1.Introduction

In the real air combat,the radio frequency stealth(RFS)technology can heavily improve the penetration ability,survival ability and fighting efficiency of the fighter plane[1].Nowadays,the electromagnetic wave launched by the airborne fire-control radar has become an important threat to the comprehensive stealth,because the characteristics of the electromagnetic wave can be used to judge the position and intention of the fighter plane which launches the electromagnetic wave[2].Therefore,the researches on the RFS technology and the improvement radar stealth performance are required urgently.Reference[3]puts forward a power classification criterion and adopts the particle swarm optimization(PSO)to control the radar power.Reference[4]advances the strategies of minimizing power and irradiation time and builds the optimal radiation control method.Reference[5]establishes the radio frequency(RF)radiation risk estimation and indicator system with the related indicator computation model.Reference[6]denotes the signal interception as the indicator of the RFS performance and proposes the solution of RFS design.Reference[7]researches the control method of the beam width,radiation time and average power on the airborne phased array radar for RFS.Reference[8]puts forward the target tracking algorithm of multiple in put multiple-output(MIMO)radar RFS performance optimization.Reference[9]describes the RFS algorithm of the MIMO radar in the searching mode.Reference[10]designs the RF stealth to control the frequency hopping periods and the frequency hopping intervals based on the maximum conditional entropy.Reference[11]proposes the design of sampling periods based on RFS for target tracking.Reference[12]proposes a sampling period and radiation power control method based on RFS.Reference[13]puts forward a real-time control method of single radiation energy in radar tracking state based on RFS.

When the fire-control radar tracks the target,the radar radiates according to the fixed power and the interval in the past,to which the electromagnetic passive detection system is sensitive.The phased array radar(PAR),which uses the electronic scanning system,has abilities of scanning fast,changing waveform and controlling program[14].Therefore,we can optimize the detection performance through controlling the parameter reasonably.This paper aims to improve the RFS ability of PAR during the course of tracking.

The main idea of RFS during the course of tracking in this paper is to reduce the cumulative probability of interception(CPI),which denotes the probability of the electromagnetic passive detection of the opponent discovering the PAR during the course of tracking.There are two methods to reduce the CPI,one of which is to reduce the proba-bility of interception(PI)of the once irradiation,the other is to reduce the number of irradiation times.However,the two methods for RFS must ensure that the tracking task should be accomplished, because they may affect the tracking ability of the PAR and the precision of the target information.We need to balance the relationship between the RFS and the effects of tracking.In order to minimize CPI,two methods are combined under the condition of ensuring the completion of the tracking task.

The idea in this paper is that the single radiation power control model is first established to minimize PI for once radiation and then the self-adaption radiation interval method is proposed to minimize the number of irradiation times during the course of tracking.

This paper is divided in to eight sections.Section 1 is the introduction.In Section 2,we put forward the calculation model of PI during the course of tracking.In Section 3,we put forward the CPI to describe the effect of RFS in tracking and map out two solutions to improve the ability of RFS.In Section 4,we propose the self-adapting radiation energy control method(SARE)to minimize PI with once radiation during the course of tracking.In Section 5,we propose the self-adapting radiation interval control method(SARI)to minimize radiation times during the course of tracking.In Section6,we propose the SARE-SARI(SAEI)control method through combining SARE with SARI.In Section 7,we carry out the computer simulation and analyze the results.In Section 8,we conclude the paper and discuss the future work.

2.Interception probability

In this paper,we assume that the target tracked is equipped with the electromagnetic passive detection system(EPDS).In order to make effective use of RF signals,the EPDS must intercept and capture the RF signal in the power,airspace,frequency and time fields[15].We denote the probability of capturing the RF signal in the power field as pd,in the airspace field as ps,in the frequency field as pfand in the time field as pt.According to the characteristic of tracking,we can establish the calculation model of the interception probability.

We assume that the incoming signal of the EPDS consists of the radar RF signal s(t)and a zero-mean white Guassian process n(t)with the variance ε2.As the result of the uncertainties of noise signals,the power intercept probability pdis the detection probability pDat the given false-alarm probability denoted as pfa,which is the probability of the EPDS mistakenly detecting the radar signal.We assume that the radar signal is a sinusoidal wave with the amplitude A,and then the probability that the signal exceeds the threshold electrical level denoted as VT[15]is

where I0(β)is the zero order modified Bessel function.

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The single pulse signal noise radio(SNR)is

and

Equation(1)can be converted to the equation below:

原语翻译生态环境，广义上是指包括原语文本在内的原语语言、社会、经济、文化等宏观环境；狭义上是指原语文本的语言特点和文化特征。本文仅对狭义上的原语翻译生态环境进行分析，即寒山诗文本的白话文语言特点及其反映出的中国佛、道文化。

where

The radar radiation power Piinputtingthe EPDS[16]is

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where Ptis the peal power of the PAR;λ is the PAR wavelength;R is the distance between the radar and EPDS;Gti is the radar transmitting antenna gain at the direction of the EPDS;Giis the receiving antenna gain at the direction of the radar;GIPis the EPDS processor net gain.

Noise power of the EPDS is

where k is a Boltzmann constant which is equal to 1.38×10-23J/K;T0is the noise temperature;BRiis the EPDS receiver bandwidth.

SNR at the EPDS receiver outlet[11]is

where NFiis the noise ratio of the EPDS receiver.

According to the given false-alarm probability pfa,we can obtain the single pulse detection probability of pD from(1).Thus the power interception probability becomes

The airspace interception condition can be attained,when the wave beam of radar shines the EPDS.We consider the airspace intercept probability as ps=1,because the radar shines the EPDS all the time during the course of tracking.

The frequency and time intercept conditions are considered together,because the scanning in the airspace and frequency is simultaneous.

为了降低信任值的上升速度,防止恶意节点利用持续的正常行为来迅速提高自身信任值,本文选取上一周期的综合信任值CTi,j(t-1)作为历史因素纳入CTi,j(t)的计算中。则综合信任CTi,j(t)可按如下公式计算:

The EPDS monitors the total frequency band and airspace discontinuously.The intercept probability mainly depends on the possibility that the EPDS tunes in the right frequency and the right direction,because of the relative position between the radar and the EPDS,the frequencies of the radar and EPDS are changing.Then the EPDS needs time to cover the detection airspace and signal frequency but the radiation time of the radar beam denoted as TOTis very short.

We denote the number of channels and beam positions,which the EPDS can detect,as NLin TOT.And we denote the time that the EPDS spends in scanning a channel and beam position as tL[17].

Then we denote the total setting number of channel and beam position of the EPDS as Nfand NBrespectively.Thus the total setting search time of the EPDS can be described as

The time-frequency intercept probability is

The model(30)can be described as

As a general rule,TOT＜TI.Then

In conclusion,the probability that EPDS intercepts the electromagnetic radiation during the course of tracking can be denoted as

3.Cumulative PI

The airborne PAR tracks the target with the scanning and tracking(SAT)mode or the continuous track(CT)mode.Note that the PAR in both SAT and CT modes can intermittently irradiate the electromagnetic waves to track the target.We describe the process of tracking the single target in Fig.1.

（四）注重输出实践。这里所指的语言输出是指在运用语言时，将重点放在话语的内容上而非形式上，以意义表达为核心。笔者认为，英语口语教学的研究重心应是输出练习的设计。英语口语练习原则上应遵循先易后难，先简后繁循序渐进的次序，语言形式先于语言内容；语言形式和语言内容先于交际规则；语言形式的流利性先于准确性和多样性；言之有物先于言之有理；交际规则中通用交际规则先于跨文化交际规则。

Fig.1 The process of tracking the single target

The PAR starts to irradiate electromagnetic waves to track the target at time t0.After a time interval,the PAR radiates again at time t1.During the course of tracking,the total number of radiation times by the PAR is denoted as N.At every time,there is a PI denoted as pi(k),k=0,1,...,N,for the EPDS on the target.The CPI can be denoted as

where pCPIis the cumulative probability of interception.Then we can take the CPI as the measurement of the RFS capability for the whole process of tracking.Accordingly,our RFS-controlled tracking methods are designed to minimize CPI.From(16),we can achieve two solutions to minimize CPI.One is to minimize pi(k)for every time irradiation and the other is to minimize N denoted as follows:

(i)The solution to minimize pi(k)

From(15),we know the parameters Ptand TOTare utilizable.How to control Ptand TOTto minimize PI at every time radiation and ensure that the echo of the target can be detected by the PAR is a main problem.

(ii)The solution to minimize N

N depends on the time interval between two radiations denoted as Δtk.When increasing Δtk,N will decrease.However,Δtkdenotes the projection time for the filter and the large Δtkmay lead to the worse precision of tracking,even losing the target.How to control Δtkto minimize N with a certain precision requirement is another problem.

Next in Sections 3 and 4,we describe the two solutions in detail.

4.SARE

4.1 Energy control model

Then,Ptcan be described as

Through the radar equation,the single pulse SNRscan be denoted as

We assume that the residence time is the same as the irradiation time TOT.The pulse repetition period is denoted as Tr.

Substituting(18)into(19),we obtain(20)as follows:

The single pulse SNR and nppulses SNR are denoted as SNRsand SNRCIrespectively:

where Gtis the PAR transmitting antenna gain in the target direction;Gris the radar receiving antenna gain in the target direction;GRPis the radar receiver net gain;BR is the radar receiver bandwidth;NFis the radar receiver noise ratio.Substituting(20)into(21),we obtain(22)as follows:

The phased array antenna main lobe gain Gtis denoted as

然后对评价结果向量B进行处理,参考使用模糊向量单值法[13]。给4个信任等级依次赋予数值(c1=0.2,c2=0.5,c3=0.7,c4=0.95),需要满足c1

where Ntis the number of phased array antenna elements;ψ is the angle between the beam pointing and the normal of the phased array antenna plane;η is the antenna efficiency which is usually 0.6–0.9;Gtiis equal to Gtduring the course of tracking;σ is the radar cross section(RCS).

According to the given PAR false-alarm probability pRfaand the given radar detection probability pRD,we can know the single pulse minimum detection SNR denoted as SNRo min.SNRCImust meet(24)for PAR detecting the target:

The simplified in equation is

The coherent accumulation time TOTis limited by the motion features of the target and the distance resolution[18]:

where ΔR is the distance resolution;τ0is the pulse width;v is the relative velocity between the PAR and EPDS;c is the speed of light.

One of the solutions of minimizing CPI is that we control the parameters of the PAR by minimizing PI at every time radiation.

Thus we take(15)as the objective function of the control model:

where Cois the constant:

In conclusion,the control model is

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The control parameters such as radiation peak power Pt and radiation time TOTcan be obtained through solving(30).

Because of τ0 ≈ 1/BR,then

As the single pulse radiation power is denoted as E0=Ptτ0,the total radiation energy during the radiation time is

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ANSYS FLUENT模拟结果给出了腔体内红外辐射的能量场分布,根据探测面的能量场分布,可以得到探测面的辐射功率,于是本文定义了功率吸收效率来评价气体传感器对CO2气体的吸收能力。设气室内不存在吸收气体CO2时,探测面接收到的辐射功率为气室内有CO2气体时,探测面接收到的辐射功率为Pe,则CO2气体传感器的红外辐射功率吸收效率可表示为:

We call the model(33)as the energy control model.

4.2 Solving the energy control model

In order to realize the least radiation energy,the PAR receiver always keeps the minimum detection SNR to detect the signal.Therefore,the first inequation constraint of the model(33)becomes

Many pulses of PAR are reflected during once irradiation.The SNR of PAR can be increased by accumulating all pulses during once irradiation.As the PAR knows the parameters of pulses,it could adopt the coherent accumulation technology to increase its SNR.

Substituting(35)into(28),we obtain(36)as follows:

Thus,the radiation energy control model can be converted to

The relative distance R,speed v,angle ψ and RCS σ in the model(37)can be obtained by the echo information.And the objection function(37)is the monadic function on the radiation time TOT.The key to solve the model is to calculate the integration Q.As the analytical solution is very complex,the numerical integration method is adopted.Reference[19]puts forward a good numerical integration method,as follows:

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We can gain the approximate value of the PI,by the recursive computations(38)–(43)until βn ＞ 103given α-1=0,β0=0.5,β-1=0 and TOT ∈ [Tr,0.5cτ0/v].We take the genetic algorithm(GA)to globally stochastically search the minimum PI.We take an example to describe the method in Section 4.3.

4.3 An example

4.3.1 The parameters setting

上海某燃气锅炉每小时燃烧500 Nm3天然气，根据公式（9）得每小时烟气可回收的显热量为1 443.240MJ。

The parameters of PAR and EPDS are shown in Table 1 and Table 2 respectively.

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Table 1 Parameters of PAR

Parameter Value Parameter Value pfa 10-6 λ/m 0.03 pd 70% Tr/s 5×10-4 N 2 200 τ0/s 1×10-6 η 0.8 Br/MHz 1 GRP 28 NF/dB 3

Table 2 Parameters of EPDS

Parameter Value Parameter Value pf 10-8 GIP 2 NFi/dB 6 TI/s 2 Gi 10 T0/K 290

According to the false-alarm probability and the detection probability of radar,we can know that the minimum detection SNR of the PAR is 12.31 dB[17].The SNR should become 15.31 dB,because of the fluctuation characteristics of the Swerling II target model.We obtain C0=1.254 5×1010and C1=4.853 0×1022based on(29)and(26).We assume that R=200 km,v=200 m/s,ψ =45° and σ =5 m2.

Then,we take all above parameters to the model(35):

It is the optimization problem to solve the model(44).

4.3.2 Searching the minimum PI based on GA

Step 2 Producing the initial population

Step 1 Encoding

The variable TOTas the real number should be encoded to the binary string{0,1}.The size of the string depends on the accuracy required.If we set the accuracy as 10-6,the closed interval[5×10-4,0.8]should be divided into 0.499 5×106parts.And the size of binary string need 19 bit at least,because

The transformational relationship between the real number TOTand the binary string(u19u18u17 ···u0)2are

by(46)and(47),we get

are denoted as the left end point 5×104and the right end point 0.8,respectively.

Now we use the GA[20]to solve the model(44).

The initial chromosome population consists of 100 stochastic binary strings at the size of 20 bit.

Step 3 Computing the fitness

We take the opposite number of the objective function as the fitness function:

According to numerical integration methods(38)–(43)and(44),we can acquire the approximate value of-pi.

Step 4 Selection

According to the fitness of parents,we adopt the roulette method to select the individual.

Step 5 Crossover

We adopt the one-point crossover method to produce the new individual at the crossover probability denoted as pc=0.2.

Step 6 Variation

We take the variation step at the variation probability pm=0.08.

After producing the initial population Step 2,take 50 times looping execution Steps 3–6 and then output the best individual and optimal solution to end.

4.3.3 The results

As the parameters in Table 1 and Table 2 are fixed in certain systems,we take the computer to calculate two groups of measurement information for validating the model in Table 3.

Table 3 Measurement information

Group R/km v/(m/s) ψ σ/m2 1 200 200 45° 50 2 60 50 0° 1

In Fig.2,we show the GA result of Group 1 of measurement information.In Fig.2(a),we show the distribution and fitness of 100 chromosomes at the original,3rd and 50th generation respectively.We may know the shape of the objective function from Fig.2(a),which is a monotone decreasing function and whose extreme value is at the bound of TOT.The evolution process based on the GA is showed in Fig.2(b)and the best results are TOT=0.0011 s and-pi=-0.000 5.

Fig.2 The result of Group 1

Next,in Fig.3,like Fig.2,we show the GA results of Group 2 of measurement information and the best results are TOT=2.997 8 s and-pi=-0.000 3.Then we could gain each group of Ptby(35).Comparing with the traditional minimum radiation power control method,we denote the radiation time in this paper as fixed 0.1 s in Table 4.The minimum radiation power control method is abbreviated to the MRP control method.

Fig.3 The result of Group 2

Table 4 Results of SARE and MRP

Group Method Pt/W TOT/s pi SARE 9 177 0.001 1 0.000 5 1 MRP 101 0.1 0.045 8 SARE 0.681 8 2.997 8 0.000 3 2 MRP 20.43 0.1 0.050 0

The PI with once radiation is a small real number such as 0.000 3 and 0.000 5,although,the CPI with a long time of tracking will be a high probability.For example,we assume that the PI with each radiation is 0.000 5,the time of tracking is 30 min and the repetition period is 1 s,and then the number of radiation times is 1 800 and the CPI will reach 90%.Thus we also need to reduce the number of radiation times as possible as we can for a lower CPI.

5.SARI

5.1 Tracking algorithm

We assume that the PAR tracks the EPDS in a two dimensional space to simplify the problem of RFS.The PAR gives measurements of the target’s relative distance R and the relative azimuth angle ψ.We denote the direction of head of aircraft with PAR as Y axis and the direction of its normal as X in Fig.4.

Fig.4 Relationship among the parameters

Thus the state space model can be expressed with equations of the following form:

where xkis the state of the aircraft with EPDS at the time k,ykis the measurement at the time k,qk-1∼N(0,Qk-1)is the process noise at the time k-1,rk∼N(0,Rk)is the measurement noise at the time k,Ak-1 is the transition matrix of the dynamic model,H is the measurement model function as the following equation:

Ak-1is discretized by the equation of the following form:

where F is the continuous transition matrix,Δtk-1is equal to tk-tk-1which is the stepsize of the discretization.Thus Δtk-1is the radiation interval between tkand tk-1when the PAR detects the target.Thus Δtk-1affects the precision of tracking by Ak-1.Others that affect the precision are the tracking algorithms and the movement models.As H is a nonlinear function,the state model(49)is nonlinear.At present,the main filter algorithms for the nonlinear are extended Kalman filter(EKF)[21],unscented Kalman filter(UKF)[21]and particle filter(PF)[22].And the common movement models are constant velocity model(CV),constant acceleration model(CA)and constant turning model(CT)[21].We chose a kind of the filter algorithms and the movement models to control influence factors to investigate how Δtk-1affects the precision of tracking.Then we adopt the CV-EKF algorithm as the tracking algorithm to state the SARI control method.Then the parameters of model(49)can be described as

EKF algorithm:

where xk|k-1is the projection of state,Pk|k-1is the covariance of xk|k-1,?xkis the estimation of state xk,?Pk is the covariance of the estimation of?xkstate.Pk|k-1describes the precision of the position which the wave beam of PAR points to at time k before the measurement of xk coming.If Pk|k-1is too large,the wave beam of PAR may not cover the EPDS and lead to losing it during the course of tracking.Thus when we control Δtk-1,we should constrain Pk|k-1under a threshold denoted as the expectation covariance matrix to ensure the task of tracking normally completed.Next in section 5.2,we will discuss the expectation of the covariance matrix.

5.2 Expectation covariance matrix

In order to keep tracking the target,on the one hand the PAR needs enough energy of echo.On the other hand,the target must locate in the wave gate of PAR.We denote the event that the target’s distance R locates in the distance of the wave gate as Hrand the event that the target locates at the coverage of the azimuth of the wave beam as Hψ.Because of the uncertainty of the target movement model and the stochastic noise,there is the error between the projection of the target’s state xk|k-1and the true state xk.The probability of Hrand Hψ must meet a degree of certain brief.We assume that the target’s distance R obeys the normal distribution,i.e.,R ∼ N(Rp,σR)and denote the distance of the wave gate as Gd,where Rpis the projection of distance and σRis the variance.Then the probability of Hris

We can calculate the standard deviation:

where u0.5δ is the bilateral quantile of N(0,1)given δ.We assume the width of the wave beam of PAR is denoted as α at the azimuth.We can calculate the standard deviation of the azimuth as same as Hrby(62):

Then we can model the expectation covariance matrix in polar coordinates:

As the process of filtering in Cartesian coordinates,we need to convert Pseto Pe,which is the expectation covariance matrix in Cartesian coordinates:

where J(k-1)is the Jacobian matrix:

5.3 Radiation interval control with expectation covariance

When the expectation covariance matrix is set,the maximum radiation interval denoted as Δtmaxcan be acquired.And the minimum radiation interval denotes as Δtminis the radiation time TOTon each radiation time of the PAR,as showed in Fig.5.

Fig.5 Radiation interval

In Fig.5,TOT(k)=NkTk.The range of Δt can be got as follows:

Then in order to acquire the minimum radiation times during the course of tracking,the radiation interval of PAR at the time k should be as same as Δtmax(Pe),which ensures that the PAR tracks the EPDS without losing with a certain probability.We can calculate Δtmax(Pe)from the equations:

whereM(i,j)is the element of the ith row and the jth column of the matrix M,Δtk(1),Δtk(2)and Δtk(3)are the answers of the equations 1,2 and 3 in(69)respectively,Pe can be gained from section 5.2 and Pk|k-1can be gained from the EKF algorithm.

6.SARE-SARI tracking control method

We combine the SARE with the SARI to reduce the CPI during the course of tracking,which is denoted as the SARE-SARI tracking control method as follows:

Step 1 Initializing

Set the initial time t=tk=0,k=0 and the parameters of PAR and EPDS.

Set a certain brief degree of Hrand Hψ.

Set the initial state x0and the covariance P0of the CV-EKF tracking algorithm.

Set the initial radiation interval Δt0.Set k=1,and at time t1=t0+Δt0.

Set initial Pt(k=1)and TOT(k=1)to radiate the target.

Receive the target’s measurement information R(k=1)and ψ(k=1)and σ which we assume is fixed.

Execute the CV-EKF algorithm to gain the estimated stateand covariance

Step 2 Executing SARI

Calculate Pe(k+1)by(66).

Acquire Δtmax(Pe(k+1))by(69).

Set Δtk= Δtmax(Pe(k+1))and gain Akby(52).

Use(53)to get xk+1|k.

Control the PAR points to(x(k+1,k),y(k+1,k))at time tk+1=tk+Δtk.

Step 3 Executing SARE

Take R(k),ψ(k),v(k)and σ into the model(37)and solve the model by GA to gain TOT(k+1),Pt(k+1)and pi(k+1).

Control the PAR with TOT(k+1)and Pt(k+1)to radiate the EPDS at time tk+1.

Receive R(k+1)and ψ(k+1),execute the CV-EKF tracking algorithm.

Calculate the CPI by(16).Step 4 Setting k=k+1

Return to Step 2 and go on.

Step 5 Outputting the CPI and Ending

7.Simulation results

7.1 Purpose of simulation

In order to test and verify the SARE-SARI tracking control method which can be abbreviated as SAEI,we compare it with the fixed radiation interval and SARE which can be abbreviated as FIAE and the SARI and fixed radiation energy which can be abbreviated as AIFE in aspects of the precision of tracking and the effect of RFS.

We adopt the Monte Carlo method to carryout the simulation.We choose root mean square error(RMSE)as the index for the precision of tracking:

where M is the times of Monte Carlo(MC)simulation,Nj is the total radiation times of the jth Monte Carlo simulation,x(i)and y(i)are the true location coordinates of the target tracked,(i)and(i)are the estimated location coordinates of the target tracked.

The effect of RFS is evaluated by the average of CPI.

whereCPIis the average of CPI of the total times of Monte Carlo simulation,andis the CPI of the jth Monto Carlo simulation.

7.2 Setting parameters of simulation

We continue to use the parameters of PAR and EPDS in Table 1 and Table 2.The width of the wave beam of PAR α is 3° and the distance of wave gate Gdis 100 m.Both the brief degree of Hrand Hψare 85%.We assume the target’s RCS is 5 m2.The initial state is x(0)=(50 000,200,40 000,150)and initial covariance P(0)=diag(1 000,40,1000,40).The covariance of process noise is Q=diag(400,400).The covariance of the measurement noise is R=diag(10 000,0.000 1).The initial radiation interval is Δt0=2 s.The initial radiation power is Pt(1)=25 000 W and the time is TOT=0.02 s.The fixed radiation power of AIFE is Pt=25 000 W and the fixed radiation interval is Δt=2 s.

The target’s true motion trajectory is denoted in Table 5 and is showed in Fig.6.

Table 5 Target’s true motion trajectory

Number Time/s Type Parameter 1 0–60 CV vx=200 m/s,vy=150 m/s 2 60–100 CT ω=0.05 rad 3 100–120 CA ax=5 m/s2,ay=8 m/s2 4 120–200 CT ω=–0.03 rad

Fig.6 Target’s trajectory

We execute Monte Carlo(MC)simulations 100 times.

7.3 Results

In Fig.7,we show the effect of tracking of SAEI,AIFE and FIAE.We see that all three methods can track the target very well,but FIAE is better than others because the data rate of FIAE is higher.In Fig.8,the number of radiation times of FIAE is a constant as 100 and the numbers of AIFE and SAEI are below 100 in general.In Fig.9,the averages of radiation intervals of AIFE and SAEI in each simulation are larger than FIAE in a great measure.

Fig.7 Three methods for tracking with CV-EKF in the 50th tracking

Fig.8 Number of radiation times

Fig.9 Average of radiation interval

Thus we know that SAEI and AIFE can reduce the number of radiation times during the course of tracking.In Fig.10,we show the CPI in each simulation.As the CPI describes the effect of RFS,we can see that the effect of RFS of SAIE is the best and FIAE is better than AIFE.This also indicates the radiation energy control method is a greater contribution to RFS than the radiation interval control method.

Fig.10 CPI in each simulation

In Table 6,we record RMSE,N andpCPIof three methods to compare the effect of tracking and RFS.We can know that the effect of tracking of SAIE is the worst but the effect of RFS of SAIE is the best.Thus we conclude that we can not get the best effects of tracking and RFS at the same time and we have to make a trade-off between them.

Table 6 Results of three methods

Method RMSE/m N pCPI/%FIAE 74.78 100 57 AIFE 129.24 74 42 SAEI 141.95 73.6 23

There are many factors that affect the tracking,including the tracking algorithm,the motion model,the target maneuver,the error of observation and so on.In this paper,the main content of our research is how to improve the RFS capability of radar in the tracking state.The emphasis is on reducing the probability of interception and the cumulative probability of interception.CV-EKF is a common nonlinear tracking algorithm shared in SAEI,AIFE and FIAE methods to remove the difference selection of the nonlinear tracking algorithm.Of course,other nonlinear tracking algorithms can also be applied such as CV-UKF and CV-PF.In this paper,one of the reasons why EKF and CV can achieve better tracking is that we set the appropriate system noise covariance and measurement noise covariance as well as the initial value.And the main reason is that the noise of the target measurement is small(the standard deviation of distance R is 100m and angle ψ standard deviation is 0.01 rad).In other words,in the filtering process,the influence of the tracking algorithm and the motion model on the tracking is weakened by the accurate observation.The aim is to study the influence of Δt on the accuracy the tracking.If Δt increases,the frequency of radar measurements will be smaller and the accuracy of tracking will be significantly lower without the observation.Therefore,for FIAE,the radar uses a fixed radiation period to track the target and the tracking is obtained by using EKFCV in Fig.7 under the condition of sufficient and accurate measurement.The SAEI and FIAE adopt the adaptive control method of the radiation periods and then the total number of SAEI and FIAE measurement is smaller than FIAE.Therefore,the tracking effect of SAEI and FIAE is worse than FIAE.

8.Conclusions and future work

In this paper,the purpose is to reduce CPI of the EPDS for the radar during the course of tracking.From the point of view of radar application,we propose two solutions which are SARE and SARI and combine them in the process of tracking.Comparing with FIAE and AIFE,SAEI has both advantages and disadvantages.If we want to increase the precision of tracking with SAEI,we can enlarge the brief degree of the events Hrand Hψbut the number of radiation times will be enlarged.However,we may find an equilibrium point between the effect of tracking and RFS on basis of military requirements.The different tracking algorithms may affect the decision,because UKF and PF can acquire the better precision than EKF,which will increase the adjustment of the range.Thus this paper for emphasizing the trade-off problem chooses EKF.The parameters of PAR and EPDS also affect the results of SAEI,unfortunately,however,the parameters of EPDS can not be known very easily in war.Thus the problem is a key to the success of RFS but this does not mean failure.We need many sources to gain them such as history data,information data and friendly forces.

Future work can be carried out in the following aspects.Firstly,the problems of RFS of the multiple target tracking should be considered.This work will be more complex because it involves unknown targets and tracking correlation.Secondly,the shape of the wave beam should be controlled to reduce PI during the course of tracking.Thirdly,the new model of RFS should contain the influences of the stealth wave and frequency hopping,because they have become the main methods to avoid the detection by the enemy.There is much work to do in the future for RSF during the course of tracking.And the trade-off between the effect of tracking and RFS is still the core issue and the best method is to utilize the latest technology of the tracking algorithm and RF to enhance the precision of tracking and RFS together.

Acknowledgment

Thanks the anonymous referees to give advices, and thanks the guidance from associate professor ZHOU Zhongliang in Air Force Engineering University and the helps of other researchers.

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