CN105607045B - A kind of optimizing location method of radar network under Deceiving interference - Google Patents
A kind of optimizing location method of radar network under Deceiving interference Download PDFInfo
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- CN105607045B CN105607045B CN201610014960.2A CN201610014960A CN105607045B CN 105607045 B CN105607045 B CN 105607045B CN 201610014960 A CN201610014960 A CN 201610014960A CN 105607045 B CN105607045 B CN 105607045B
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Abstract
The invention belongs to Anti-jamming Technology for Radar field, a kind of optimizing location method of radar network under Deceiving interference is disclosed, including：Calculate radar network is spoofed probability；Construction minimizes the object function for being spoofed probability；The coverage of each node radar in radar network is calculated, constructs the object function of maximum coverage；According to the object function for minimizing the object function for being spoofed probability and maximum coverage, combined optimization object function is established；Determine the constraints at radar network cloth station；According to the constraints at radar network cloth station and combined optimization object function, the optimized-type of networking Method in Positioning of Radar under Deceiving interference is constructed；According to the optimized-type of networking Method in Positioning of Radar under Deceiving interference, the cloth station location of each node radar in radar network is obtained, to save radar system resource, is realized to the more accurate effectively inhibition of Deceiving interference.
Description
Technical field
The present invention relates to a kind of optimization cloth of radar network under Anti-jamming Technology for Radar field more particularly to Deceiving interference
It stands method, this method is spoofed that probability minimizes and the maximized combined optimization criterion of coverage carries out cloth using radar network
It stands, obtains the cloth station location of radar network, so as to improve the ability that radar network inhibits Deceiving interference.
Background technology
Deceiving interference refers to that interference signal is difficult to differentiate between with target echo signal in radar receiver, is mixed the spurious with the genuine,
Prevent radar from target information is correctly detected.The operation principle of Deceiving interference is：To the radar emission signal of reception, warp
Interference modulation is crossed, change has related parameter, then modulated signal is forwarded back to radar, with the echo-signal of guinea pig target.
Radar is made to be difficult to target of distinguishing the true from the false, so as to achieve the purpose that fascination and upset radar to true target detection and tracking.Number is penetrated
The maturation of frequency memory technology makes it possible the generation of high fidelity decoy, and thunder is given on the different position for deviateing target
Up to many decoys of generation so that the radar cannot distinguish between true and false target.Therefore, although real goal is permitted included in what is shown
Within multiple target, if but radar do not identify the ability of true and false target, necessarily using all targets all as real goal processing.
If radar can not effectively antagonize Deceiving interference, detection, tracking must be remained to false target, so as to occupy a large amount of thunders
Up to system resource, the data-handling capacity of radar is seriously affected.
Since monostatic radar visual angle is single, it is difficult to be resisted to it.And radar network afield may be constructed full side
Position, three-dimensional, multi-level fight system have the technical performances such as full frequency band, more systems, more overlap coefficients, thus with very
Strong survival ability and antijamming capability.For frequency domain confrontation, by multi-section multiband radar networking, it is impossible to an interference
Machine disturbs so wide frequency range.For space electromagnetic environment, after multi-section radar network composite, not only radiation source quantity increases, and makes
The density of signal space and distribution are more complicated, and in frequency range, signal form, parameter type and threat level etc.
Difficulty will be brought to reconnaissance system so that interference quality is very limited.
The relative position of each node radar can directly affect the correlation test to target, and then influence group in radar network
Net radar is spoofed probability, and therefore, the optimizing location for studying radar network under Deceiving interference is of great significance.By right
Radar network carries out reasonable cloth station, can be effectively reduced threat of the Deceiving interference to radar network.
The content of the invention
In view of the above-mentioned problems, the optimizing location side it is an object of the invention to provide radar network under a kind of Deceiving interference
Method to save radar system resource, is realized to the more accurate effectively inhibition of Deceiving interference.
In order to achieve the above objectives, the embodiment of the present invention, which adopts the following technical scheme that, is achieved.
A kind of optimizing location method of radar network under Deceiving interference, the radar network include multiple node radars,
The optimizing location method includes the following steps：
Step 1, calculate radar network is spoofed probability；
Step 2, according to the probability that is spoofed of the radar network, construction radar network minimum is spoofed the target of probability
Function；
Step 3, the coverage of each node radar in the radar network, and covering according to each node radar are calculated
Lid scope constructs the object function of radar network maximum coverage；
Step 4, according to the object function of the object function and maximum coverage for minimizing and being spoofed probability, build
Vertical combined optimization object function；
Step 5, according to cloth station spacing, search coverage and the radar network between adjacent node radar in the radar network
Investigative range, determine the constraints at radar network cloth station；
Step 6, according to the constraints at the radar network cloth station and the combined optimization object function, deception formula is constructed
The optimized-type of the lower networking Method in Positioning of Radar of interference；
Step 7, according to the optimized-type of networking Method in Positioning of Radar under the Deceiving interference, each node in radar network is obtained
The cloth station location of radar.
The invention has the advantages that：(1) method of distributed interference, this hair are inhibited compared to available data processing stage
The bright method by optimizing location inhibits the adverse effect of Deceiving interference, since Optimal Station position can obtain offline, therefore this
The radar system resource occupied needed for inventive method is less；(2) since the present invention not only considers radar coverage and networking thunder
Up to probability is spoofed, the factor that must take into consideration in the engineerings such as cloth station spacing and the search coverage of node radar has also been taken into account, therefore
The present invention is more advantageous to engineering practice.
Description of the drawings
It in order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is attached drawing needed in technology description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
Some embodiments of invention, for those of ordinary skill in the art, without creative efforts, can be with
Other attached drawings are obtained according to these attached drawings.
Fig. 1 is that the flow of the optimizing location method of radar network under a kind of Deceiving interference provided in an embodiment of the present invention is shown
It is intended to one；
Fig. 2 is that the flow of the optimizing location method of radar network under a kind of Deceiving interference provided in an embodiment of the present invention is shown
It is intended to two；
Fig. 3 is that the radar network obtained using the method for the present invention selects search coverage center X under the conditions of Deceiving interference0
=[0,0]T, radius R=10km, radar network cloth station result emulation schematic diagram.
Specific embodiment
Below in conjunction with the attached drawing in the embodiment of the present invention, the technical solution in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other without making creative work
Embodiment belongs to the scope of protection of the invention.
The embodiment of the present invention provides a kind of optimizing location method of radar network under Deceiving interference, the radar network bag
Containing multiple node radars, optimizing location method provided by the invention carries out under an existing radar network topological structure
Optimization.As shown in Figure 1, the optimizing location method includes the following steps：
Step 1 calculates radar network and is spoofed probability.
Step 1 specifically includes following sub-step：
(1a) establishes hypothesis testing model.
Sub-step (1a) specifically includes following sub-step：
(1a1) is by the mahalanobis distance d of any two node radar measurement errorsijAs statistical check amount,Wherein △ Z=dZi-dZjFor the difference of any two node radar measurement errors, wherein ∑ij=E
[(dZi-dZj)(dZi-dZj)T] represent the i-th node radar and jth node radar measurement errors difference covariance matrix, dZiFor
The measurement error of i-th node radar, dZjFor the measurement error of jth node radar, i, j=1,2 ..., N i ≠ j, i, j represent section
The number of point radar, N represent the number of node radar；
(1a2) hypothesis testing model is as follows：Wherein, H is remembered0Represent two measuring value ZiAnd ZjIt is corresponding
Real goal, H1Represent two measuring value ZiAnd ZjCorresponding false targets,Represent that degree of freedom is
2, significance is the chi square distribution of 1-a；
(1a3) is in H1Under conditions of establishment, it is E that the difference △ Z of two node radar measurement errors, which obey average, is assisted
Variance matrix is ∑ijNormal distribution, average
Coordinate of the decoy arrived for the i-th node radar, jth node radar detection under same rectangular coordinate system,I=1,2 ..., N, ρiFor the i-th node detections of radar range-to-go, θiFor the i-th section
Detections of radar is put to the angle of target, △ d represent deception distance；
(1a4) is in H1Under conditions of establishment, the two-dimensional probability density function f of △ Zij(x, y) is：
WhereinRepresent the precision of x-axis measurement difference,The precision of expression y-axis measurement difference, ρ=
ξ12/(σxσy) represent that x-axis measurement difference measures the related coefficient of difference, ξ with y-axis11, ξ12, ξ21, ξ22For matrix ∑ijIt is corresponding each
A matrix element.
(1b) calculates probability of miscarriage of justice of any two node radar to decoy according to the hypothesis testing model.
Illustratively, data anastomosing algorithm may be employed and calculate probability of miscarriage of justice of any two node radar to decoy.
Sub-step (1b) specifically includes following sub-step：
(1b1) i-th node radar, jth node radar are using data anastomosing algorithm to the probability of miscarriage of justice P of decoyijFor：
Wherein P (H0|H1) represent assuming that H1Under be judged to H0Probability, dijRepresent mahalanobis distance and
Wherein,
(1b2) is by probability of miscarriage of justice PijExpression formula be converted into integrated form, and simplification obtains：
WhereinRepresent the integrating range y directions upper limit,Represent integrating range y directions lower limit,For integrating range x
The upper limit on direction,Represent the upper limit of integrating range in the x direction.
(1c), to the probability of miscarriage of justice of decoy, it is general to calculate being spoofed for radar network according to any two node radar
Rate.
In the case where node radar number is more than 2, every two node radars need to differentiate decoy, then to institute
Some court verdicts carries out fusion treatment using the principle of ' take with ', obtains final identification result.
Sub-step (1c) specifically includes following sub-step：
Radar network is spoofed probabilityWherein PfRepresent radar network is spoofed probability, Pij
Represent the probability of miscarriage of justice of the i-th node radar and jth node radar to decoy, ∏ represents even to multiply symbol.
The purpose that cloth station is carried out to radar network is the influence in order to inhibit Deceiving interference to radar network, i.e. raising group
Performance of the net radar under the conditions of Deceiving interference.To realize the purpose, the optimization object function of selection is to maximize networking thunder
The investigative range reached is spoofed the intersection of probability with minimizing radar network.Specific implementation step is as follows：
Step 2, according to the probability that is spoofed of the radar network, construction minimizes the object function for being spoofed probability.
Step 2 specifically includes following sub-step：
(2a) divides the search coverage Ω of radar network, obtains multiple sub- search coverage ΩD, according to different sub- spies
The degree of danger in region is surveyed, to sub- search coverage ΩDIt is assigned to weighting coefficient w；
(2b) is spoofed probability P according to radar networkf, construct the object function F for minimizing and being spoofed probability1For：
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi,θi), ρi、θiRespectively
For distance and angle of the position in polar coordinate system of i-th of node radar, i=1,2 ..., N, N is radar network interior joint thunder
The number reached, ΩDRepresent sub- search coverage, w represents sub- search coverage ΩDCorresponding weighting coefficient, min expressions are minimized, Σ
Represent summation symbol.
Step 3, the coverage of each node radar in the radar network, and covering according to each node radar are calculated
Lid scope constructs the object function of maximum coverage.
Step 3 specifically includes following sub-step：
(3a) calculates the coverage S of each node radari=X | | | X-Xi||≤Rimax, wherein X represents target location,
XiRepresent the position of the i-th node radar, RimaxRepresent the maximum detectable range of the i-th node radar, ‖ ‖ represent 2 norms；
The object function F of (3b) maximum coverage2For：
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi,θi), ρi、θiRespectively
For distance and angle of the position in polar coordinate system of i-th of node radar, i=1,2 ..., N, N is radar network interior joint thunder
The number reached, SiIt is respectively X for each node radar site coordinate1,X2,…,XNSituation investigative range, max expression take maximum
Value, ∪ expressions take ' union '.
Step 4, according to the object function of the object function and maximum coverage for minimizing and being spoofed probability, build
Vertical combined optimization object function.
Step 4 specifically includes following sub-step：
(4a) is spoofed the object function F of probability according to minimizing1With the object function F of maximum coverage2, establish
Combined optimization object function F is：
The optimizing location problem of radar network is a multi-objective optimization question.It, can be to each to multi-objective optimization question
Majorized function is assigned to different weight coefficients, is synthesized a scalar objective function, then optimizes solution.
(4b) combined optimization object function F is converted into：
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi,θi), ρi、θiRespectively
For distance and angle of the position in polar coordinate system of i-th of node radar, i=1,2 ..., N, N is radar network interior joint thunder
The number reached, SiIt is respectively X for each node radar site coordinate1,X2,…,XNSituation investigative range, ΩDRepresent son detection
Region, w represent sub- search coverage ΩDCorresponding weighting coefficient, ∪ expressions take ' union ', and the size of 0≤λ≤1, λ characterize group
Net radar stresses degree to be spoofed probability and coverage.
Step 5, according to cloth station spacing, search coverage and the detection model between adjacent node radar in the radar network
It encloses, determines the constraints at radar network cloth station.
Cloth station is carried out to radar network, in addition to optimization object function to be considered, radar network system need to be taken into account to cloth
The constraints of station location, mainly including the following aspects：First, it is the advantage of guarantee radar network various visual angles and reception target
The distance between incoherence between signal, two node radars cannot be too near；2nd, ensure that the expectation to radar network is visited as far as possible
Survey the covering of spatial domain scope.
Step 5 specifically includes following sub-step：
(5a) is according to requirement of the radar network the cloth station spacing between adjacent node radar, i.e. any two node thunder
Constraints d (the X of cloth station spacing between reachingi,Xj)≥△Rmin, the cloth station spacing d between any two node radar
(Xi,Xj) be：
Wherein, Ri、θiThe range information and angle information of respectively i-th node radar, Rj、θjRespectively j-th of node
The range information and angle information of radar, i, j are the number of node radar, i=1,2 ..., N, j=1,2 ..., N, i ≠ j,
△RminRepresent the minimum threshold of distance between two node radars；
(5b) radar network is to the search coverage Ω of true and false targetDIn the investigative range of radar network, i.e.,
Wherein, Ri、θiThe respectively range information and angle information of the i-th node radar, R, θ are respectively the distance letter of target
Breath and angle information, Ψ be radar network cloth station scope, ΩDRepresent the sub- search coverage of radar network,Represent arbitrary, ∈
Expression belongs to.
Step 6, according to the constraints at the radar network cloth station and the combined optimization object function, deception formula is constructed
The optimized-type of the lower networking Method in Positioning of Radar of interference.
Step 6 specifically includes following sub-step：
According to the constraints at the radar network cloth station and the combined optimization object function, construct under Deceiving interference
Optimized-type Q (the X at radar network cloth station1,X2,…,XN)：
Wherein, s.t. represents constraints,Represent arbitrary.
Step 7, according to the optimized-type of networking Method in Positioning of Radar under the Deceiving interference, each node in radar network is obtained
The cloth station location of radar.
To the optimized-type Q (X at radar network cloth station1,X2,…,XN) solved, analytic solutions are obtained, rule of thumb chooses and closes
Optimal Station position of the solution of reason as radar network
(7a) is using iterative algorithm to the optimized-type Q (X at radar network cloth station1,X2,…,XN) solved, it is cheated
Under formula disturbed condition in radar network each node radar optimization position coordinatesWherein,Respectively
Range information and angle information of the i-th node radar obtained for iterative algorithm in polar coordinate system；
(7b) is by the optimization position coordinates of each node radarRectangular coordinate system is transformed into, is obtained under rectangular coordinate system
The rectangular co-ordinate of each node radarAs the Optimal Station position of radar network, wherein, the x of each node radar
Axial coordinate and y-axis coordinate are respectively：
With reference to Fig. 2 and Fig. 3 to the cloth station method of radar network under a kind of Deceiving interference provided in an embodiment of the present invention
Effect passes through following emulation further verification.
1. experiment scene：
By taking the radar network of three node radar compositions as an example, cloth station simulation analysis are optimized, without loss of generality, if visiting
Survey region ΩDFor border circular areas, and it is divided into 5 sub-regions with waiting radiuses, the weighting coefficient per sub-regions is from inside to outside successively
Increase, ΩD=X | | | X-X0| |≤R } wherein, X0Represent the center of circle of search coverage, R is its radius.Per sub-regions and its add
Weight coefficient is respectively：If can cloth station scope Ψ be
Rectangular region：X-axis excursion be -80km~-40km, y-axis excursion be -80km~80km, the ginseng of each node radar
Identical, the power range radius R of numberimax=100km, angle measurement accuracy 0.002rad, range accuracy 70m；Between two radars most
Small distance is limited to △ Rmin=10km.
2. experiment content and analysis：
Experiment one：Select search coverage center X0=[0,0]T, radius R=10km under different weights coefficient lambda, can obtain
To three radars are optimized cloth station as a result, as shown in Figure 2.
From fig. 2 it can be seen that the Optimal Station position obtained using the method for the present invention：The 1st section obtained in λ=0
The cloth erect-position of point radar is set to (- 80, -40) km, and the cloth erect-position of the 2nd node radar is set to (- 40,80) km, the 3rd node thunder
The cloth erect-position reached is set to (- 40, -80) km；The cloth erect-position of the 1st node radar obtained during λ=0.5 is set to (- 40,80) km,
The cloth erect-position of 2nd node radar is set to (- 40, -50) km, and the cloth erect-position of the 3rd node radar is set to (- 40, -80) km；λ=
The cloth erect-position of the 1st node radar obtained when 1 is set to (- 40, -80) km, the cloth erect-position of the 2nd node radar be set to (- 40, -
50) km, the cloth erect-position of the 3rd node radar are set to (- 40, -80) km；From cloth station, result can be seen that full under three circumstances
Sufficient constraints, the side light correctness of optimum results.
The above description is merely a specific embodiment, but protection scope of the present invention is not limited thereto, any
Those familiar with the art in the technical scope disclosed by the present invention, can readily occur in change or replacement, should all contain
Lid is within protection scope of the present invention.Therefore, protection scope of the present invention should be based on the protection scope of the described claims.
Claims (10)
1. a kind of optimizing location method of radar network under Deceiving interference, which is characterized in that the radar network includes multiple
Node radar, the optimizing location method include the following steps：
Step 1, calculate radar network is spoofed probability；
Step 2, according to the probability that is spoofed of the radar network, construction radar network minimum is spoofed the target letter of probability
Number；
Step 3, the coverage of each node radar in the radar network is calculated, and according to the covering model of each node radar
It encloses, constructs the object function of radar network maximum coverage；
Step 4, according to the object function of the object function and maximum coverage for minimizing and being spoofed probability, connection is established
Close optimization object function；
Step 5, according to the spy of cloth station spacing, search coverage and radar network between adjacent node radar in the radar network
Scope is surveyed, determines the constraints at radar network cloth station；
Step 6, according to the constraints at the radar network cloth station and the combined optimization object function, Deceiving interference is constructed
The optimized-type of lower networking Method in Positioning of Radar；
Step 7, according to the optimized-type of networking Method in Positioning of Radar under the Deceiving interference, each node radar in radar network is obtained
Cloth station location.
2. the optimizing location method of radar network under a kind of Deceiving interference according to claim 1, which is characterized in that step
Rapid 1 specifically includes following sub-step：
(1a) establishes hypothesis testing model；
(1b) calculates probability of miscarriage of justice of any two node radar to decoy according to the hypothesis testing model；
(1c) even multiplies any two node radar probability of miscarriage of justice of decoy, and it is general to obtain being spoofed for radar network
Rate.
3. the optimizing location method of radar network under a kind of Deceiving interference according to claim 2, which is characterized in that son
Step (1a) specifically includes following sub-step：
(1a1) is by the mahalanobis distance d of any two node radar measurement errorsijAs statistical check amount,
Wherein Δ Z=dZi-dZjFor the difference of any two node radar measurement errors, wherein ∑ij=E [(dZi-dZj)(dZi-dZj)T]
Represent the covariance matrix of the difference of the i-th node radar and jth node radar measurement errors, dZiIt is missed for the measurement of the i-th node radar
Difference, dZjFor the measurement error of jth node radar, i, j=1,2 ..., Ni ≠ j, i, j represent the number of node radar, N expression sections
The number of point radar；
(1a2) hypothesis testing model is as follows：Wherein, H is remembered0Represent two measuring value ZiAnd ZjCorresponding true mesh
Mark, H1Represent two measuring value ZiAnd ZjCorresponding false targets,It is 2 to represent degree of freedom, conspicuousness
Level is the chi square distribution of 1-a；
(1a3) is in H1Under conditions of establishment, the difference Δ Z of two node radar measurement errors obeys average as E, association side
Poor matrix is ∑ijNormal distribution, average For
Coordinate of the decoy that i-th node radar, jth node radar detection are arrived under same rectangular coordinate system,ρiFor the i-th node detections of radar range-to-go, θiFor the i-th section
Detections of radar is put to the angle of target, Δ d represents deception distance；
(1a4) is in H1Under conditions of establishment, the two-dimensional probability density function f of Δ Zij(x, y) is：
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WhereinRepresent the precision of x-axis measurement difference,Represent the precision of y-axis measurement difference, ρ=ξ12/(σx
σy) represent that x-axis measurement difference measures the related coefficient of difference, ξ with y-axis11, ξ12, ξ21, ξ22For matrix ∑ijCorresponding each matrix
Element.
4. the optimizing location method of radar network under a kind of Deceiving interference according to claim 3, which is characterized in that son
Step (1b) specifically includes following sub-step：
(1b1) i-th node radar, jth node radar are using data anastomosing algorithm to the probability of miscarriage of justice P of decoyijFor：
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<mi>H</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>K</mi>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>x</mi>
<msub>
<mi>&sigma;</mi>
<mi>x</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mi>x</mi>
<msub>
<mi>&sigma;</mi>
<mi>x</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mfrac>
<mi>y</mi>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>y</mi>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<mi>&delta;</mi>
<mo>|</mo>
<msub>
<mi>H</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein P (H0|H1) represent assuming that H1Under be judged to H0Probability, dijRepresent mahalanobis distance and
<mrow>
<msub>
<mi>d</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>&Delta;Z</mi>
<mi>T</mi>
</msup>
<msup>
<mi>&Sigma;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mi>&Delta;</mi>
<mi>Z</mi>
<mo>=</mo>
<msup>
<mi>K</mi>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>x</mi>
<msub>
<mi>&sigma;</mi>
<mi>x</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<mi>&rho;</mi>
<mo>(</mo>
<mfrac>
<mi>x</mi>
<msub>
<mi>&sigma;</mi>
<mi>x</mi>
</msub>
</mfrac>
<mo>)</mo>
<mo>(</mo>
<mfrac>
<mi>y</mi>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mfrac>
<mo>)</mo>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>y</mi>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
(1b2) is by probability of miscarriage of justice PijExpression formula be converted into integrated form, and simplification obtains：
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>H</mi>
<mn>0</mn>
</msub>
<mo>|</mo>
<msub>
<mi>H</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<msqrt>
<mi>&delta;</mi>
</msqrt>
<mo>/</mo>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mrow>
<mrow>
<msqrt>
<mi>&delta;</mi>
</msqrt>
<mo>/</mo>
<msub>
<mi>&sigma;</mi>
<mi>y</mi>
</msub>
</mrow>
</msubsup>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<msub>
<mi>g</mi>
<mrow>
<mi>l</mi>
<mi>o</mi>
<mi>w</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>g</mi>
<mrow>
<mi>u</mi>
<mi>p</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>x</mi>
<mi>d</mi>
<mi>y</mi>
</mrow>
WhereinRepresent the integrating range y directions upper limit,Represent integrating range y directions lower limit,For integrating range x
The upper limit on direction,Represent the upper limit of integrating range in the x direction；
(1b3) radar network is spoofed probabilityWherein PfRepresent radar network is spoofed probability, Pij
Represent the probability of miscarriage of justice of the i-th node radar and jth node radar to decoy, Π represents even to multiply symbol.
5. the optimizing location method of radar network under a kind of Deceiving interference according to claim 1, which is characterized in that step
Rapid 2 specifically include following sub-step：
(2a) divides the search coverage Ω of radar network, obtains multiple sub- search coverage ΩD, according to different sub- detecting areas
The degree of danger in domain, to sub- search coverage ΩDIt is assigned to weighting coefficient w；
(2b) is spoofed probability P according to radar networkf, construct the object function F for minimizing and being spoofed probability1For：
<mrow>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>N</mi>
</mrow>
</munder>
<munder>
<mo>&Sigma;</mo>
<msub>
<mi>&Omega;</mi>
<mi>D</mi>
</msub>
</munder>
<mi>w</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>P</mi>
<mi>f</mi>
</msub>
</mrow>
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi, θi), ρi、θiRespectively i-th
Distance and angle of the position of a node radar in polar coordinate system, i=1,2 ..., N, N are radar network interior joint radar
Number, ΩDRepresent sub- search coverage, w represents sub- search coverage ΩDCorresponding weighting coefficient, min expressions are minimized, and ∑ represents
Summation symbol.
6. the optimizing location method of radar network under a kind of Deceiving interference according to claim 1, which is characterized in that step
Rapid 3 specifically include following sub-step：
(3a) calculates the coverage S of each node radari=X | | | X-Xi||≤Rimax, wherein X represents target location, XiIt represents
The position of i-th node radar, RimaxRepresent the maximum detectable range of the i-th node radar, | | | | represent 2 norms；
The object function F of (3b) maximum coverage2For：
<mrow>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
<mrow>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>N</mi>
</mrow>
</munder>
<munderover>
<mrow>
<mi></mi>
<mo>&cup;</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>S</mi>
<mi>i</mi>
</msub>
</mrow>
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi, θi), ρi、θiRespectively i-th
Distance and angle of the position of a node radar in polar coordinate system, i=1,2 ..., N, N are radar network interior joint radar
Number, SiIt is respectively X for each node radar site coordinate1, X2..., XNSituation investigative range, max expression be maximized,
∪ expressions take ' union '.
7. the optimizing location method of radar network under a kind of Deceiving interference according to claim 1, which is characterized in that step
Rapid 4 specifically include following sub-step：
(4a) is spoofed the object function F of probability according to minimizing1With the object function F of maximum coverage2, establish joint
Optimization object function F is：
<mrow>
<mi>F</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<munder>
<mi>min</mi>
<mrow>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<mi>N</mi>
</mrow>
</munder>
<munder>
<mi>&Sigma;</mi>
<msub>
<mi>&Omega;</mi>
<mi>D</mi>
</msub>
</munder>
<mi>w</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>P</mi>
<mi>f</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<munder>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
<mrow>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>N</mi>
</mrow>
</munder>
<munderover>
<mrow>
<mi></mi>
<mo>&cup;</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>S</mi>
<mi>i</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
(4b)) combined optimization object function F is converted into：
<mrow>
<mi>F</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mi>min</mi>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
</munder>
<mrow>
<mo>(</mo>
<mi>&lambda;</mi>
<munder>
<mo>&Sigma;</mo>
<msub>
<mi>&Omega;</mi>
<mi>D</mi>
</msub>
</munder>
<mi>w</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>p</mi>
<mi>f</mi>
</msub>
<mo>-</mo>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>-</mo>
<mi>&lambda;</mi>
</mrow>
<mo>)</mo>
<munderover>
<mrow>
<mi></mi>
<mo>&cup;</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>S</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
Wherein, optimized variable is position coordinates X of all node radars under polar coordinate systemi=(ρi, θi), ρi、θiRespectively i-th
Distance and angle of the position of a node radar in polar coordinate system, i=1,2 ..., N, N are radar network interior joint radar
Number, SiIt is respectively X for each node radar site coordinate1, X2..., XNSituation investigative range, ΩDRepresent sub- search coverage,
W represents sub- search coverage ΩDCorresponding weighting coefficient, ∪ expressions take ' union ', and the size of 0≤λ≤1, λ characterize radar network
Stress degree to be spoofed probability and coverage.
8. the optimizing location method of radar network under a kind of Deceiving interference according to claim 6, which is characterized in that step
Rapid 5 specifically include following sub-step：
(5a) according to requirement of the radar network the cloth station spacing between adjacent node radar, i.e., any two node radar it
Between cloth station spacing constraints d (Xi, Xj)≥ΔRmin, the cloth station spacing d (X between any two node radari, Xj)
For：
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>j</mi>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mi>X</mi>
<mi>j</mi>
</msub>
<mo>|</mo>
<mo>|</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msubsup>
<mi>R</mi>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>R</mi>
<mi>j</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>R</mi>
<mi>i</mi>
</msub>
<msub>
<mi>R</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msqrt>
<mo>,</mo>
</mrow>
Wherein, Ri、θiThe range information and angle information of respectively i-th node radar, Rj、θjRespectively j-th of node radar
Range information and angle information, i, j are the number of node radar, i=1,2 ..., N, j=1,2 ..., N, i ≠ j, Δ Rmin
Represent the minimum threshold of distance between two node radars；
(5b) radar network is to the search coverage Ω of true and false targetDIn the investigative range of radar network, i.e.,
<mrow>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<mi>X</mi>
<mo>|</mo>
<mo>|</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msubsup>
<mi>R</mi>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<mi>R</mi>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>R</mi>
<mi>i</mi>
</msub>
<mi>R</mi>
<mi> </mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<mi>&theta;</mi>
<mo>)</mo>
</mrow>
</mrow>
</msqrt>
<mo>&le;</mo>
<msub>
<mi>R</mi>
<mrow>
<mi>i</mi>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>,</mo>
<mo>&ForAll;</mo>
<mi>X</mi>
<mo>&Element;</mo>
<msub>
<mi>&Omega;</mi>
<mi>D</mi>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>&Element;</mo>
<mi>&Psi;</mi>
</mrow>
Wherein, Ri、θiThe respectively range information and angle information of the i-th node radar, R, θ be respectively target range information and
Angle information, Ψ be radar network cloth station scope, ΩDRepresent the sub- search coverage of radar network,Represent arbitrary, ∈ is represented
Belong to.
9. the optimizing location method of radar network under a kind of Deceiving interference according to claim 8, which is characterized in that step
Rapid 6 specifically include：
According to the constraints at the radar network cloth station and the combined optimization object function, networking under Deceiving interference is constructed
Optimized-type Q (the X of Method in Positioning of Radar1, X2..., XN)：
<mrow>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
</munder>
<mi>F</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>N</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&GreaterEqual;</mo>
<msub>
<mi>&Delta;R</mi>
<mi>min</mi>
</msub>
<mo>,</mo>
<mo>&ForAll;</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>N</mi>
<mo>,</mo>
<mi>i</mi>
<mo>&NotEqual;</mo>
<mi>j</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mtd>
</mtr>
<mtr>
<mtd>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<mi>X</mi>
<mo>|</mo>
<mo>|</mo>
<mo>&le;</mo>
<msub>
<mi>R</mi>
<mrow>
<mi>i</mi>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>,</mo>
<mo>&ForAll;</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>&Element;</mo>
<msub>
<mi>&Omega;</mi>
<mi>D</mi>
</msub>
<mo>,</mo>
<mi>X</mi>
<mo>&Element;</mo>
<mi>&Psi;</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>.</mo>
</mrow>
10. the optimizing location method of radar network under a kind of Deceiving interference according to claim 1, which is characterized in that
Step 7 specifically includes：
(7a) is to the optimized-type Q (X of networking Method in Positioning of Radar under Deceiving interference1, X2..., XN) solved, it obtains deception formula and does
Disturb the optimization position coordinates of each node radar in lower radar networkWherein,Respectively i-th section
Range information and angle information of the point radar in polar coordinate system；
(7b) is by the optimization position coordinates of each node radarRectangular coordinate system is transformed into, is obtained each under rectangular coordinate system
The rectangular co-ordinate of a node radarWherein, the x-axis coordinate of each node radar and y-axis coordinate are respectively：
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