D$ Expression is by vector

Line up M * N matrix, I _MNMean MN dimension unit matrix, symbol diag means diagonalization;

2) utilize double iterative algorithm to solve formula (2), obtain

And p _{θ 1}.Solution procedure is as follows:

2.1) given initial value

It is carried out to normalization Symbol || || mean delivery.

2.2) calculating p _{θ 0}

Wherein L means sample number, X _lThree-dimensional sampled data while meaning l sky.

2.3) calculate

2.4) judgement iteration stopping formula

Whether set up, wherein ε means given iteration stopping parameter.If set up, iteration stops; If be false, will

Be updated to

Then repeated execution of steps 2.2) and 2.3), until this internal layer iteration stops, last, output is optimum to be weighed

And p _{θ 1}=p _{θ 0}

3) fixing weight vector

And p _{θ 1}, solve weight vector p _D1

Order

Set up following cost function:

$\left\{\begin{array}{c}\min E{\left|\right|p_{\mathrm{\&theta;}1}^{\mathrm{<figure-callout\; id="1"\; label="H\; Yp\; d"\; filenames="HDA0000371543780000031.png"\; state="\{\{state\}\}">H</figure-callout>}}{\mathrm{Yp}}_d{1}_{}\left|\right|}^2\\ s.t.p_{\mathrm{\&theta;}1}^{\mathrm{<figure-callout\; id="1"\; label="H\; bp\; d"\; filenames="HDA0000371543780000031.png"\; state="\{\{state\}\}">H</figure-callout>}}{\mathrm{bp}}_d{1}_{}=1\end{array}\right.---\left(5\right)$

Wherein Y=diag (Y (1) ..., Y (D)), Utilize method of Lagrange multipliers to try to achieve:

${\mathrm{<figure-callout\; id="1"\; label="p\; d"\; filenames="HDA0000371543780000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_d=\frac{{\left(\frac{1}{L}\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right){\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right)}^H\right)}^{-1}{b}^H{p}_{\mathrm{\&theta;}1}}{\left|\right|{\left(\frac{1}{L}\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right){\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right)}^H\right)}^{-1}{b}^H{p}_{\mathrm{\&theta;}1}\left|\right|}---\left(6\right)$

Y wherein _iMean l training sample;

4) judgement iteration stopping formula || p _D1-p _D0||<z, whether set up 0<z<<1, and wherein z means given iteration stopping parameter.If setting up iteration stops; If be false, by p _D0Be updated to p _D1, repeated execution of steps 1), 2), 3) until external iteration stop.Finally, output optimum solution p _d,

And p _θ.

5. by the variable p tried to achieve _d, And p _θOptimum solution p _D1,

And p _{θ 1}, three-dimensional filter weight vector while recovering clearancen according to step 3, in order to clutter reduction.

Effect of the present invention further illustrates by following l-G simulation test:

1. simulated conditions

Experiment is adopted the rectangle plane battle array as shown in Figure 1, and wherein M=4 means the array element line number, and N=8 means the array element columns, and V=90m/s means carrier aircraft speed, and R=500km means range, θ _t=90 ° mean azimuth of target,

Mean the target angle of pitch, ψ _t=90 ° mean target face angle, v _t=36m/s means the radial velocity of target with respect to carrier aircraft.Carrier aircraft height 8km, wavelength 0.3m, array element distance is 0.15m, PRF is 1200Hz, in a CPI, umber of pulse is 16, controlling antenna wave beam to point front normal direction, and emission pitching peacekeeping azimuth dimension all adds the fixing power of 30dB, single array element input miscellaneous noise ratio (CNR) is 60dB, signal to noise ratio (snr) is 0dB, array element amplitude phase error 2%, three-dimensional filter weight vector while adopting 2 groups of dual the direct product decompositions to approach sky in emulation, even D=2, wherein D≤r ₁, r ₁Mean weight matrix W ₁Order.

2. emulation content

Emulation 1, when (sample number is 100), in the present invention, in dual double iterative algorithm, during the internal layer iteration, output signal-to-noise ratio improves the change curve with iterations, as Fig. 3 when small sample.

Emulation 2, when (sample number is 100), in the present invention, in dual double iterative algorithm, during external iteration, output signal-to-noise ratio improves the change curve with iterations, as Fig. 4 when small sample.

Emulation 3, when (sample number is 100), the present invention and EFA method output signal-to-noise ratio improve the performance comparison diagram, result such as Fig. 5 when small sample.

Emulation 4, when (sample number is 100), when the present invention and EFA method were processed emulated data, residual spur normalization output power comparison diagram, as Fig. 6 when small sample.

Emulation 5, the output signal-to-noise ratio of the present invention and EFA method improves the change curve comparison diagram with sample number, as Fig. 7.

3. simulation analysis

As can be seen from Figure 3, under Small Sample Size, when having the array element error, in the present invention, the internal layer iteration of dual double iterative algorithm can restrain through 5 steps, as can be seen from Figure 4, in the present invention, the external iteration of dual double iterative algorithm can restrain through 2 steps, and dual double iterative algorithm can be restrained through 10 steps altogether, and total computation complexity is O (10 (8) ³+ 10 (16) ³+ 2 (32) ³).And the EFA method need to be inverted to the covariance matrix of 96 dimensions, computation complexity is O (96 ³).Computation complexity O (10 (8) of the present invention ³+ 10 (16) ³+ 2 (32) ³The O of)<<(96 ³), with the EFA method, compare and greatly reduce computation complexity, saved calculated amount.

As can be seen from Figure 5, when small sample (sample number is 100), the present invention compares with the EFA method, and the performance improvement of 5dB left and right is arranged.In the situation that 100 samples, due to sample number, be less than 2 times of covariance matrix dimension in the EFA method, namely 100<<192, the now EFA method self-adaptation profit and loss is larger, and in the present invention, the highest dimension of covariance matrix is 32, (32 * 2)<<100, so the self-adaptation profit and loss of the present invention is very little.In addition, due to the existence of amplitude phase error, when the EFA method is carried out doppler filtering in time domain, there is more clutter to enter main lobe, make algorithm performance descend, and the present invention carry out iteration associating self-adaptive processing in time domain, spatial domain azimuth dimension, spatial domain pitching dimension, and error is had to better robustness.Therefore, the present invention has better moving-target detection performance.

As can be seen from Figure 6, in the how general rate frequency of normalization, be 0.4 o'clock, the present invention compares with the EFA method, and at the have an appointment performance improvement of 5dB of each range gate, the present invention has better moving-target and detects performance.

As can be seen from Figure 7, the present invention starts to approach convergence when 64 samples, and the EFA method starts to approach convergence when 192 samples, illustrate that the present invention can restrain under Small Sample Size, and the EFA method just can restrain in full-page proof given figure situation.And, in the clutter environment of true airborne radar fast time variant, meeting independent identically distributed sample less, the present invention has better practicality.

Be more than preferred embodiment of the present invention, do not form any limitation of the invention, obviously without departing from the principles of the present invention, equal available not three-dimensional filter weight vectors when dual the direct product decomposition approaches sky on the same group, but these are all at the row of protection of the present invention.

Claims (2)

1. three-dimensional clutter suppression method during the airborne radar space based on dual the direct product decomposition, at first, during by sky, the three-dimensional filter weight vector carries out a subdirect product decomposition, and while obtaining sky, the three-dimensional filter weight vector is about the direct product form of the composition of time domain weight vector group, spatial domain weight vector group; Secondly, spatial domain weight vector group is carried out to two subdirect product decompositions, obtain the direct product form of the composition of spatial domain weight vector group about spatial domain azimuth dimension weight vector group, spatial domain pitching right-safeguarding set of vectors; Three-dimensional filter weight vector when then, the multipair time domain weight vector group of intercepting, spatial domain azimuth dimension weight vector group and spatial domain pitching right-safeguarding set of vectors are approached sky; Then, utilize dual double iterative algorithm to solve multipair time domain weight vector group, spatial domain azimuth dimension weight vector group and the spatial domain pitching right-safeguarding set of vectors of intercepting; Finally, three-dimensional filter weight vector while recovering clearancen by the multipair weight vector group of trying to achieve, be used for clutter reduction, and the specific implementation process is as follows:

Suppose that airborne radar array is M * N rectangular surfaces battle array, M presentation surface battle array line number wherein, N presentation surface number of arrays, the umber of pulse in relevant processing time CPI is K;

During (1) to sky, three-dimensional filter weight vector w carries out a subdirect product decomposition

By w=[w _1,1,1W _{M, 1,1}W _1,2,1W _{M, 2,1}W _{1, N, K}W _{M, N, K}] ^TBe arranged in the weight matrix W of a K * MN dimension ₁, wherein symbol T means transposition, utilizes the full rank decomposition method by weight matrix W ₁Be decomposed into R wherein ₁Mean weight matrix W ₁Order, u _nMean spatial domain weight vector group, d _nMean time domain weight vector group, symbol H means conjugate transpose; By equation The right and left column vector, obtain a subdirect product decomposition

Wherein symbol * means to get conjugation, symbol

Mean direct product;

(2) to spatial domain weight vector group u _nCarry out two subdirect product decompositions

By spatial domain weight vector group u _nLine up M * N and tie up matrix U _n, utilize the full rank decomposition method by weight matrix U _nBe decomposed into

R wherein ₂Mean U _nOrder, θ _nMean spatial domain pitching right-safeguarding set of vectors,

Mean spatial domain azimuth dimension weight vector group; By equation

The right and left column vector, obtain two subdirect product decompositions

(3) by the subdirect product decomposition of the weight vector w that obtains in step (1)

And the spatial domain weight vector group u obtained in step (2) _nTwo subdirect product decompositions

Carry out truncation, the dual the direct product decomposition form of three-dimensional filter weight vector w while obtaining sky:

D<<r ₁, symbol<<mean much smaller than;

(4) respectively by d _n,

And θ _nColumn vector, n=1 wherein ..., D, form column vector $p_d={[d_1^T,...,d_D^T]}^T,$

$p_{\mathrm{\&theta;}}={[{\mathrm{\&theta;}}_1^T,...,{\mathrm{\&theta;}}_D^T]}^T,$ And utilize dual double iterative algorithm, solve column vector p _d,

And p _θ

(5) according to the column vector p tried to achieve _d,

And p _θWhile recovering clearancen, three-dimensional filter weight vector w, carry out the clutter inhibition.

2. three-dimensional clutter suppression method during the airborne radar space based on dual the direct product decomposition according to claim 1, is characterized in that described step (4) utilizes dual double iterative algorithm to ask optimum solution, specifically carries out as follows:

Suppose x=[x _1,1,1X _{M, 1,1}X _1,2,1X _{M, 2,1}X _{1, N, K}X _{M, N, K}] ^TThe three-dimensional sampled data vector that receives during for sky, target time domain steering vector is a _d, target spatial domain azimuth dimension steering vector is

Target spatial domain pitching dimension steering vector is a _θ, the goal orientation vector is

1) fixing p _d, ask p _θOptimum solution, given initial value

Set up following cost function:

Wherein $X=\mathrm{diag}(\mathrm{reshape}((d_1^H\&CircleTimes;I_{\mathrm{MN}})x,M,N),...,\mathrm{reshape}((d_D^H\&CircleTimes;I_{\mathrm{MN}})x,M,N)),$ Symbol E means to ask expectation, $\mathrm{reshape}((d_i^H\&CircleTimes;I_{\mathrm{MN}})x,M,N),i=1,...,D$ Expression is by vector

Line up M * N matrix, I _MNMean MN dimension unit matrix, symbol diag means diagonalization;

2) utilize double iterative algorithm to solve formula (1), obtain

And p _{θ 1}, solution procedure is as follows:

2.1) given initial value It is carried out to normalization

Symbol || || mean delivery;

2.2) calculating p _{θ 0}

Wherein L means sample number, X _lThree-dimensional sampled data while meaning l sky;

2.3) calculate

2.4) judgement iteration stopping formula

Whether set up, wherein ε, if set up, iteration stops if meaning given iteration stopping parameter; If be false, will

Be updated to

Then repeated execution of steps 2.2) and 2.3), until this internal layer iteration stops; The optimum power of output

And p _{θ 1}=p _{θ 0}

3) fixing weight vector And p _{θ 1}, solve weight vector p _D1,

Order Set up following cost function:

$\left\{\begin{array}{c}\min E{\left|\right|p_{\mathrm{\&theta;}1}^H{\mathrm{Yp}}_{d1}\left|\right|}^2\\ s.t.p_{\mathrm{\&theta;}1}^H{\mathrm{bp}}_{d1}=1\end{array}\right.---\left(4\right)$

Wherein Y=diag (Y (1) ..., Y (D)), Utilize method of Lagrange multipliers to try to achieve:

$p_{d1}=\frac{{\left(\frac{1}{L}\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right){\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right)}^H\right)}^{-1}{b}^H{p}_{\mathrm{\&theta;}1}}{\left|\right|{\left(\frac{1}{L}\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right){\left(Y_l^H{p}_{\mathrm{\&theta;}1}\right)}^H\right)}^{-1}{b}^H{p}_{\mathrm{\&theta;}1}\left|\right|}---\left(5\right)$

Y wherein _iMean l training sample;

4) judgement iteration stopping formula || p _D1-p _D0||<z, whether set up 0<z<<1, and wherein z means given iteration stopping parameter, stops if set up iteration; If be false, by p _D0Be updated to p _D1, repeated execution of steps 1), 2), 3) until external iteration stop, last, output optimum solution p _d,

And p _θ.

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