Waveform design for radar and extended target in the environment of electronic warfare

WANG Yuxi,HUANG Guoce,and LI Wei,2

1.Information and Navigation College,Air Force Engineering University,Xi’an 710077,China;

2.Collaborative Innovation Center of Information Sensing and Understanding,Xidian University,Xi’an 710071,China

1.Introduction

Radar transmit waveform is critical to radar system performance.Different from the traditional signal processing methods,which are receiver-centric,optimizing the transmit waveform with the knowledge learned from the envi-ronment and targets by prior received echoes can make full use of the degree of transmit waveform and achieve the characteristic of adaptivity.For the last few years,there have been a fruit of achievements about adaptive waveform design with different constraints.Most of the existing waveform design methods are based on the assumption that radar is smart and the target is dumb.However,in the modern electronic warfare,the competition between radar and target is increasingly intense.Not only radar can adaptively optimize its waveform,but also the target which is equipped with countermeasure system can intelligently interfere with radar for self-protection.Motivated by the development of both radar waveform design and jamming techniques,this paper focuses on the waveform design for radar and the extended target in the environment of electronic warfare.

The existing wave form design approaches can be classified into three categories according to the criteria adopted:maximizing mutual information(MI),minimum mean square error(MMSE)and maximizing signal-to-noise ratio(SNR).The information theory was firstly used to design the matched radar waveform for a known target in[1]with maximizing the MI between the target’s response and the received echoes.A relationship between the estimation theory’s MMSE and information theory’s MI in Gaussian noise was given by[2].In[3],these two waveform design criteria were further extended to the statistical multiple input and multiple output(MIMO)radar and a conclusion was made that in white Gaussian noise,the optimized waveforms by the criteria of MI and MMSE were same.However,in the colored noise condition,another conclusion that optimal waveforms achieved by MI and MMSE were different was given by[4]and[5]respectively.With the increasement of radar waveform bandwidth,range resolution is improved.Consequently,the assumption that targets have in finite wide band response,i.e.,point targets,is not suitable.Waveform design for extended targets has recently drawn much attention.In[6],the collocated MIMO radar’s transmit sequence for the extended target in the presence of clutter was optimized with the principle of maximizing SNR.Contributions to waveform design for known and statistical extended targets with signal-to-interference-plus-noise ratio (SINR) and MI principles were given in[7],and the relationship between these two optimal criteria was recovered.Based on these achievements,a potentially cognitive transmit signal design method jointed the optimization of the receiver filter in signal-dependent clutter was proposed in[8].Inspired by the design method of[8],a few other waveform design methods which considered the peak-to-averageratio(PAR)and similarity constraints in the presence of clutter were proposed in[9,10].In addition,with the flexibility of MIMO radar’s transmit waveform,joint optimization of transmit sequence and receive filter for point or the extended target becomes a hot topic recently[11,12].These methods can optimize the transmit sequence with the prior knowledge of the environment and target.

All the above waveform design methods are based on the assumption that the target is dumb.However,in the environment of modern electronic warfare,some targets are equipped with the jamming system and can intelligently interfere with radar to protect themselves.These smart targets can even adaptively adjust the jamming spectrum based on the estimation of radar’s waveform parameters.Adaptive jamming techniques and the interaction between adaptive jamming and anti-jammingin the communication system have long been the hot topics[13,14].Intelligent jamming methods for the communication system were proposed in[15,16].Motivated by these jamming methods for communication,jamming techniques which were used against synthetic aperture radar(SAR)were designed in[17,18].MIMO radar and jammer games based on MI criterion were studied in[19],but it only focused on the point target and did not consider the effect of clutter and the fact that the variance of target’s response or propagation gains for each antenna are different.In fact,the radar’s waveform optimization strategy depends on the clutter and the response of the target.Even when the transmit signal is strong,the effect of noise on power’s allocation can be ignored compared with the effect of the clutter.This paper focuses on the wave form design for radar and the extended target in the environment of electronic warfare.With the prior knowledge of the power spectral density(PSD)of target’s response,noise and clutter,three different countermeasure models of smart radar and dumb target,dumb radar and smart target,and smart radar and smart target are proposed.Based on the criterion of SINR,the optimal waveform spectrum for smart radar and smart target is obtained respectively.Especially for the scenario when radar and target are all smart,a novel two-step water- filling method based on SINR is proposed in the presence of clutter to achieve the optimal waveform and jamming signal.

The rest of this paper is organized as follows.In Section 2,the model of radar and the extended target in presence of clutter is introduced.Three different countermeasure models of smart radar and dumb target,dumb radar and smart target,and smart radar and smart target are proposed,and optimum waveform in the presence of clutter for radar and target are studied in Section 3.Section 4 shows the simulation results of three different models and gives the corresponding analysis.Finally,a conclusion is obtained in Section 5.

2.Signal model for extended target

We assume that the complex-valued baseband target impulse response and transmit waveform are h(t)and x(t),respectively.Let H(f)and X(f)denote the Fourier transforms of h(t)and x(t).Let r(t)be the complex-valued receive filter impulse response and n(t)be a complex-valued,zero-mean channel noise process with PSD Snn(f),which is non-zero over the waveform bandwidth W.Let the homogeneous clutter c(t)be a complex-valued,zero-mean Gaussian random process and its uniform PSD is Scc(f),which means that Scc(f)is a constant within W.Suppose the jamming signal released by the target is j(t)and its PSD is J(f).The variables in boldface letters denote random process whereas others are deterministic.Fig.1 shows the block diagram of the signal model for SINR-based waveform design.

Fig.1 Signal model for SINR-based waveform design

According to Fig.1,the signal y(t)at the output of the receive filter is

where the operator∗denotes convolution.Let ycnj(t)=r(t)∗(x(t)∗c(t)+j(t)+n(t))and ys(t)=r(t)∗(x(t)∗h(t))denote the interference and desired signal.The output SINR at time t0is denoted[7]as

if and only if

(2)is achieved and SINR is maximized.Suppose the target can estimate radar’s signal spectrum precisely and adjust its jamming bandwidth to be the same with the radar’s signal to improve the efficiency of jamming power.Therefore the output SINR expression(2)can be expressed as

where W is the bandwidth of radar and jamming signal,K is the number of samples in frequency and Δf is the sampled frequency interval.

If the impulse response of the target is a finite-duration stochastic model,the target’s response can be supposed to be h(t)=a(t)g(t),where g(t)is a wide-sense stationary and a(t)is a rectangular window of duration Th.Thus h(t)is a finite-duration random process having support only in[0,Th]and locally stationary within Th.The energy spectral variance(ESV)of h(t)can be defined[7]as

where H(f)is the Fourier transform of h(t)and μh(f)is the mean value of H(f),which is supposed to be 0.The output SINR for the stochastic target model[7]is

Compared with(3),the difference in(5)is that|H(f)|2is replaced by the ESV.For simplicity,this paper just considers the known target signal model,but the achieved conclusions are also suitable to the stochastic target model.

3.SINR-based waveform optimization in the environment of electronic warfare

In the environment of electronic warfare,interaction between radar and target is more and more complicated.Different countermeasure models mean different waveform optimization strategies.With the extended target signal model,this section gives the countermeasure models of smart radar and dumb target,smart target and dumb radar and smart radar and smart target respectively.And for each case,the waveform optimization strategy for the smart one is obtained.

3.1 Smart radar and dumb target

Suppose that radar is smart and can adaptively optimize its transmit waveform according to the knowledge of the environment and target,the previous information can be learned by some cognitive methods[8],while the target is dumb and cannot intelligently optimize its jamming waveform.Therefore,with conservativeness and rationality,the dumb target can only release white Gaussian noise jamming signal within radar’s bandwidth W.For this situation,radar will choose the following strategy to optimize its transmit waveform.

where Psand PJare the power constraints of radar and target.Note that for the coherence of mathematical expression,we still use Scc(fk)to denote the uniform PSD of the clutter at frequency point fkin this and the following sections.Because the jamming spectrum within the bandwidth is white,the objective function(6)only depends on|X(fk)|2.With the known target’s jamming power strategy,the function

3.2 Smart target and dumb radar

When radar is dumb and target is smart,i.e.,the radar is a general one and the target is equipped with an intelligent countermeasure system.In order to minimize the output SINR and prevent radar from operating as well as it might,the target will optimize its jamming signal according to the reconnoitered parameters of radar’s transmit waveform.Without losing generality,suppose radar’s waveform spectrum is white within the bandwidth W,so the waveform design strategy of the smart target is

Obviously,with the known|X(fk)|2,the kernel of(10)is convex and the power constraint of J(fk)is linear,the optimized jamming waveform can also be achieved by Lagrange multipliers,i.e.,

Taking the derivate of L(J(fk),γ)with respect to J(fk)and setting it to 0,the optimized jamming waveform is

From the result(12),we can see that the optimal jamming signal obtained by the water- filling method depends not only on radar’s waveform spectrum and noise but also on the clutter.Even when radar’s transmit power is large,the clutter is strong enough so that the noise can be ignored.

3.3 Smart radar and smart target

In the environment of modern warfare,the most possible scenario is that radar and target are all smart.For example,the target is a fighter equipped with countermeasure system which can reconnoiter the radar waveform parameters and release the jamming signal according to the reconnoitered radar signal,and the radar is a modern air defense early warning radar which can adaptively optimize its waveform according to the information about the target and environment as well as the received jamming signal.In this scenario,radar knows that its signal could be intercepted and interfered by the target,so radar will select a conservative strategy to optimize the possible worst case.This situation is similar to the case that radar is the leader in the Stackelberg game model.The conservative radar system will choose its waveform design strategy

According to Sion’s minimax theorem,the optimization problem(13)can be reformulated as

Because the smart radar can optimally react to its opponent’s jamming signal,the optimized|X(fk)|2of(6)can be applied as the first step.Based on(9),(14)is reduced to

For rationality considerations we know that in order to improve the efficiency of the limited jamming power,the target will not pour its power to the frequency point where there is no radar signal.Thus|X(fk)|2＞0 and we have

On this basis,(15)can be simplified to

From(17)we can find that the corresponding items(·)+in the objective function and the constraint are always active simultaneously.Since radar and target are all smart,they can intelligently and timely change their waveform,in order to solve the optimization problem(17),we need to prove and find the equilibrium between radar and target firstly.

There are four characteristics of optimal solutions that guarantee a water- filling solution of J(fk)with total jamming power constraint.

(i)If|X(fk)|2=0,we have J(fk)=0;ifwe have

(ii)For any two frequency sub-bands fmand fn,if J(fm)＞0 and J(fn)＞0,

(iii)For any two frequency sub-bands fmand fn,if J(fm)＞0 and J(fn)=0,then we have J(fm)+Snn(fm)＜Snn(fn).

When radar and target achieve the equilibrium,from rationality considerations we know that if there is no signal power on a certain frequency sub-band fk,the smart target will not allocate any jamming power on this sub-band in case the limited jamming power is wasted.In order to optimize the jamming performance,the target will allocate its jamming power on the sub-bands which are the allocated signal power by radar.Thus the characteristic(i)is verified.The following focuses on the proof of the second one.

When J(fk)＞0,according to(i)we have

Let xk=J(fk)+Snn(fk),then the contribution of xkto the objective function(17)at sub-band fkis

From(21)we can see that there is always possible to find a positive Δ,wheresatisfying

Obviouslygm(xm)andgn(xn)cannot be the optimal solu-tions and this contradicts the assumption.ii)and iii)can be similarly proved and characteristic(ii)holds true.In a similar way,characteristics(iii)and(iv)can be also verified by contradiction.These characteristics guarantee a water-filling solution of J(fk)with a total jamming power constraint.

With the above proof,(17)can be solved by the second step water- filling algorithm.The optimized J(fk)is given by

Substitute(23)into(9),and the value of|X(fk)|2can be also obtained.Finally,the results of the original minmax optimization problem(13)are as follows:

From the above derivation process,we can see that the minmax optimization problem can be solved by the two step water- filling method.Particularly,for the first stage target pours its jamming power to each frequency point within the bandwidth according to the noise and the ratio of|H(fk)|and Scc(fk)of each frequency point;and then radar allocates its signal power according to the jamming,noise as well as the clutter to maximize the output SINR.

4.Simulation results and analysis

In this section,we simulate the above three different models and give corresponding analysis.Suppose radar’s transmitted signal bandwidth is W=100 MHz.In order to recover the power allocations of radar and target at each frequency point clearly,for simplicity and without losing generality we divide the whole bandwidth into five equal sub-bands and each sub-band’s bandwidth is and 20 MHz.Denote{|H(fk)|2}={6.5,5.3,4,4,1.6}and{Snn(fk)}={2.2,5.4,4.6,7,3.5}as the PSD for each sub-band and the subscripts k=1,2,...,5 correspond to the five different sub-bands.For the practical application scenarios such as plant ground or sea surface,the clutter response can be set as a constant within the bandwidth and without losing generality,let Scc(fk)=1 for each sub band.

4.1 Jamming power fixed

In this example,PJis fixed to 20 dB and PScan be changed from 10 dB to 30 dB.Fig.2 shows the optimized jamming strategy against the radar’s signal power PSwhen jammer is smart and radar is dumb. Because radar is dumb and the signal spectrum within bandwidth is uniform,when PSis small and the clutter is not strong,the target will allocate its jamming power for each sub-band.However,with the increasement of PS,the clutter becomes strong and the target will focus its limited jamming power on the sub-bands with bigger|H(fk)|2to minimize the output SINR.Note that in Fig.2,the allocated jamming power of sub-band 3 is bigger than that of sub-band 4,although these two sub-bands have the same|H(fk)|2,the reason is that sub-band 3 has a smaller noise PSD and as a result the allocated jamming power is smaller than that of sub-band4,which conforms to(12).To the contrary,when radar is smart and target is dumb,Fig.3 shows radar’s optimal power allocation at each sub-band against PS.We can see that radar’s power allocation strategy is contrast to the strategy of smart target in Fig.2.When PSis small,radar will focus its limited signal power on the sub-bands with bigger|H(fk)|2.With the increasement of PS,clutter,jamming and noise determine radar’s power allocation strategy simultaneously and radar optimizes its power allocation to make sure the output SINR is maximized.

Fig.2 Smart target’s jamming optimization strategy against dumb radar’s PS

Fig.3 Smart radar’s waveform optimization strategy against PS with dumb target

When both radar and target are all smart,Fig.4 and Fig.5 show their power allocation respectively.Fig.4 shows that target’s jamming power allocation strategy is determined by both|H(fk)|2and Snn(fk).For a sub-band with good performance which means that the sub-band has big|H(fk)|2and small Snn(fk),the target will allocate more jamming power and the allocated jamming power on this sub-band can have the best jamming effect.Different from other two cases,since radar knows that its waveform design strategy can be reconnoitered by the target,radar will not pourits most power on the sub-band with best performance.To the contrary,it will allocate its power in a certain more conservative way just as Fig.5 shows.Fig.6 shows the out put SINRs of three different cases against PS.Obviously,the out put SINR that corresponds to smart radar and smart target lies between other two cases.

Fig.4 Smart target’s jamming power allocation with smart radar

Fig.5 Smart radar’s waveform optimization strategy against PS with smart target

Fig.6 Output SINRs of three different countermeasure models against PS

Compared with the case that both radar and target are smart,the dumb radar or the dumb target of other two cases cannot intelligently optimize their waveforms and cannot consequently make full use of the limited power.

4.2 Radar waveform power fixed

In this section,radar’s waveform power is fixed to 20 dB and the jamming power can be changed from 0 dB to 30 dB.Fig.7 shows the optimal jamming power allocation strategy when the target is smart and radar is dumb.From Fig.7 we can see that when jamming power is small compared with radar’s signal power,the target will focus its jamming power on the sub-band with the biggest|H(fk)|2.When PJis big enough,the target can allocate a part of jamming power on other sub-bands so that the jamming power is made full use and the output SINR is minimized.However,when the target is dumb and radar is smart,with the increasement of the uniform jamming level within bandwidth,radar’s optimal power allocation will also change against PJas shown in Fig.8.Note that when PJ≤10 dB,sub-band 1 which has the biggest|H(fk)|2does not get the most allocated power. The reason is that when jamming power is small,the sub-band with good performance will easily achieve saturated mode.The contribution of a certain amount of signal power allocated to a saturated sub-band is smaller than that of the same amount of signal power allocated to other non-saturated sub-bands.However,when target’s jamming power is bigger than radar’s waveform power,compared with the clutter,the jamming signal is dominant.Thus if the jamming power is big enough, smart radar will focus its signal power on the sub-bands with bigger|H(fk)|2.

Fig.7 Smart target’s jamming optimization strategy against PJ with dumb radar

Fig.8 Smart radar’s waveform optimization strategy against PJ with dumb target

When radar and target are all smart,the optimal jammingpower allocation strategy and the optimal radar waveform power allocation strategy are shown in Fig.9 and Fig.10,respectively.Obviously,Fig.9 is similar with Fig.7,but in this scenario radar is smart and radar’s power allocation is not uniform within bandwidth,the optimal jamming power allocation of these two cases are not exactly the same.The analysis of Fig.7 is also suitable to Fig.9.From Fig.10 we can see that when jamming power is small,the optimized radar’s waveform power allocation is similar to Fig.8,however when jamming power increases,the smart radar will not focus all the power on the sub-bands with bigger|H(fk)|2,to the contrary it will select a conservative strategy to optimize its spectrum because it knows that the smart target can reconnoiter its transmit waveform and design the jamming signal which may result in a worse performance.Fig.11 shows the output SINRs of three cases against jamming power PJ.

Fig.9 Smart target’s jamming optimization strategy against PJ with smart radar

Fig.10 Smart radar’s waveform optimization strategy against PJ with smart target

It is obvious that the output SINR of the scenario when radar and target are all smart outperforms the output SINR of the case that radar is dumb and target is smart,which means that the worst case is optimized through radar’s conservative waveform design strategy.

5.Conclusions

In the environment of modern electronic warfare,the competition between radar and target is becoming more and more intense.This paper studies three different scenarios with smart target and dumb radar,smart radar and dumb target,smart radar and smart target,respectively.Based on the SINR criterion,the waveform spectrum is optimized for the smart participant of each scenario through the water- filling method in the presence of clutter.Especially for the case of smart radar and smart target,the equilibrium of two confrontation sides is analytically derived which guarantees that the optimization problem can be solved by the two-step water- filling method.Simulation results under different power constraints are given and the optimal waveform spectrum strategies of smart radar or target in different scenarios are analyzed respectively.In this paper we only consider the waveform power optimization for radar and target,the further optimization of waveform phases for radar and target will be our future work.

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