CN102385049A - Two-dimensional coherent signal direction estimation method based on double parallel arrays - Google Patents

Abstract

The invention discloses a two-dimensional coherent signal direction estimation method based on double parallel arrays. Two azimuths are solved by a cross covariance matrix between two linear arrays, the geometrical characteristic of the arrays and the rotational invariance of even linear arrays. In addition, the two azimuths do not need to be solved at the same time, and therefore calculation time is saved. According to the geometrical characteristic of an array group and the rotational invariance of the even arrays, an incident signal is decorrelated by the subarray smoothing technology; and the cross covariance matrix of the two arrays is used for lightening influence of nearby noise on zero space estimation. The two-dimensional coherent signal direction estimation method does not need eigenvalue decomposition, and the pairing of an azimuth and an angle of pitch can be prevented.

Description

Based on the relevant sense method of estimation of the two dimension of two parallel arrays

Technical field

The invention belongs to the estimation technique field of 2-d direction finding, the relevant sense method of estimation of particularly a kind of two dimension based on two parallel arrays.

Background technology

Array Signal Processing is developed rapidly in recent years, and its application relates to numerous military affairs and national economy fields such as radar, communication and sonar.In the Array Signal Processing field, estimate that the problem that many narrow band signals incide the 2-d direction finding that a sensor array lists is a gordian technique.Recently; Based on the 2-d direction finding estimation of the array of simple and geometry in particular, mostly the array of use is two or more uniform arrays, has received researcher's special concern; And in reality; Estimating in the time of the both direction angle also needs, particularly when incoming signal be when being concerned with, but estimation when this technical method also fails to accomplish the both direction angle at present; On the other hand, some subspace methods based on characteristic value decomposition appear in the 2-d direction finding estimation, use two parallel uniform arrays, can realize the automatic coupling at the position angle and the elevation angle.Yet the 2-d direction finding of coherent signal is estimated in these methods, also not to be considered.

Summary of the invention

In order to overcome the deficiency that above-mentioned prior art exists; The object of the present invention is to provide the relevant sense method of estimation of a kind of two dimension based on two parallel arrays; This method can be implemented efficiently; Add the geometrical property of having utilized array group and the invariable rotary characteristic of uniform array, come incoming signal decoherence property through the subarray smoothing technique, and utilize the cross covariance matrix of two arrays alleviate near the influence of noise in kernel is estimated; The present invention does not need characteristic value decomposition, and has avoided the pairing between the position angle and the angle of pitch.

In order to achieve the above object, the technical scheme that the present invention adopted is:

The relevant sense method of estimation of a kind of two dimension based on two parallel arrays, step is following:

Step 1: shown in accompanying drawing; Level to the surface level of vertical formation; Also be on the x-y plane, be provided with the uniform array of two mutual horizontal parallel in advance, these two uniform arrays are respectively first uniform array and second uniform array; Wherein first uniform array has M isotropic sensor, is followed successively by x ₁, x ₂..., x _M, second uniform array also has M isotropic sensor, is followed successively by y ₁, y ₂..., y _M, the distance in the distance between two uniform arrays and each uniform array between the neighboring sensors all is d, as the far field coherent signal { s that p wavelength is λ _k(n) } respectively from { α _k, β _kDirection incides on first uniform array and second uniform array, α wherein _kAnd β _kBe respectively far field coherent signal { s to be asked _k(n) } incident direction and level to and vertically between angle, the signal that such first uniform array and the sampling of second uniform array receive is expressed as formula (1) and formula (2) respectively:

y(n)＝A(α)s(n)+w _y(n) (1)

x(n)＝A(α)D(β)s(n)+w _x(n) (2)

Y (n)=[y wherein ₁(n), y ₂(n) ..., y _M(n)] ^T,

x(n)＝[x ₁(n)，x ₂(n)，...，x _M(n)] ^T，

s(n)＝[s ₁(n)，s ₂(n)，...，s _p(n)] ^T，

w _y(n)＝[w _y1(n)，w _y2(n)，...，w _yM(n)] ^T，

w _x(n)＝[w _x1(n)，w _x2(n)，...，w _xM(n)] ^T，

A(a)＝[a(α ₁)，a(α ₂)，...，a(α _p)]，

$a\left({\mathrm{\α}}_k\right)={[1,e^{j{\mathrm{\τ}}_k},...,e^{j(M-1){\mathrm{\τ}}_k}]}^T,$

$D\left(\mathrm{\β}\right)=\mathrm{diag}(e^{{\mathrm{j\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$

y ₁(n), y ₂(n) ..., y _M(n) represent the y that is followed successively by of second uniform array respectively ₁, y ₂..., y _MThe sampled signal of isotropic sensor, x ₁(n), x ₂(n) ..., x _M(n) represent the x that is followed successively by of first uniform array respectively ₁, x ₂..., x _MThe sampled signal of isotropic sensor, τ _k=2 π dcos α _k/ λ, γ _k=2 π dcos β _k/ λ, p and M are natural number and the M>=2p more than or equal to 1, and n is the signal sampling number, w _Y1(n), w _Y2(n) ..., w _YM(n) expression adds y respectively ₁(n), y ₂(n) ..., y _M(n) noise jamming signal, w _X1(n), w _X2(n) ..., w _XM(n) expression adds x respectively ₁(n), x ₂(n) ..., x _M(n) noise jamming signal, k are the natural number smaller or equal to p;

Step 2: the angle [alpha] of estimating relative y axle _k, be about to these two uniform arrays and at first be divided into p overlapping subarray, wherein each subarray comprises m=M-p+1 array element, and the sampled signal of l subarrays is expressed as the second even subarray sampled signal y of l subarrays _l(n)=[y _l(n), y _L+1(n) ..., y _L+m-1(n)] ^TThe first even subarray sampled signal x with the l subarrays _l(n)=[x _l(n), x _L+1(n) ..., x _L+m-1(n)] ^T, derive the first cross covariance matrix then

The second cross covariance matrix

The 3rd cross covariance matrix

With the 4th cross covariance matrix

The cross covariance matrix of four m * M, respectively suc as formula shown in (3), formula (4), formula (5) and the formula (6):

$R_{y_{l}x}=E\left\{y_l\left(n\right)x^H\left(n\right)\right\}$ (3)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)R_s{D}^{*}\left(\mathrm{\β}\right)A^H\left(\mathrm{\α}\right)$

$R_{x_{l}y}=E\left\{x_l\left(n\right)y^H\left(n\right)\right\}$ (4)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)D\left(\mathrm{\β}\right)R_s{A}^H\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{y_{l}x}=E\left\{J_m{y}_l^{*}\left(n\right)x^T\left(n\right)\right\}$ (5)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right){R_s}^{*}D\left(\mathrm{\β}\right)A^T\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{x_{l}y}=E\left\{J_m{x}_l^{*}\left(n\right)y^T\left(n\right)\right\}$ (6)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right)D^{*}\left(\mathrm{\β}\right){R_s}^{*}{A}^H\left(\mathrm{\α}\right)$

Wherein $D\left(\mathrm{\α}\right)=\mathrm{Diag}(e^{{\mathrm{J\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$ Be the submatrix of m * p, by capable composition of preceding m of A (α), J _nBe the opposition angular unit battle array of n * n, then, through combining these first cross covariance matrixes The second cross covariance matrix

The 3rd cross covariance matrix

With the 4th cross covariance matrix

L=1,2 ..., p, the expansion cross covariance matrix of derivation m * 4pM, shown in (7) and formula (8):

$R=[R_{y_{1}x},...,R_{y_{p}x},R_{x_{1}y},...,R_{x_{p}y},{\stackrel{\‾}{R}}_{y_{1}x},...,{\stackrel{\‾}{R}}_{y_{p}x},{\stackrel{\‾}{R}}_{x_{1}y},...,{\stackrel{\‾}{R}}_{x_{p}y}]---\left(7\right)$

According to M>=2p;

is divided into two submatrixs

$\stackrel{\‾}{A}\left(\mathrm{\α}\right)={\left[\begin{array}{c}{A}_1\left(\mathrm{\α}\right)\\ A_2\left(\mathrm{\α}\right)\end{array}\right]}_{\}m-q}^{\}q}---\left(8\right)$

Derive the linear operation matrix P of p * (m-p) from formula (3)～formula (7), shown in (9):

$P=A_1^{-H}\left(\mathrm{\α}\right)A_2^H\left(\mathrm{\α}\right)={\left(R_1{R}_1^H\right)}^{-1}{R}_1{R}_2^H---\left(9\right)$

R wherein ₁And R ₂Capable and the last m-p of preceding p that comprises R in the formula (7) respectively is capable, structural matrix Q=[P ^T,-I _M-p] ^T, according to Q ^HA (α)=O _{(m-p) * p}, I wherein _nAnd O _{N * q}The unit matrix of expression n * n and the null matrix of n * q;

Be limited according to sampling number n in reality again; Angle is obtained by the formula of minimizing (10)

$f\left(\mathrm{\α}\right)={\stackrel{\‾}{a}}^H\left(\mathrm{\α}\right)\widehat{\mathrm{\Π}}\stackrel{\‾}{a}\left(\mathrm{\α}\right)---\left(10\right)$

Wherein finding the solution of

used the matrix inversion lemma simplification;

is the estimation of variable x; P and M are natural number and the M>=2p more than or equal to 1; N is the signal sampling number, and k is the natural number smaller or equal to p;

Step 3: the angle beta of estimating relative x axle _k,, construct two vectors and be respectively secondary vector promptly according to the l subarrays

And primary vector

The mix vector of constructing two signal vectors and array acceptance vector again is respectively $z_l\left(n\right)={[y_l^T\left(n\right),x_l^T\left(n\right)]}^T$ With

Obtain the cross covariance matrix of two 2m * 2 (p-1) thus

With

Respectively suc as formula shown in (11) and the formula (12):

(11)

(12)

Wherein $\stackrel{\‾}{A}(\mathrm{\α},\mathrm{\β})={[{\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right),D\left(\mathrm{\β}\right){\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right)]}^T,$

$b_k\left(\mathrm{\α}\right)={[e^{j(k-1){\mathrm{\τ}}_1},e^{j(k-1){\mathrm{\τ}}_2},...,e^{j(k-1){\mathrm{\τ}}_p}]}^T,$ The expansion cross covariance matrix that obtains a 2m * 4p (p-1) then is suc as formula shown in (13):

Also obtain linear operation matrix thus suc as formula shown in (14):

$\stackrel{\‾}{P}=A_1^{-H}\left(\mathrm{\α}\right)A_2^H(\mathrm{\α},\mathrm{\β})={\left({\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_1^H\right)}^{-1}{\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_2^H---\left(14\right)$

Wherein, and

is the submatrix of

, and the Estimation of Spatial Spectrum of two dimension is by equation (15)

Derive; Wherein,

derive angle through solving equation (15) subsequently and angle

from the formula of minimizing (16)

Obtain, k=1 wherein, 2 ..., p.

Through utilizing two cross covariance matrixes between the linear array; Find the solution two position angles with the geometrical property of array and the invariable rotary characteristic of even linear array; And do not need to find the solution simultaneously two position angles, on calculating, saved the time, add the geometrical property of having utilized array group and the invariable rotary characteristic of uniform array; Come the incoming signal decorrelationization through the subarray smoothing technique; And the cross covariance matrix that utilizes two arrays alleviate near the influence of noise in kernel is estimated, the present invention does not need characteristic value decomposition, and azimuthal collaborative estimation problem has been avoided.

Description of drawings

Accompanying drawing is a work structuring principle schematic of the present invention.

Embodiment

Below in conjunction with accompanying drawing the present invention is done more detailed explanation.

Based on the relevant sense method of estimation of the two dimension of two parallel arrays, step is following:

Step 1: shown in accompanying drawing; Level to the surface level of vertical formation; Also be on the x-y plane, be provided with the uniform array of two mutual horizontal parallel in advance, these two uniform arrays are respectively first uniform array and second uniform array; Wherein first uniform array has M isotropic sensor, is followed successively by x ₁, x ₂..., x _M, second uniform array also has M isotropic sensor, is followed successively by y ₁, y ₂..., y _M, the distance in the distance between two uniform arrays and each uniform array between the neighboring sensors all is d, as the far field coherent signal { s that p wavelength is λ _k(n) } respectively from { α _k, β _kDirection incides on first uniform array and second uniform array, α wherein _kAnd β _kBe respectively far field coherent signal { s to be asked _k(n) } incident direction and level to and vertically between angle, the signal that such first uniform array and the sampling of second uniform array receive is expressed as formula (1) and formula (2) respectively:

y(n)＝A(α)s(n)+w _y(n) (1)

x(n)＝A(α)D(β)s(n)+w _x(n) (2)

Y (n)=[y wherein ₁(n), y ₂(n) ..., y _M(n)] ^T,

x(n)＝[x ₁(n)，x ₂(n)，...，x _M(n)] ^T，

s(n)＝[s ₁(n)，s ₂(n)，...，s _p(n)] ^T，

w _y(n)＝[w _y1(n)，w _y2(n)，...，w _yM(n)] ^T，

w _x(n)＝[w _x1(n)，w _x2(n)，...，w _xM(n)] ^T，

A(a)＝[a(α ₁)，a(α ₂)，...，a(α _p)]，

$a\left({\mathrm{\α}}_k\right)={[1,e^{j{\mathrm{\τ}}_k},...,e^{j(M-1){\mathrm{\τ}}_k}]}^T,$

$D\left(\mathrm{\β}\right)=\mathrm{diag}(e^{{\mathrm{j\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$

y ₁(n), y ₂(n) ..., y _M(n) represent the y that is followed successively by of second uniform array respectively ₁, y ₂..., y _MThe sampled signal of isotropic sensor, x ₁(n), x ₂(n) ..., x _M(n) represent the x that is followed successively by of first uniform array respectively ₁, x ₂..., x _MThe sampled signal of isotropic sensor, τ _k=2 π dcos α _k/ λ, γ _k=2 π dcos β _k/ λ, p and M are natural number and the M>=2p more than or equal to 1, and n is the signal sampling number, w _Y1(n), w _Y2(n) ..., w _YM(n) expression adds y respectively ₁(n), y ₂(n) ..., y _M(n) noise jamming signal, w _X1(n), w _X2(n) ..., w _XM(n) expression adds x respectively ₁(n), x ₂(n) ..., x _M(n) noise jamming signal, k are the natural number smaller or equal to p;

Step 2: the angle [alpha] of estimating relative y axle _k, be about to these two uniform arrays and at first be divided into an overlapping subarray of p, wherein each subarray comprises m=M-p+1 array element, and the sampled signal of l subarrays is expressed as the second even subarray sampled signal y of l subarrays _l(n)=[y _l(n), y _L+1(n) ..., y _L+m-1(n)] ^TThe first even subarray sampled signal x with the l subarrays _l(n)=[x _l(n), x _L+1(n) ..., x _L+m-1(n)] ^T, derive the first cross covariance matrix then

The second cross covariance matrix

The 3rd cross covariance matrix

With the 4th cross covariance matrix

The cross covariance matrix of four m * M, respectively suc as formula shown in (3), formula (4), formula (5) and the formula (6):

$R_{y_{l}x}=E\left\{y_l\left(n\right)x^H\left(n\right)\right\}$ (3)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)R_s{D}^{*}\left(\mathrm{\β}\right)A^H\left(\mathrm{\α}\right)$

$R_{x_{l}y}=E\left\{x_l\left(n\right)y^H\left(n\right)\right\}$ (4)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)D\left(\mathrm{\β}\right)R_s{A}^H\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{y_{l}x}=E\left\{J_m{y}_l^{*}\left(n\right)x^T\left(n\right)\right\}$ (5)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right){R_s}^{*}D\left(\mathrm{\β}\right)A^T\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{x_{l}y}=E\left\{J_m{x}_l^{*}\left(n\right)y^T\left(n\right)\right\}$ (6)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right)D^{*}\left(\mathrm{\β}\right){R_s}^{*}{A}^H\left(\mathrm{\α}\right)$

Wherein $D\left(\mathrm{\α}\right)=\mathrm{Diag}(e^{{\mathrm{J\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$

Be the submatrix of m * p, by capable composition of preceding m of A (α), J _nBe the opposition angular unit battle array of n * n, then, through combining these first cross covariance matrixes

The second cross covariance matrix

The 3rd cross covariance matrix

With the 4th cross covariance matrix

L=1,2 ..., p, the expansion cross covariance matrix of derivation m * 4pM, shown in (7) and formula (8):

$R=[R_{y_{1}x},...,R_{y_{p}x},R_{x_{1}y},...,R_{x_{p}y},{\stackrel{\‾}{R}}_{y_{1}x},...,{\stackrel{\‾}{R}}_{y_{p}x},{\stackrel{\‾}{R}}_{x_{1}y},...,{\stackrel{\‾}{R}}_{x_{p}y}]---\left(7\right)$

According to M>=2p;

is divided into two submatrixs

$\stackrel{\‾}{A}\left(\mathrm{\α}\right)={\left[\begin{array}{c}{A}_1\left(\mathrm{\α}\right)\\ A_2\left(\mathrm{\α}\right)\end{array}\right]}_{\}m-q}^{\}q}---\left(8\right)$

Derive the linear operation matrix P of p * (m-p) from formula (3) and formula (7), shown in (9):

$P=A_1^{-H}\left(\mathrm{\α}\right)A_2^H\left(\mathrm{\α}\right)={\left(R_1{R}_1^H\right)}^{-1}{R}_1{R}_2^H---\left(9\right)$

R wherein ₁And R ₂Capable and the last m-p of preceding p that comprises R in the formula (7) respectively is capable, structural matrix Q=[P ^T,-I _M-p] ^T, according to Q ^HA (α)=O _{(m-p) * p}, I wherein _nAnd O _{N * q}The unit matrix of expression n * n and the null matrix of n * q;

Be limited according to sampling number n in reality again; Angle

is obtained by the formula of minimizing (10)

$f\left(\mathrm{\α}\right)={\stackrel{\‾}{a}}^H\left(\mathrm{\α}\right)\widehat{\mathrm{\Π}}\stackrel{\‾}{a}\left(\mathrm{\α}\right)---\left(10\right)$

Wherein finding the solution of

used the matrix inversion lemma simplification; is the estimation of variable x; P and M are natural number and the M>=2p more than or equal to 1; N is the signal sampling number, and k is the natural number smaller or equal to p;

Step 3: the angle beta of estimating relative x axle _k,, construct two vectors and be respectively secondary vector promptly according to the l subarrays

And primary vector

The mix vector of constructing two signal vectors and array acceptance vector again is respectively $z_l\left(n\right)={[y_l^T\left(n\right),x_l^T\left(n\right)]}^T$ With

Obtain the cross covariance matrix of two 2m * 2 (p-1) thus With

Respectively suc as formula shown in (11) and the formula (12):

(11)

(12)

Wherein $\stackrel{\‾}{A}(\mathrm{\α},\mathrm{\β})={[{\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right),D\left(\mathrm{\β}\right){\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right)]}^T,$

$b_k\left(\mathrm{\α}\right)={[e^{j(k-1){\mathrm{\τ}}_1},e^{j(k-1){\mathrm{\τ}}_2},...,e^{j(k-1){\mathrm{\τ}}_p}]}^T,$ The expansion cross covariance matrix that obtains a 2m * 4p (p-1) then is suc as formula shown in (13):

Also obtain linear operation matrix

thus suc as formula shown in (14):

$\stackrel{\‾}{P}=A_1^{-H}\left(\mathrm{\α}\right)A_2^H(\mathrm{\α},\mathrm{\β})={\left({\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_1^H\right)}^{-1}{\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_2^H---\left(14\right)$

Wherein, and

is the submatrix of

, and the Estimation of Spatial Spectrum of two dimension is by equation (15)

Derive; Wherein,

derive angle

through solving equation (15) subsequently and angle from the formula of minimizing (16)

Obtain, k=1 wherein, 2 ..., p.

Claims (1)

1. one kind based on the relevant sense method of estimation of the two dimension of two parallel arrays, it is characterized in that step is following:

Step 1: shown in accompanying drawing; Level to the surface level of vertical formation; Also be on the x-y plane, be provided with the uniform array of two mutual horizontal parallel in advance, these two uniform arrays are respectively first uniform array and second uniform array; Wherein first uniform array has M isotropic sensor, is followed successively by x ₁, x ₂..., x _M, second uniform array also has M isotropic sensor, is followed successively by y ₁, y ₂..., y _M, the distance in the distance between two uniform arrays and each uniform array between the neighboring sensors all is d, as the far field coherent signal { s that p wavelength is λ _k(n) } respectively from { α _k, β _kDirection incides on first uniform array and second uniform array, α wherein _kAnd β _kBe respectively far field coherent signal { s to be asked _k(n) } incide on first uniform array and second uniform array with respect to horizontal direction with longitudinally along counterclockwise and the angle of measuring, the signal that such first uniform array and the sampling of second uniform array receive is expressed as formula (1) and formula (2) respectively:

y(n)＝A(α)s(n)+w _y(n) (1)

x(n)＝A(α)D(β)s(n)+w _x(n) (2)

Y (n)=[y wherein ₁(n), y ₂(n) ..., y _M(n)] ^T,

x(n)＝[x ₁(n)，x ₂(n)，...，x _M(n)] ^T，

s(n)＝[s ₁(n)，s ₂(n)，...，s _p(n)] ^T，

w _y(n)＝[w _y1(n)，w _y2(n)，...，w _yM(n)] ^T，

w _x(n)＝[w _x1(n)，w _x2(n)，...，w _xM(n)] ^T，

A(a)＝[a(α ₁)，a(α ₂)，...，a(α _p)]，

$a\left({\mathrm{\α}}_k\right)={[1,e^{j{\mathrm{\τ}}_k},...,e^{j(M-1){\mathrm{\τ}}_k}]}^T,$

$D\left(\mathrm{\β}\right)=\mathrm{diag}(e^{{\mathrm{j\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$

y ₁(n), y ₂(n) ..., y _M(n) represent the y that is followed successively by of second uniform array respectively ₁, y ₂..., y _MThe sampled signal of isotropic sensor, x ₁(n), x ₂(n) ..., x _M(n) represent the x that is followed successively by of first uniform array respectively ₁, x ₂..., x _MThe sampled signal of isotropic sensor, τ _k=2 π dcos α _k/ λ, γ _k=2 π dcos β _k/ λ, p and M are natural number and the M>=2p more than or equal to 1, and n is the signal sampling number, w _Y1(n), w _Y2(n) ..., w _YM(n) expression adds y respectively ₁(n), y ₂(n) ..., y _M(n) noise jamming signal, w _X1(n), w _X2(n) ..., w _XM(n) expression adds x respectively ₁(n), x ₂(n) ..., x _M(n) noise jamming signal, k are the natural number smaller or equal to p;

Step 2: the angle [alpha] of estimating relative y axle _k, be about to these two uniform arrays and at first be divided into p overlapping subarray, wherein each subarray comprises m=M-p+1 array element, and the sampled signal of l subarrays is expressed as the second even subarray sampled signal y of l subarrays _l(n)=[y _l(n), y _L+1(n) ..., y _L+m-1(n)] ^TThe first even subarray sampled signal x with the l subarrays _l(n)=[x _l(n), x _L+1(n) ..., x _L+m-1(n)] ^T, derive the first cross covariance matrix then

The second cross covariance matrix The 3rd cross covariance matrix

With the 4th cross covariance matrix

The cross covariance matrix of four m * M, respectively suc as formula shown in (3), formula (4), formula (5) and the formula (6):

$R_{y_{l}x}=E\left\{y_l\left(n\right)x^H\left(n\right)\right\}$ (3)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)R_s{D}^{*}\left(\mathrm{\β}\right)A^H\left(\mathrm{\α}\right)$

$R_{x_{l}y}=E\left\{x_l\left(n\right)y^H\left(n\right)\right\}$ (4)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{l-1}\left(\mathrm{\α}\right)D\left(\mathrm{\β}\right)R_s{A}^H\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{y_{l}x}=E\left\{J_m{y}_l^{*}\left(n\right)x^T\left(n\right)\right\}$ (5)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right){R_s}^{*}D\left(\mathrm{\β}\right)A^T\left(\mathrm{\α}\right)$

${\stackrel{\‾}{R}}_{x_{l}y}=E\left\{J_m{x}_l^{*}\left(n\right)y^T\left(n\right)\right\}$ (6)

$=\stackrel{\‾}{A}\left(\mathrm{\α}\right)D^{-(m+l-2)}\left(\mathrm{\α}\right)D^{*}\left(\mathrm{\β}\right){R_s}^{*}{A}^H\left(\mathrm{\α}\right)$

Wherein $D\left(\mathrm{\α}\right)=\mathrm{Diag}(e^{{\mathrm{J\γ}}_1},e^{j{\mathrm{\γ}}_2},...,e^{j{\mathrm{\γ}}_p}),$ Be the submatrix of m * p, by capable composition of preceding m of A (α), J _nBe the opposition angular unit battle array of n * n, then, through combining these first cross covariance matrixes

The second cross covariance matrix

The 3rd cross covariance matrix

With the 4th cross covariance matrix

L=1,2 ..., p, the expansion cross covariance matrix of derivation m * 4pM, shown in (7) and formula (8):

$R=[R_{y_{1}x},...,R_{y_{p}x},R_{x_{1}y},...,R_{x_{p}y},{\stackrel{\‾}{R}}_{y_{1}x},...,{\stackrel{\‾}{R}}_{y_{p}x},{\stackrel{\‾}{R}}_{x_{1}y},...,{\stackrel{\‾}{R}}_{x_{p}y}]---\left(7\right)$

According to M>=2p;

is divided into two submatrixs

$\stackrel{\‾}{A}\left(\mathrm{\α}\right)={\left[\begin{array}{c}{A}_1\left(\mathrm{\α}\right)\\ A_2\left(\mathrm{\α}\right)\end{array}\right]}_{\}m-q}^{\}q}---\left(8\right)$

Derive the linear operation matrix P of p * (m-p) from formula (3) and formula (7), shown in (9):

$P=A_1^{-H}\left(\mathrm{\α}\right)A_2^H\left(\mathrm{\α}\right)={\left(R_1{R}_1^H\right)}^{-1}{R}_1{R}_2^H---\left(9\right)$

R wherein ₁And R ₂Capable and the last m-p of preceding p that comprises R in the formula (7) respectively is capable, structural matrix Q=[P ^T,-I _M-p] ^T, according to Q ^HA (α)=O _{(m-p) * p}, I wherein _nAnd O _{N * q}The unit matrix of expression n * n and the null matrix of n * q;

Be limited according to sampling number n in reality again; Angle

is obtained by the formula of minimizing (10)

$f\left(\mathrm{\α}\right)={\stackrel{\‾}{a}}^H\left(\mathrm{\α}\right)\widehat{\mathrm{\Π}}\stackrel{\‾}{a}\left(\mathrm{\α}\right)---\left(10\right)$

Wherein finding the solution of

used the matrix inversion lemma simplification; is the estimation of variable x; P and M are natural number and the M>=2p more than or equal to 1; N is the signal sampling number, and k is the natural number smaller or equal to p;

Step 3: the angle beta of estimating relative x axle _k,, construct two vectors and be respectively secondary vector promptly according to the l subarrays

And primary vector

The mix vector of constructing two signal vectors and array acceptance vector again is respectively $z_l\left(n\right)={[y_l^T\left(n\right),x_l^T\left(n\right)]}^T$ With

Obtain the cross covariance matrix of two 2m * 2 (p-1) thus With

Respectively suc as formula shown in (11) and the formula (12):

(11)

(12)

Wherein $\stackrel{\‾}{A}(\mathrm{\α},\mathrm{\β})={[{\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right),D\left(\mathrm{\β}\right){\stackrel{\‾}{A}}^T\left(\mathrm{\α}\right)]}^T,$

$b_k\left(\mathrm{\α}\right)={[e^{j(k-1){\mathrm{\τ}}_1},e^{j(k-1){\mathrm{\τ}}_2},...,e^{j(k-1){\mathrm{\τ}}_p}]}^T,$ The expansion cross covariance matrix that obtains a 2m * 4p (p-1) then is suc as formula shown in (13):

Also obtain linear operation matrix

thus suc as formula shown in (14):

$\stackrel{\‾}{P}=A_1^{-H}\left(\mathrm{\α}\right)A_2^H(\mathrm{\α},\mathrm{\β})={\left({\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_1^H\right)}^{-1}{\stackrel{\‾}{R}}_1{\stackrel{\‾}{R}}_2^H---\left(14\right)$

Wherein,

and is the submatrix of , and the Estimation of Spatial Spectrum of two dimension is by equation (15)

Derive; Wherein,

derive angle through solving equation (15) subsequently and angle

from the formula of minimizing (16)

Obtain, k=1 wherein, 2 ..., p.

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