CN105974366A - Four-order cumulant sparse representation-based MIMO (multiple-input-multiple-output) radar direction of arrival estimation method under mutual coupling condition - Google Patents

Abstract

The invention belongs to the monostatic MIMO (multiple-input-multiple-output) radar system technical field and relates to a four-order cumulant sparse representation-based MIMO (multiple-input-multiple-output) radar direction of arrival estimation method under a mutual coupling condition. The method of the invention comprises the following steps that: a transmitting array transmits mutually-orthogonal phase encoding signals, a receiving end carries out matched filtering on the phase encoding signals so as to obtain receiving data, and the influence of unknown mutual coupling is eliminated through linear transform based on the strap-shaped symmetric Toeplitz structures of the mutual coupling matrixes of the transmitting array and the receiving array; and a dimension reduction conversion matrix is constructed to carry out dimension reduction processing on mutual coupling-eliminated data, and a four-order cumulant matrix of a special form is constructed based on a new data matrix. According to the method of the invention, since the four-order cumulant technique and a weighted sparse representation framework are adopted, colored noises are successfully inhibited. The method of the invention can achieve accurate direction of arrival estimation under a Gaussian color noise condition, and has higher angular resolution and better angle estimation performance.

Description

Under array mutual-coupling condition, MIMO radar direction of arrival based on fourth order cumulant rarefaction representation is estimated Meter method

Technical field

The invention belongs to single base MIMO radar system technical field, be specifically related under array mutual-coupling condition based on fourth order cumulant The MIMO radar Wave arrival direction estimating method of rarefaction representation.

Background technology

Multiple-input and multiple-output (MIMO) radar is made up of novel array structure, launches mutually orthogonal waveform, due to than Tradition phased-array radar has a lot of advantage such as higher resolution and more preferable parameter recognizability, and attracts increasing closing Note.MIMO radar can be divided into two classes, statistics MIMO radar and relevant MIMO radar (IEEESignalProcessingMagazi Ne, 2007,24 (5): 106-114).Statistics MIMO radar obtains spatial gain by big sky line-spacing, and relevant MIMO radar is led to Cross close sky line-spacing and form virtual array large aperture, it is thus achieved that higher spatial resolution and more degree of freedom.Relevant MIMO Radar includes bistatic and single base MIMO radar, and the former emission array and receiving array are separated from each other, and the latter is tight Put altogether.In the present invention, what we studied is the Mutual coupling problem in the MIMO radar of single base.

Angle estimation has important function in signal processing and radar application, utilizes MIMO radar and traditional array signal The similarity of model, in a large number about MIMO radar DOA estimate document emerge, the most representational be based on The subspace angle estimating method of angle searching or invariable rotary characteristic, specifically includes MUSIC, Capon and ESPRIT method etc.. MUSIC and Capon has the biggest computation complexity and is unfavorable for real time signal processing.ESPRIT algorithm has high operation efficiency With more preferable angle estimation performance.Derive from method include dimensionality reduction Capon, dimensionality reduction ESPRIT (RD-ESPRIT, ElectronicsLetters, 2011,47 (4): 283-284) and tenth of the twelve Earthly Branches ESPRIT, RD-ESPRIT and ESPRIT at the tenth of the twelve Earthly Branches compares ESPRIT Method has lower amount of calculation.Launching beam territory energy concentration techniques (IEEETransactionsonSignalProcessin G, 2011,59 (6): 2669-2682), make signal to noise ratio (SNR) gain be maximized.But, when considering mutual coupling, above son Space-wise hydraulic performance decline even lost efficacy.In order to eliminate mutual coupling, certain methods is suggested, similar MUSIC Angle is estimated by (SignalProcessing, 2012,92 (2): 517-522) by angle searching, and rooting MUSIC has Lower computation complexity, similar ESPRIT method (SignalProcessing, 2012,92 (12): 3039-3048) has good Good angle estimation performance, avoids spatial peaks search simultaneously.

In Array Signal Processing, it is frequently encountered by situation rather than the white noise of space gauss heat source model, side mentioned above The Mutual coupling performance of method declines further.Higher Order Cumulants application in DOA estimation can solve the problem that this problem, because of The Gaussian noise of any variance can be suppressed for fourth order cumulant (FOC).Based on this characteristic, FOC-MUSIC, FOC-ESPRIT It is suggested etc. method.Under array mutual-coupling condition, a kind of FOC-MUSIC method of improvement (SignalProcessing, 2009,89: 1839-1843) all there is under the conditions of white Gaussian noise and coloured noise good estimation performance.

Emerging rarefaction representation attracts increasing concern in signal analysis field, by finding the most sparse table of data In showing that can apply it to DOA estimates.All simulation results confirm, compared with conventional subspace method, and sparse representation method There is remarkable advantage, it is possible to better adapt to special circumstances, more high angular resolution is provided in addition and is less dependent on incidence The prior information of signal number.In order to estimate that direction of arrival certain methods is suggested, such as l₁-SVD, FOCUSS, l₁-SRACV And W-l₁-SRACV etc..W-l₁-SRACV (IEEESignalProcessingLetters, 2012,19 (3): 155-158) passes through The weighting l of array of designs covariance vector₁Norm minimum framework and there is good resolution.Real-valued rarefaction representation algorithm (IEEEAntennasandWirelessPropagationLetters, 2013,12:376-379) reduces amount of calculation.Mutual coupling Under the conditions of, the l of improvement₁-SVD (IEEEAntennasandWirelessPropagationLetters, 2012,11:1210- 1213) angle estimating method is suggested, and unknown mutual coupling is eliminated.But, it is Gauss when considering mutual coupling or noise During coloured noise, the estimation hydraulic performance decline of above method.

Summary of the invention

It is an object of the invention to provide MIMO radar based on fourth order cumulant rarefaction representation under a kind of array mutual-coupling condition Arrival direction estimating method.

The object of the present invention is achieved like this:

MIMO radar Wave arrival direction estimating method based on fourth order cumulant rarefaction representation under array mutual-coupling condition, including walking as follows Rapid:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving terminal obtains after carrying out matched filtering process and connects Receive data, and utilize and launch and the mutual coupling matrix banding symmetry Toeplitz construction features that all has of receiving array, by linearly Conversion eliminates the impact of unknown mutual coupling；

(2) data after eliminating mutual coupling are carried out dimension-reduction treatment by structure dimensionality reduction transition matrix, and then based on new data square Battle array constructs the fourth order cumulant matrix with specific form；

(3) fourth order cumulant observing matrix is carried out dimension-reduction treatment, it is thus achieved that the corresponding model under framework of sparse representation, and profit By steering vector and the orthogonality of corresponding noise subspace, design weight matrix is to strengthen sparse solution；

(4) design weighting l₁The framework of sparse representation that norm constraint minimizes, utilizes programming software bag SOC computational methods, Obtain and recover matrix, find the non-zero row recovered in matrix, it is achieved to MIMO radar system under the conditions of white Gaussian noise and coloured noise The accurate estimation of target DOA in system.

Step (1) utilizes single base MIMO radar mutual coupling matrix banding symmetry Toeplitz construction features, by linearly Conversion eliminates the impact of unknown mutual coupling as follows:

(1.1) taking soon, receiving data is

X (t)=CAs (t)+n (t)

WhereinC_tAnd C_rIt is to launch and the mutual coupling matrix of receiving array respectively,It is the long-pending behaviour of Kronecker Make.A=[a (θ₁),...,a(θ_P)] andP=1,2 ... P, P are far field objects sums.WithRespectively be launch and reception lead To vector, θ_pBe pth target DOA, M and N respectively be launch and reception antenna number；S (t)=[s₁(t),...,s_P(t)]^T Being non-Gaussian signal, n (t) is zero mean Gaussian white noise or coloured noise；

(1.2) eliminating unknown mutual coupling, construct selection matrix J, new data vector is

WhereinI_QRepresent that Q × Q ties up unit Matrix,K+1 is non-zero the mutual coupling coefficient number.It is new noise, D is Comprise the diagonal matrix of unknown mutual coupling information.

Data after eliminating mutual coupling are carried out dimension-reduction treatment, Jin Erji by structure dimensionality reduction transition matrix in described step (2) Construct in new data matrix and there is the fourth order cumulant matrix of specific form as follows:

(2.1) data matrix after eliminating mutual coupling is carried out dimension-reduction treatment, construct dimensionality reduction transition matrix J₃As follows

WhereinAnd

(2.2) by dimensionality reduction transition matrix J₃Being multiplied with data vector y (t), new data vector is

WhereinB=[b (θ₁),...,b (θ_P)] and It it is the new noise vector after removing mutual coupling and lengthy and jumbled row；

(2.3) construct there is the fourth order cumulant matrix of specific form:

WhereinRepresentMiddle kth₂Row and kth₁The element of row, by collectingJ snap, it is thus achieved that Fourth order cumulant matrixSample estimateIn each item

WhereinIt isKth_iIndividual element,Under the conditions of collecting J snap,

Described step (3) carries out dimension-reduction treatment to fourth order cumulant observing matrix, it is thus achieved that under framework of sparse representation Corresponding model, and utilize steering vector and the orthogonality of corresponding noise subspace, design weight matrix is to strengthen sparse solution by such as Lower step:

(3.1) fourth order cumulant observing matrix is carried out dimension-reduction treatment:

Wherein V_sBeing made up of the right singular vector of corresponding P big singular value, U and V is respectively by the left and right of singular value decomposition Singular vector forms,D_mcAnd C_sRespectively comprise singular value, unknown mutual coupling With the diagonal matrix of signal fourth order cumulant,S_FBIn the i-th row jth column element be S_FB(i, j)=| [Fb(θ_j)]_i|²[Fb(θ_j)]_i, [Fb (θ_j)]_iIt is Fb (θ_j) i-th value.

(3.2) orthogonality of steering vector and noise subspace is utilized, design weight matrix reinforcement sparse solution:

WhereinIt is the complete dictionary under framework of sparse representation, For the discrete sample grid of all DOA interested, L ＞ ＞ P,Comprise complete dictionaryThe middle possible corresponding true DOAs of target P steering vector. ByIn remaining steering vector composition；U_nFor noise subspace, by U (P+1) arriveRow composition；Comprise potentiallyWith I-th value be in W I-th row 2-norm,

Design weighting l in described step (4)₁The framework of sparse representation that norm constraint minimizes, utilizes programming software bag SOC Computational methods, it is thus achieved that recover matrix, find the non-zero row recovered in matrix, it is achieved under the conditions of white Gaussian noise and coloured noise In MIMO radar system, the accurately estimation of target DOA is as follows:

Design weighting l₁The framework of sparse representation that norm constraint minimizes,

WhereinIt isThe sample of actual value is estimated；It is L × 1 dimensional vector,Sparse matrixMeetη regularization parameter is chosen as, on the basis of fourth order cumulant variance of estimaion error uniforms, to be hadThe higher limit of card side's distribution of degree of freedom high probability 1-ε confidence interval, ε=0.001 is enough；Utilize formulaCalculate η；Finally, calculated by SOC program, mappingFind P peak value and obtain target DOAs.

The beneficial effects of the present invention is:

1, due to the fact that fourth-order cumulant approach and the application of weighting framework of sparse representation, successfully inhibit color to make an uproar Sound, in the case of gauss heat source model, the present invention provides accurate Mutual coupling, the similar ESPRIT of ratio, FOC-MUSIC and l₁-SVD method has higher angular resolution and more preferable angle estimation performance.

2, due to the fact that the application of weighting rarefaction representation technology, weight matrix enhances sparse solution, and no matter noise is high This white noise or coloured noise, the present invention estimates that performance is superior to similar ESPRIT, FOC-MUSIC and l₁-SVD method, has Lower SNR threshold value, it is provided that higher angular resolution；

3, due to the fact that application and the specific form of designed fourth order cumulant matrix of dimensionality reduction switch technology, calculate Amount is substantially reduced, and the computation complexity of the present invention is more reasonable than FOC-MUSIC method.

Accompanying drawing explanation

The general frame figure of Fig. 1 present invention；

During Fig. 2 mutual coupling of the present invention K=2 and K=3, lower three angle on targets of SNR=-10dB and SNR=0dB different situations The spatial spectrum estimated；

During Fig. 3 distinct methods mutual coupling white noise, three angle on targets estimate root-mean-square error and Between Signal To Noise Ratio；

During Fig. 4 distinct methods mutual coupling coloured noise, three angle on targets estimate root-mean-square error and Between Signal To Noise Ratio；

During Fig. 5 distinct methods mutual coupling white noise, two angle on targets estimate root-mean-square error and angle spaced relationship；

During Fig. 6 distinct methods mutual coupling coloured noise, two angle on targets estimate root-mean-square error and angle spaced relationship；

During Fig. 7 distinct methods mutual coupling white noise, three angle on targets estimate root-mean-square error and fast umber of beats relation；

During Fig. 8 distinct methods mutual coupling coloured noise, three angle on targets estimate root-mean-square error and fast umber of beats relation；

During Fig. 9 distinct methods mutual coupling white noise, three angle on targets estimate resoluting probability and Between Signal To Noise Ratio；

During Figure 10 distinct methods mutual coupling coloured noise, three angle on targets estimate resoluting probability and Between Signal To Noise Ratio.

Detailed description of the invention

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

The present invention provides single base based on fourth order cumulant rarefaction representation multi input under the conditions of a kind of unknown mutual coupling error Multi output (multiple-input multiple-output, MIMO) radar direction of arrival (direction of arrival, It is called for short DOA) method of estimation, primarily to solve at present under framework of sparse representation, the angle in the MIMO radar system of single base Degree method of estimation existence is not suitable for gauss heat source model and mutual coupling error situation, the shortcomings such as estimated accuracy is undesirable.First pass through Linear transformation eliminates mutual coupling, and structure dimensionality reduction transition matrix reduces computation complexity.Then fourth order cumulant matrix is utilized Favored form, design weighting l₁Norm constraint minimizes framework of sparse representation and obtains the DOA of target.Its process is: obtain single base The reception signal of ground MIMO radar system, utilizes the special construction launching the mutual coupling matrix (MCM) all having with receiving array, logical Cross linear transformation and eliminate the impact of unknown mutual coupling；Data after eliminating mutual coupling are carried out dimensionality reduction by then structure dimensionality reduction transition matrix, And then based on new data matrix structure fourth order cumulant observing matrix, and carry out dimension-reduction treatment further；Obtain rarefaction representation Model, utilizes the orthogonality of steering vector and noise subspace, designs weight matrix；Finally design weighting l₁Norm constraint is minimum Change framework of sparse representation, it is thus achieved that recover matrix, find the non-zero row recovered in matrix, it is achieved to white Gaussian noise or coloured noise feelings The accurate estimation of target DOA in MIMO radar system under condition.Estimate with the mutual coupling error condition base MIMO radar direction of arrival that places an order Count traditional similar ESPRIT, FOC-MUSIC and l₁-SVD method is compared, and the present invention is under the conditions of white Gaussian noise and coloured noise All provide accurate DOA to estimate, there is higher angular resolution and more preferable angle estimation performance simultaneously.

It is an object of the invention to overcome the defect of said method, it is provided that under the conditions of a kind of new unknown mutual coupling error based on Single base MIMO radar Wave arrival direction estimating method of fourth order cumulant rarefaction representation.The present invention utilizes transmitting and receiving array Mutual coupling matrix (MCM) the banding symmetry Toeplitz structure having, eliminates the impact of unknown mutual coupling；It is then based on new reception number According to, construct dimensionality reduction transition matrix, and then construct the fourth order cumulant matrix with specific form to reduce what sparse signal recovered Computation complexity；Finally design weighting l₁The framework of sparse representation that norm constraint minimizes, by finding recover in matrix non- Zero row obtains the DOA of target.Compared with the traditional method of base MIMO radar angle estimation that places an order with unknown mutual coupling error condition, by In make use of fourth-order cumulant approach and weighting framework of sparse representation, the present invention carries under the conditions of white Gaussian noise and coloured noise Estimate that there is higher angular resolution and more preferable angle estimation performance simultaneously for accurate DOA.Direction of arrival of the present invention is estimated Count and mainly include the following aspects:

1, the structure of data is received and that transmitting, receiving array all have is mutual according to place an order base MIMO radar of array mutual-coupling condition Coupling matrix banding symmetry Toeplitz construction features, eliminates the impact of unknown mutual coupling by linear transformation.

Considering list base, arrowband MIMO radar system, emission array and receiving array are respectively by M and N number of antenna sets Become, array be half-wavelength space away from uniform linear array (ULAs).Single base MIMO radar is launched and receiving array Closely put altogether, therefore a far field objects is considered as them and there is identical angle (i.e. direction of arrival (DOA)).Launch Array utilizes M antenna to launch M the different arrowbands quadrature wave with same band and mid frequency, it is assumed that target numbers is P, θ_pRepresent the DOA of pth target.In the case of having K=k+1 non-zero the mutual coupling coefficient and min{M, N} ＞ 2k, emission array The impact of mutual coupling is all considered with receiving array.Then, it is expressed as in certain output taking receiving terminal matched filtering device soon

WhereinIt is non-Gaussian signal,It is There is white Gaussian noise or the coloured noise vector of zero-mean.WithIt is to launch and receive steering vector respectively, p=1,2 ... P.Wherein C_tAnd C_rIt is to launch and receiving array respectively Mutual coupling matrix, mutual coupling matrix is banding symmetric Toeptlitz matrix and is expressed as

Wherein c_ij(i=t, r；J=0,1 ..., k) it is the mutual coupling coefficient, and the distance dependent between two antennas, and meet 0<|c_ik|<...<|c_i1|<|c_i0|=1.Based onConstruction features and Kronecker amass the characteristic of operation, unknown array mutual-coupling condition The reception data of lower MIMO radar are

WhereinIt is to guide matrix, andAbout signal and noise introducing hypothesis below:

1) signal is the steady non-Gaussian signal of zero-mean, is mutually independent；

2) noise is zero-mean Gaussian noise, may be white noise or space correlation coloured noise；

3) noise and signal are independent mutually.

Sparse representation method lost efficacy due to the existence of unknown mutual coupling matrix.For constructing effective fourth order cumulant sparse table Show framework, it is necessary first to eliminate and launch and the mutual coupling of receiving array.Utilize mutual coupling matrix banding in formula (2) symmetrical Toeplitz construction features, for C_ta_t(θ_p), p=1,2 ..., P, definitionDimension selection matrixTo select C_tCentreOK.OrderHave

WhereinIt is scalar, andDefinition is another One selection matrixSimilar formula (4), then

WhereinDerive the most further

WhereinOrderJ is taken The left side of x (t), obtains

Wherein y (t) is newly received data vector, diagonal matrix D=diag [| c (θ₁)|,...,|c(θ_P) |] comprise the unknown The information of mutual coupling.WithIt is newly to guide matrix and noise respectively.Due to mutual coupling Matrix C is transformed into for diagonal matrix, and therefore mutual coupling is no longer on guiding matrix existence impact, i.e. mutual coupling error shadow in formula (7) Ring and be compensated.

Data after eliminating mutual coupling are carried out dimension-reduction treatment by 2, structure dimensionality reduction transition matrix, and then based on new data square Battle array constructs the fourth order cumulant matrix with specific form.

Data after eliminating mutual coupling are carried out dimension-reduction treatment by structure dimensionality reduction transition matrix, as follows

After eliminating unknown mutual coupling, in order to reduce the complexity that following fourth order cumulant calculates, based on single base The special construction structure dimensionality reduction conversion of MIMO radar.Pth rowFor

It appeared thatIn comprise a lot of duplicate keys, and meet

Wherein G and b (θ_p) expression as follows

WhereinAnd Duplicate keys will cause substantial amounts of lengthy and jumbled information in following fourth order cumulant calculates, and then at sparse letter Number recover in cause the biggest amount of calculation.In order to solve this problem, structure dimensionality reduction conversion is as follows

Therefore, by dimensionality reduction transition matrix J₃It is multiplied with data vector y (t), has

Wherein WithIt is to eliminate mutual coupling and removal respectively New data vector sum noise after lengthy and jumbled row.Meanwhile, F is calculated as

WhereinDimensionality reduction conversion not only by receive data matrix dimension fromSubstantially reduce ArriveAnd avoiding additional space coloured noise, J is fast umber of beats.

The fourth order cumulant matrix with specific form is constructed based on new data matrix, as follows

After compensate for unknown mutual coupling error impact and reducing reception data dimension, by collectingJ fast Clap, dataFourth order cumulantIt is defined as

WhereinIt isKth_iIndividual element,Under the conditions of J snap,Estimation in each item obtain as follows

According to formula (14), design fourth order cumulant matrixAs follows

WhereinRepresentMiddle kth₂Row and kth₁The element of row.Characteristic based on fourth order cumulant and signal s_i (i=1,2 ..., P) between separate grade it is assumed that derive

WhereinWithIt is b (θ respectively_pKth in)₂And kth₁Individual element,It it is letter Number s_pFourth order cumulant.Structure fourth order cumulant matrixAfter,Individual value is reasonably arranged inIn.Due toIn comprise k₁And k₂All possible permutation, deriveAs follows

WhereinD_mc=diag (| c (θ₁)|⁴,...,|c(θ_P)|⁴) and C_s=diag (β₁,..., β_P) it is diagonal matrix.Meanwhile, S_FBIn the i-th row and jth column element be

S_FB(i, j)=| [Fb (θ_j)]_i|²[Fb(θ_j)]_i(19)

Wherein1≤j≤P, [Fb (θ_j)]_iIt is Fb (θ_jI-th value in).

3, fourth order cumulant observing matrix is carried out dimension-reduction treatment, it is thus achieved that the corresponding model under framework of sparse representation, and profit By steering vector and the orthogonality of corresponding noise subspace, design weight matrix is to strengthen sparse solution.

Fourth order cumulant observing matrix is carried out dimension-reduction treatment, as follows

The computation complexity of sparse representation method, fourth order cumulant observing matrix is estimated in order to reduce DOA furtherEnergy Enough fromDimension is converted intoDimension, as follows

WhereinComprise the signal energy of the overwhelming majority, can be used to replaceDOAs is estimated accurately Meter.V_sIt is made up of the right singular vector that correspond to big singular value, additionally,U and V is respectively by left singular vector and the right side Singular vector forms,It is diagonal matrix, andIt it is singular value.OrderD_mcAnd C_sIt is diagonal matrix.Some row of recovering in matrix model original when not considering mutual coupling is During null vector, the corresponding row in T is the most necessarily zero, though D_mcWithComposition be totally unknown.This shows D_mc,And V_s Introducing will not change sparse solution, after therefore eliminating mutual coupling by the linear transformation of formula (7), existing at formula (20) In form, direction of arrival can successfully be estimated by sparse representation method.

Obtain the corresponding model under framework of sparse representation, as follows

In order to apply sparse representation method to estimate DOA, orderDiscrete sample grid for all DOAs interested.May DOA number be far longer than target numbers, i.e. L ＞ ＞ P, so be possible DOAs construct complete dictionaryOrderMeetWhen spatial domain point is with the presence of real goal Time,Middle corresponding behavior non-vanishing vector and other are null vector.Therefore,With original matrix T, there is identical row support, we DOA estimation problem can be converted into and find correspondence in complete dictionaryThe position of non-zero row.For estimating the minimum of non-zero row Number, direct sparse measurement is l₀-norm is punished, but l₀-norm minimum be non-convex optimization and NP difficulty and cannot solve. l₁The punishment of-norm can be used for reasonably solving this problem.For obtaining sparse matrixIt is constructed as follows l₁-norm constraint is minimum Change framework

WhereinAndη is regularization parameter, by collecting J snapIt is true ValueSample estimate, from formula (14) (15) (16) and (20) acquisition.

Utilize steering vector and the orthogonality of corresponding noise subspace, design weight matrix to strengthen sparse solution, following institute Show

In order to strengthen l₁-norm is punished to have better access to l₀-norm is punished, it is proposed that weight l as follows₁-Norm minimum Change sparse representation method.By complete dictionaryThe P being divided into two parts, a part to comprise the possible corresponding true DOAs of target by row is individual Steering vector, another part is recovered matrix by remaining correspond toThe steering vector composition of middle zero row.Therefore,Can represent ForThen select in U from (P+1) toBe total toArrange as noise empty Between U_n,It is expressed as

Wherein as J → ∞, it is proved to W₁In each item go to zero, therefore based on this characteristic, design one and add Weight matrix is as follows

WhereinLittle, and Simultaneously as J → ∞, WithIt is respectivelyWithI-th and J item.Comprise two parts potentially, thereforeComprise potentiallyWithW_rIt is capable of the punishment of big weights dilute Dredge vectorMiddle may be zero element, the item that little weights retention is bigger.In other words, the introducing of weighting matrix can be strengthened Sparse solution, improves the degree of accuracy that DOA estimates.

4, design weighting l₁The framework of sparse representation that norm constraint minimizes, utilizes programming software bag SOC (second order cone) to count Calculation method, it is thus achieved that recover matrix, finds the non-zero row recovered in matrix, it is achieved to MIMO under the conditions of white Gaussian noise and coloured noise The accurate estimation of target DOA in radar system.

Structure weighting framework of sparse representation is as follows

Wherein η is the regularization parameter of step-up error amount, and the selection of η numerical value is played critically important in last DOA estimates Effect.OrderFor fourth order cumulant estimation difference, in order to obtainDistribution character, first push away LeadDistribution, and It it is actual valueSample estimate, thereforeBased onElement structure, it is believed thatIt is a series ofFourth order cumulant estimated value.According to document (Signals, SystemsandComputers, 1993,2:1186-1190), for statistic processesFourth order cumulant is estimatedMiddle institute There is itemHaveMeet the characteristic of asymptotic normality distribution.Additionally,In the case of the most snaps,Be have zero-mean asymptotic just State is distributed, i.e.

Wherein AsN (μ, V) represents that average is μ, and covariance matrix isAsymptotic normality distribution. And it is as follows to understand fourth order cumulant estimation error covariance according to document

Wherein τ=k₁,k₂,k₃,k₄, ρ=k₁',k'₂,k₃',k'₄, abbreviation represents Q₄₄=Q₄₄(τ, ρ), m₂(λ)=m₂(k₁, λ), and m₂(λ ')=m₂(k₁',λ').According toCan obtain and estimate Evaluation, andk₁₁=k₂,k₁₂= k₃,k₁₃=k₄；k₂₁=k₃,k₂₂=k₂,k₂₃=k₄；k₃₁=k₄,k₃₂=k₃,k₃₃=k₂, in like manner k' has identical definition.Meanwhile, Formula (15) is utilized to obtain k rank sample moment

Work as k₂=k₃=k₄And k'₂=k₃'=k'₄, V byStructure Become,Come fromDerive i and k₁, k₂Relation as follows

Wherein []_remExpression valueRemainder, []_ceilRepresent not less than value? Little integer.J and k₁', k'₂Relation identical with formula (27).Due toUtilize the characteristic of vectoring operations, Can derive

WhereinBased on formula (28) andThe linear transformation of middle asymptotic normality distribution Invariant feature, derivesIn about averageAnd covariance matrixConclusion as follows

WhereinI.e.Meeting average is zero and covariance matrix is E_vV(E_v)^HGradually Nearly normal distribution.Therefore, according to document (IEEETransactionsonSignalProcessing, 2005,53 (8): 3010- 3022), in the present invention regularization parameter η be chosen as fourth order cumulant variance of estimaion error uniform on the basis of, haveThe higher limit of card side's distribution of degree of freedom high probability 1-ε confidence interval, ε=0.001 is enough.Utilize Matlab Software, formulaCan be used to calculate η.On the other hand, formula (24) can pass through SOC (two Rank are bored) programme software bundle calculating, such as SeDuMi and CVX.Finally, by solving formula (24), mappingObtain target DOAs。

Embodiment 1

Step one, set up unknown mutual coupling error condition and place an order the reception signal model of base MIMO radar, according to mutual coupling square Battle array construction features, utilizes linear transformation to eliminate unknown mutual coupling.

Considering list base, arrowband MIMO radar system, emission array and receiving array are respectively by M and N number of antenna sets Become, array be half-wavelength space away from uniform linear array.Single base MIMO radar is launched and receiving array is closely total to Put, therefore a far field objects is considered as them and there is identical angle (i.e. direction of arrival (DOA)).Emission array profit M the different arrowbands quadrature wave with same band and mid frequency is launched, it is assumed that target numbers is P, θ with M antenna_pRepresent The DOA of pth target.In the case of having K=k+1 non-zero the mutual coupling coefficient and min{M, N} ＞ 2k, emission array and reception Array all considers the impact of mutual coupling.Then, it is expressed as in certain output taking receiving terminal matched filtering device soon

WhereinIt is non-Gaussian signal,It is There is white Gaussian noise or the coloured noise vector of zero-mean.WithIt is to launch and receive steering vector respectively, p=1,2 ... P.Wherein C_tAnd C_rIt is to launch and receiving array respectively Mutual coupling matrix, mutual coupling matrix is banding symmetric Toeptlitz matrix and is expressed as

Wherein c_ij(i=t, r；J=0,1 ..., k) it is the mutual coupling coefficient, 0 < | c_ik|<...<|c_i1|<|c_i0|=1, with antenna Spacing is relevant.Based onStructure and Kronecker amass characteristic, and under array mutual-coupling condition, MIMO radar reception data are

WhereinIt is to guide matrix, andMake the following assumptions about signal and noise: signal is the steady non-gaussian of zero-mean Signal, is mutually independent；Noise is zero-mean Gaussian noise, may be white noise or space correlation coloured noise；Noise and letter Number mutual independence.

Sparse representation method lost efficacy due to the existence of unknown mutual coupling matrix.For constructing effective fourth order cumulant sparse table Show framework, it is necessary first to eliminate and launch and the mutual coupling of receiving array.Utilize mutual coupling matrix banding in formula (31) symmetrical Toeplitz construction features, for C_ta_t(θ_p), p=1,2 ..., P, definitionDimension selection matrixTo select C_tCentreOK.OrderHave

WhereinIt is scalar, andDefine another choosing Select matrixSimilar formula (33), can obtain

WhereinDerive the most further

Wherein c (θ_p)=c_t(θ_p)c_r(θ_p),OrderJ is taken advantage of To the left side of x (t), obtain

Wherein y (t) is newly received data vector, diagonal matrix D=diag [| c (θ₁)|,...,|c(θ_P) |] comprise the unknown Mutual coupling information.WithIt is newly to guide matrix and noise.Mutual coupling matrix is transformed to diagonal angle Matrix, therefore mutual coupling is no longer on guiding matrix existence impact, and in formula (36), mutual coupling error is compensated.

Data after eliminating mutual coupling are carried out dimension-reduction treatment by step 2, structure dimensionality reduction transition matrix.

After eliminating unknown mutual coupling, in order to reduce the complexity that following fourth order cumulant calculates, based on single base The special construction structure dimensionality reduction conversion of MIMO radar.Pth rowFor

It appeared thatIn comprise a lot of duplicate keys, and meet

Wherein G and b (θ_p) expression as follows

WhereinAnd Duplicate keys will cause substantial amounts of lengthy and jumbled information in following fourth order cumulant calculates, and then at sparse letter Number recover in cause the biggest amount of calculation.In order to solve this problem, structure dimensionality reduction conversion is as follows

Therefore, by dimensionality reduction transition matrix J₃It is multiplied with data vector y (t), obtains new dataAs follows

Wherein It it is new noise vector.Meanwhile, F is calculated as

WhereinDimensionality reduction conversion not only by receive data matrix dimension fromSubstantially reduce ArriveAnd avoiding additional space coloured noise, J is fast umber of beats.

Step 3, construct the fourth order cumulant matrix with specific form based on new data matrix.

After compensate for unknown mutual coupling error impact and reducing reception data dimension, by collectingJ fast Clap, dataFourth order cumulantIt is defined as

WhereinIt isKth_iIndividual element,Under the conditions of collecting J snap,Sample estimate in each item obtain as follows

According to formula (43), design fourth order cumulant matrixAs follows

WhereinRepresentMiddle kth₂Row and k₁The element of row.Based on signal and noise it is assumed that derive

WhereinWithIt is b (θ respectively_pKth in)₂And kth₁Individual element,It it is letter Number s_pFourth order cumulant.Structure fourth order cumulant matrixAfter,Individual value is reasonably arranged inIn.By InIn comprise k₁And k₂All possible permutation, deriveAs follows

WhereinD_mc=diag (| c (θ₁)|⁴,...,|c(θ_P)|⁴) and C_s=diag (β₁,..., β_P) it is diagonal matrix.Meanwhile, S_FBIn the i-th row and jth column element be

S_FB(i, j)=| [Fb (θ_j)]_i|²[Fb(θ_j)]_i(48)

Wherein1≤j≤P, [Fb (θ_j)]_iIt is Fb (θ_jI-th value in).

Step 4, fourth order cumulant observing matrix is carried out dimension-reduction treatment.

The computation complexity of sparse representation method, fourth order cumulant observing matrix is estimated in order to reduce following DOA furtherCan be fromDimension is converted intoDimension, as follows

WhereinComprise most signal energy, can be used to replaceDOA is accurately estimated.V_sBy Correspond to the right singular vector composition of big singular value, additionally,U and V is made up of left and right singular vector respectively,It is diagonal matrix,It it is singular value.Order D_mcAnd C_sIt is diagonal matrix.When not considering mutual coupling, some row in original recovery matrix model is null vector, right in T Should go is the most necessarily zero, though D_mcWithComposition totally unknown.This shows D_mc,And V_sIntroducing will not change sparse Solve, therefore after eliminating mutual coupling by formula (36), in formula (49) existing form, sparse representation method can be successfully to DOA Estimate.

Step 5, the corresponding model obtained under framework of sparse representation, just utilizing steering vector and corresponding noise subspace The property handed over, design weight matrix is to strengthen sparse solution.

In order to apply sparse representation method to estimate DOA, orderDiscrete sample grid for all DOAs interested.May DOAs number be far longer than target numbers, i.e. L ＞ ＞ P, so be possible DOAs construct complete dictionaryOrderMeetWhen spatial domain point is with the presence of real goal Time,Middle corresponding behavior non-vanishing vector and other are null vector.Therefore,With original matrix T, there is support of going together mutually, Wo Menneng Enough being converted into by DOA estimation problem finds correspondence in complete dictionaryThe position of non-zero row.For estimating the minimum number of non-zero row Mesh, direct sparse measurement is l₀-norm is punished, but l₀-norm minimum be non-convex optimization and NP difficulty and cannot solve.l₁- Norm punishment can be used for reasonably solving this problem.In order to strengthen l₁-norm is punished to have better access to l₀-norm is punished, It is proposed that weight l as follows₁-norm minimum sparse representation method.By complete dictionaryIt is divided into two parts, part bag by row Containing P the steering vector of the possible corresponding true DOAs of target, another part is recovered matrix by remaining correspond toMiddle zero row Steering vector forms.Therefore,Can be expressed asThen select in U from (P+1) to's AltogetherRow are as noise subspace U_n,It is expressed as

Wherein as J → ∞, it is proved to W₁In each item go to zero, therefore based on this characteristic, design a power Value matrix is as follows

WhereinLittle, and Simultaneously as J → ∞, WithIt is respectivelyWithI-th and J item.Comprise two parts potentially, thereforeComprise potentiallyWithW_rAchieve big weights and punish those Sparse vectorMiddle may be zero element, the item that little weights retention is bigger.In other words, the introducing of weighting matrix can add Strong sparse solution, improves DOA and estimates degree of accuracy.

Step 6, design weighting l₁Norm constraint minimizes framework of sparse representation, it is thus achieved that recover matrix, finds and recovers matrix In non-zero row, it is thus achieved that the accurate DOA of target.

Structure weighting framework of sparse representation is as follows

WhereinAndη is regularization parameter, by collecting J snapIt is true ValueSample estimate, from formula (43) (44) (45) and (49) acquisition.η is the regularization parameter of step-up error amount, η numerical value Select to play critically important effect in last DOA estimates.

OrderFor fourth order cumulant estimation difference, in order to obtainDistribution character, we are first First deriveDistribution, and It it is actual valueSample estimate, therefore Based onElement structure, it is believed thatIt is a series ofFourth order cumulant estimated value.According to document (Signals, SystemsandComputers, 1993,2:1186-1190), for statistic processes And fourth order cumulant estimationIn all itemsTool HaveMeet the characteristic of asymptotic normality distribution, in the case of the most snaps,It is the asymptotic normality distribution with zero-mean, i.e.

Wherein AsN (μ, V) represents that average is μ, and covariance matrix isAsymptotic normality distribution. And it is as follows to understand fourth order cumulant estimation error covariance according to document

Wherein τ=k₁,k₂,k₃,k₄, ρ=k₁',k'₂,k₃',k'₄, abbreviation represents Q₄₄=Q₄₄(τ, ρ), m₂(λ)=m₂(k₁, λ), and m₂(λ ')=m₂(k₁',λ').According toCan obtain and estimate Evaluation, andk₁₁=k₂,k₁₂= k₃,k₁₃=k₄；k₂₁=k₃,k₂₂=k₂,k₂₃=k₄；k₃₁=k₄,k₃₂=k₃,k₃₃=k₂, in like manner k' has identical definition.Meanwhile, Formula (44) is utilized to obtain k rank sample moment

Work as k₂=k₃=k₄And k'₂=k₃'=k'₄, V byStructure Become,Come fromDerive i and k₁, k₂Relation as follows

Wherein []_remExpression valueRemainder, []_ceilRepresent not less than value? Little integer.J and k₁', k'₂Relation identical with formula (55).Due toUtilize the characteristic of vectoring operations, Can derive

WhereinBased on formula (56) andThe linear transformation of middle asymptotic normality distribution Invariant feature, derivesIn about averageAnd covariance matrixConclusion as follows

WhereinI.e.Meeting average is zero and covariance matrix is E_vV(E_v)^HGradually Nearly normal distribution.Therefore, according to document (IEEETransactionsonSignalProcessing, 2005,53 (8): 3010- 3022), in the present invention regularization parameter η be chosen as fourth order cumulant variance of estimaion error uniform on the basis of, haveThe higher limit of card side's distribution of degree of freedom high probability 1-ε confidence interval, ε=0.001 is enough.Utilize Matlab Software, formulaCan be used to calculate η.On the other hand, formula (52) can pass through SOC (two Rank are bored) programme software bundle calculating, such as SeDuMi and CVX.Finally, by solving formula (52), mappingObtain target DOAs。

The effect of the present invention can be illustrated by herein below:

(1) computational complexity analysis:

The main amount of calculation of the present invention is the calculating of fourth order cumulant and obtains extensive by SOC program bag solution formula (52) The sparse solution of complex matrix.The former has only to due to the specific form of designed fourth order cumulant matrix The latter needsWherein L₁Being discrete sample grid sum, therefore the main computational complexity of the present invention isBut, in FOC-MUSIC method, due to all of (k₁,k₂,k₃,k₄) in Matrix C_xIn all Need by random permutation so that amount of calculation beDimensionality reduction is changed Application and advantageous fourth order cumulant matrix form make computation amount, therefore the computation complexity of present invention ratio FOC-MUSIC method is more reasonable.

(2) simulated conditions and content:

We present some simulation results to confirm effectiveness of the invention and advantage.Use similar ESPRIT, FOC-MUSIC And l₁-SVD method contrasts with the present invention.The root-mean-square error (RMSE) of angle estimation is used for evaluation perspective and estimates performance, It is defined as

WhereinIt is that i & lt MonteCarlo tests DOA actual value θ_pEstimation, Q be MonteCarlo test total degree, Simulations below selects Q=500.Consider list base, the arrowband MIMO radar system with M transmitting antenna and N number of reception antenna System, launch and receiving array be half-wavelength space away from uniform linear array.Additionally, the reception data in Fang Zhen are to be mixed with height This noise and binary phase shift keying (BPSK) signal of mutual coupling existing between elements impact, in most cases, it is assumed that uncorrelated target Number P be known, SNR is defined asS' and N' is that J takes signal soon and makes an uproar Sound matrix.Additionally, for the present invention and FOC-MUSIC method and l₁-SVD method, discrete sample grid is all from-90 ° to 90 ° Conversion, uniform intervals is 0.05 °.

(3) simulation result:

1, during mutual coupling K=2 and K=3 of the present invention, lower three angle on targets of SNR=-10dB and SNR=0dB different situations are estimated The spatial spectrum of meter

When figure two illustrates the present invention for non-zero the mutual coupling coefficient K=2 and K=3, SNR=-10dB with SNR=0dB is different In the case of spatial spectrum, wherein the DOAs of M=N=7, J=2000, three uncorrelated target is respectively θ₁=-20 °, θ₂=0 ° and θ₃=20 °.As K=3, the non-zero the mutual coupling coefficient of emission array is [c_t0,c_t1,c_t2]=[1,0.6+j0.2,0.02+j0.1], Receiving array is [c_r0,c_r1,c_r2]=[1,0.5+j0.3,0.01+j0.2].As K=2, they are [c_t0,c_t1]=[1, 0.0174+j0.0377] and [c_r0,c_r1]=[1,0.0521-j0.1029], and apply this case to simulations below In.As shown in Figure 2, the spatial peaks of the present invention is very sharp-pointed, and sidelobe is the lowest, even if increasing at k, and the situation that SNR reduces Under.This shows that the present invention can provide fabulous estimation performance.

2, during distinct methods mutual coupling white noise, three angle on targets estimate root-mean-square error and Between Signal To Noise Ratio

Under the conditions of figure three illustrates white Gaussian noise, the relation of RMSE and SNR that distinct methods DOA estimates, wherein three Uncorrelated target is θ₁=-20 °, θ₂=0 ° and θ₃=20 °, M=N=7 simultaneously, J=4000.From figure three it can be seen that low SNR region l₁-SVD method has better performance than additive method.But, due to the fact that and make use of designed quadravalence to tire out The specific form of accumulated amount matrix and weighting framework of sparse representation, when SNR exceedes about 0dB, the present invention and similar ESPRIT, FOC-MUSIC and l₁-SVD method is compared, it is provided that best angle estimation performance.

3, during distinct methods mutual coupling coloured noise, three angle on targets estimate root-mean-square error and Between Signal To Noise Ratio

In the case of figure four illustrates gauss heat source model, the relation of RMSE and SNR that DOA estimates, other conditions are protected with figure three Hold consistent.With l₁-SVD, similar ESPRIT with FOC-MUSIC method compares, and figure four clearly illustrates the estimation of the present invention Can be all best in all SNR regions.Because make use of the special nature of fourth order cumulant, the present invention successfully inhibits color Noise, but at l₁-SVD is with in similar ESPRIT method, and coloured noise result in the decline of performance, especially in low SNR region.

4, distinct methods mutual coupling is white or two angle on targets estimate root-mean-square errors and angle spaced relationship during coloured noise

Figure five and figure six illustrate RMSE and the relation at angle interval, wherein M=N=7, J=4000, consider height in figure five This white noise and SNR=5dB, and figure six is it is considered that gauss heat source model and SNR=0dB.Every kind of method has two not phases Closing target, DOA is respectively θ₁=0 ° and θ₂=0 °+Δ θ, and Δ θ is from 2 ° to 14 ° of changes.From figure five, when Δ θ is at certain Time within value, the estimation performance of the present invention is close with similar ESPRIT method, and all than FOC-MUSIC and l₁-SVD is deemed-to-satisfy4 Can be good.When angle interval exceedes about 6 °, present invention performance in all methods is best.As shown in figure 6, lower of coloured noise Invent the angle estimation performance best for the offer of space phase close-target, i.e. the present invention has the highest space angle resolution.

5, distinct methods mutual coupling is white or three angle on targets estimate root-mean-square errors and fast umber of beats relation during coloured noise

Figure seven and figure eight illustrate RMSE and the relation of fast umber of beats, the M=N=7 that distinct methods DOA estimates.Figure seven is made an uproar Sound is white Gaussian noise and SNR=5dB, and figure eight is gauss heat source model and SNR=0dB.Consider three and there is different DOA not Related objective is θ₁=-11.5 °, θ₂=0 ° and θ₃=11.5 °.From figure seven and figure eight it can be seen that for white Gaussian noise and color Noise, the present invention provides in all snap regions and well estimates performance, additionally, the present invention is compared to similar ESPRIT, FOC-MUSIC and l₁The estimation advantage of-SVD method is more and more prominent along with the increase of J.

6, distinct methods mutual coupling is white or three angle on targets estimate resoluting probabilities and Between Signal To Noise Ratio during coloured noise

Under the conditions of figure nine and figure ten respectively show white Gaussian noise and gauss heat source model, the target of distinct methods is differentiated general Rate and the relation of SNR, wherein M=N=7, J=2000.The DOAs of three uncorrelated targets is respectively θ₁=-11 °, θ₂=0 ° and θ₃=11 °.Additionally, when the absolute value of three target all DOA estimation differences is all within 0.1 °, target is considered successfully to divide Distinguish.As it can be seen, when the snr is high enough, all methods provide the target resoluting probability of 100%, but, the resolution of every kind of method Probability begins to decline in certain point, is defined as SNR marginal value.From figure nine with figure ten it is clear that the present invention is to similar ESPRIT, FOC-MUSIC and l₁-SVD method is compared has minimum SNR marginal value, and they show no matter noise is high simultaneously This white noise or coloured noise, the present invention provides higher target resoluting probability.

Claims (5)

1. MIMO radar Wave arrival direction estimating method based on fourth order cumulant rarefaction representation under array mutual-coupling condition, it is characterised in that: Comprise the steps:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving terminal obtains after carrying out matched filtering process and receives number According to, and utilize and launch and the mutual coupling matrix banding symmetry Toeplitz construction features that all has of receiving array, pass through linear transformation Eliminate the impact of unknown mutual coupling；

(2) data after eliminating mutual coupling are carried out dimension-reduction treatment by structure dimensionality reduction transition matrix, and then based on new data matrix structure Make the fourth order cumulant matrix with specific form；

(3) fourth order cumulant observing matrix is carried out dimension-reduction treatment, it is thus achieved that the corresponding model under framework of sparse representation, and utilization is led To the orthogonality of the corresponding noise subspace of vector, design weight matrix is to strengthen sparse solution；

(4) design weighting l₁The framework of sparse representation that norm constraint minimizes, utilizes programming software bag SOC computational methods, it is thus achieved that extensive Complex matrix, finds the non-zero row recovered in matrix, it is achieved to mesh in MIMO radar system under the conditions of white Gaussian noise and coloured noise The accurate estimation of mark DOA.

Under array mutual-coupling condition the most according to claim 1, MIMO radar direction of arrival based on fourth order cumulant rarefaction representation is estimated Meter method, it is characterised in that: step (1) utilize single base MIMO radar mutual coupling matrix banding symmetry Toeplitz structure special Point, eliminates the impact of unknown mutual coupling as follows by linear transformation:

(1.1) taking soon, receiving data is

X (t)=CAs (t)+n (t)

WhereinC_tAnd C_rIt is to launch and the mutual coupling matrix of receiving array respectively,It it is the long-pending operation of Kronecker.A= [a(θ₁),...,a(θ_P)] andP is far field objects sum.WithRespectively be launch and reception lead To vector, θ_pBe pth target DOA, M and N respectively be launch and reception antenna number；S (t)=[s₁(t),...,s_P(t)]^T Being non-Gaussian signal, n (t) is zero mean Gaussian white noise or coloured noise；

(1.2) eliminating unknown mutual coupling, construct selection matrix J, new data vector is

$y\left(t\right)=Jx\left(t\right)=\tilde{A}Ds\left(t\right)+\tilde{n}\left(t\right)$

WhereinI_QRepresent that Q × Q ties up unit square Battle array,K+1 is non-zero the mutual coupling coefficient number.It is new noise, D is Comprise the diagonal matrix of unknown mutual coupling information.

Under array mutual-coupling condition the most according to claim 1, MIMO radar direction of arrival based on fourth order cumulant rarefaction representation is estimated Meter method, it is characterised in that: structure dimensionality reduction transition matrix in described step (2), the data after eliminating mutual coupling are carried out dimensionality reduction Process, and then construct based on new data matrix and there is the fourth order cumulant matrix of specific form as follows:

(2.1) data matrix after eliminating mutual coupling is carried out dimension-reduction treatment, construct dimensionality reduction transition matrix J₃As follows

$J_3={\left(G^{H}G\right)}^{(-\frac{1}{2})}{G}^H$

WhereinAnd

(2.2) by dimensionality reduction transition matrix J₃Being multiplied with data vector y (t), new data vector is

$\tilde{y}\left(t\right)=J_{3}y\left(t\right)=FBDs\left(t\right)+\stackrel{\&OverBar;}{\tilde{n}}\left(t\right)$

WhereinB=[b (θ₁),...,b(θ_P)] And It it is the new noise vector after removing mutual coupling and lengthy and jumbled row；

(2.3) construct there is the fourth order cumulant matrix of specific form:

$C_{\tilde{y}}(k_2,k_1)=cum\{{\tilde{y}}_{k_2},{\tilde{y}}_{k_1}^{*},{\tilde{y}}_{k_1},{\tilde{y}}_{k_1}^{*}\}$

WhereinRepresentMiddle kth₂Row and kth₁The element of row, by collectingJ snap, it is thus achieved thatQuadravalence Cumulant matricesSample estimateIn each item

$\begin{array}{c}cum({\tilde{y}}_{k_1},{\tilde{y}}_{k_2}^{*},{\tilde{y}}_{k_3},{\tilde{y}}_{k_4}^{*})=E\left({\tilde{y}}_{k_1}{\tilde{y}}_{k_2}^{*}{\tilde{y}}_{k_3}{\tilde{y}}_{k_4}^{*}\right)-E\left({\tilde{y}}_{k_1}{\tilde{y}}_{k_2}^{*}\right)E\left({\tilde{y}}_{k_3}{\tilde{y}}_{k_4}^{*}\right)\\ -E\left({\tilde{y}}_{k_1}{\tilde{y}}_{k_3}\right)E\left({\tilde{y}}_{k_2}^{*}{\tilde{y}}_{k_4}^{*}\right)-E\left({\tilde{y}}_{k_1}{\tilde{y}}_{k_4}^{*}\right)E\left({\tilde{y}}_{k_2}^{*}{\tilde{y}}_{k_3}\right)\end{array}$

WhereinIt isKth_iIndividual element,Under the conditions of collecting J snap,

Under array mutual-coupling condition the most according to claim 1, MIMO radar direction of arrival based on fourth order cumulant rarefaction representation is estimated Meter method, it is characterised in that: described step (3) carries out dimension-reduction treatment to fourth order cumulant observing matrix, it is thus achieved that sparse table Showing the corresponding model under framework, and utilize steering vector and the orthogonality of corresponding noise subspace, design weight matrix is to strengthen Sparse solution is as follows:

(3.1) fourth order cumulant observing matrix is carried out dimension-reduction treatment:

${\tilde{C}}_{\tilde{y}}=C_{\tilde{y}}{V}_s={\mathrm{FBD}}_{mc}{C}_s{S}_{FB}^H{V}_s$

Wherein V_sBe made up of the right singular vector of corresponding P big singular value, U and V respectively by the left and right of singular value decomposition unusual to Amount composition,D_mcAnd C_sRespectively comprise singular value, unknown mutual coupling and signal The diagonal matrix of fourth order cumulant,S_FBIn the i-th row jth column element be S_FB(i, j)=| [Fb (θ_j)]_i|²[Fb(θ_j)]_i, [Fb (θ_j)]_iIt is Fb (θ_j) i-th value.

(3.2) orthogonality of steering vector and noise subspace is utilized, design weight matrix reinforcement sparse solution:

$W_r=diag\&lsqb;{\left(W_1^{\left(l_2\right)}\right)}^T,{\left(W_2^{\left(l_2\right)}\right)}^T\&rsqb;/max\left(W_2^{\left(l_2\right)}\right)$

WhereinIt is the complete dictionary under framework of sparse representation,For institute There is a discrete sample grid of DOA interested, L ＞ ＞ P,Comprise complete dictionaryP of the middle possible corresponding true DOAs of target Steering vector. ByIn remaining steering vector composition；U_nFor noise subspace, by U (P+1) ArriveRow composition；Comprise potentiallyWith I-th value be the i-th row in W 2-norm,

Under array mutual-coupling condition the most according to claim 1, MIMO radar direction of arrival based on fourth order cumulant rarefaction representation is estimated Meter method, it is characterised in that: design weighting l in described step (4)₁The framework of sparse representation that norm constraint minimizes, utilizes and compiles Journey software kit SOC computational methods, it is thus achieved that recover matrix, find the non-zero row recovered in matrix, it is achieved to white Gaussian noise and color Under noise conditions, in MIMO radar system, the accurate of target DOA is estimated as follows:

Design weighting l₁The framework of sparse representation that norm constraint minimizes,

$\begin{array}{cc}\underset{T_{\widehat{\mathrm{\&theta;}}}}{\min }\left|\right|W_r{T}_{\widehat{\mathrm{\&theta;}}}^{\left(l_2\right)}|{|}_1,& s.t.\left|\right|{\hat{\tilde{C}}}_{\tilde{y}}-{\tilde{B}}_{\widehat{\mathrm{\&theta;}}}{T}_{\widehat{\mathrm{\&theta;}}}|{|}_F\&le;\sqrt{\mathrm{\&eta;}}\end{array}$

WhereinIt isThe sample of actual value is estimated；It is L × 1 dimensional vector,Sparse matrixMeetη regularization parameter is chosen as, on the basis of fourth order cumulant variance of estimaion error uniforms, to be hadThe higher limit of card side's distribution of degree of freedom high probability 1-ε confidence interval, ε=0.001 is enough；Utilize formulaCalculate η；Finally, calculated by SOC program, mappingFind P peak value and obtain target DOAs.