CN105824016B - The steady space-time adaptive processing method of motion platform detections of radar treetop level target - Google Patents
The steady space-time adaptive processing method of motion platform detections of radar treetop level target Download PDFInfo
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Abstract
The present invention disclose a kind of steady space-time adaptive processing method of motion platform detections of radar treetop level target, mainly solve " mirror image " false target that treetop level target in a multi-path environment generates pollute training sample and caused by performance loss problem.Implementation step is：1. 4 constraint dot frequencies of design optimization；2. build Doppler domain constraints；3. build spatial domain constraints；4. solve optimal weight vector；5. obtain output data；The present invention with inequality constraints by replacing the equality constraint of original recipe in Doppler domain, be combined the amplitude constraint constrained instead of original recipe in spatial domain with width, it can realize that main lobe is conformal on the premise of degree of freedom is not lost, realize the robustness of space-time adaptive processing method, the performance of motion platform detections of radar treetop level target is improved, available for detection of the motion platform thunder to treetop level target.
Description
Technical field
The invention belongs to Radar Technology fields, space-time adaptive signal processing technology are further related to, available for multipath
The monitoring and early warning of extreme low-altitude moving target under environment.
Background technology
Motion platform radar is provided to aerial or ground moving object monitoring and following function, have it is important military and
Civilian value.However, depending on facing the ground clutter problem of doppler spread during work under motion platform radar so that faint mesh
Mark signal will drown out in clutter background.Space-time adaptive handle STAP joint spaces and the two-dimensional signal of time, it can be achieved that
Dim moving target detection under strong clutter background, provides effective detection to low-level penetration target, is subject to domestic and foreign scholars
Extensive concern.
In conventional motion platform radar STAP methods, the covariance matrix of clutter and interference is neighbouring by unit to be detected
What the training data of range gate was estimated, when the clutter in training unit and unit to be detected and interference meet independent same distribution
Condition, and number of training be more than 2 times of degree of freedom in systems when, performance loss be less than 3dB.For positive side view missile-borne radar,
It is linear coupling between the spatial domain frequency and Doppler frequency of clutter, and the coupled relation is not change with distance,
So the training sample that different distance obtains can be approximately considered and meet independent same distribution characteristic.
With the development of vehicle technology, the flying height for anti-target of dashing forward can reach below 100m.At this point, extreme low-altitude mesh
There are coupling effect and multipath clutter between mark and environment, " mirror image " false target is formed.And " mirror image " target and true mesh
Mark has different space angles, Doppler and distance, and causing real goal, there are angle-Doppler in space-time two-dimensional plane
Two-dimensional expansion.Once false target is comprised in training sample, the response distortion of STAP methods main lobe will be caused, causes target phase
Disappear, cause the penalty of STAP methods.For STAP performance loss problems caused by goal constraint inaccuracy, traditional is linear
Least mean-square error LCMV Beamforming Methods are constrained by increasing multiple linear restrictions near main lobe come broadening Wave beam forming
The main lobe response of device.The shortcomings that LCMV methods is the degree of freedom caused due to the increase of linear restriction for clutter reduction and interference
It reduces, causes that the secondary lobe of Beam-former becomes higher or the null of interference radiating way shoals.In fact, cause STAP hydraulic performance declines
Factor not only has the inaccuracy of goal constraint, but also includes the error of clutter covariance matrix estimation.It is non-for forward sight array etc.
Positive side-looking mode, clutter also there are serious distance dependencies, cause clutter covariance matrix to estimate there are error, so as to cause
The performance of STAP methods drastically declines.
The content of the invention
It is an object of the invention to propose at a kind of steady space-time adaptive of motion platform detections of radar treetop level target
Reason method to solve above-mentioned the deficiencies in the prior art, improves missile-borne radar detection performance.
The present invention basic ideas be：By in target to be detected and Doppler domain on more adjacent and spatial domain
Adjacent 2 points amplitude and the phase combining constraints using optimization realize that the main lobe of STAP two dimension responses is conformal, realization side
Case includes as follows：
1) 4 constraint dot frequencies of design optimization：
If target is the 0th obligatory point, the Doppler frequency of the point isSpace Angle frequency isThe Doppler frequency of 1st obligatory point isSpace Angle frequency is2nd
The Doppler frequency of a obligatory point isSpace Angle frequency isThe Doppler of 3rd obligatory point
Frequency isSpace Angle frequency isWherein, v is platform movement velocity, and λ is radar emission signal
Wavelength, θ0For the azimuth of target, φ0For the pitch angle of target, fPRFFor the pulse recurrence frequency of radar emission signal, M is
Array elements number, K are umber of pulse；
2) Doppler domain constraints is built：
2a) by Doppler domain optimization problemIn constraints | s0 Hw|2≥1
and|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 s3
The steering vector during sky of respectively the 0th obligatory point and the 3rd obligatory point, j represent imaginary number, ρ0, ρ1, φ is respectively to be asked
Binary vectorIn 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 of the 3rd obligatory point
s3It obtains space-time two-dimensional and is oriented to matrix S=(s0,s3)；
2b2) according to the covariance matrix R of sampled dataX, space-time two-dimensional be oriented to matrix S, be calculatedWherein,Represent generalized inverse, r0, r1, r2, β is according to quaternion matrixIn
The different mediant of four numerical value that four elements obtain；
2b3) according to r0, r1, r2, β obtains：
φ=- β+π,
3) spatial domain constraints is built：
3a) build airspace optimization problem：
Wherein, s1, s2The space-time two-dimensional steering vector of 1st obligatory point and the 2nd obligatory point respectively in step 1), α0,
α1, α2, β0, β1, β2For ternary vector to be askedIn 6 unknown parameters；
3b) according to s1, s0, s2The parameter for obtaining constraints in the optimization problem is：
Wherein, M be array elements number, d represent array element spacing, λ be radar emission signal wavelength, ψiRepresent i-th about
Spot is with the space cone angle of radar array axial direction；
4) optimal weight vector is solved：
4a) according to step 2a) and 3a) structure spatial domain Doppler domain combines constrained optimization problem：
Wherein, C=(s1,s0,s2,s3) it is that new space-time two-dimensional is oriented to matrix,For constraint vector；
4b) according to C, u obtains optimal weight vector and is：
W=RX -1C(CHRX -1C)-1u
5) the output data Y=w of motion platform detections of radar treetop level target is obtained according to optimal weight vector wTX, wherein X
Represent sampled data, T represents transposition operation.
The present invention has the following advantages compared with prior art：
The present invention is connected by replacing the equality constraint of original recipe with inequality constraints in Doppler domain in spatial domain with width
Contract beam replaces the amplitude constraint of original recipe, can realize that main lobe is conformal on the premise of degree of freedom is not lost, so as to overcome
Mirror target pollution training sample and the problem of cause detection target capabilities loss, realize space-time adaptive processing method
Robustness improves the performance of motion platform detections of radar treetop level target.
Description of the drawings
Fig. 1 is the usage scenario figure of the present invention；
Fig. 2 is the realization flow chart of the present invention；
Fig. 3 is the Wave beam forming figure with inventive method detection treetop level target；
Fig. 4 is graph of the performance improvement factor pair than optimal performance improvement factor that target is detected with the method for the present invention.
Specific embodiment
The embodiment of the present invention and effect are described in further detail below in conjunction with the accompanying drawings.
With reference to Fig. 1, usage scenario of the invention is：Using motion platform as radar platform, podium level H, movement
Speed is v.Object height is h, movement velocity vs.Radar configuration mode be one-dimensional uniform line-array, array number M, array axes
For line perpendicular to radar motion direction, the pulse recurrence frequency of array emitter signal is fPRF。
With reference to Fig. 2, the equality constraint of the invention by replacing original recipe with inequality constraints in Doppler domain, in spatial domain
Be combined the amplitude constraint constrained instead of original recipe with width, to realize the robustness of space-time adaptive processing method, improves fortune
The performance of moving platform detections of radar treetop level target.Implementation step is as follows：
Step 1,4 constraint dot frequencies of design optimization.
If target is the 0th obligatory point, the Doppler frequency of the point isSpace Angle frequency is
fs 0=sin θ0cosφ0；
The Doppler frequency of 1st obligatory point isSpace Angle frequency is
The Doppler frequency of 2nd obligatory point isSpace Angle frequency is
The Doppler frequency of 3rd obligatory point isSpace Angle frequency is
Wherein, v be platform movement velocity, λ be radar emission signal wavelength, θ0For the azimuth of target, φ0For target
Pitch angle, fPRFFor the pulse recurrence frequency of radar emission signal, M is array elements number, and K is umber of pulse.
Step 2, Doppler domain constraints is built.
2a) calculate the space-time two-dimensional steering vector of i-th of obligatory point.
According to the pulse recurrence frequency f of radar emission signalPRF, array elements number M, umber of pulse K, the i-th obligatory point it is how general
Strangle frequencyAnd Space Angle frequencyObtain the spatial domain steering vector of i-th of obligatory pointAnd time domain steering vector
Wherein i=0,1,2,3, T represents transposition operation；
2b) according to step 2a) in i-th of obligatory point spatial domain steering vectorAnd time domain steering vector
To the space-time two-dimensional steering vector of i-th of obligatory pointWherein,Represent that Kronecker multiplies
Product；
2c) according to step 2b) in i-th of obligatory point space-time two-dimensional steering vectorIt obtains from the 0th constraint
O'clock to the 3rd obligatory point space-time two-dimensional steering vector：
2d) by Doppler domain optimization problemIn constraints | s0 Hw|2≥1
and|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 s3
Respectively step 2c) in the sky of the 0th obligatory point and the 3rd obligatory point when steering vector, j represents imaginary number, ρ0, ρ1, φ difference
For binary vector to be askedIn three unknown parameters；
2e) calculation procedure 2d) in three unknown parameter ρ0, ρ1, φ：
2e1) according to step 2c) in the 0th obligatory point space-time two-dimensional steering vector s0, two during the sky of the 3rd obligatory point
Tie up steering vector s3It obtains space-time two-dimensional and is oriented to matrix S=(s0,s3)；
2e2) according to step 2e1) it is hollow when two-dimensional guide matrix S, the covariance matrix R of sampled dataX, it is calculatedWherein,Represent generalized inverse, -1 representing matrix inversion operation, r0, r1, r2, β is root
According to quaternion matrixIn the different mediant of obtained four numerical value of four elements；
2e3) according to step 2e2) in r0, r1, r2, β obtains required φ, ρ0, ρ1：
φ=- β+π,
Above-mentioned steps 2d) in Doppler domain constrained optimization problem replaced in Doppler domain with inequality constraints in original method
Equality constraint, can not loss system degree of freedom, so as to which the secondary lobe that avoids Beam-former becomes higher or interference radiating way
Null shoals.
Step 3, spatial domain constraints is built.
3a) build airspace optimization problem：
Wherein, s1, s2Respectively step 2c) in the space-time two-dimensional steering vector of the 1st obligatory point and 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, s2The parameter for obtaining constraints in the optimization problem is：
Wherein, d represents array element spacing, ψiRepresent space cone angle of i-th of obligatory point with radar array axial direction；
Spatial domain constraint is mutually constrained using 3 points of joint width, and amplitude and phase restriction match with echo signal, therefore
It is lost in performance smaller.
Step 4, optimal weight vector is solved.
4a) according to step 2d) in Doppler domain restricted problemWith step 3a) in spatial domain
Restricted problemStructure spatial domain Doppler domain joint constrained optimization is asked
Topic.
Since the width phase constraints of the 0th obligatory point in Doppler domain restricted problem is wHs0=ρ0, spatial domain restricted problem
In the width phase constraints of the 0th obligatory point beIt for this purpose, need to be by the 0th obligatory point in the restricted problem of spatial domain
Constraints be converted into Doppler domain constrain it is identical, airspace optimization problem is write as therefore：
Then, spatial domain Doppler domain joint constrained optimization problem becomes：
Wherein, C=(s1,s0,s2,s3) it is that new space-time two-dimensional is oriented to matrix,For constraint vector；
4b) according to step 4a) in C and u obtain optimal weight vector and be：
W=RX -1C(CHRX -1C)-1U,
Wherein, -1 representing matrix inversion operation, H represent conjugate transposition operation.
Step 5, according to step 4b) in optimal weight vector w obtain the output number of motion platform detections of radar treetop level target
According to Y=wTX, wherein X represent sampled data, and T represents transposition operation；
The effect of the present invention is described further below by emulation experiment.
1. simulation parameter：
If the reference carrier frequency f of motion platform radar0=10GHz, pulse recurrence frequency fPRF=60KHz, radar array are battle array
First spacing be half-wavelength uniform line-array, array number M=20, umber of pulse K=16, the movement velocity v=400m/s of radar platform,
Podium level H=1080m, object height h=80m, target velocity vs=100m/s, clutter noise ratio 60dB, signal noise
Than for 25dB.
2. emulation content：
Emulation 1, under above-mentioned simulation parameter, the Wave beam forming of treetop level target, result figure such as Fig. 3 are carried out with inventive method
It is shown.
As seen from Figure 3, the Wave beam forming directional diagram of the method for the present invention is not distorted in main lobe region, it is seen that the present invention
Method is effective.
Emulation 2 under above-mentioned simulation parameter, compares optimal performance with the performance improvement factor pair of the method for the present invention detection target
Improvement factor curvilinear motion, the results are shown in Figure 4.
From fig. 4, it can be seen that the improvement factor of the method for the present invention is only than optimal decline 5dB or so, it is seen that the method for the present invention
Better performances.
The correctness of this simulating, verifying present invention, validity and reliability.
Claims (3)
1. the steady space-time adaptive processing method of motion platform detections of radar treetop level target, including：
1) 4 constraint dot frequencies of design optimization：
If target is the 0th obligatory point, the Doppler frequency of the point isSpace Angle frequency is fs 0=
sinθ0cosφ0, the Doppler frequency of the 1st obligatory point isSpace Angle frequency is2nd about
The Doppler frequency of spot isSpace Angle frequency isThe Doppler frequency of 3rd obligatory point
ForSpace Angle frequency is fs 3=fs 0, wherein, v is platform movement velocity, and λ is the ripple of radar emission signal
It is long, θ0For the azimuth of target, φ0For the pitch angle of target, fPRFFor the pulse recurrence frequency of radar emission signal, M is array
Array number, K are 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, RXFor the covariance matrix of sampled data, s0And s3Respectively
For the sky of the 0th obligatory point and the 3rd obligatory point when steering vector, j represents imaginary number, ρ0, ρ1, φ is respectively binary to be asked
VectorIn 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 point3
Matrix S=(s are oriented to space-time two-dimensional0,s3)；
2b2) according to the covariance matrix R of sampled dataX, space-time two-dimensional be oriented to matrix S, be calculatedWherein,Represent generalized inverse, r0, r1, r2, β is according to quaternion matrixIn
The different mediant of four numerical value that four elements obtain；
2b3) according to r0, r1, r2, β obtains：
φ=- β+π,
3) spatial domain constraints is built：
3a) build airspace optimization problem：
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Wherein, M be array elements number, d represent array element spacing, λ be radar emission signal wavelength, ψiRepresent i-th of obligatory point with
The space cone angle of radar array axial direction；
4) optimal weight vector is solved：
4a) according to step 2a) and 3a) structure spatial domain Doppler domain combines constrained optimization problem：Wherein,
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Constraint vector；
4b) according to C, u obtains optimal weight vector and is：
W=RX -1C(CHRX -1C)-1u
5) the output data Y=w of motion platform detections of radar treetop level target is obtained according to optimal weight vector wTX, wherein X are represented
Sampled data, T represent transposition operation.
2. the steady space-time adaptive processing method of motion platform detections of radar treetop level target according to claim 1,
Wherein step 2a) in calculate the space-time two-dimensional steering vector s of the 0th obligatory point and the 3rd obligatory point0, s3, as follows
It carries out：
2a1) according to the pulse recurrence frequency f of radar emission signalPRF, array elements number M, umber of pulse K, the sky of k-th of obligatory point
Between angular frequency fs kAnd the Doppler frequency of k-th of obligatory pointObtain the spatial domain steering vector a (f of k-th of obligatory points k) with
And time domain steering vector
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<mi>s</mi>
<mi>k</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>M</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>k</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>k</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>f</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</msup>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>k</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>f</mi>
</mrow>
</msub>
</mfrac>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
Wherein, k=0,3；
2a2) according to step 2a1) in a (fs k) andObtain the space-time two-dimensional steering vector of k-th of obligatory point：
<mrow>
<msub>
<mi>s</mi>
<mi>k</mi>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>k</mi>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>k</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>k</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>&CircleTimes;</mo>
<mi>b</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>k</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
Wherein,Represent Kronecker product；
2a3) according to step 2a2) inObtain the space-time two-dimensional steering vector of the 0th obligatory point
The space-time two-dimensional steering vector of 3rd obligatory point
3. the steady space-time adaptive processing method of motion platform detections of radar treetop level target according to claim 1,
Wherein step 3a) in calculate the space-time two-dimensional steering vector s of the 1st obligatory point and the 2nd obligatory point1, s2, as follows
It carries out：
3a1) according to the pulse recurrence frequency f of radar emission signalPRF, array elements number M, umber of pulse K, the sky of l-th of obligatory point
Between angular frequencyAnd the Doppler frequency of l-th of obligatory pointObtain the spatial domain steering vector of l-th of obligatory pointAnd
Time domain steering vector
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>l</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<mi>d</mi>
<mi>&lambda;</mi>
</mfrac>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>l</mi>
</msubsup>
</mrow>
</msup>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<mi>d</mi>
<mi>&lambda;</mi>
</mfrac>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>l</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>M</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>l</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>l</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>f</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</msup>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<mn>2</mn>
<mi>&pi;</mi>
<mfrac>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>l</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>f</mi>
</mrow>
</msub>
</mfrac>
<mrow>
<mo>(</mo>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
Wherein, l=1,2；
3a2) according to step 3a1) inAndObtain the space-time two-dimensional steering vector of l-th of obligatory point：
<mrow>
<msub>
<mi>s</mi>
<mi>l</mi>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>l</mi>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>l</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>s</mi>
<mi>l</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>&CircleTimes;</mo>
<mi>b</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>f</mi>
<mi>d</mi>
<mi>l</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
Wherein,Represent Kronecker product；
3a3) according to step 3a2) inObtain the space-time two-dimensional steering vector of the 1st obligatory point
The space-time two-dimensional steering vector of 2nd obligatory point
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