CN105824016A - Robust space-time adaptive processing method for super-low-altitude target detection by moving platform radar - Google Patents

Abstract

The invention discloses a robust space-time adaptive processing method for super-low-altitude target detection by a moving platform radar, which mainly solves the problem of performance loss caused as a training sample is polluted by a mirror false target generated by the super-low-altitude target in a multipath environment. The method comprises steps: 1, optimized frequencies of four constraint points are designed; 2, a Doppler domain constraint condition is built; 3, an airspace constraint condition is built; 4, an optimal weight vector is solved; and 5, output data are obtained. Through replacing an equality constraint in the previous method by an inequality constraint in the Doppler domain, an amplitude-phase joint constraint in the airspace is used for replacing an amplitude constraint in the previous method, main lobe shape preserving is realized on the premise of not losing degree of freedom, robustness of the space-time adaptive processing method is realized, the performance of super-low-altitude target detection by the moving platform radar is improved, and the method of the invention can be used for detecting the super-low-altitude target by the moving platform radar.

Description

The sane space-time adaptive processing method of motion platform detections of radar treetop level target

Technical field

The invention belongs to Radar Technology field, further relate to space-time adaptive signal processing technology, can be used for supervision and the early warning of extreme low-altitude moving target under multi-path environment.

Background technology

Motion platform radar provides the aerial or supervision of ground moving object and following function, has important dual-use value.But, face the ground clutter problem of doppler spread time under motion platform radar depending on work so that weak target signal will drown out in clutter background.Space-time adaptive processes the two-dimensional signal of STAP joint space and time, may be implemented in the Dim moving target detection under strong clutter background, it is provided that the effective detection to low-level penetration target, by the extensive concern of Chinese scholars.

In conventional motion platform radar STAP method, the covariance matrix of clutter and interference is to estimate to obtain by the training data of unit adjacency door to be detected, when the clutter in training unit with unit to be detected and interference meet independent same distribution condition, and number of training more than 2 times of degree of freedom in systems time, performance loss be less than 3dB.For positive side-looking missile-borne radar, it is linear coupling between spatial domain frequency and the Doppler frequency of clutter, and this coupled relation does not changes with distance, so the training sample that different distance obtains can be approximately considered and meet independent same distribution characteristic.

Along with the development of vehicle technology, the flying height of prominent anti-target can reach below 100m.Now, between treetop level target and environment, there is coupling effect and multipath clutter, form " mirror image " false target.And " mirror image " target has different space angle, Doppler and distances from real goal, real goal is caused to there is angle-Doppler's two-dimensional expansion in space-time two-dimensional plane.During once false target is comprised in training sample, will result in the response distortion of STAP method main lobe, cause target cancellation, cause the penalty of STAP method.For the inaccurate STAP performance loss problem caused of goal constraint, traditional linear restriction least mean-square error LCMV Beamforming Method carrys out the main lobe of broadening Beam-former and responds by increasing the multiple linear restrictions near main lobe.The shortcoming of LCMV method be due to the increase of linear restriction cause for clutter reduction and interference degree of freedom reduce, cause the secondary lobe of Beam-former to uprise or interference radiating way zero fall into shoal.It practice, cause the factor of STAP hydraulic performance decline not only have the inaccurate of goal constraint, and include the error that clutter covariance matrix is estimated.For non-working side patterns such as forward sight arrays, clutter there is also serious distance dependencies, causes clutter covariance matrix to estimate to there is error, thus causes the performance of STAP method drastically to decline.

Summary of the invention

It is an object of the invention to propose the sane space-time adaptive processing method of a kind of motion platform detections of radar treetop level target, to solve above-mentioned the deficiencies in the prior art, improve missile-borne radar detection performance.

The basic ideas of the present invention are: use the amplitudes optimized and phase combining to retrain by more adjacent in target to be detected and Doppler domain and spatial domain adjacent 2, it is achieved the main lobe of STAP two dimension response is conformal, and its implementation includes the following:

1) 4 obligatory point frequencies of design optimization:

If target is the 0th obligatory point, the Doppler frequency of this point isSpace Angle frequency isThe Doppler frequency of the 1st obligatory point isSpace Angle frequency isThe Doppler frequency of the 2nd obligatory point isSpace Angle frequency isThe Doppler frequency of the 3rd obligatory point isSpace Angle frequency isWherein, v is platform movement velocity, and λ is the wavelength of radar emission signal, θ₀For the azimuth of target, φ₀For the angle of pitch of target, f_PRFFor the pulse recurrence frequency of radar emission signal, M is array elements number, and K is umber of pulse；

2) Doppler domain constraints is built:

2a) by Doppler domain optimization problemIn constraints | s₀ ^Hw|²≥1and|s₃ ^Hw|²>=1 is write as

Wherein, w is optimal solution to be asked, and H represents conjugate transposition operation, and RX is the covariance matrix of sampled data, s₀And s₃Steering vector when being respectively the 0th obligatory point and the 3rd obligatory point empty, j represents imaginary number, ρ₀, ρ₁, binary vector that φ is to be askedIn three unknown parameters；

2b) calculation procedure 2a) in three unknown parameter ρ₀, ρ₁, φ:

2b1) according to the space-time two-dimensional steering vector s of the 0th obligatory point₀, the space-time two-dimensional steering vector s of the 3rd obligatory point₃Obtain space-time two-dimensional and guide matrix S=(s₀,s₃)；

2b2) according to the covariance matrix R of sampled data_X, space-time two-dimensional guides matrix S, is calculatedWherein,Represent generalized inverse, r₀, r₁, r₂, β is according to quaternion matrixIn the different mediant of four numerical value obtaining of four elements；

2b3) according to r₀, r₁, r₂, β obtains:

φ=-β+π,

3) spatial domain constraints is built:

3a) build airspace optimization problem:

$\left\{\begin{array}{c}\underset{w}{min}{w}^H{R}_{X}w\\ \begin{array}{cc}s.t.& w^H(s_1,s_0,s_2)=({\mathrm{\α}}_1{e}^{{\mathrm{j\β}}_1},{\mathrm{\α}}_0{e}^{{\mathrm{j\β}}_0},{\mathrm{\α}}_2{e}^{{\mathrm{j\β}}_2})\end{array}\end{array}\right.$

Wherein, s₁, s₂Be respectively step 1) in the 1st obligatory point and the space-time two-dimensional steering vector of the 2nd obligatory point, α₀, α₁, α₂, β₀, β₁, β₂For ternary vector to be askedIn 6 unknown parameters；

3b) according to s₁, s₀, s₂Obtaining the parameter of constraints in this optimization problem is:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\left|{s_1}^H{s}_1\right|\\ {\mathrm{\β}}_1=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_1\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_0=\left|{s_0}^H{s}_0\right|\\ {\mathrm{\β}}_0=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_0\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_2=\left|{s_2}^H{s}_2\right|\\ {\mathrm{\β}}_2=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_2\right)\end{array}\right.$

Wherein, M is array elements number, and d represents array element distance, and λ is the wavelength of radar emission signal, ψ_iRepresent that i-th obligatory point is with the axial space cone angle of radar array；

4) optimum weight vector is solved:

4a) according to step 2a) and 3a) build spatial domain Doppler domain associating constrained optimization problems:Wherein, C=(s₁,s₀,s₂,s₃) it is that new space-time two-dimensional guides matrix,For constraint vector；

4b) according to C, u obtains optimum weight vector and is:

W=R_X ^-1C(C^HR_X ^-1C)^-1u

5) output data Y=w of motion platform detections of radar treetop level target are obtained according to optimum weight vector w^TX, wherein X represents that sampled data, T represent that transposition operates.

The present invention compared with prior art has the advantage that

The present invention is by replacing the equality constraint of original recipe in Doppler domain inequality constraints, it is combined at spatial domain width and retrains the amplitude constraint of replacement original recipe, main lobe can be realized conformal on the premise of degree of freedom is not lost, thus overcome mirror target and pollute training sample and cause the problem that target capabilities is lost that detects, achieve the robustness of space-time adaptive processing method, improve the performance of motion platform detections of radar treetop level target.

Accompanying drawing explanation

Fig. 1 is the use scene graph of the present invention；

Fig. 2 is the flowchart of the present invention；

Fig. 3 is the Wave beam forming figure with inventive method detection treetop level target；

Fig. 4 is the curve chart than optimal performance improvement factor of the performance improvement factor pair by the inventive method detection target.

Detailed description of the invention

Below in conjunction with the accompanying drawings the embodiment of the present invention and effect are described in further detail.

With reference to Fig. 1, the use scene of the present invention is: using motion platform as radar platform, podium level is H, and movement velocity is v.Object height is h, and movement velocity is v_s.Radar configuration mode is one-dimensional uniform line-array, and array number is M, and array axis is perpendicular to radar motion direction, and the pulse recurrence frequency of array emitter signal is f_PRF。

With reference to Fig. 2, the present invention is by replacing the equality constraint of original recipe in Doppler domain inequality constraints, it is combined at spatial domain width and retrains the amplitude constraint of replacement original recipe, to realize the robustness of space-time adaptive processing method, improve the performance of motion platform detections of radar treetop level target.Implementation step is as follows:

Step 1,4 obligatory point frequencies of design optimization.

If target is the 0th obligatory point, the Doppler frequency of this point isSpace Angle frequency is f_s ⁰=sin θ₀cosφ₀；

The Doppler frequency of the 1st obligatory point isSpace Angle frequency is

The Doppler frequency of the 2nd obligatory point isSpace Angle frequency is

The Doppler frequency of the 3rd obligatory point isSpace Angle frequency is

Wherein, v is platform movement velocity, and λ is the wavelength of radar emission signal, θ₀For the azimuth of target, φ₀For the angle of pitch of target, f_PRFFor the pulse recurrence frequency of radar emission signal, M is array elements number, and K is umber of pulse.

Step 2, builds Doppler domain constraints.

2a) calculate the space-time two-dimensional steering vector of i-th obligatory point.

Pulse recurrence frequency f according to radar emission signal_PRF, array elements number M, umber of pulse K, the Doppler frequency of the i-th obligatory pointAnd Space Angle frequencyObtain the spatial domain steering vector of i-th obligatory pointAnd time domain steering vector

$a\left(f_s^i\right)={(1,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^i},...,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^i(M-1)})}^T$

$b\left(f_d^i\right)={(1,e^{j2\mathrm{\π}\frac{f_d^i}{f_{prf}}},...,e^{j2\mathrm{\π}\frac{f_d^i}{f_{prf}}(K-1)})}^T$

Wherein i=0,1,2,3, T represents that transposition operates；

2b) according to step 2a) in the spatial domain steering vector of i-th obligatory pointAnd time domain steering vectorObtain the space-time two-dimensional steering vector of i-th obligatory pointWherein,Represent Kronecker product；

2c) according to step 2b) in the space-time two-dimensional steering vector of i-th obligatory pointObtain the space-time two-dimensional steering vector from the 0th obligatory point to the 3rd obligatory point:

2d) by Doppler domain optimization problemIn constraints | s₀ ^Hw|²≥1and|s₃ ^Hw|²>=1 is write as

Wherein, w is optimal solution to be asked, and H represents conjugate transposition operation, and RX is the covariance matrix of sampled data, s₀And s₃Be respectively step 2c) in the 0th obligatory point and the 3rd obligatory point empty time steering vector, j represents imaginary number, ρ₀, ρ₁, binary vector that φ is to be askedIn three unknown parameters；

2e) calculation procedure 2d) in three unknown parameter ρ₀, ρ₁, φ:

2e1) according to step 2c) in the space-time two-dimensional steering vector s of the 0th obligatory point₀, the space-time two-dimensional steering vector s of the 3rd obligatory point₃Obtain space-time two-dimensional and guide matrix S=(s₀,s₃)；

2e2) according to step 2e1) hollow time two-dimensional guide matrix S, the covariance matrix R of sampled data_X, it is calculatedWherein,Represent generalized inverse ,-1 representing matrix inversion operation, r₀, r₁, r₂, β is according to quaternion matrixIn the different mediant of four numerical value obtaining of four elements；

2e3) according to step 2e2) in r₀, r₁, r₂, β obtains required φ, ρ₀, ρ₁:

φ=-β+π,

${\mathrm{\ρ}}_0=\left\{\begin{array}{cc}1,& \frac{r_2}{r_0}\≤1\\ \frac{r_2}{r_0},& \frac{r_2}{r_0}>1\end{array}\right.,$

${\mathrm{\ρ}}_1=\left\{\begin{array}{cc}1,& \frac{r_2}{r_1}\≤1\\ \frac{r_2}{r_1},& \frac{r_2}{r_1}>1\end{array}\right.;$

Above-mentioned steps 2d) in Doppler domain constrained optimization problems replace the equality constraint in original method with inequality constraints at Doppler domain, can not the degree of freedom of loss system, thus avoid the secondary lobe of Beam-former to uprise or zero the falling into and shoal of interference radiating way.

Step 3, builds spatial domain constraints.

3a) build airspace optimization problem:

$\left\{\begin{array}{c}\underset{w}{min}{w}^H{R}_{X}w\\ \begin{array}{cc}s.t.& w^H(s_1,s_0,s_2)=({\mathrm{\α}}_1{e}^{{\mathrm{j\β}}_1},{\mathrm{\α}}_0{e}^{{\mathrm{j\β}}_0},{\mathrm{\α}}_2{e}^{{\mathrm{j\β}}_2})\end{array}\end{array}\right.$

Wherein, s₁, s₂Be respectively step 2c) in the 1st obligatory point and the space-time two-dimensional steering vector of the 2nd obligatory point, α₀, α₁, α₂, β₀, β₁, β₂For ternary vector to be askedIn 6 unknown parameters；

3b) according to step 2c) in s₁, s₀, s₂Obtaining the parameter of constraints in this optimization problem is:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\left|{s_1}^H{s}_1\right|\\ {\mathrm{\β}}_1=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_1\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_0=\left|{s_0}^H{s}_0\right|\\ {\mathrm{\β}}_0=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_0\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_2=\left|{s_2}^H{s}_2\right|\\ {\mathrm{\β}}_2=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_2\right)\end{array}\right.$

Wherein, d represents array element distance, ψ_iRepresent that i-th obligatory point is with the axial space cone angle of radar array；

Spatial domain constraint uses the associating width of 3 to retrain mutually, and its amplitude and phase restriction match with echo signal, therefore lose less in performance.

Step 4, solves optimum weight vector.

4a) according to step 2d) in the restricted problem of Doppler domainWith step 3a) in the restricted problem in spatial domainBuild spatial domain Doppler domain associating constrained optimization problems.

Owing in Doppler domain restricted problem, the width phase constraints of the 0th obligatory point is w^Hs₀=ρ₀, in the restricted problem of spatial domain, the width phase constraints of the 0th obligatory point isTo this end, the constraints of the in the restricted problem of spatial domain the 0th obligatory point need to be converted into Doppler domain constraint is identical, for this, airspace optimization problem is write as:

$\left\{\begin{array}{c}\underset{w}{min}{w}^H{R}_{X}w\\ \begin{array}{cc}s.t.& w^H(s_1,s_0,s_2)=({\mathrm{\α}}_1\frac{{\mathrm{\ρ}}_0}{{\mathrm{\α}}_0}{e}^{j({\mathrm{\β}}_1-{\mathrm{\β}}_0)},{\mathrm{\ρ}}_0,{\mathrm{\α}}_2\frac{{\mathrm{\ρ}}_0}{{\mathrm{\α}}_0}{e}^{j({\mathrm{\β}}_2-{\mathrm{\β}}_0)})\end{array}\end{array}\right.$

Then, spatial domain Doppler domain associating constrained optimization problems becomes:

$\left\{\begin{array}{c}\underset{W}{min}{w}^H{R}_{X}w\\ \begin{array}{cc}s.t.& w^{H}C=u\end{array}\end{array}\right.,$

Wherein, C=(s₁,s₀,s₂,s₃) it is that new space-time two-dimensional guides matrix,For constraint vector；

4b) according to step 4a) in C and u obtain optimum weight vector and be:

W=R_X ^-1C(C^HR_X ^-1C)^-1U,

Wherein ,-1 representing matrix inversion operation, H represents conjugate transposition operation.

Step 5, according to step 4b) in optimum weight vector w obtain output data Y=w of motion platform detections of radar treetop level target^TX, wherein X represents that sampled data, T represent that transposition operates；

Below by emulation experiment, the effect of the present invention is described further.

1. simulation parameter:

If the reference carrier frequency f of motion platform radar₀=10GHz, pulse recurrence frequency f_PRF=60KHz, radar array be array element distance be the uniform line-array of half-wavelength, array number M=20, umber of pulse K=16, movement velocity v=400m/s of radar platform, podium level H=1080m, object height h=80m, target velocity v_s=100m/s, clutter noise ratio is 60dB, and signal noise ratio is 25dB.

2. emulation content:

Emulation 1, under above-mentioned simulation parameter, carries out the Wave beam forming of treetop level target by inventive method, and result figure is as shown in Figure 3.

As seen from Figure 3, the Wave beam forming directional diagram of the inventive method is not distorted in main lobe region, it is seen that the inventive method is effective.

Emulation 2, under above-mentioned simulation parameter, with the performance improvement factor pair of the inventive method detection target than optimal performance improvement factor curvilinear motion, result is as shown in Figure 4.

From fig. 4, it can be seen that the improvement factor of the inventive method only optimum decline about the 5dB of ratio, it is seen that the better performances of the inventive method.

This simulating, verifying correctness of the present invention, validity and reliability.

Claims (3)

1. the sane space-time adaptive processing method of motion platform detections of radar treetop level target, including:

1) 4 obligatory point frequencies of design optimization:

If target is the 0th obligatory point, the Doppler frequency of this point isSpace Angle frequency isThe Doppler frequency of the 1st obligatory point isSpace Angle frequency isThe Doppler frequency of the 2nd obligatory point isSpace Angle frequency isThe Doppler frequency of the 3rd obligatory point isSpace Angle frequency isWherein, v is platform movement velocity, and λ is the wavelength of radar emission signal, θ₀For the azimuth of target, φ₀For the angle of pitch of target, f_PRFFor the pulse recurrence frequency of radar emission signal, M is array elements number, and K is umber of pulse；

2) Doppler domain constraints is built:

2a) by Doppler domain optimization problemIn constraints | s₀ ^Hw|²≥1and|s₃ ^Hw|²>=1 is write as

Wherein, w is optimal solution to be asked, and H represents conjugate transposition operation, R_XFor the covariance matrix of sampled data, s₀And s₃Steering vector when being respectively the 0th obligatory point and the 3rd obligatory point empty, j represents imaginary number, ρ₀, ρ₁, binary vector that φ is to be askedIn three unknown parameters；

2b) calculation procedure 2a) in three unknown parameter ρ₀, ρ₁, φ:

2b1) according to the space-time two-dimensional steering vector s of the 0th obligatory point₀, the space-time two-dimensional steering vector s of the 3rd obligatory point₃Obtain space-time two-dimensional and guide matrix S=(s₀,s₃)；

2b2) according to the covariance matrix R of sampled data_X, space-time two-dimensional guides matrix S, is calculatedWherein,Represent generalized inverse, r₀, r₁, r₂, β is according to quaternion matrixIn the different mediant of four numerical value obtaining of four elements；

2b3) according to r₀, r₁, r₂, β obtains:

φ=-β+π,

3) spatial domain constraints is built:

3a) build airspace optimization problem:

$\left\{\begin{array}{c}\underset{w}{min}{w}^H{R}_{X}w\\ \begin{array}{cc}s.t.& w^H(s_1,s_0,s_2)=({\mathrm{\α}}_1{e}^{{\mathrm{j\β}}_1},{\mathrm{\α}}_0{e}^{{\mathrm{j\β}}_0},{\mathrm{\α}}_2{e}^{{\mathrm{j\β}}_2})\end{array}\end{array}\right.$

Wherein, s₁, s₂Be respectively step 1) in the 1st obligatory point and the space-time two-dimensional steering vector of the 2nd obligatory point, α₀, α₁, α₂, β₀, β₁, β₂For ternary vector to be askedIn 6 unknown parameters；

3b) according to s₁, s₀, s₂Obtaining the parameter of constraints in this optimization problem is:

$\left\{\begin{array}{c}{\mathrm{\α}}_1=\left|{s_1}^H{s}_1\right|\\ \{{\mathrm{\β}}_1=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}cos\left({\mathrm{\ψ}}_1\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_0=\left|{s_0}^H{s}_0\right|\\ {\mathrm{\β}}_0=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}\cos \left({\mathrm{\ψ}}_0\right)\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\α}}_2=\left|{s_2}^H{s}_2\right|\\ {\mathrm{\β}}_2=2\mathrm{\π}\frac{M-1}{2}\frac{d}{\mathrm{\λ}}\cos \left({\mathrm{\ψ}}_2\right)\end{array}\right.$

Wherein, M is array elements number, and d represents array element distance, and λ is the wavelength of radar emission signal, ψ_iRepresent that i-th obligatory point is with the axial space cone angle of radar array；

4) optimum weight vector is solved:

4a) according to step 2a) and 3a) build spatial domain Doppler domain associating constrained optimization problems:Wherein, C=(s₁,s₀,s₂,s₃) it is that new space-time two-dimensional guides matrix,For constraint vector；

4b) according to C, u obtains optimum weight vector and is:

W=R_X ^-1C(C^HR_X ^-1C)^-1u

5) output data Y=w of motion platform detections of radar treetop level target are obtained according to optimum weight vector w^TX, wherein X represents that sampled data, T represent that transposition operates.

The sane space-time adaptive processing method of motion platform detections of radar treetop level target the most according to claim 1, wherein step 2a) in calculate the 0th obligatory point and the space-time two-dimensional steering vector s of the 3rd obligatory point₀, s₃, carry out as follows:

2a1) according to pulse recurrence frequency f of radar emission signal_PRF, array elements number M, umber of pulse K, the Space Angle frequency of kth obligatory pointAnd the Doppler frequency of kth obligatory pointObtain the spatial domain steering vector of kth obligatory pointAnd time domain steering vector

$a\left(f_s^k\right)={(1,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^k},...,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^k(M-1)})}^T$

$b\left(f_d^k\right)={(1,e^{j2\mathrm{\π}\frac{f_d^k}{f_{prf}}},...,e^{j2\mathrm{\π}\frac{f_d^k}{f_{prf}}(M-1)})}^T$

Wherein, k=0,3；

2a2) according to step 2a1) inAndObtain the space-time two-dimensional steering vector of kth obligatory point:

$s_k(f_s^k,f_d^k)=a\left(f_s^k\right)\⊗b\left(f_d^k\right)$

Wherein,Represent Kronecker product.

2a3) according to step 2a2) inObtain the space-time two-dimensional steering vector of the 0th obligatory pointThe space-time two-dimensional steering vector of the 3rd obligatory point

The sane space-time adaptive processing method of motion platform detections of radar treetop level target the most according to claim 1, wherein step 3a) in calculate the 1st obligatory point and the space-time two-dimensional steering vector s of the 2nd obligatory point₁, s₂, carry out as follows:

3a1) according to pulse recurrence frequency f of radar emission signal_PRF, array elements number M, umber of pulse K, the Space Angle frequency of l obligatory pointAnd the Doppler frequency of l obligatory pointObtain the spatial domain steering vector of l obligatory pointAnd time domain steering vector

$a\left(f_s^l\right)={(1,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^l},...,e^{j2\mathrm{\π}\frac{d}{\mathrm{\λ}}{f}_s^l(M-1)})}^T$

$a\left(f_d^l\right)={(1,e^{j2\mathrm{\π}\frac{f_d^l}{f_{prf}}},...,e^{j2\mathrm{\π}\frac{f_d^l}{f_{prf}}(K-1)})}^T$

Wherein, l=1,2；

3a2) according to step 3a1) inAndObtain the space-time two-dimensional steering vector of l obligatory point:

$s_l(f_s^l,f_d^l)=a\left(f_s^l\right)\⊗b\left(f_d^l\right)$

3a3) according to step 3a2) inObtain the space-time two-dimensional steering vector of the 1st obligatory pointThe space-time two-dimensional steering vector of the 2nd obligatory point