CN104865556A - MIMO radar system DOA estimation method based on real domain weighting minimization l1-norm method - Google Patents

Abstract

The invention relates to the technical field of an MIMO radar system, especially relates to applications of MIMO radar system DOA estimations, and specifically relates to an MIMO radar system DOA estimation method based on a real domain weighting minimization l1-norm method. The method comprises following steps: reducing dimensions of received data by the use of dimension reduction matrixes; decomposing singular values and acquiring a model corresponding to a sparse representation framework; designing a weight matrix in which diagonal elements correspond to real domain MUSIC spectrums according to the orthogonality of real domain steering vectors and corresponding noise subspaces so as to solve an MMV problem; and estimating object DOA in the MIMO radar system. According to the invention, SNR gains are enhanced through dimension reduction transition, the designed weighting l1-norm better approaches the l0-norm, and sparse solutions are enhanced. Compared with an l1-SVD algorithm and an RV l1-SVD algorithm, the method can achieve a higher resolution.

Description

Based on real domain weight minimization l 1the MIMO radar system DOA estimation method of Norm Method

Technical field

The present invention relates to MIMO radar system technical field, the application of particularly MIMO radar system DOA estimation, is a kind of based on real domain weight minimization l specifically ₁the MIMO radar system DOA estimation method of Norm Method.

Background technology

In recent years, due to multiple-input and multiple-output (multiple-input multiple-output, MIMO) array radar system (IEEE Signal Processing Magazine, 2007,24 (5): 106-114) compared to traditional phased array radar system potential advantage and paid close attention to greatly.In MIMO radar system, angle estimation is a critical problem.For this problem, some are based on the method for subspace, such as MUSIC algorithm (IEEE Trans.Antennas and propagation, 1986,34 (3): 276-280) and ESPRIT algorithm (IEEE Trans.Signal Process., 1989,37 (7): 984-995), in the angle estimation of MIMO radar system, application is obtained.On the other hand, utilize the special construction of MIMO radar system, RD-ESPRIT (Electronics Letters:2011 has been proposed in DOA estimates, 47 (4): 283-284) and conjugation ESPRIT (C-ESPRIT) (Signal Process., 2013,93:2070-2075) algorithm.Angle estimation performance is improve based on the proposition transmitting array beam dimensional energy concentration techniques (IEEE Transations on Signal Processing, 2011,59 (6): 2669-2682) algorithm.But the performance of these methods, at low SNR, normally can not meet the requirements of when limited fast umber of beats or object space tight distribution.

In recent years, the appearance in rarefaction representation field was estimated to provide new viewpoint to the DOA in Array Signal Processing, had proposed some sparse representation method in the related art.A kind of l proposed is estimated for DOA ₁-svd algorithm (IEEE Trans.Signal Process., 2005,53 (8): 3010-3022), utilizes l ₁norm punishment is close to l ₀norm is punished, and pays close attention to immediate data.L ₁-SRACV algorithm (IEEE Trans.Signal Process., 2011,59 (2): 629-638) and CMSR algorithm (IEEE Trans.Aerosp.Electron.Syst., 2013,49 (3)) not utilize immediate data but openness based on array covariance vector.On the other hand, real domain l ₁-SVD (RV l ₁-SVD) algorithm (IEEE Antennas Wireless Propag.Lett., 2013,12:376-379), compared to l ₁-svd algorithm has lower computation complexity, better angle estimation performance.Method mentioned above is all based on l ₁norm is punished, l ₁norm punishment can not better close to l ₀norm is punished.A kind of iterative algorithm proposed in (Journal of fourier analysis and applications, 2008,14 (5): 887-905), weight minimization l ₁norm Method, better close to l ₀norm is punished.But this faces two large problems: 1) be only applicable to single vector of measuring and recover problem.But, in MIMO array system DOA estimates, relate to vector of measuring more recover problem; 2) in MIMO radar system, need two-dimentional complete dictionary to recover thinned array, perhaps this lost efficacy when recovering sparse matrix.

Summary of the invention

The object of the invention is to the defect overcoming said method, propose a kind of newly based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of norm.

The object of the present invention is achieved like this:

Comprise the steps:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving end obtains after carrying out matched filtering process and receives data, and utilizes dimensionality reduction matrix to carry out dimension-reduction treatment to reception data;

(2) utilize unitary transformation matrix, the augmented sample matrix receiving data after dimensionality reduction is become real domain, carry out svd and corresponding model under obtaining framework of sparse representation;

(3) utilize the orthogonality of real domain steering vector noise subspace corresponding to it, design a diagonal entry and compose corresponding weight matrix to solve MMV problem with real domain MUSIC;

(4) real domain weight minimization l is designed ₁norm framework, utilizes programming software bag SOC second order cone computing method, obtains and recovers matrix, finds the non-zero row recovered in matrix, realizes the estimation to target DOA in MIMO radar system.

As follows dimension-reduction treatment is carried out to reception data in described step (1):

(1.1) receive-launch the structure of steering vector according to single base MIMO radar system, the transmitting-reception steering vector of MIMO radar system meets:

$a_t\left(\mathrm{\θ}\right)\⊗a_r\left(\mathrm{\θ}\right)=\mathrm{Gb}\left(\mathrm{\θ}\right),$

A in formula _t(θ) and a _r(θ) be respectively transmitting steering vector and receive steering vector, q=M+N – 1 is transition matrix and one dimension steering vector respectively,

$G=\left[\begin{array}{c}{J}_0\\ J_1\\ \·\\ \·\\ \·\\ J_{M-1}\end{array}\right],$

$b\left(\mathrm{\θ}\right)={[1,e^{\mathrm{j\π}\sin {\mathrm{\θ}}_p},\·\·\·,e^{\mathrm{j\π}(Q-1)\sin {\mathrm{\θ}}_p}]}^T,$

Wherein,

J _m＝[0 _N×m,I _N,0 _N×(M-m-1)]，m＝0,1,…,M-1，

By utilizing matrix G ^hthe individual different element of corresponding Q, two-dimensional guide vector can be converted to one dimension steering vector and namely carry out dimension-reduction treatment;

(1.2) according to transition matrix, dimensionality reduction matrix is W=F ^-1/2g ^h, wherein

(1.3) utilize W to obtain dimensionality reduction and receive data then have

$\stackrel{\‾}{X}=\mathrm{WAS}+\mathrm{WN}=F^{1/2}\mathrm{BS}+\mathrm{WN}=\stackrel{\‾}{B}S+\stackrel{\‾}{N};$

In formula meet B=[b (θ ₁), b (θ ₂) ..., b (θ _p)],

Described step utilizes unitary transformation matrix in (2) as follows, and the augmented sample matrix receiving data after dimensionality reduction is become real domain, carries out svd and corresponding model under obtaining framework of sparse representation:

(2.1) augmented sample matrix is considered wherein Γ _qbe that to have anti-diagonal element be 1, other elements are Q × Q switching matrix of 0, () ^*represent conjugate operation, Y is center Hermite Matrix and can be converted to a real domain matrix,

$Y=U_Q^H\left[\begin{array}{cc}\stackrel{\‾}{X}& {\mathrm{\Γ}}_Q\stackrel{\‾}{X}\end{array}*{\mathrm{\Γ}}_J\right]U_{2J},$

Wherein, unitary transformation matrix is

$U_{2K+1}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc}{I}_K& 0& j{I}_K\\ 0^T& \sqrt{2}& 0^T\\ {\mathrm{\Π}}_K& 0& -j{\mathrm{\Π}}_K\end{array}\right],$

Guiding matrix meet ${\mathrm{\Γ}}_Q{\left(\stackrel{\‾}{B}\mathrm{\Φ}\right)}^{*}=\stackrel{\‾}{B}\mathrm{\Φ},\mathrm{\Φ}=\mathrm{diag}(e^{-\mathrm{j\π}(Q-1)/2\sin {\mathrm{\θ}}_1},\·\·\·,e^{-\mathrm{j\π}(Q-1)2\sin {\mathrm{\θ}}_P}),$ Diag [] represents diagonalization operation, and the linear array wherein about reception data is centrosymmetric after dimensionality reduction conversion, and after unitary transformation, real domain guiding matrix is

Wherein S _Υ=[Φ ^*s Φ S ^*Γ _j] U _2Jreal domain signal matrix, it is real domain noise matrix;

(2.2) to Y _Υapplication svd SVD technology, has

Wherein v _sby the Y that correspond to P maximum singular value _Υthe right singular vector of real domain form;

(2.3) apply framework of sparse representation, the complete dictionary of real domain one dimension can be expressed as:

In formula ${\stackrel{\‾}{B}}^{\widehat{\mathrm{\θ}}}=[b\left({\widehat{\mathrm{\θ}}}_1\right),b\left({\widehat{\mathrm{\θ}}}_2\right),\·\·\·,b\left({\widehat{\mathrm{\θ}}}_L\right)],{\mathrm{\Φ}}^{\widehat{\mathrm{\θ}}}=\mathrm{diag}(e^{-\mathrm{j\π}(Q-1)/2\sin \widehat{{\mathrm{\θ}}_1}},\·\·\·,e^{-\mathrm{j\π}(Q-1)/2\sin \widehat{{\mathrm{\θ}}_L}}),$ Under framework of sparse representation,

Utilize the orthogonality of real domain steering vector noise subspace corresponding to it in described step (3) as follows, design diagonal entry composes corresponding weight matrix with real domain MUSIC:

(3.1) complete for real domain dictionary is divided into two parts: then have

In formula by the real domain steering vector that correspond to possibility target p=1,2 ..., P forms, by dictionary remaining real domain steering vector composition, V _nreal domain noise subspace, by Y _Υcarry out svd can obtain;

(3.2) according to the orthogonality of real domain steering vector and corresponding noise subspace, as J → ∞, W _{1, i}→ 0, W _{2, i}> 0, definition weight matrix

$W=\mathrm{diag}\left(\right[W_1^T,W_2^T\left]\right)/\max \left(W_2\right)$

Due to W _{1, i}/ max (W ₂) < W _{2, i}/ max (W ₂).

Real domain weight minimization l is designed as follows in described step (4) ₁norm framework, utilizes programming software bag SOC second order cone computing method, obtains and recovers matrix, finds the non-zero row recovered in matrix, estimates the target DOA in MIMO radar system:

Real domain weight minimization l ₁norm is

In formula be regularization parameter, utilize programming software bag SOC second order cone to calculate, by mapping

Beneficial effect of the present invention is:

The present invention is strengthened by dimensionality reduction conversion SNR gain, the weighting l simultaneously ₁norm is better close to l ₀norm and enhance sparse solution, compares l ₁-SVD and RV l ₁-svd algorithm has higher resolution; The present invention, due to the application of real domain switch technology, is better than l at low SNR regional perspective estimated performance ₁-SVD and RV l ₁-SVD, and due to above-mentioned advantage of the present invention it close to CRB; The present invention is owing to comprising front and back Search Space Smoothing and weighting l ₁norm technology, the change for related coefficient has robustness, and compares l ₁-SVD and RV l ₁-SVD has better angle estimation performance.The invention solves based on weighting l ₁there is shortcomings such as not being suitable for many measurement vector recovery problems in the Wave arrival direction estimating method of norm, and can be applicable to lower fast umber of beats well.

Accompanying drawing explanation

Fig. 1 is general frame figure of the present invention;

Fig. 2 algorithms of different differentiates the relation of the probability of success and angle intervals for two uncorrelated targets;

Fig. 3 algorithms of different is for three uncorrelated root-mean-square errors of angle on targets estimation and the relation of signal to noise ratio (S/N ratio);

Fig. 4 algorithms of different estimates the relation of root-mean-square error and related coefficient for two angle on targets;

Fig. 5 algorithms of different is for three uncorrelated root-mean-square errors of angle on targets estimation and the relation of fast umber of beats.

Specific embodiments

Frame diagram below in conjunction with Mutual coupling is described in more detail the present invention

The invention provides a kind of based on real domain weight minimization l ₁multiple-input and multiple-output (the multiple-input multiple-output of norm, MIMO) radar system target direction of arrival (Direction of arrival, be called for short DOA) method of estimation, mainly there is the shortcomings such as the high and estimated accuracy of dictionary complexity is undesirable in order to solve in current MIMO radar system based on the Wave arrival direction estimating method of rarefaction representation.First according to the feature of MIMO radar system, utilize dimensionality reduction change and unitary transformation technology, make reception data be transformed into low-dimensional and be real domain.Then according to real domain MUSIC spectrum, for obtaining weight minimization l ₁norm Frame Design weight matrix, and then by finding the non-zero row recovered in matrix to estimate DOA.Its process is: the Received signal strength model setting up single base MIMO radar system, and structure dimensionality reduction matrix carries out dimension-reduction treatment; Then utilize unitary transformation matrix the augmented sample matrix receiving data after dimensionality reduction to be become real domain, carry out svd and corresponding model under obtaining framework of sparse representation; According to the orthogonality of real domain steering vector noise subspace corresponding to it, design a diagonal entry and compose corresponding weight matrix with real domain MUSIC, and then design weight minimization l ₁norm framework, makes its sparse solution closer to l ₀norm thus greatly strengthen openness; Finally be restored matrix, and find the non-zero row recovered in matrix, realizes the estimation to target DOA in MIMO radar system.With the existing DOA estimation method l based on rarefaction representation ₁-SVD and real domain l ₁-svd algorithm is compared, and the present invention has higher resolution, in low SNR situation and for uncorrelated target and related objective, all have better angle estimation performance, and can be applicable to lower fast umber of beats well.

The present invention according to the feature of MIMO radar system, utilize dimensionality reduction change and unitary transformation technology, make reception data be transformed into low-dimensional and be real domain; Then according to the orthogonality of real domain steering vector noise subspace corresponding to it, design a diagonal entry and compose corresponding weight matrix with real domain MUSIC, and then design weight minimization l ₁norm framework, makes its sparse solution closer to l ₀norm thus greatly strengthen openness; Finally be restored matrix, and by finding the non-zero row recovered in matrix to estimate DOA.With the existing DOA estimation method l based on rarefaction representation ₁-SVD and real domain l ₁-svd algorithm is compared, and the present invention has higher resolution, in low SNR situation and for uncorrelated target and related objective, all have better angle estimation performance, and can be applicable to lower fast umber of beats well.DOA estimation method of the present invention mainly comprises the following aspects:

1, set up the Received signal strength model of single base MIMO radar, and design dimensionality reduction matrix to reception data carry out dimension-reduction treatment.

Suppose list base, an arrowband MIMO radar system with M emitting antenna and N number of receiving antenna, they are that array element distance is from the space uniform linear array (ULA) for half wavelength.In MIMO radar system, utilize M transmission antennas transmit M the quadrature wave with same band and centre frequency.Suppose have P target to be positioned at the same scope in array far field.θ _prepresent the Bo Dajiao (DOA) of p target.After using matched filter, the output of receiving array can be expressed as

x(t)＝As(t)+n(t) (1)

Wherein receive data vector, signal data vector, middle β _p(t) and f _preflection coefficient and Doppler frequency respectively.

launch-receive guiding matrix, wherein represent the long-pending operation of Kronecker, $a_t\left({\mathrm{\&theta;}}_p\right)={[1,e^{\mathrm{j\&pi;}\sin {\mathrm{\&theta;}}_p},\&CenterDot;\&CenterDot;\&CenterDot;,e^{\mathrm{j\&pi;}(M-1)\sin {\mathrm{\&theta;}}_p}]}^T$ Launch steering vector, $a_r\left({\mathrm{\&theta;}}_p\right)={[1,e^{\mathrm{j\&pi;}\sin {\mathrm{\&theta;}}_p},\&CenterDot;\&CenterDot;\&CenterDot;,e^{\mathrm{j\&pi;}(N-1)\sin {\mathrm{\&theta;}}_p}]}^T$ Receive steering vector. be one and there is zero-mean and covariance matrix is σ ²i _mNadditional random white complex gaussian noise vector.By collecting J snap, the reception data in formula (1) become

X＝AS+N (2)

Wherein X=[x (t ₁) ..., x (t _j)], S=[s (t ₁) ..., s (t _j)], N=[n (t ₁) ..., n (t _j)].

According to reception-transmitting steering vector structure known, launch-receive steering vector meet

$a_t\left(\mathrm{\&theta;}\right)\&CircleTimes;a_r\left(\mathrm{\&theta;}\right)=\mathrm{Gb}\left(\mathrm{\&theta;}\right)---\left(3\right)$

Wherein with be transition matrix and one dimension steering vector respectively, be expressed as

$G=\left[\begin{array}{c}{J}_0\\ J_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ J_{M-1}\end{array}\right]---\left(4\right)$

$b\left(\mathrm{\&theta;}\right)={[1,e^{\mathrm{j\&pi;}\sin {\mathrm{\&theta;}}_p},\&CenterDot;\&CenterDot;\&CenterDot;,e^{\mathrm{j\&pi;}(Q-1)\sin {\mathrm{\&theta;}}_p}]}^T---\left(5\right)$

Wherein J _m=[0 _{n × m}, I _n, 0 _{n × (M-m-1)}], m=0,1 ..., M-1.According to formula (4), we define matrix F=G ^hg, as follows

According to formula (3) and formula (6), by utilizing matrix G ^hthe individual different element of corresponding Q, two-dimensional guide vector can be converted to one dimension steering vector, but, utilize G ^hcoloured noise will be increased.In order to avoid addition color noise, dimensionality reduction matrix can be defined as W=F ^-1/2g ^h, meet WW ^h=I _q.Therefore, utilize W can obtain dimensionality reduction and receive data as follows

$\stackrel{\&OverBar;}{X}=\mathrm{WAS}+\mathrm{WN}=F^{1/2}\mathrm{BS}+\mathrm{WN}=\stackrel{\&OverBar;}{B}S+\stackrel{\&OverBar;}{N}---\left(7\right)$

Wherein meet B=[b (θ ₁), b (θ ₂) ..., b (θ _p)],

2, utilize unitary transformation matrix, the augmented sample matrix receiving data after dimensionality reduction is become real domain, carry out svd and corresponding model under obtaining framework of sparse representation.

The augmented sample matrix of the reception data after dimensionality reduction become real domain, as follows

According to formula (7), dimensionality reduction receives data and correspond to containing weight matrix F ^1/2linear array.Therefore we consider an augmented sample matrix wherein Γ _qbe that to have anti-diagonal element be 1, other elements are Q × Q switching matrix of 0, () ^*represent conjugate operation.Y is center Hermite Matrix and can be converted to a real domain matrix

$Y=U_Q^H\left[\begin{array}{cc}\stackrel{\&OverBar;}{X}& {\mathrm{\&Gamma;}}_Q\stackrel{\&OverBar;}{X}\end{array}*{\mathrm{\&Gamma;}}_J\right]U_{2J}---\left(8\right)$

Wherein, unitary transformation matrix is defined as

$U_{2K+1}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc}{I}_K& 0& j{I}_K\\ 0^T& \sqrt{2}& 0^T\\ {\mathrm{\&Pi;}}_K& 0& -j{\mathrm{\&Pi;}}_K\end{array}\right]---\left(9\right)$

By omitting U _2K+1central row and central series be easy to obtain U _2K.As can be seen from formula (8), owing to merging forward backward averaging value, sample size adds one times from J to 2J.On the other hand, the guiding matrix in formula (7) meet wherein diag [] represents diagonalization operation.Result shows, in formula (7) about receive data linear array dimensionality reduction conversion after be Central Symmetry array.After unitary transformation, real domain guiding matrix can be expressed as therefore, through simple algebraic operation, formula (8) can be write as following form

Wherein S _Υ=[Φ ^*s Φ S ^*Γ _j] U _2Jreal domain signal matrix, it is real domain noise matrix.

Carry out svd and corresponding model under obtaining framework of sparse representation, as follows

To Y _Υapplication svd (SVD) technology, has

Wherein v _sby the Y that correspond to P maximum singular value _Υthe right singular vector of real domain form.

Based on corresponding to the openness of whole extraterrestrial target, the signal model in formula (2) can be transformed into a sparse representation model.A series of possible position is represented with Ω, (L>>P) grid covering Ω is represented.Therefore, launch complete dictionary and receive standby dictionary and be expressed as constructing complete dictionary is under framework of sparse representation, the signal model in formula (2) can be write as into

$X=A^{\widehat{\mathrm{\&theta;}}}{S}^{\widehat{\mathrm{\&theta;}}}+N---\left(12\right)$

Wherein with S, there is identical row to support, i.e. matrix sparse.In order to estimate sparse representation model in formula (12) can be seen as and minimize l ₁norm problem, is expressed as

$\min {\left|\right|{\left(S^{\widehat{\mathrm{\&theta;}}}\right)}^{\left(l_2\right)}\left|\right|}_{1}s.t.{\left|\right|X-A^{\widehat{\mathrm{\&theta;}}}{S}^{\widehat{\mathrm{\&theta;}}}\left|\right|}_2\&le;\mathrm{\&eta;}---\left(13\right)$

Wherein || || ₁with || || ₂represent l respectively ₁norm and l ₂norm.

The framework of sparse representation of similar formula (12), the complete dictionary of real domain one dimension can be expressed as wherein ${\stackrel{\&OverBar;}{B}}^{\widehat{\mathrm{\&theta;}}}=[b\left({\widehat{\mathrm{\&theta;}}}_1\right),b\left({\widehat{\mathrm{\&theta;}}}_2\right),\&CenterDot;\&CenterDot;\&CenterDot;,b\left({\widehat{\mathrm{\&theta;}}}_L\right)],{\mathrm{\&Phi;}}^{\widehat{\mathrm{\&theta;}}}=\mathrm{diag}(e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_1}},\&CenterDot;\&CenterDot;\&CenterDot;,e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_L}}).$ Therefore, under framework of sparse representation, formula (11) can be written as

Wherein with there is identical row to support, this means matrix sparse.

3, utilize real domain steering vector ( each row) orthogonality of noise subspace corresponding to it, design a diagonal entry and compose corresponding weight matrix to solve MMV problem with real domain MUSIC.

The complete dictionary of real domain can be divided into two parts: wherein by the real domain steering vector that correspond to possibility target composition, by dictionary remaining real domain steering vector composition.Therefore, have

Wherein, V _nreal domain noise subspace, by Y _Υcarry out svd can obtain.As J → ∞, W _{1, i}→ 0, W _{2, i}> 0.Therefore, we define weight matrix

$W=\mathrm{diag}\left(\right[W_1^T,W_2^T\left]\right)/\max \left(W_2\right)---\left(16\right)$

Due to W _{1, i}/ max (W ₂) < W _{2, i}/ max (W ₂), for MMV problem, weight matrix W can realize well large weights be used for punishing may in sparse matrix be more zero those, and little weights are used for storing larger item, this and for the iteration weight minimization l of SMV problem ₁the research method of norm is consistent.

4, real domain weight minimization l is designed ₁norm framework, utilizes programming software bag SOC (second order cone) computing method, obtains and recovers matrix, finds the non-zero row recovered in matrix, realizes the estimation to target DOA in MIMO radar system.

Real domain weight minimization l ₁norm problem becomes

Wherein be regularization parameter, formula (17) can utilize programming software bag SOC (second order cone) to calculate, such as SeDuMi.By mapping dOA estimates can obtain from solution formula (17) process.

In formula (17), regularization parameter step-up error amount and play important effect in last DOA estimated performance.Regularization parameter selection rely on be due to W and U _k(k=Q, 2J) is all orthogonal matrix, noise matrix N _Υalso real domain Gaussian distribution is met.Therefore, noise matrix approximate real domain Gaussian distribution, this is because V _sjust N _Υa function.Therefore pass through variance standardization, be approximated to the χ that one degree of freedom is (M+N-1) P ²distribution.By having 99% fiducial interval higher limit select regularization parameter of the present invention

Step one, set up the Received signal strength model of single base MIMO radar.

Suppose list base, an arrowband MIMO radar system with M emitting antenna and N number of receiving antenna, they are that array element distance is from the space uniform linear array (ULA) for half wavelength.In MIMO radar system, utilize M transmission antennas transmit M the quadrature wave with same band and centre frequency.Suppose have P target to be positioned at the same scope in array far field.θ _prepresent the Bo Dajiao (DOA) of p target.After using matched filter, the output of receiving array can be expressed as

x(t)＝As(t)+n(t) (18)

Wherein receive data vector, signal data vector, middle β _p(t) and f _preflection coefficient and Doppler frequency respectively.

launch-receive guiding matrix, wherein represent the long-pending operation of Kronecker, launch steering vector, receive steering vector. be one and there is zero-mean and covariance matrix is σ ²i _mNthe random complicated white Gaussian noise vector of complexity.By collecting J snap, the reception data in formula (18) become

X＝AS+N (19)

Wherein X=[x (t ₁) ..., x (t _j)], S=[s (t ₁) ..., s (t _j)], N=[n (t ₁) ..., n (t _j)].

Step 2, design dimensionality reduction matrix carry out dimension-reduction treatment to reception data.

According to reception-transmitting steering vector structure known, launch-receive steering vector meet

$a_t\left(\mathrm{\&theta;}\right)\&CircleTimes;a_r\left(\mathrm{\&theta;}\right)=\mathrm{Gb}\left(\mathrm{\&theta;}\right)---\left(20\right)$

Wherein with (Q=M+N-1) be transition matrix and one dimension steering vector respectively, be expressed as

$G=\left[\begin{array}{c}{J}_0\\ J_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ J_{M-1}\end{array}\right]---\left(21\right)$

$b\left(\mathrm{\&theta;}\right)={[1,e^{\mathrm{j\&pi;}\sin {\mathrm{\&theta;}}_p},\&CenterDot;\&CenterDot;\&CenterDot;,e^{\mathrm{j\&pi;}(Q-1)\sin {\mathrm{\&theta;}}_p}]}^T---\left(22\right)$

Wherein J _m=[0 _{n × m}, I _n, 0 _{n × (M-m-1)}], m=0,1 ..., M-1.According to formula (21), we define matrix F=G ^hg, as follows

According to formula (20) and formula (23), by utilizing matrix G ^hthe individual different element of corresponding Q, two-dimensional guide vector can be converted to one dimension steering vector, but, utilize G ^hcoloured noise will be increased.In order to avoid addition color noise, dimensionality reduction matrix can be defined as W=F ^-1/2g ^h, meet WW ^h=I _q.Therefore, utilize W can obtain dimensionality reduction and receive data as follows

$\stackrel{\&OverBar;}{X}=\mathrm{WAS}+\mathrm{WN}=F^{1/2}\mathrm{BS}+\mathrm{WN}=\stackrel{\&OverBar;}{B}S+\stackrel{\&OverBar;}{N}---\left(24\right)$

Wherein meet B=[b (θ ₁), b (θ ₂) ..., b (θ _p)],

Step 3, utilize unitary transformation matrix, the augmented sample matrix receiving data after dimensionality reduction is become real domain.

According to formula (24), dimensionality reduction receives data and correspond to containing weight matrix F ^1/2linear array.Therefore we consider an augmented sample matrix wherein Γ _qbe that to have anti-diagonal element be 1, other elements are Q × Q switching matrix of 0, () ^*represent conjugate operation.Y is center Hermite Matrix and can be converted to a real domain matrix

$Y=U_Q^H\left[\begin{array}{cc}\stackrel{\&OverBar;}{X}& {\mathrm{\&Gamma;}}_Q\stackrel{\&OverBar;}{X}\end{array}*{\mathrm{\&Gamma;}}_J\right]U_{2J}---\left(25\right)$

Wherein, unitary transformation matrix is defined as

$U_{2K+1}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc}{I}_K& 0& j{I}_K\\ 0^T& \sqrt{2}& 0^T\\ {\mathrm{\&Pi;}}_K& 0& -j{\mathrm{\&Pi;}}_K\end{array}\right]---\left(26\right)$

By omitting U _2K+1central row and central series be easy to obtain U _2K.As can be seen from formula (25), owing to merging forward backward averaging value, sample size adds one times from J to 2J.On the other hand, the guiding matrix in formula (24) meet wherein diag [] represents diagonalization operation.Result shows, in formula (24) about receive data linear array dimensionality reduction conversion after be Central Symmetry array.After unitary transformation, real domain guiding matrix can be expressed as therefore, through simple algebraic operation, formula (25) can be write as following form

Wherein S _Υ=[Φ ^*s Φ S ^*Γ _j] U _2Jreal domain signal matrix, it is real domain noise matrix.

Step 4, svd is carried out to real domain augmented sample matrix and corresponding model under obtaining framework of sparse representation.

To Y _Υapplication svd (SVD) technology, has

Wherein v _sby a corresponding P maximum singular value, Y _Υthe right singular vector of real domain form.

Based on corresponding to the openness of whole extraterrestrial target, the signal model in formula (19) can be transformed into a sparse representation model.A series of possible position is represented with Ω, (L>>P) grid covering Ω is represented.Therefore, launch complete dictionary and receive standby dictionary and be expressed as constructing complete dictionary is under framework of sparse representation, the signal model in formula (19) can be write as into

$X=A^{\widehat{\mathrm{\&theta;}}}{S}^{\widehat{\mathrm{\&theta;}}}+N---\left(29\right)$

Wherein with S, there is identical row to support, i.e. matrix sparse.In order to estimate sparse representation model in formula (29) can be seen as and minimize l ₁norm problem, is expressed as

$\min {\left|\right|{\left(S^{\widehat{\mathrm{\&theta;}}}\right)}^{\left(l_2\right)}\left|\right|}_{1}s.t.{\left|\right|X-A^{\widehat{\mathrm{\&theta;}}}{S}^{\widehat{\mathrm{\&theta;}}}\left|\right|}_2\&le;\mathrm{\&eta;}---\left(30\right)$

Wherein || || ₁with || || ₂represent l respectively ₁norm and l ₂norm.

The framework of sparse representation of similar formula (29), the complete dictionary of real domain one dimension can be expressed as wherein ${\stackrel{\&OverBar;}{B}}^{\widehat{\mathrm{\&theta;}}}=[b\left({\widehat{\mathrm{\&theta;}}}_1\right),b\left({\widehat{\mathrm{\&theta;}}}_2\right),\&CenterDot;\&CenterDot;\&CenterDot;,b\left({\widehat{\mathrm{\&theta;}}}_L\right)],{\mathrm{\&Phi;}}^{\widehat{\mathrm{\&theta;}}}=\mathrm{diag}(e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_1}},\&CenterDot;\&CenterDot;\&CenterDot;,e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_L}}).$ Therefore, under framework of sparse representation, formula (28) can be written as

Wherein with there is identical row to support, this means matrix sparse.

Step 5, for MMV problem, utilize the orthogonality of real domain steering vector noise subspace corresponding to it, design weight matrix.

The complete dictionary of real domain can be divided into two parts: wherein by the real domain steering vector that correspond to possibility target composition, by dictionary remaining real domain steering vector composition.Therefore, have

Wherein, V _nreal domain noise subspace, by Y _Υcarry out svd can obtain.As J → ∞, W _{1, i}→ 0, W _{2, i}> 0.Therefore, we define weight matrix

$W=\mathrm{diag}\left(\right[W_1^T,W_2^T\left]\right)/\max \left(W_2\right)---\left(33\right)$

Due to W _{1, i}/ max (W ₂) < W _{2, i}/ max (W ₂), for MMV problem, weight matrix W can realize well large weights be used for punishing may in sparse matrix be more zero those, and little weights are used for storing larger item, this and for the iteration weight minimization l of SMV problem ₁the research method of norm is consistent.

Step 6, design real domain weight minimization l ₁norm framework, obtains and recovers matrix, finds the non-zero row recovered in matrix, realizes the estimation to target DOA in MIMO radar system.

Real domain weight minimization l ₁norm problem becomes

Wherein be regularization parameter, formula (34) can utilize programming software bag SOC (second order cone) to calculate, such as SeDuMi.By mapping dOA estimates can obtain from solution formula (34) process.

In formula (34), regularization parameter step-up error amount and play important effect in last DOA estimated performance.Regularization parameter selection rely on be due to W and U _k(k=Q, 2J) is all orthogonal matrix, noise matrix N _Υalso real domain Gaussian distribution is met.Therefore, noise matrix approximate real domain Gaussian distribution, this is because V _sjust N _Υa function.Therefore pass through variance standardization, be approximated to the χ that one degree of freedom is (M+N-1) P ²distribution.By having 99% fiducial interval higher limit select regularization parameter of the present invention

Effect of the present invention illustrates by following emulation:

(1) simulated conditions and content:

By the present invention and l ₁-SVD, RV l ₁-SVD and CRB contrasts.Suppose single base MIMO radar system, M=N=5, emission array and receiving array are all that array element distance is from the space uniform linear array for half wavelength.For all methods in emulation, all hypothetical target number is known.In all methods, space lattice to be all unification be 0.1 ° from-90 ° to 90 ° scope, and select the fiducial interval of regularization parameter to be 99%.

(2) simulation result

1, algorithms of different differentiates the relation of the probability of success and angle intervals for two uncorrelated targets

Fig. 2 illustrates the relation for two uncorrelated target resolutions and the angle of departure.Wherein fast umber of beats J=50, SNR=5dB and suppose that uncorrelated target comes from θ ₁=0 °, θ ₂=0 °+Δ θ, Δ θ be change from 1 ° to 10 °.If there are at least two spikes in spatial spectrum, and meet wherein θ _iestimation, so these two targets can be regarded as and have been solved.As shown in Figure 2, RV l ₁-svd algorithm is better than l ₁-svd algorithm.In addition, the present invention is strengthened by dimensionality reduction conversion SNR gain, the weighting l simultaneously ₁norm is better close to l ₀norm and enhance sparse solution.Therefore, the present invention compares l ₁-SVD and RV l ₁-svd algorithm has higher resolution.

2, algorithms of different is for three uncorrelated root-mean-square errors of angle on targets estimation and the relation of signal to noise ratio (S/N ratio)

Fig. 3 illustrates the relation of root-mean-square error (RMSE) and the SNR estimated for three uncorrelated angle on targets, wherein fast umber of beats J=50, and supposes that three uncorrelated targets come from θ ₁=-20 °, θ ₂=-10 °, θ ₃=10 °.As can be seen from Figure 3, due to real domain switch technology, at low SNR region RV l ₁-SVD compares l ₁-svd algorithm has better angle estimation.In addition, angle estimation performance of the present invention is better than l ₁-SVD and RV l ₁-SVD, and due to above-mentioned advantage of the present invention it close to CRB.

3, algorithms of different estimates the relation of root-mean-square error and related coefficient for two angle on targets

Fig. 4 illustrates the relation two angle on targets being estimated to RMSE and related coefficient.Wherein fast umber of beats J=50, SNR=5dB.Suppose that two targets come from θ ₁=-20 °, θ ₂=-10 ° and the variation range of related coefficient is 0 to 1.As shown in Figure 4, the present invention has robustness for the change of related coefficient, and compares l ₁-SVD and RV l ₁-SVD has better angle estimation performance.This is because the present invention comprises front and back Search Space Smoothing and weighting l ₁norm technology.

4, the root-mean-square error of algorithms of different for three uncorrelated angle on targets estimations and the relation of fast umber of beats

Fig. 5 illustrates the relation of RMSE and the fast umber of beats estimated for three uncorrelated angle on targets.Suppose SNR=5dB, three uncorrelated targets come from θ ₁=-20 °, θ ₂=-10 °, θ ₃=10 °.As shown in Figure 5, angle estimation Performance Ratio l of the present invention ₁-SVD and RV l ₁-svd algorithm is all good, close to CRB within the scope of all snaps, this means that the present invention can be applicable to lower fast umber of beats well.

Claims (5)

1. based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of Norm Method, is characterized in that, comprise the steps:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving end obtains after carrying out matched filtering process and receives data, and utilizes dimensionality reduction matrix to carry out dimension-reduction treatment to reception data;

(2) utilize unitary transformation matrix, the augmented sample matrix receiving data after dimensionality reduction is become real domain, carry out svd and corresponding model under obtaining framework of sparse representation;

(3) utilize the orthogonality of real domain steering vector noise subspace corresponding to it, design a diagonal entry and compose corresponding weight matrix to solve MMV problem with real domain MUSIC;

(4) real domain weight minimization l is designed ₁norm framework, utilizes programming software bag SOC second order cone computing method, obtains and recovers matrix, finds the non-zero row recovered in matrix, realizes the estimation to target DOA in MIMO radar system.

2. according to claim 1 based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of Norm Method, is characterized in that: carry out dimension-reduction treatment to reception data as follows in described step (1):

(1.1) receive-launch the structure of steering vector according to single base MIMO radar system, the transmitting-reception steering vector of MIMO radar system meets:

$a_t\left(\mathrm{\&theta;}\right)\&CircleTimes;a_r\left(\mathrm{\&theta;}\right)=\mathrm{Gb}\left(\mathrm{\&theta;}\right),$

A in formula _t(θ) and a _r(θ) be respectively transmitting steering vector and receive steering vector, q=M+N – 1 is transition matrix and one dimension steering vector respectively,

$G=\left[\begin{array}{c}{J}_0\\ J_1\\ .\\ .\\ .\\ J_{M-1}\end{array}\right],$

$b\left(\mathrm{\&theta;}\right)=1,{[1,e^{{\mathrm{j\&pi;}\sin \mathrm{\&theta;}}_p},...,e^{{\mathrm{j\&pi;}(Q-1)\sin \mathrm{\&theta;}}_p}]}^T,$

Wherein,

$J_m=[0_{N\&times;M},I_N,0_{N\&times;(M-m-1)}],m=\mathrm{0,1},...,M-1,$

By utilizing matrix G ^hthe individual different element of corresponding Q, two-dimensional guide vector can be converted to one dimension steering vector and namely carry out dimension-reduction treatment;

(1.2) according to transition matrix, dimensionality reduction matrix is W=F ^-1/2g ^h, wherein

(1.3) utilize W to obtain dimensionality reduction and receive data then have

$\stackrel{\&OverBar;}{X}=\mathrm{WAS}+\mathrm{WN}=F^{1/2}\mathrm{BS}+\mathrm{WN}=\stackrel{\&OverBar;}{B}S+\stackrel{\&OverBar;}{N};$

In formula meet B=[b (θ ₁), b (θ ₂) ..., b (θ _p)],

3. according to claim 1 based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of Norm Method, it is characterized in that: described step utilizes unitary transformation matrix in (2) as follows, the augmented sample matrix receiving data after dimensionality reduction is become real domain, carries out svd and corresponding model under obtaining framework of sparse representation:

(2.1) augmented sample matrix is considered $Y=\left[\begin{array}{cc}\stackrel{\&OverBar;}{X}& {\mathrm{\&Gamma;}}_Q{\stackrel{\&OverBar;}{X}}^{*}{\mathrm{\&Gamma;}}_J\end{array}\right],$ Wherein Γ _qbe that to have anti-diagonal element be 1, other elements are Q × Q switching matrix of 0, () ^*represent conjugate operation, Y is center Hermite Matrix and can be converted to a real domain matrix,

$Y=U_Q^H\left[\begin{array}{cc}\stackrel{\&OverBar;}{X}& {\mathrm{\&Gamma;}}_Q{\stackrel{\&OverBar;}{X}}^{*}\end{array},{\mathrm{\&Gamma;}}_J\right]U_{2J},$

Wherein, unitary transformation matrix is

$U_{2K+1}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc}{I}_K& 0& {\mathrm{jI}}_K\\ 0^T& \sqrt{2}& 0^T\\ {\mathrm{\&Pi;}}_K& 0& -J{\mathrm{\&Pi;}}_K\end{array}\right],$

Guiding matrix meet ${\mathrm{\&Gamma;}}_Q{\left(\stackrel{\&OverBar;}{B}\mathrm{\&Phi;}\right)}^{*}=\stackrel{\&OverBar;}{B}\mathrm{\&Phi;},\mathrm{\&Phi;}=\mathrm{diag}(e^{-\mathrm{j\&pi;}(Q-1)/2{\sin \mathrm{\&theta;}}_1},...,e^{-\mathrm{j\&pi;}(Q-1)/2{\sin \mathrm{\&theta;}}_P}),$ Diag [] represents diagonalization operation, and the linear array wherein about reception data is centrosymmetric after dimensionality reduction conversion, and after unitary transformation, real domain guiding matrix is

Wherein S _Υ=[Φ ^*s Φ S ^*Γ _j] U _2Jreal domain signal matrix, it is real domain noise matrix;

(2.2) to Y _Υapplication svd SVD technology, has

Wherein v _sby the Y that correspond to P maximum singular value _Υthe right singular vector of real domain form;

(2.3) apply framework of sparse representation, the complete dictionary of real domain one dimension can be expressed as:

In formula ${\stackrel{\&OverBar;}{B}}^{\widehat{\mathrm{\&theta;}}}=[b\left({\widehat{\mathrm{\&theta;}}}_1\right),b\left({\widehat{\mathrm{\&theta;}}}_2\right),...,b\left({\widehat{\mathrm{\&theta;}}}_L\right)],{\mathrm{\&Phi;}}^{\widehat{\mathrm{\&theta;}}}=\mathrm{diag}(e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_1}},...,e^{-\mathrm{j\&pi;}(Q-1)/2\sin \widehat{{\mathrm{\&theta;}}_L}}),$ Under framework of sparse representation,

4. according to claim 1 based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of Norm Method, it is characterized in that: the orthogonality utilizing real domain steering vector noise subspace corresponding to it in described step (3) as follows, design diagonal entry composes corresponding weight matrix with real domain MUSIC:

(3.1) complete for real domain dictionary is divided into two parts: then have

In formula by the real domain steering vector that correspond to possibility target p=1,2 ..., P forms, by dictionary remaining real domain steering vector composition, V _nreal domain noise subspace, by Y _Υcarry out svd can obtain;

(3.2) according to the orthogonality of real domain steering vector and corresponding noise subspace, as J → ∞, W _{1, i}→ 0, W _{2, i}> 0, definition weight matrix

$W=\mathrm{diag}\left(\right[W_1^T,W_2^T)/\max \left(W_2\right)$

Due to W _{1, i}/ max (W ₂) < W _{2, i}/ max (W ₂).

5. according to claim 1 based on real domain weight minimization l ₁the MIMO radar system DOA estimation method of Norm Method, is characterized in that: design real domain weight minimization l in described step (4) as follows ₁norm framework, utilizes programming software bag SOC second order cone computing method, obtains and recovers matrix, finds the non-zero row recovered in matrix, estimates the target DOA in MIMO radar system:

Real domain weight minimization l ₁norm is

In formula be regularization parameter, utilize programming software bag SOC second order cone to calculate, by mapping