CN104251989A - Compression spatial spectrum-based single base MIMO (Multiple Input Multiple Output) radar target DOA (Direction of Arrival) estimation method - Google Patents

Abstract

The invention relates to the technical field of radars, in particular to the application of a single base MIMO (Multiple Input Multiple Output) radar system and a compression spatial spectrum-based single base MIMO radar target DOA (Direction of Arrival) estimation method. The compression spatial spectrum-based single base MIMO radar target DOA estimation method comprises the steps of enabling an emission array to emit phase-coded signals which are mutually orthorhombic and enabling a receiving terminal to perform matched filtering processing through a receiving array matching filter; performing dimensionality reduction processing on J receiving data subjected to snapshotting matched filtering through a dimensionality reduction array; calculating a covariance matrix R of the receiving data Y subjected to dimensionality reduction processing and working out an intersection subspace of a noise subspace and a conjugate subspace thereof; constructing a compression spatial spectrum function and searching the compression spatial spectrum function; excluding a false DOA and obtaining a real DOA of a target. The compression spatial spectrum-based single base MIMO radar target direction of arrival estimation method avoids a combined search of a two-dimension DOA of a traditional MUSIC algorithm when searches the airspace DOA, and only needs one-dimensional space search and the algorithm complexity is reduced.

Description

Based on single base MIMO radar object wave arrival direction estimating method of compression stroke spectrum

Technical field

The present invention relates to Radar Technology field, particularly the application of single base MIMO radar system, be specifically related to a kind of single base MIMO radar object wave arrival direction estimating method based on compression stroke spectrum.

Background technology

In recent years, multiple-input and multiple-output (Multiple-Input Multiple-Output, MIMO) radar is owing to having more potential advantages than traditional phased-array radar, cause the extensive concern of field of radar expert, and become rapidly a popular research topic of current radar circle.According to the different configuration modes of MIMO radar emission array and receiving array, MIMO radar can be divided into two large classes: a class is statistics MIMO radar (IEEE Signal Processing Magazine, 2008,25 (1): 116-129), another kind of is relevant MIMO radar (IEEE Signal Processing Magazine, 2007,24 (5): 106-114) the bistatic MIMO radar of bistatic, is comprised.Statistics MIMO radar utilizes spatial domain diversity technique to improve the detection performance of radar, and relevant MIMO radar utilizes waveform diversity technology to obtain the virtual aperture larger than true aperture, and therefore it can obtain accurate Mutual coupling.The present invention mainly relates to single base and to be concerned with the target direction of arrival of MIMO radar.

In the practical application of MIMO radar, the Mutual coupling of target is an importance.Some DOA estimation algorithms have been proposed at present in MIMO radar, such as invariable rotary subspace (Estimation of Signal Parameters via Rotational Invariance Technique, ESPRIT) algorithm (Electronics Letters:2008, 44 (12): 770-771), RD-ESPRIT (Reduced-Dimensional ESPRIT, RD-ESPRIT) algorithm (Electronics Letters:2011, 47 (4): 283-284) and based on Wave arrival direction estimating method (the IEEE Transations on Signal Processing:2011 of emitted energy concentration techniques, 59 (6): 2669-2682).These methods utilize the direction of arrival of the invariable rotary characteristic estimating target of the virtual steering vector of MIMO radar, and often lose partial virtual array aperture, what cause final goal direction of arrival estimates at certain deviation.And multiple signal classification method (Multiple Signal Classification, MUSIC) (IEEE Transations on Antennas and Propagation, 1986,34 (3), the Mutual coupling performance of suboptimum 276-280) can be obtained, but be applied directly in the Mutual coupling of MIMO radar and face two large problems: 1) need 2-d direction finding parametric joint to search for; 2) direction of arrival search is carried out to whole spatial domain.Therefore traditional MUSIC algorithm often causes computational complexity too high in the application of MIMO radar, cannot meet the demand of real time signal processing in actual environment.

Summary of the invention

The object of the invention is to the defect overcoming said method, propose a kind of Wave arrival direction estimating method of the many MIMO radar in single base based on compression stroke spectrum newly.

The realization of the inventive method, comprises the steps:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving end utilizes receiving array matched filter to carry out matched filtering process, until fast umber of beats reaches J;

Involved receiving array exports expression formula:

$x\left(t\right)=[a_t\left({\mathrm{\θ}}_1\right)\⊗a_r\left({\mathrm{\θ}}_1\right),...,a_t\left({\mathrm{\θ}}_p\right)\⊗a_r\left({\mathrm{\θ}}_p\right)]s\left(t\right)+n\left(t\right)$

In formula, P represents incoherent number of targets, and has P=1,2,3 ...; θ _pfor the direction of arrival of respective objects; a _t(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Μ-1) sin θ _p)] ^tfor launching steering vector, a _r(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Ν-1) sin θ _p)] ^tfor receiving steering vector; $s_p\left(t\right)={\mathrm{\β}}_p\left(t\right)e^{j2\mathrm{\π}{f}_{p}t},$ β _p(t) and f _prepresent reflection coefficient and Doppler frequency respectively; represent that zero-mean and covariance matrix are σ ²i _mNwhite Gaussian noise vector;

Take soon at J, the reception data obtained after matched filtering process can be expressed as:

X＝AS+N

In formula, x=[x (t ₁) ..., x (t _j)], S=[s (t ₁) ..., s (t _j)], N=[n (t ₁) ..., n (t _j)] be white Gaussian noise matrix;

(2) utilize dimensionality reduction matrix, dimension-reduction treatment is carried out to the reception data that J obtains after taking matched filtering process soon, after obtaining dimension-reduction treatment, receives data Y;

The expression formula receiving data Y after involved dimension-reduction treatment is:

Y＝WX＝F ^-(1/2)FBS+WN＝F ^(1/2)BS+WN

In formula, W is dimensionality reduction matrix, and has W=F ^-(1/2)g ^h; X is the reception data that receiving array matched filter exports;

B=[b (θ ₁), b (θ ₂) ..., b (θ _p)], wherein b (θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (M+ Ν-2) sin θ _p)] ^t;

(3) receive the covariance matrix R of data Y after calculating dimension-reduction treatment, utilize Eigenvalues Decomposition to obtain noise subspace, and calculate noise subspace and its conjugation intersection subspace collected works space;

The involved expression formula receiving the covariance matrix R of data Y is:

$R=\frac{1}{J}Y{Y}^H$

The expression formula of covariance matrix being carried out to Eigenvalues Decomposition is:

$R=U_s{\mathrm{\Λ}}_s{U}_s^H\text{+}{U}_n{\mathrm{\Λ}}_n{U}_n^H$

In formula, U _sfor the signal subspace be made up of P large eigenwert characteristic of correspondence vector, U _nfor the noise subspace be made up of M+N-1-P little eigenwert characteristic of correspondence vector, Λ _sfor diagonal matrix, and its diagonal element is made up of P large eigenwert; Λ _nfor diagonal matrix, its diagonal element M+N-1-P little eigenwert composition;

Involved noise subspace U _nand conjugation subspace common factor subspace expression formula is:

${\stackrel{\‾}{U}}_n=U_{n}E(:,1:M+M-1-2P)$

In formula, E is the matrix of left singularity characteristics vector composition, and E meets svd expression formula:

$D=E\stackrel{\‾}{\mathrm{\Λ}}{V}^H$

Wherein, v is the matrix of left singularity characteristics vector composition, for the diagonal matrix of singular value composition;

(4) construct compression stroke spectral function, compression stroke spectral function is searched for, and utilize spatial spectrum conjugate symmetry property to obtain the true direction of arrival of target with false direction of arrival

The expression formula of compression stroke spectral function is:

$P_c=\frac{1}{{\left[F^{(1/2)}b\left(\mathrm{\θ}\right)\right]}^H{\stackrel{\‾}{U}}_n{\stackrel{\‾}{U}}_n^H{F}^{(1/2)}b\left(\mathrm{\θ}\right)}$

(5) utilize true direction of arrival steering vector and noise subspace orthogonal property, get rid of false direction of arrival, obtain the true direction of arrival of target;

True direction of arrival steering vector and noise subspace orthogonal property expression formula are:

${\left|\right|{\left[F^{(1/2)}b\left({\mathrm{\θ}}_i\right)\right]}^H{\stackrel{\‾}{U}}_n\left|\right|}^2=0$

Be met minimum P value of above formula, then corresponding is exactly real target direction of arrival.

The present invention has following characteristics compared with prior art:

1, the present invention is in MIMO radar when carrying out spatial domain direction of arrival search, avoids traditional MUSIC algorithm 2-d direction finding Syndicating search, only needs the one-dimensional space to compose search, reduces computational complexity;

2, the present invention is when searching for spatial domain direction of arrival, traditional MUSIC algorithm is avoided to search for whole observation, only need search for half observation spatial domain and utilize the symmetry characteristic of direction of arrival directly to obtain direction of arrival, reduce computational complexity further;

3, the present invention has Mutual coupling performance more better than RD-ESPRIT, has similar Mutual coupling performance to traditional MUSIC algorithm simultaneously.

Accompanying drawing explanation

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

Fig. 2 is the spatial domain spectrogram of the present invention and MUSIC algorithm.

The Mutual coupling mean square deviation error of Fig. 3 algorithms of different and Between Signal To Noise Ratio figure.

The direction of arrival resoluting probability of Fig. 4 algorithms of different and Between Signal To Noise Ratio figure.

The operation time of Fig. 5 algorithms of different and the graph of a relation of transmitting and receiving array number.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is further elaborated:

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

Consider that single base MIMO radar is made up of M emitting antenna and N number of receiving antenna, its emission array and receiving array are array element and form apart from for the even linear array of half wavelength.In the MIMO radar of single base, consider that emission array is together with receiving array close-packed arrays, therefore target is the same relative to emission array with the direction of arrival of receiving array, the unified direction of arrival (Direction Of Arrival, DOA) being designated as target.At the transmitting terminal of MIMO radar, emission array launches the orthogonal narrow band signal that a group has same band and centre frequency.Consider that P is positioned at the inside uncorrelated target in far field of a distance, wherein θ _p(p=1,2 ..., P) and represent the DOA of p target relative to emission array and receiving array, then receiving array matched filter exports and can be represented as

$x\left(t\right)=[a_t\left({\mathrm{\θ}}_1\right)\⊗a_r\left({\mathrm{\θ}}_1\right),...,a_t\left({\mathrm{\θ}}_p\right)\⊗a_r\left({\mathrm{\θ}}_p\right)]s\left(t\right)+n\left(t\right)---\left(1\right)$

In formula $s_p\left(t\right)={\mathrm{\β}}_p\left(t\right)e^{j2\mathrm{\π}{f}_{p}t},$ β _p(t) and f _prepresent reflection coefficient and Doppler frequency a respectively _t(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Μ-1) sin θ _p)] ^tfor launching steering vector, a _r(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Ν-1) sin θ _p)] ^tfor receiving steering vector. represent that zero-mean and covariance matrix are σ ²i _mNwhite Gaussian noise vector.Definition received signal strength then in formula (1) can be expressed as x (t)=As (t)+n (t).Take soon at J, Received signal strength can be expressed as

X＝AS+N (2)

Wherein S=[s (t ₁) ..., s (t _j)], N=[n (t ₁) ..., n (t _j)] be white Gaussian noise matrix.

According to send-receive steering vector structure, then have

Wherein z=exp (j π sin θ).Can know from formula (3), the send-receive guiding of single base MIMO radar, only containing the individual different element of M+N-1, namely only has the Virtual array that M+N-1 different.Then send-receive steering vector can be expressed as

$a_t\left(\mathrm{\θ}\right)\⊗a_r\left(\mathrm{\θ}\right)=\mathrm{Gb}\left(\mathrm{\θ}\right)---\left(4\right)$

In formula, G and b (θ) is transformation matrix and corresponding virtual steering vector,

b(θ)＝[1,exp(jπsinθ _p),...,exp(jπ(Μ+Ν-2)sinθ _p)] ^T

In formula

J _m＝[0 _N×m,I _N,0 _N×(M-m-1)]，m＝0,1,...,M-1 (6)

According to the architectural characteristic of transformation matrix in formula (5), we define matrix F=G ^hg is as follows

In order to avoid coloured noise, definition dimensionality reduction matrix is W=F ^-(1/2)g ^h, meet WW ^h=I _m+N-1.Utilize dimensionality reduction matrix to dock and take in row relax, then have

Y＝WX＝F ^-(1/2)FBS+WN＝F ^(1/2)BS+WN (8)

Wherein B=[b (θ ₁), b (θ ₂) ..., b (θ _p)].Known according to formula (8), it is F that the reception data after dimensionality reduction are equivalent to weights ^(1/2)homogenous linear virtual array.

Step 2, calculating dimensionality reduction receive the covariance matrix of data, utilize Eigenvalues Decomposition technology to obtain noise subspace, and calculate noise subspace and conjugation intersection subspace collected works space thereof.

Calculate the covariance matrix of dimensionality reduction matrix, as follows

$R=\frac{1}{J}Y{Y}^H---\left(9\right)$

Eigenvalues Decomposition is carried out to covariance matrix, as follows

$R=U_s{\mathrm{\Λ}}_s{U}_s^H\text{+}{U}_n{\mathrm{\Λ}}_n{U}_n^H---\left(10\right)$

U in formula _sfor signal subspace is made up of P large eigenwert characteristic of correspondence vector, U _nfor noise subspace is made up of M+N-1-P little eigenwert characteristic of correspondence vector, Λ _sand Λ _nbe diagonal matrix, its diagonal element is made up of P large eigenwert and M+N-1-P little eigenwert respectively.

Noise subspace U is obtained by formula (10) _n, utilize SVD technology to solve U _nwith common factor subspace.First a matrix is defined then svd is carried out to it

$D=E\stackrel{\‾}{\mathrm{\Λ}}{V}^H---\left(11\right)$

In formula, E and V is respectively the matrix of left singularity characteristics vector composition, for the diagonal matrix of singular value composition.Noise subspace and conjugation intersection subspace collected works space thereof can be obtained by following formula

${\stackrel{\‾}{U}}_n=U_{n}E(:,1:M+M-1-2P)---\left(12\right)$

Same up-to-date style (12) obtain Un and common factor subspace

Step 3, construct compression stroke according to the biorthogonality of common factor subspace and steering vector and conjugation steering vector and compose search function, spatial spectrum function is searched for and utilizes spatial spectrum conjugate symmetry property to obtain the true direction of arrival of target and false direction of arrival.

It is known according to formula (12), for Un and common factor subspace.Therefore,

$\mathrm{span}\left({\stackrel{\‾}{U}}_n\right)=\mathrm{span}\left(U_n\right),\mathrm{span}\left(U_n^{*}\right)=\mathrm{span}\left({\stackrel{\‾}{U}}_n\right)---\left(13\right)$

According to orthogonality and the weight matrix F of steering vector and noise subspace ^(1/2)real-valued property, then have

$F^{(1/2)}\left(\mathrm{b\θ}\right)\⊥U_n,F^{(1/2)}{b}^{*}\left(\mathrm{\θ}\right)\⊥U_n^{*}---\left(14\right)$

Known according to formula (14), to each real direction of arrival θ _p(p=1,2 ..., P) in conjugation noise subspace, all there is a false direction of arrival-θ _p(p=1,2 ..., P).Convolution (13) and (14), then have

$F^{(1/2)}b\left(\mathrm{\θ}\right)\⊥{\stackrel{\‾}{U}}_n,F^{(1/2)}{b}^{*}\left(\mathrm{\θ}\right)\⊥{\stackrel{\‾}{U}}_n---\left(15\right)$

Known according to formula (15), for noise subspace and its conjugation intersection subspace collected works space steering vector and conjugation steering vector are all orthogonal to common factor subspace therefore compression stroke spectral function is constructed as follows

$P_c=\frac{1}{{\left[F^{(1/2)}b\left(\mathrm{\θ}\right)\right]}^H{\stackrel{\‾}{U}}_n{\stackrel{\‾}{U}}_n^H{F}^{(1/2)}b\left(\mathrm{\θ}\right)}---\left(16\right)$

If carry out whole spatial domain to formula (16) to carry out whole observation spatial domain observation [-90 °, 90 °], according to common factor subspace just give the characteristic of steering vector and conjugation steering vector, then can obtain the true direction of arrival of target and false direction of arrival simultaneously.Notice real goal and false symmetry characteristic, here we only need to carry out [0 ° to half observation spatial domain, 90 °] or [-90 °, 0 °] carry out volume-search coverage, then utilize conjugate property to obtain other direction of arrival, namely obtain P true direction of arrival and P false direction of arrival.

Step 4, utilize true direction of arrival steering vector and noise subspace orthogonal property, get rid of false direction of arrival, obtain the true direction of arrival of target.

2P direction of arrival θ is obtained by the conjugate property carried out formula (16) between the search in half observation spatial domain and direction of arrival _p(p=1 ..., 2P) (P true direction of arrival and P false direction of arrival).Steering vector corresponding to true direction of arrival is orthogonal to noise subspace, then have

${\left|\right|{\left[F^{(1/2)}b\left({\mathrm{\θ}}_i\right)\right]}^H{\stackrel{\‾}{U}}_n\left|\right|}^2=0---\left(17\right)$

Find out P value minimum in formula (17), then corresponding is exactly required target direction of arrival, thus realizes target Mutual coupling of the present invention.

Effect of the present invention illustrates by following computational complexity analysis and simulation:

(1) computational complexity analysis

Computational complexity of the present invention mainly focuses on for the calculating of covariance matrix R and Eigenvalues Decomposition, the Eigenvalues Decomposition of matrix D and the search of compression stroke spectrum, and therefore computational complexity of the present invention is o{ (M+N-1) ²j+ (M+N-1) ³+ (M+N-1-P) ³+ L/2 [(M+N) (M+N-1+2P)] }.And the computational complexity of MUSIC algorithm mainly concentrates on covariance matrix R _x=1/LXX ^hcalculating and Eigenvalues Decomposition, and spatial spectrum search is carried out in whole observation spatial domain, and therefore the computational complexity of MUSIC algorithm is o{ (MN) ²j+ (MN) ³+ L [(MN+1) (MN-P) }, wherein L is the cut-point sum in whole observation spatial domain.Therefore the present invention has lower computational complexity than having than MUSIC algorithm.

Simulated conditions and content:

Here consider that single base MIMO radar has 8 to launch array element and 8 reception array elements.Emission array and receiving array are formed apart from the even linear array being half wavelength by array element.Suppose to there is P=3 irrelevant target, its direction of arrival is respectively θ ₁=10 °, θ ₂=20 ° and θ ₃=40 °, sampling umber of beats is 200.In following great majority experiment, contrast with RD-ESPRIT, MUSIC algorithm and institute of the present invention extracting method.MUSIC algorithm and volume-search coverage step-length of the present invention are 0.01 °.The root-mean-square error of Mutual coupling is defined as

$(1/P){\mathrm{\Σ}}_{P=1}^P\sqrt{(1/Q){\mathrm{\Σ}}_{i=1}^Q{({\widehat{\mathrm{\θ}}}_{P,i}-{\mathrm{\θ}}_p)}^2}---\left(18\right)$

In formula that i-th Monte Carlo test ripple reaches angle θ _pmutual coupling, Q=200 is the number of times of Monte Carlo experiment.

(3) simulation result

1, the whole Spatial Spectrum of the present invention and MUSIC

From shown in Fig. 2 being the spatial domain spectrogram that the present invention and MUSIC algorithm are searched for whole spatial domain.As we know from the figure, MUSIC algorithm estimates the direction of arrival of P target accurately, namely has P spike.And 2P spike is appearring in the present invention, and these spikes are symmetrical relative to 0 ° of degree, namely have P real goal direction of arrival and P false direction of arrival.Therefore the present invention can obtain P candidate's direction of arrival by carrying out search to half observation space spectrum [0 °, 90 °] or [-90 °, 0 °], then utilizes the symmetry characteristic of direction of arrival to obtain other P candidate's direction of arrival.Steering vector corresponding to true direction of arrival and noise subspace orthogonal property is finally utilized to get rid of false direction of arrival.

2, the mean square deviation error of algorithms of different and the relation of signal to noise ratio (S/N ratio)

As can be seen from Figure 3, Mutual coupling performance of the present invention is more superior than RD-ESPRIT, and has similar Mutual coupling performance to MUSIC algorithm.But the computational complexity of algorithm of the present invention well below MUSIC, will have better signal real-time characteristic, has better prospect in practical application.

3, the direction of arrival resoluting probability of algorithms of different and SNR relation

As can be seen from Figure 4, direction of arrival resoluting probability of the present invention is far longer than RD-ESPRIT algorithm, has similar resoluting probability with MUSIC algorithm simultaneously.

4, the operation time of algorithms of different and the relation of transmitting and receiving array number

As can be seen from Figure 5, the operation time of MUSIC algorithm is along with launching array element and receiving the increase of array number and increase rapidly, and the present invention presents the process steadily increased, operation time is far less than MUSIC algorithm, this and the theoretical analysis result of computational complexity are corresponding corresponding, and therefore the present invention has good real-time characteristic.

Claims (1)

1., based on single base MIMO radar object wave arrival direction estimating method of compression stroke spectrum, it is characterized in that, comprise the steps:

(1) emission array launches mutually orthogonal phase-coded signal, and receiving end utilizes receiving array matched filter to carry out matched filtering process, until fast umber of beats reaches J;

Involved receiving array exports expression formula:

$x\left(t\right)=[a_t\left({\mathrm{\θ}}_1\right)\⊗a_r\left({\mathrm{\θ}}_1\right),\·\·\·,a_t\left({\mathrm{\θ}}_p\right)\⊗a_r\left({\mathrm{\θ}}_p\right)]s\left(t\right)+n\left(t\right)$

In formula, P represents incoherent number of targets, and has P=1,2,3 ...; θ _pfor the direction of arrival of respective objects; a _t(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Μ-1) sin θ _p)] ^tfor launching steering vector, a _r(θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (Ν-1) sin θ _p)] ^tfor receiving steering vector; $s_p\left(t\right)={\mathrm{\β}}_p\left(t\right)e^{j2\mathrm{\π}{f}_{p}t},$ β _p(t) and f _prepresent reflection coefficient and Doppler frequency respectively; represent that zero-mean and covariance matrix are σ ²i _mNwhite Gaussian noise vector;

Take soon at J, the reception data obtained after matched filtering process can be expressed as:

X＝AS+N

In formula, $A=[a_t\left({\mathrm{\θ}}_1\right)\⊗a_r\left({\mathrm{\θ}}_1\right),\·\·\·,a_t\left({\mathrm{\θ}}_p\right)\⊗a_r\left({\mathrm{\θ}}_p\right)],$ X=[x (t ₁) ..., x (t _j)], S=[s (t ₁) ..., s (t _j)], N=[n (t ₁) ..., n (t _j)] be white Gaussian noise matrix;

(2) utilize dimensionality reduction matrix, dimension-reduction treatment is carried out to the reception data that J obtains after taking matched filtering process soon, after obtaining dimension-reduction treatment, receives data Y;

The expression formula receiving data Y after involved dimension-reduction treatment is:

Y＝WX＝F ^-(1/2)FBS+WN＝F ^(1/2)BS+WN

In formula, W is dimensionality reduction matrix, and has W=F ^-(1/2)g ^h; X is the reception data that receiving array matched filter exports;

B=[b (θ ₁), b (θ ₂) ..., b (θ _p)], wherein b (θ _p)=[1, exp (j π sin θ _p) ..., exp (j π (M+ Ν-2) sin θ _p)] ^t;

(3) receive the covariance matrix R of data Y after calculating dimension-reduction treatment, utilize Eigenvalues Decomposition to obtain noise subspace, and calculate noise subspace and its conjugation intersection subspace collected works space;

The involved expression formula receiving the covariance matrix R of data Y is:

$R=\frac{1}{J}{\mathrm{YY}}^H$

The expression formula of covariance matrix being carried out to Eigenvalues Decomposition is:

$R=U_s{\mathrm{\Λ}}_s{U}_s^H+U_n{\mathrm{\Λ}}_n{U}_n^H$

In formula, U _sfor the signal subspace be made up of P large eigenwert characteristic of correspondence vector, U _nfor the noise subspace be made up of M+N-1-P little eigenwert characteristic of correspondence vector, Λ _sfor diagonal matrix, and its diagonal element is made up of P large eigenwert; Λ _nfor diagonal matrix, its diagonal element M+N-1-P little eigenwert composition;

Involved noise subspace U _nand conjugation subspace common factor subspace expression formula is:

${\stackrel{\‾}{U}}_n=U_{n}E(:,1:M+M-1-2P)$

In formula, E is the matrix of left singularity characteristics vector composition, and E meets svd expression formula:

$D=E\stackrel{\‾}{\mathrm{\Λ}}{V}^H$

Wherein, v is the matrix of left singularity characteristics vector composition, for the diagonal matrix of singular value composition;

(4) construct compression stroke spectral function, compression stroke spectral function is searched for, and utilize spatial spectrum conjugate symmetry property to obtain the true direction of arrival of target ${\widehat{\mathrm{\θ}}}_P(P=\mathrm{1,2},...,P)$ With false direction of arrival $-{\widehat{\mathrm{\θ}}}_P(P=\mathrm{1,2},...,P);$

The expression formula of compression stroke spectral function is:

$P_c=\frac{1}{{\left[F^{(1/2)}b\left(\mathrm{\θ}\right)\right]}^H{\stackrel{\‾}{U}}_n{\stackrel{\‾}{U}}_n^H{F}^{(1/2)}b\left(\mathrm{\θ}\right)}$

(5) utilize true direction of arrival steering vector and noise subspace orthogonal property, get rid of false direction of arrival, obtain the true direction of arrival of target;

True direction of arrival steering vector and noise subspace orthogonal property expression formula are:

${\left|\right|{\left[F^{(1/2)}b\left({\mathrm{\θ}}_i\right)\right]}^H{\stackrel{\‾}{U}}_n\left|\right|}^2=0$

Be met minimum P value of above formula, then corresponding is exactly real target direction of arrival.