CN105607045B - A kind of optimizing location method of radar network under Deceiving interference - Google Patents

Abstract

The invention belongs to the technical field of radar anti-jamming, and discloses a method for optimizing station deployment of networked radars under deceptive interference, including: calculating the spoofed probability of the networked radar; constructing an objective function that minimizes the spoofed probability; calculating the networking Construct the objective function of maximizing the coverage of the coverage of each node radar in the radar; establish a joint optimization objective function according to the objective function of minimizing the probability of being deceived and the objective function of maximizing the coverage; determine the constraints of networked radar station deployment conditions; according to the constraint conditions of networked radar station deployment and the joint optimization objective function, the optimal formula for networked radar station deployment under deceptive jamming is constructed; according to the optimal formula for networked radar station deployment under deceptive jamming, the network radar The station location of each node radar is used to save radar system resources and achieve more accurate and effective suppression of deceptive interference.

Description

A method for optimal deployment of networked radar under deceptive jamming

technical field

The present invention relates to the field of radar anti-jamming technology, and in particular to a method for optimizing station deployment of networked radars under deceptive interference. The location of the networked radar is obtained, so as to improve the ability of the networked radar to suppress deceptive interference.

Background technique

Deceptive interference refers to the indistinguishability between the interference signal and the target echo signal in the radar receiver, so that the radar cannot correctly detect the target information. The working principle of deceptive jamming is: transmit signals to the received radar, after interference modulation, change the relevant parameters, and then forward the modulated signal back to the radar to simulate the echo signal of the radar target. It makes it difficult for the radar to distinguish between true and false targets, thereby achieving the purpose of confusing and disrupting the radar's detection and tracking of real targets. The maturity of digital radio frequency memory technology makes it possible to generate high-fidelity false targets. Many false targets are generated for the radar at different positions away from the target, so that the radar cannot distinguish between real and false targets. Therefore, although real targets are included in many of the displayed targets, if the radar does not have the ability to distinguish real and false targets, all targets must be treated as real targets. If the radar cannot effectively fight against deceptive jamming, it must always detect and track false targets, which will occupy a large amount of radar system resources and seriously affect the data processing capability of the radar.

Due to the single point of view of monostatic radar, it is difficult to fight against it. The networked radar can form an all-round, three-dimensional, and multi-level combat system on the battlefield. It has technical performances such as full frequency bands, multiple systems, and multiple overlapping coefficients, so it has strong survivability and anti-interference capabilities. As far as frequency domain countermeasures are concerned, it is impossible to use a jammer to jam such a wide frequency band with a network of multiple multi-band radars. As far as the space electromagnetic environment is concerned, after multiple radars are networked, not only the number of radiation sources will increase, making the density and distribution of the signal space more complex, but also the reconnaissance system will be affected in terms of frequency range, signal form, parameter type and threat level. It is difficult to make the quality of interference is greatly limited.

The relative position of each node radar in the networked radar will directly affect the correlation test of the target, and then affect the spoofing probability of the networked radar. Therefore, it is of great significance to study the optimal layout of the networked radar under deceptive jamming. The threat of deceptive interference to the networked radar can be effectively reduced by rationally distributing the stations of the networked radar.

Contents of the invention

In view of the above problems, the object of the present invention is to provide an optimized site layout method for networked radars under deceptive jamming, so as to save radar system resources and achieve more accurate and effective suppression of deceptive jamming.

In order to achieve the above purpose, the embodiments of the present invention adopt the following technical solutions to achieve.

A method for optimizing station deployment of networked radars under deceptive interference, wherein the networked radars include a plurality of node radars, and the method for optimizing station deployment includes the following steps:

Step 1, calculate the probability of being deceived by the networked radar;

Step 2, according to the spoofed probability of the networked radar, constructing an objective function for the networked radar to minimize the spoofed probability;

Step 3, calculating the coverage of each node radar in the networked radar, and constructing an objective function for maximizing the coverage of the networked radar according to the coverage of each node radar;

Step 4, according to the objective function of minimizing the probability of being deceived and the objective function of maximizing coverage, establish a joint optimization objective function;

Step 5, according to the station layout distance between adjacent node radars in the network radar, the detection area and the detection range of the network radar, determine the constraint conditions for the network radar station deployment;

Step 6, according to the constraints of the networked radar station deployment and the joint optimization objective function, construct an optimal formula for the networked radar station deployment under deceptive interference;

In step 7, according to the optimal formula for deploying radar stations in the network under deceptive interference, obtain the station deployment positions of the radars of each node in the radar network.

The present invention has the following advantages: (1) Compared with the existing method for suppressing distributed interference in the data processing stage, the present invention suppresses the adverse effects of deceptive interference by optimizing station deployment, since the optimal station deployment position can be obtained offline, Therefore, the radar system resources required by the method of the present invention are less; (2) because the present invention not only considers the radar coverage area and the probability of being deceived by the networked radar, but also takes into account the station spacing of node radars and the necessary detection areas in projects such as Considering factors, so the present invention is more conducive to engineering practice.

Description of drawings

In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. Those skilled in the art can also obtain other drawings based on these drawings without creative work.

FIG. 1 is a schematic flow diagram 1 of an optimal station deployment method for a networked radar under deceptive interference provided by an embodiment of the present invention;

Fig. 2 is a schematic flow diagram 2 of an optimal station deployment method for a networked radar under deceptive interference provided by an embodiment of the present invention;

Fig. 3 is a simulation schematic diagram of networked radar station layout results obtained by using the method of the present invention under the condition of deceitful interference, selecting the detection area center X ₀ =[0,0] ^T , radius R=10km.

Detailed ways

The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some, not all, embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

An embodiment of the present invention provides a method for optimal deployment of networked radars under deceptive interference. The networked radars include multiple node radars. The optimized station deployment method provided by the present invention is based on an existing networked radar topology optimized below. As shown in Figure 1, the method for optimizing station layout includes the following steps:

Step 1. Calculate the spoofing probability of the networking radar.

Step 1 specifically includes the following sub-steps:

(1a) Build a hypothesis testing model.

Substep (1a) specifically includes the following substeps:

(1a1) Take the Mahalanobis distance d _ij of the radar measurement error of any two nodes as the statistical test quantity, where △Z=dZ _i -dZ _j is the difference between the measurement errors of any two radar nodes, where ∑ _ij =E[(dZ _i -dZ _j )(dZ _i -dZ _j ) ^T ] means that the i-th node radar and the j-th node The covariance matrix of the difference between node radar measurement errors, dZ _i is the measurement error of the i-th node radar, dZ _j is the measurement error of the j-th node radar, i, j=1, 2,..., N i≠j, i, j represents the number of the node radar, and N represents the number of node radars;

(1a2) The hypothesis testing model is as follows: Among them, H ₀ means that the two measured values Z _i and Z _j correspond to real targets, H ₁ means that the two measured values Z _i and Z _j correspond to deceptive false targets, Indicates a chi-square distribution with 2 degrees of freedom and a significance level of 1-a;

(1a3) Under the condition that H ₁ is established, the difference △ Z of the radar measurement errors of the two nodes obeys the mean value E, the covariance matrix is a normal distribution of ∑ _ij , and the mean value is the coordinates of the false target detected by the i-th node radar and the j-th node radar in the same rectangular coordinate system, i=1,2,...,N, _ρi is the distance of the target detected by the i-th node radar, θ _i is the angle of the i-th node radar detected target, △d represents the deception distance;

(1a4) Under the condition that H ₁ is established, the two-dimensional probability density function f _ij (x, y) of △Z is:

in Indicates the precision with which the x-axis measures the difference, Indicates the accuracy of the y-axis measurement difference, ρ=ξ ₁₂ /(σ _x σ _y ) represents the correlation coefficient between the x-axis measurement difference and the y-axis measurement difference, ξ ₁₁ , ξ ₁₂ , ξ ₂₁ , ξ ₂₂ are the corresponding matrix Σ _ij individual matrix elements.

(1b) According to the hypothesis testing model, calculate the misjudgment probability of any two node radars for false targets.

Exemplarily, a data fusion algorithm may be used to calculate the misjudgment probability of any two node radars for a false target.

Substep (1b) specifically includes the following substeps:

(1b1) The misjudgment probability P _ij of the i-th node radar and the j-th node radar using the data fusion algorithm for false targets is:

Among them, P(H ₀ |H ₁ ) represents the probability of being judged as H ₀ under the hypothesis H ₁ , d _ij represents the Mahalanobis distance and

in,

(1b2) Transform the expression of misjudgment probability P _ij into integral form, and simplify to get:

in Indicates the upper limit of the integration interval in the y direction, Indicates the lower limit of the integration interval in the y direction, is the upper limit of the integration interval in the x direction, Indicates the upper limit of the integration interval in the x direction.

(1c) Calculate the deception probability of the networked radar according to the misjudgment probability of the false target by the radars of any two nodes.

When the number of node radars is greater than 2, every two node radars need to identify false targets, and then use the principle of 'take and' to fuse all the judgment results to obtain the final identification result.

Substep (1c) specifically includes the following substeps:

The probability of being deceived by the networked radar Among them, P _f represents the probability of being deceived by the networked radar, P _ij represents the misjudgment probability of the i-th node radar and the j-th node radar on the false target, and ∏ represents the multiplication symbol.

The purpose of deploying the networked radar is to suppress the influence of deceptive jamming on the networked radar, that is, to improve the performance of the networked radar under the condition of deceptive jamming. To achieve this goal, the optimal objective function selected is the intersection of maximizing the detection range of the networked radar and minimizing the probability of being deceived by the networked radar. The specific implementation steps are as follows:

Step 2, according to the deception probability of the networked radar, construct an objective function that minimizes the deception probability.

Step 2 specifically includes the following sub-steps:

(2a) Divide the detection area Ω of the networked radar to obtain multiple sub-detection areas Ω _D , and assign a weighting coefficient w to the sub-detection area Ω _D according to the degree of danger of different sub-detection areas;

(2b) According to the deception probability P _f of the networked radar, construct the objective function F ₁ that minimizes the deception probability as:

Among them, the optimization variable is the position coordinates X _i = (ρ _i , θ _i ) of all node radars in the polar coordinate system, and ρ _i and θ _i are the distance and angle of the position of the i-th node radar in the polar coordinate system, respectively , i=1,2,...,N,N is the number of node radars in the networked radar, Ω _D represents the sub-detection area, w represents the weighting coefficient corresponding to the sub-detection area Ω _D , min represents the minimum value, Σ represents Summation symbol.

Step 3, calculating the coverage of each node radar in the networked radar, and constructing an objective function for maximizing the coverage according to the coverage of each node radar.

Step 3 specifically includes the following sub-steps:

(3a) Calculate the coverage area S _i ={X|||XX _i ||≤R _imax } of each node radar, where X represents the target position, X _i represents the position of the i-th node radar, and R _imax represents the i-th node radar The maximum detection distance of , ‖‖ means 2 norm;

(3b) The objective function _F2 for maximizing coverage is:

Among them, the optimization variable is the position coordinates X _i = (ρ _i , θ _i ) of all node radars in the polar coordinate system, and ρ _i and θ _i are the distance and angle of the position of the i-th node radar in the polar coordinate system, respectively , i=1,2,...,N,N is the number of node radars in the networked radar, S _i is the detection range of the situation where the position coordinates of each node radar are X ₁ , X ₂ ,...,X _N , max means to take the maximum value, and ∪ means to take the 'union'.

Step 4, according to the objective function of minimizing the probability of being deceived and the objective function of maximizing coverage, establish a joint optimization objective function.

Step 4 specifically includes the following sub-steps:

(4a) According to the objective function F ₁ that minimizes the probability of being deceived and the objective function F ₂ that maximizes coverage, establish a joint optimization objective function F as:

The optimal station placement problem of networked radar is a multi-objective optimization problem. For multi-objective optimization problems, different weight coefficients can be assigned to each optimization function, which can be synthesized into a scalar objective function, and then optimized for solution.

(4b) The joint optimization objective function F is transformed into:

Among them, the optimization variable is the position coordinates X _i = (ρ _i , θ _i ) of all node radars in the polar coordinate system, ρ _i , θ _i are the distance and Angle, i=1,2,...,N,N is the number of node radars in the networked radar, S _i is the detection range of the situation where the position coordinates of each node radar are X ₁ , X ₂ ,...,X _N , Ω _D represents the sub-detection area, w represents the weighting coefficient corresponding to the sub-detection area Ω _D , ∪ represents the 'union', 0≤λ≤1, and the size of λ represents the focus of the networked radar on the probability of being deceived and the coverage degree.

Step 5, according to the station deployment distance, detection area and detection range between adjacent node radars in the network radar, determine the constraint conditions for network radar station deployment.

For networked radar station deployment, in addition to considering the optimization of the objective function, it is necessary to take into account the constraints of the networked radar system on the location of the station, mainly including the following aspects: The incoherence between the target signals, the distance between the two node radars should not be too close; Second, try to ensure the coverage of the expected detection airspace range of the networked radars.

Step 5 specifically includes the following sub-steps:

(5a) According to the requirements of networked radars on the station spacing between adjacent node radars, that is, the constraint condition d(X _i , X _j )≥△R _min for the station spacing between any two node radars, The distance d(X _i , X _j ) between any two radar nodes is:

Among them, R _i and θ _i are the distance information and angle information of the i-th node radar respectively, R _j and θ _j are the distance information and angle information of the j-th node radar respectively, and i and j are the numbers of the node radar, i=1,2,…,N, j=1,2,…,N, i≠j, △R _min represents the minimum threshold value of the distance between two node radars;

(5b) The detection area Ω _D of the networked radar for real and false targets is within the detection range of the networked radar, that is

Among them, R _i and θ _i are the distance information and angle information of the i-th node radar respectively, R and θ are the distance information and angle information of the target respectively, Ψ is the range of networked radar stations, Ω _D represents the range of networked radar sub-detection area, means any, and ∈ means belongs to.

Step 6: According to the constraint conditions of the networked radar station deployment and the joint optimization objective function, an optimization formula for the networked radar station deployment under deceptive interference is constructed.

Step 6 specifically includes the following sub-steps:

According to the constraints of the networked radar station deployment and the joint optimization objective function, the optimal formula Q(X ₁ ,X ₂ ,...,X _N ) for the networked radar station deployment under deceptive interference is constructed:

Among them, st represents the constraint condition, means arbitrary.

In step 7, according to the optimal formula for deploying radar stations in the network under deceptive interference, obtain the station deployment positions of the radars of each node in the radar network.

Solve the optimal formula Q(X ₁ ,X ₂ ,…,X _N ) for networked radar station deployment to obtain an analytical solution, and select a reasonable solution based on experience as the optimal station deployment location for networked radar

(7a) Use the iterative algorithm to solve the optimal formula Q(X ₁ ,X ₂ ,…,X _N ) of networked radar stations, and obtain the optimal position coordinates of each node radar in the networked radar under the condition of deceptive interference in, are the distance information and angle information of the i-th node radar in the polar coordinate system obtained by the iterative algorithm, respectively;

(7b) The optimal position coordinates of each node radar Convert to the Cartesian coordinate system to obtain the Cartesian coordinates of each node radar in the Cartesian coordinate system As the optimal deployment position of the networked radar, the x-axis coordinates and y-axis coordinates of each node radar are respectively:

2 and 3, the effect of a networked radar station deployment method under deceptive interference provided by the embodiment of the present invention is further verified by the following simulation.

1. Experimental scene:

Taking the networked radar composed of three node radars as an example, the simulation analysis of optimized station layout is carried out. Without loss of generality, the detection area Ω _D is assumed to be a circular area, and it is divided into 5 sub-areas with equal radii. The weighting coefficient of each sub-area Increasing from inside to outside, Ω _D ={X|||XX ₀ ||≤R} where X ₀ represents the center of the detection area, and R is its radius. Each sub-region, and its weighting coefficients are: Assume that the station deployment range Ψ is a rectangular area: the change range of the x-axis is -80km to -40km, the change range of the y-axis is -80km to 80km, the parameters of the radars of each node are the same, the power range radius R _imax = 100km, and the angle measurement accuracy 0.002rad, the ranging accuracy is 70m; the minimum distance between the two radars is limited to △R _min =10km.

2. Experimental content and analysis:

Experiment 1: Select the center of the detection area X ₀ =[0,0] ^T , the radius R=10km, and under different weighting coefficients λ, the results of optimizing the site layout of the three radars can be obtained, as shown in Figure 2.

As can be seen from Fig. 2, the optimal site deployment position obtained by the method of the present invention: the site layout position of the first node radar obtained at λ=0 is (-80,-40) km, the second node radar The deployment position of the radar of the third node is (-40,80) km, the deployment position of the third node radar is (-40,-80) km; when λ=0.5, the deployment position of the first node radar is (- 40,80)km, the location of the second node radar is (-40,-50)km, and the location of the third node radar is (-40,-80)km; obtained when λ=1 The location of the first node radar is (-40,-80)km, the location of the second node radar is (-40,-50)km, and the location of the third node radar is (- 40,-80)km; from the station layout results, it can be seen that the constraint conditions are met in the three cases, which indirectly shows the correctness of the optimization results.

The above is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Anyone skilled in the art can easily think of changes or substitutions within the technical scope disclosed in the present invention. Should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the protection scope of the claims.

Claims (10)

1. An optimized station distribution method for a networking radar under deceptive interference is characterized in that the networking radar comprises a plurality of node radars, and the optimized station distribution method comprises the following steps:

step 1, calculating the deception probability of the networking radar;

step 2, constructing a target function of the networking radar for minimizing the deception probability according to the deception probability of the networking radar;

step 3, calculating the coverage area of each node radar in the networking radar, and constructing a target function of the networking radar for maximizing the coverage area according to the coverage area of each node radar;

step 4, establishing a joint optimization objective function according to the objective function of the minimized deception probability and the objective function of the maximized coverage range;

step 5, determining constraint conditions of networking radar station arrangement according to the station arrangement distance between adjacent node radars in the networking radar, the detection area and the detection range of the networking radar;

step 6, constructing an optimization formula of the networking radar station arrangement under deceptive interference according to the constraint condition of the networking radar station arrangement and the joint optimization objective function;

and 7, obtaining the station distribution position of each node radar in the networking radar according to the optimized formula of the networking radar station distribution under the deceptive jamming.

2. The station distribution optimizing method for the networking radar under deceptive jamming according to claim 1, wherein the step 1 specifically includes the following substeps:

(1a) establishing a hypothesis testing model;

(1b) calculating the misjudgment probability of any two node radars to a false target according to the hypothesis test model;

(1c) and multiplying the false judgment probabilities of the random two node radars to the false target to obtain the deceived probability of the networking radar.

3. The station distribution optimizing method for the networking radar under deceptive interference according to claim 2, wherein the sub-step (1a) specifically comprises the following sub-steps:

(1a1) mahalanobis distance d of radar measurement errors of any two nodes_ijAs a statistical test quantity, the quantity of,wherein Δ Z ═ dZ_i-dZ_jThe difference between the errors is measured for any two node radars, where ∑_ij＝E[(dZ_i-dZ_j)(dZ_i-dZ_j)^T]Covariance matrix, dZ, representing the difference between the measurement errors of the i-th and j-th node radars_iFor the measurement error, dZ, of the i-th node radar_jThe measurement error of the jth node radar is represented by i, j being 1,2, …, Ni being not equal to j, i and j representing the number of the node radar, and N representing the number of the node radar;

(1a2) the hypothesis testing model is as follows:wherein, note H₀Representing two measured values Z_iAnd Z_jCorresponding to the true target, H₁Representing two measured values Z_iAnd Z_jIn response to a spoofed decoy,chi-square distribution with a degree of freedom of 2 and a significance level of 1-a;

(1a3) at H₁Under the condition of being established, the mean value of the difference delta Z between the radar measurement errors of the two nodes is E, and the covariance matrix is sigma_ijNormal distribution, mean value of The coordinates of the false targets detected by the ith node radar and the jth node radar under the same rectangular coordinate system,ρ_idistance, θ, to target detected for ith node radar_iFor the angle of the target detected by the ith node radar, delta d represents the deception distance;

(1a4) at H₁When true, the two-dimensional probability density function f of Delta Z_ij(x, y) is:

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whereinIndicating the accuracy of the x-axis measurement difference,indicating the accuracy of the y-axis measurement difference, p ξ₁₂/(σ_xσ_y) Representing the correlation coefficient of the x-axis measurement difference with the y-axis measurement difference, ξ₁₁，ξ₁₂，ξ₂₁，ξ₂₂Is a matrix sigma_ijCorresponding respective matrix elements.

4. The station distribution optimizing method for the networking radar under deceptive interference according to claim 3, wherein the sub-step (1b) comprises the following sub-steps:

(1b1) misjudgment probability P of the ith node radar and the jth node radar to the false target by adopting a data fusion algorithm_ijComprises the following steps:

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <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> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;le;</mo> <mi>&amp;delta;</mi> <mo>|</mo> <msub> <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>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <msub> <mi>&amp;sigma;</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mi>&amp;rho;</mi> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <msub> <mi>&amp;sigma;</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>&amp;delta;</mi> <mo>|</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

wherein P (H)₀|H₁) Is shown in hypothesis H₁The lower quilt is judged to be H₀Probability of (d)_ijRepresents the Mahalanobis distance and

<mrow> <msub> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>&amp;Delta;Z</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Sigma;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&amp;Delta;</mi> <mi>Z</mi> <mo>=</mo> <msup> <mi>K</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <msub> <mi>&amp;sigma;</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mi>&amp;rho;</mi> <mo>(</mo> <mfrac> <mi>x</mi> <msub> <mi>&amp;sigma;</mi> <mi>x</mi> </msub> </mfrac> <mo>)</mo> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mfrac> <mo>)</mo> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow>

wherein,

(1b2) probability of false positive P_ijThe expression of (c) is converted to integral form and simplified to yield:

<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>&amp;Integral;</mo> <mrow> <mo>-</mo> <msqrt> <mi>&amp;delta;</mi> </msqrt> <mo>/</mo> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mrow> <mrow> <msqrt> <mi>&amp;delta;</mi> </msqrt> <mo>/</mo> <msub> <mi>&amp;sigma;</mi> <mi>y</mi> </msub> </mrow> </msubsup> <msubsup> <mo>&amp;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>

whereinRepresents the upper limit of the integration interval in the y direction,represents the lower limit of the integration interval in the y direction,is the upper limit in the direction of the integration interval x,represents the upper limit of the integration interval in the x direction;

(1b3) spoofing probability of networking radarWherein P is_fRepresenting the probability of being spoofed, P, of a networked radar_ijIndicating an ith node mineAnd the misjudgment probability of the radar of the reach and j node on the false target is represented by pi, which is a continuous multiplication symbol.

5. The station distribution optimizing method for the networking radar under deceptive jamming according to claim 1, wherein the step 2 specifically comprises the following substeps:

(2a) dividing a detection region omega of the networking radar to obtain a plurality of sub-detection regions omega_DAccording to the danger degree of different sub-detection areas, the sub-detection area omega_DAssigning a weighting coefficient w;

(2b) spoofed probability P based on networking radar_fConstructing an objective function F that minimizes the probability of being spoofed₁Comprises the following steps:

<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>&amp;Sigma;</mo> <msub> <mi>&amp;Omega;</mi> <mi>D</mi> </msub> </munder> <mi>w</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>P</mi> <mi>f</mi> </msub> </mrow>

wherein, the optimization variable is the position coordinate X of all node radars under the polar coordinate system_i＝(ρ_i，θ_i)，ρ_i、θ_iAre respectively the ith node thunderThe distance and angle of the position reached in the polar coordinate system are 1,2, …, N, N is the number of node radars in the networking radar, omega_DDenotes a sub-detection region, w denotes a sub-detection region Ω_DAnd corresponding weighting coefficients, min represents the minimum value, and Σ represents the summation sign.

6. The station distribution optimizing method for the networking radar under deceptive jamming according to claim 1, wherein the step 3 specifically comprises the following sub-steps:

(3a) calculating the coverage area S of each node radar_i＝{X|||X-X_i||≤R_imaxWhere X denotes the target position, X_iIndicating the location of the i-th node radar, R_imaxThe maximum detection distance of the ith node radar is represented, and | | represents a 2-norm;

(3b) maximizing the coverage of the objective function F₂Comprises the following steps:

<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>&amp;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, the optimization variable is the position coordinate X of all node radars under the polar coordinate system_i＝(ρ_i，θ_i)，ρ_i、θ_iRespectively the distance and angle of the position of the ith node radar in a polar coordinate system, i is 1,2, …, N, N is the number of the node radars in the networking radar, S_iThe radar position coordinates of each node are respectively X₁，X₂，…，X_NMax denotes taking the maximum value, and ∪ denotes taking the 'union'.

7. The station distribution optimizing method for the networking radar under deceptive jamming according to claim 1, wherein the step 4 specifically includes the following sub-steps:

(4a) according to an objective function F which minimizes the probability of being spoofed₁And an objective function F to maximize coverage₂Establishing a joint optimization objective function F as follows:

<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>&amp;Sigma;</mi> <msub> <mi>&amp;Omega;</mi> <mi>D</mi> </msub> </munder> <mi>w</mi> <mo>&amp;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>&amp;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) joint optimization objective function F translates 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>&amp;lambda;</mi> <munder> <mo>&amp;Sigma;</mo> <msub> <mi>&amp;Omega;</mi> <mi>D</mi> </msub> </munder> <mi>w</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>p</mi> <mi>f</mi> </msub> <mo>-</mo> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;lambda;</mi> </mrow> <mo>)</mo> <munderover> <mrow> <mi></mi> <mo>&amp;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, the optimization variable is the position coordinate X of all node radars under the polar coordinate system_i＝(ρ_i，θ_i)，ρ_i、θ_iRespectively the distance and angle of the position of the ith node radar in a polar coordinate system, i is 1,2, …, N, N is the number of the node radars in the networking radar, S_iThe radar position coordinates of each node are respectively X₁，X₂，…，X_NThe detection range of the case of (2), omega_DDenotes a sub-detection region, w denotes a sub-detection region Ω_DThe corresponding weighting coefficient, ∪, is represented by taking 'union', 0 ≦ λ ≦ 1, and the size of λ represents the network radar pair spoofed probability and the degree of the coverage area emphasis.

8. The station distribution optimizing method for the networked radar under deceptive jamming according to claim 6, wherein the step 5 specifically includes the following sub-steps:

(5a) according to the requirement of the networking radar on the station arrangement distance between the adjacent node radars, namely the constraint condition d (X) of the station arrangement distance between any two node radars_i，X_j)≥ΔR_minThe random ofStationing distance d (X) between two node radars_i，X_j) Comprises the following steps:

<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>&amp;theta;</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>j</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>,</mo> </mrow>

wherein R is_i、θ_iRespectively distance information and angle information, R, of the ith node radar_j、θ_jDistance information and angle information of the jth node radar are respectively shown, i and j are serial numbers of the node radars, i is 1,2, …, N, j is 1,2, …, N, i is not equal to j, and delta R_minA minimum threshold value representing a distance between two node radars;

(5b) detection area omega of networking radar for true and false targets_DIn the detection range of networked radar, 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>&amp;theta;</mi> <mi>i</mi> </msub> <mo>-</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>&amp;le;</mo> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>X</mi> <mo>&amp;Element;</mo> <msub> <mi>&amp;Omega;</mi> <mi>D</mi> </msub> <mo>,</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>&amp;Element;</mo> <mi>&amp;Psi;</mi> </mrow>

Wherein R is_i、θ_iRespectively are distance information and angle information of an ith node radar, R and theta are respectively distance information and angle information of a target, psi is a range of a networking radar station arrangement, and omega is_DRepresenting the sub-detection zones of the networked radar,denotes arbitrary, and e denotes belonging.

9. The station distribution optimizing method for the networked radar under deceptive jamming according to claim 8, wherein the step 6 specifically includes:

according to the groupConstructing an optimized Q (X) formula of the networking radar station distribution under deceptive interference according to the constraint conditions of the networking radar station distribution and the joint optimization objective function₁，X₂，…，X_N)：

<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>&amp;GreaterEqual;</mo> <msub> <mi>&amp;Delta;R</mi> <mi>min</mi> </msub> <mo>,</mo> <mo>&amp;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>&amp;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>&amp;le;</mo> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo>&amp;ForAll;</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>&amp;Element;</mo> <msub> <mi>&amp;Omega;</mi> <mi>D</mi> </msub> <mo>,</mo> <mi>X</mi> <mo>&amp;Element;</mo> <mi>&amp;Psi;</mi> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>

10. The station distribution optimizing method for the networking radar under deceptive jamming according to claim 1, wherein the step 7 specifically includes:

(7a) optimized Q (X) for arranging station of networking radar under deceptive jamming₁，X₂，…，X_N) Solving is carried out to obtain the optimized position coordinates of each node radar in the networking radar under deceptive jammingWherein,respectively obtaining distance information and angle information of the ith node radar in a polar coordinate system;

(7b) optimizing the position coordinates of each node radarConverting the radar nodes into a rectangular coordinate system to obtain rectangular coordinates of each node radar in the rectangular coordinate systemThe x-axis coordinate and the y-axis coordinate of each node radar are respectively as follows: