CN103777179B - A kind of submatrix dimension reduction method for the three-dimensional conformal array of radar - Google Patents

Abstract

The invention belongs to radar array antenna signal processing technology field, disclose a kind of submatrix dimension reduction method for the three-dimensional conformal array of radar.The method can be used for the multiple conformal array of solid with turning axle, as frustum battle array, cylindrical array or semicylinder battle array, spherical array etc.Method of the present invention is: first carry out Subarray partition to the three-dimensional conformal array of radar; Then three-dimensional rotation transformation equation is utilized to adjust each submatrix beam position; Finally calculate dimensionality reduction matrix.The present invention can correspondingly adjust its beam position according to the three-dimensional rotation of each submatrix, to reach the object ensureing that the beam position of whole front is still consistent after dimensionality reduction.

Description

A kind of submatrix dimension reduction method for the three-dimensional conformal array of radar

Technical field

The invention belongs to radar array antenna signal processing technology field, relate to a kind of submatrix dimension reduction method for the three-dimensional conformal array of radar.The method can be used for the multiple conformal array of solid with turning axle, as frustum battle array, cylindrical array or semicylinder battle array, spherical array etc.

Background technology

In modern antennas technical field, array antenna is applied more and more widely in the radio systems such as communication, radar, sonar, exploration, navigation, radio astronomy and biomedical engineering.Compared to traditional antenna, array antenna has the advantage such as wave beam control flexibly, higher signal gain, extremely strong antijamming capability and high-precision space hyperresolution, thus receives the very big concern of people.Simultaneously along with the fast development of microelectric technique, Digital Signal Processing, parallel processing technique, the function of array antenna is increasingly powerful, and range of application is increasingly extensive.

Array antenna is broadly divided into from array structure: linear array, planar array and three-dimensional array.There is many defects in the linear array generally adopted at present and planar array, as: beam scanning narrow range; Mutual coupling effect between antenna element is relevant to scan angle.Three-dimensional array not only can effectively avoid above-mentioned shortcoming, and have conformal with carrier surface, do not affect carrier gas dynamic performance, simplify the advantages such as astronomical cycle.

But the element number of array of large stereo array is hundreds of even several thousand often, if directly carry out the process of array element level, one be need the receiving cable identical with array element quantity, A/D converts and weighting process etc., system architecture complexity, hardware cost is high; Two is when carrying out self-adaptive processing, needs more fast umber of beats, makes the operand of the correlation computations such as the adaptive algorithm of array huge, be difficult to requirement of real time.

Mostly traditional submatrix dimension-reduction treatment is for linear array and planar array, but three-dimensional conformal array can not be applied to, because there is following problem demanding prompt solution in it: compared to planar array dimensionality reduction, not only Existential Space translation between the submatrix in three-dimensional conformal array dimensionality reduction, also there is three-dimensional rotation, and traditional dimension reduction method does not adjust the beam position of submatrix, therefore after cannot ensureing dimensionality reduction, the beam position of whole front is still consistent.

Summary of the invention

The object of the present invention is to provide a kind of submatrix dimension reduction method for the three-dimensional conformal array of radar, traditional submatrix dimensionality reduction technology can be solved and can only be used for linear array and face battle array and the problem that can not be used for three-dimensional conformal array, the method can correspondingly adjust its beam position according to the three-dimensional rotation of each submatrix, to reach the object ensureing that the beam position of whole front is still consistent after dimensionality reduction.

General thought of the present invention is: first carry out Subarray partition to the three-dimensional conformal array of radar; Then three-dimensional rotation transformation equation is utilized to adjust each submatrix beam position; Finally calculate dimensionality reduction matrix.

For a submatrix dimension reduction method for the three-dimensional conformal array of radar, the conformal array of wherein said solid has turning axle, it is characterized in that comprising the following steps:

(1) Subarray partition is carried out to the three-dimensional conformal array of radar, determine submatrix number and each submatrix array number, and the three-dimensional system of coordinate that settles the standard, original submatrix, initial three-dimensional coordinate system, relatively submatrix, relative dimensional coordinate system;

(2) Criterion three-dimensional system of coordinate (XYZO), its Z axis is by the turning axle of the conformal array of described solid, and the pitching of setting beam position in standard three dimensional coordinate system (XYZO), position angle are respectively θ ₀, translation standard three dimensional coordinate system (XYZO) obtains the initial three-dimensional coordinate system (X that initial point is positioned at original submatrix phase center ₁y ₁z ₁o ₁), wherein angle ∠ O ₁oZ is the pitching angle theta of original submatrix ₁, setting initial point O ₁be D at the intersection point of the XOY plane of standard three dimensional coordinate system (XYZO) ₁point, wherein angle ∠ XOD ₁for the position angle of original submatrix if for beam position is at initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in direction vector, be known quantity, wherein P ₁point is at initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in coordinate be (x ₁, y ₁, z ₁); Original submatrix and initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) rotate around the turning axle of the conformal array of solid, obtain corresponding submatrix and relative dimensional coordinate system (X relatively. simultaneously ₂y ₂z ₂o ₂), if initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in P ₁point obtains relative dimensional coordinate system (X after rotating ₂y ₂z ₂o ₂) in P ₂point, therefore P ₂point is at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in coordinate be also (x ₁, y ₁, z ₁), namely definition angle ∠ O ₂oZ is the pitching angle theta of relative submatrix ₂, setting initial point O ₂be D at the intersection point of the XOY plane of standard three dimensional coordinate system (XYZO) ₂point, wherein angle ∠ XOD ₂for the position angle of relative submatrix if for beam position is at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector, be unknown quantity; Definition θ, for relative submatrix and the pitching of original submatrix, the difference of orientation angles, i.e. θ=θ ₂-θ ₁,

(3) at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in, if P ₂point is to plane X ₂o ₂y ₂intersection point be E1, definition for vector position angle, ∠ P ₂o ₂z ₂=θ ₁' be vector the angle of pitch; If Q ₂point is to plane X ₂o ₂y ₂intersection point be E ₂, definition for vector position angle, ∠ Q ₂o ₂z ₂=θ ' ₂for vector the angle of pitch; Definition θ ', be respectively vector with vector pitching, orientation angles difference, i.e. θ '=θ ' ₂-θ ₁',

(4) utilize relative submatrix beam position adjustment three-dimensional rotation transformation equation group, calculate the beam position of relative submatrix at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector

Wherein, relative submatrix beam position adjustment three-dimensional rotation transformation equation group:

$z_2=z_1\cos {\mathrm{\θ}}^{\′}-\sqrt{x_1^2+y_1^2}\text{sin}{\mathrm{\θ}}^{\′}$

Wherein (x ₁, y ₁, z ₁) be vector at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in coordinate, be known quantity; θ ', for vector with vector pitching, orientation angles difference, be known quantity; for relative submatrix beam position is at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector, be unknown quantity;

Further, utilize the beam position of relative submatrix at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector according to following following formulae discovery steering vector:

$a_{\mathrm{sub}}(\mathrm{\θ},\mathrm{\φ})=[e^{-j2\mathrm{\π}{f}_0{t}_{11}},e^{-j2\mathrm{\π}{f}_0{t}_{12}},...e^{-j2\mathrm{\π}{f}_0{t}_{1N}},...e^{-j2\mathrm{\π}{f}_0{t}_{M1}},e^{-j2\mathrm{\π}{f}_0{t}_{M2}},...e^{-j2\mathrm{\π}{f}_0{t}_{\mathrm{MN}}}]$

Wherein t _ij=(x _ij* x ₂+ y _ij* y ₂+ z _ij* z ₂)/c, (x _ij, y _ij, z _ij) be relative submatrix i-th row, the three-dimensional coordinate of jth array unit in standard three dimensional coordinate system (XYZO), i=1,2, M; J=1,2, N; C is the light velocity, f ₀for signal frequency;

(5) repeat step (4), calculate the direction vector of beam position in the relative dimensional coordinate system of its correspondence of each relative submatrix in the three-dimensional conformal array of radar; And the steering vector that the beam position calculating each relative submatrix is corresponding;

(6) dimensionality reduction matrix is built

Steering vector corresponding for the beam position of each relative submatrix is substituted into following formula and forms dimensionality reduction matrix:

Wherein, a _kfor the steering vector of a kth submatrix wherein k=1,2, L.

The present invention is used for the submatrix dimension reduction method of the three-dimensional conformal array of radar, solve traditional submatrix dimensionality reduction technology and can only be used for linear array and face battle array and the problem that can not be used for three-dimensional conformal array, the method can correspondingly adjust its beam position according to the three-dimensional rotation of each submatrix, to reach the object ensureing that the beam position of whole front is still consistent after dimensionality reduction; And there is following following characteristics and advantage:

(1) utilize three-dimensional rotation transformation equation to adjust each submatrix beam position, after ensureing whole array dimensionality reduction, beam position is still consistent, thus realizes three-dimensional conformal array dimensionality reduction.

(2) the conformal formation of multiple solid is applicable to, as frustum battle array, cylindrical array or semicylinder battle array, spherical array etc.; The Subarray partition of the various ways such as array number in a burst of type of anyon, arbitrarily submatrix number, arbitrarily submatrix, submatrix array element is overlapping or not overlapping can be carried out to three-dimensional conformal array row according to subsequent treatment and application demand.

Accompanying drawing explanation

Below in conjunction with the drawings and specific embodiments, the present invention is described in further detail.

Fig. 1 is algorithm flow chart of the present invention;

Fig. 2 is for the coordinate system of frustum battle array, submatrix definition schematic diagram in the present invention;

Fig. 3 is three-dimensional rotation transformation equation group derivation schematic diagram in the present invention;

Fig. 4 is the frustum battle array evenly not overlapping schematic diagram being divided into 16 submatrixs;

Fig. 5 is the frustum battle array evenly not overlapping beam pattern being divided into 16 submatrix dimensionality reductions;

Fig. 6 is the non-homogeneous not overlapping schematic diagram being divided into 8 submatrixs of frustum battle array;

Fig. 7 is the non-homogeneous not overlapping beam pattern being divided into 8 submatrix dimensionality reductions of frustum battle array;

Fig. 8 is the schematic diagram that the even overlap of frustum battle array is divided into 16 submatrixs;

Fig. 9 is the beam pattern that the even overlap of frustum battle array is divided into 16 submatrix dimensionality reductions;

Embodiment

With reference to Fig. 1, be algorithm flow chart of the present invention, comprise three processes by flow diagram algorithm of the present invention: the three-dimensional conformal array of radar carries out Subarray partition process; Three-dimensional rotation transformation equation is utilized to calculate each submatrix beam position process; Calculate dimensionality reduction matrix process.Wherein utilizing three-dimensional rotation transformation equation to adjust each submatrix beam position process is core of the present invention.The concrete methods of realizing of each process is described in detail below in conjunction with Fig. 2 to Fig. 9.

1, the three-dimensional conformal array of radar carries out Subarray partition process

1. Subarray partition is carried out to three-dimensional conformal array row, determine submatrix number and each submatrix array number, and the three-dimensional system of coordinate that settles the standard, original submatrix, initial three-dimensional coordinate system, relatively submatrix, relative dimensional coordinate system;

2. as shown in Figure 2, Criterion three-dimensional system of coordinate (XYZO), its Z axis is by the turning axle of the conformal array of described solid, and the pitching of setting beam position in standard three dimensional coordinate system (XYZO), position angle are respectively θ ₀, translation standard three dimensional coordinate system (XYZO) obtains the initial three-dimensional coordinate system (X that initial point is positioned at original submatrix phase center ₁y ₁z ₁o ₁), wherein angle ∠ O ₁oZ is the pitching angle theta of original submatrix ₁, setting initial point O ₁be D at the intersection point of the XOY plane of standard three dimensional coordinate system (XYZO) ₁point, wherein angle ∠ XOD ₁for the position angle of original submatrix if for beam position is at initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in direction vector, be known quantity, wherein P ₁point is at initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in coordinate be (x ₁, y ₁, z ₁); Original submatrix and initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) rotate around the turning axle of the conformal array of solid, obtain corresponding submatrix and relative dimensional coordinate system (X relatively. simultaneously ₂y ₂z ₂o ₂), if initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in P ₁point obtains relative dimensional coordinate system (X after rotating ₂y ₂z ₂o ₂) in P ₂point, therefore P ₂point is at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in coordinate be also (x ₁, y ₁, z ₁), namely definition angle ∠ O ₂oZ is the pitching angle theta of relative submatrix ₂, setting initial point O ₂be D at the intersection point of the XOY plane of standard three dimensional coordinate system (XYZO) ₂point, wherein angle ∠ XOD ₂for the position angle of relative submatrix if for beam position is at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector, be unknown quantity; Definition θ, for relative submatrix and the pitching of original submatrix, the difference of orientation angles, i.e. θ=θ ₂-θ ₁,

3. as shown in Figure 3, at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in, if P ₂point is to plane X ₂o ₂y ₂intersection point be E ₁, definition for vector position angle, ∠ P ₂o ₂z ₂=θ ₁' be vector the angle of pitch; If Q ₂point is to plane X ₂o ₂y ₂intersection point be E ₂, definition for vector position angle, ∠ Q ₂o ₂z ₂=θ ' ₂for vector the angle of pitch; Definition θ ', for vector with vector pitching, orientation angles difference, i.e. θ '=θ ' ₂-θ ₁',

2, relative submatrix Pointing calculation process

(this sentences frustum battle array is example as shown in Figure 2, other formations have similar situation), for the conformal array of solid, not only Existential Space translation also Existential Space rotation between each submatrix, therefore, if the beam position of whole front is still identical after making dimensionality reduction, needs the beam position adjusting each submatrix, namely needed vector to vector sensing adjustment.

Be illustrated in figure 3 the derivation schematic diagram of three-dimensional rotation transformation equation group, and relative dimensional coordinate system (X in Fig. 2 ₂y ₂z ₂o ₂) corresponding.According to Givens rotational transform principle, make vector be transformed to vector rotational transform be: wherein it is known quantity; it is unknown quantity; T is three-dimensional Givens matrix.So three-dimensional Givens matrix T only need be known, vector can be obtained but Givens matrix there is no closed solutions in three-dimensional system of coordinate, therefore need derivation three-dimensional rotation transformation equation group.

As shown in Figure 3, if vector pitching, position angle be respectively θ ₁', vector pitching, position angle be respectively θ ' ₂, pitching between two vectors, the difference of orientation angles are θ '=θ ' ₂-θ ₁', because vector be the direction vector of beam position, be unit vector, so modulus value is equal, therefore have:

Abbreviation obtains:

Wherein substitution above formula obtains:

In like manner can obtain:

$z_2=z_1\cos {\mathrm{\θ}}^{\′}-\sqrt{x_1^2+y_1^2}\text{sin}{\mathrm{\θ}}^{\′}$

Consider vector for:

Substitute into above-mentioned equation to obtain vector:

From Space Rotating relation: vector with vector pitching, orientation angles difference θ ', equal the pitching of relative submatrix and original submatrix, the difference θ of orientation angles, i.e. θ '=θ, and vector at relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in pitching, azimuth angle theta ₁', equal vector at initial three-dimensional coordinate system (X ₁y ₁z ₁o ₁) in pitching, position angle, also equal the pitching of beam position in standard three dimensional coordinate system, azimuth angle theta ₀, so above formula can be written as:

By the pitching of beam position in standard three dimensional coordinate system, azimuth angle theta ₀, the pitching of relative submatrix and original submatrix, the difference θ of orientation angles, bring above formula into and calculate vector

3, dimensionality reduction matrix computations process

Utilize above-mentioned required beam position at each relative dimensional coordinate system (X ₂y ₂z ₂o ₂) in direction vector according to following formulae discovery steering vector:

$a_{\mathrm{sub}}(\mathrm{\θ},\mathrm{\φ})=[e^{-j2\mathrm{\π}{f}_0{t}_{11}},e^{-j2\mathrm{\π}{f}_0{t}_{12}},...e^{-j2\mathrm{\π}{f}_0{t}_{1N}},...e^{-j2\mathrm{\π}{f}_0{t}_{M1}},e^{-j2\mathrm{\π}{f}_0{t}_{M2}},...e^{-j2\mathrm{\π}{f}_0{t}_{\mathrm{MN}}}]$

Wherein t _ij=(x _ij* x ₂+ y _ij* y ₂+ z _ij* z ₂)/c, (x _ij, y _ij, z _ij) be relative submatrix i-th row, the three-dimensional coordinate of jth array unit in standard three dimensional coordinate system (XYZO), for the direction vector of beam position in relative dimensional coordinate system, c is the light velocity, f ₀for signal frequency;

Each steering vector is substituted into following formula and forms dimensionality reduction matrix:

Wherein a _kfor the steering vector of a kth submatrix wherein k=1,2, L.

4, dimensionality reduction matrix use-case

Utilize above-mentioned dimensionality reduction matrix T to carry out dimensionality reduction to three-dimensional conformal array row and to form directional diagram, method is as follows:

1. set array element level weights as W _ele, Subarray weights are W _sub, then W _sub=T ^h* W _ele.

2. array element level is set to scan as A _ele, Subarray scanning is A _sub, then A _sub=T ^h* A _ele.

3. beam pattern Pattern is:

Wherein T ^hthe conjugate transpose of representing matrix T, represent that Subarray weights are W _subconjugate transpose.

With reference to Fig. 4, be for frustum battle array, carry out the three-dimensional plot of evenly not overlapping Subarray partition.On this frustum battle array bus, array number is 4, and each circumferentially array number is 48,4 layers circumferentially every 3 array elements divide 16 submatrixs, beam position orientation, pitching are respectively 0 °, 30 °.Fig. 5 is the beam pattern after dimensionality reduction, can be obtained by Fig. 5: crest is oriented to 0 °, orientation, pitching 30 °.Proof this method achieves the solid array dimensionality reduction that classic method cannot realize accurately and effectively.

With reference to Fig. 6, be for frustum battle array, carry out the three-dimensional plot of non-homogeneous not overlapping Subarray partition.On this frustum battle array bus, array number is 4, and each circumferentially array number is 48.The submatrix number divided is 8, and each submatrix is respectively 3,4,6,8,9,5,6,7 at each array number circumferentially.Beam position orientation, the angle of pitch are respectively 0 °, 30 °.Fig. 7 is the beam pattern after dimensionality reduction.Can be obtained by Fig. 7: crest is oriented to 0 °, orientation, pitching 30 °.Proof this method achieves the solid array dimensionality reduction that classic method cannot realize accurately and effectively.

With reference to Fig. 8, be for frustum battle array, carry out the three-dimensional plot of even overlapping Subarray partition.On this frustum battle array bus, array number is 4, and each circumferentially array number is 48.The submatrix number divided is 16, and each submatrix 4 row 4 arranges, 1 array unit overlapping with adjacent submatrix.Beam position orientation, the angle of pitch are respectively 0 °, 30 °.Fig. 9 is the beam pattern after dimensionality reduction.Can be obtained by Fig. 9: crest is oriented to 0 °, orientation, pitching 30 °.Proof this method achieves the solid array dimensionality reduction that classic method cannot realize accurately and effectively.

Obviously, those skilled in the art can carry out various change and modification to the present invention and not depart from the spirit and scope of the present invention.Like this, if these amendments of the present invention and modification belong within the scope of the claims in the present invention and equivalent technologies thereof, then the present invention is also intended to comprise these change and modification.

Claims (1)

1., for a submatrix dimension reduction method for the three-dimensional conformal array of radar, the conformal array of wherein said solid has turning axle, it is characterized in that comprising the following steps:

(1) Subarray partition is carried out to the three-dimensional conformal array of radar, determine submatrix number and each submatrix array number, and the three-dimensional system of coordinate that settles the standard, original submatrix, initial three-dimensional coordinate system, relatively submatrix, relative dimensional coordinate system;

(2) Criterion three-dimensional system of coordinate XYZO, its Z axis is by the turning axle of the conformal array of described solid, and the pitching of setting beam position in standard three dimensional coordinate system XYZO, position angle are respectively θ ₀, translation standard three dimensional coordinate system XYZO obtains the initial three-dimensional coordinate system X that initial point is positioned at original submatrix phase center ₁y ₁z ₁o ₁, wherein angle ∠ O ₁oZ is the pitching angle theta of original submatrix ₁, setting initial point O ₁be D at the intersection point of the XOY plane of standard three dimensional coordinate system XYZO ₁point, wherein angle ∠ XOD ₁for the position angle of original submatrix if for beam position is at initial three-dimensional coordinate system X ₁y ₁z ₁o ₁in direction vector, be known quantity, wherein P ₁point is at initial three-dimensional coordinate system X ₁y ₁z ₁o ₁in coordinate be (x ₁, y ₁, z ₁); Original submatrix and initial three-dimensional coordinate system X ₁y ₁z ₁o ₁turning axle simultaneously around the conformal array of solid rotates, and obtains corresponding submatrix and relative dimensional coordinate system X relatively ₂y ₂z ₂o ₂if, initial three-dimensional coordinate system X ₁y ₁z ₁o ₁in P ₁point obtains relative dimensional coordinate system X after rotating ₂y ₂z ₂o ₂in P ₂point, therefore P ₂point is at relative dimensional coordinate system X ₂y ₂z ₂o ₂in coordinate be also (x ₁, y ₁, z ₁), namely definition angle ∠ O ₂oZ is the pitching angle theta of relative submatrix ₂, setting initial point O ₂be D at the intersection point of the XOY plane of standard three dimensional coordinate system XYZO ₂point, wherein angle ∠ XOD ₂for the position angle of relative submatrix if for beam position is at relative dimensional coordinate system X ₂y ₂z ₂o ₂in direction vector, be unknown quantity; Definition for relative submatrix and the pitching of original submatrix, the difference of orientation angles, namely

(3) at relative dimensional coordinate system X ₂y ₂z ₂o ₂in, if P ₂point is to plane X ₂o ₂y ₂intersection point be E ₁, definition for vector position angle, ∠ P ₂o ₂z ₂=θ ' ₁for vector the angle of pitch; If Q ₂point is to plane X ₂o ₂y ₂intersection point be E ₂, definition for vector position angle, ∠ Q ₂o ₂z ₂=θ ' ₂for vector the angle of pitch; Definition be respectively vector with vector pitching, orientation angles difference, namely

(4) utilize relative submatrix beam position adjustment three-dimensional rotation transformation equation group, calculate the beam position of relative submatrix at relative dimensional coordinate system X ₂y ₂z ₂o ₂in direction vector

Wherein, relative submatrix beam position adjustment three-dimensional rotation transformation equation group:

Wherein (x ₁, y ₁, z ₁) be vector at relative dimensional coordinate system X ₂y ₂z ₂o ₂in coordinate, be known quantity; for vector with vector pitching, orientation angles difference, be known quantity; for relative submatrix beam position is at relative dimensional coordinate system X ₂y ₂z ₂o ₂in direction vector, be unknown quantity;

Further, utilize the beam position of relative submatrix at relative dimensional coordinate system X ₂y ₂z ₂o ₂in direction vector according to following formulae discovery steering vector:

Wherein t _ij=(x _ij* x ₂+ y _ij* y ₂+ z _ij* z ₂)/c, (x _ij, y _ij, z _ij) be relative submatrix i-th row, the three-dimensional coordinate of jth array unit in standard three dimensional coordinate system XYZO, i=1,2 ..., M; J=1,2 ..., N; C is the light velocity, f ₀for signal frequency;

(5) repeat step (4), calculate the direction vector of beam position in the relative dimensional coordinate system of its correspondence of each relative submatrix in the three-dimensional conformal array of radar; And the steering vector that the beam position calculating each relative submatrix is corresponding;

(6) dimensionality reduction matrix is built

Steering vector corresponding for the beam position of each relative submatrix is substituted into following formula and forms dimensionality reduction matrix:

Wherein, a _kfor the steering vector of a kth submatrix wherein k=1,2 ..., L.