# MI-Based Robust Waveform Design in Radar and Jammer Games

Authors: Bin Wang; Xu Chen; Fengming Xin; Xin Song
Journal: Complexity
 (2019)

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## Abstract

Due to the uncertainties of the radar target prior information in the actual scene, the waveform designed based on the radar target prior information cannot meet the needs of parameter estimation. To improve the performance of parameter estimation, a novel transmitted waveform design method under the hierarchical game model of radar and jammer, which maximizes the mutual information (MI) between the radar target echo and the random target spectrum response, is proposed. In the hierarchical game model of radar and jammer, the radar is in a leading position while the jammer is in a following position. The strategy of the jammer is optimized based on the radar transmitted waveform of previous moment, then the radar selects its own strategy based on the strategy of the jammer. It is generally assumed that the radar and the jammer have intercepted the real target spectrum and then the optimal jamming and the optimal transmitted waveform spectrum are obtained. However, the exact characteristic of the real target spectrum is hard to capture accurately in actual scenes. To simulate this, the real target spectrum is considered to be within an uncertainty range which is confined by known upper and lower bounds. Then, the minimax robust jamming and the maximin robust transmitted waveform are designed successively based on the MI criteria, which optimizes the performance under the most unfavorable condition of the radar and the jammer, respectively. Simulation results demonstrate that the robust transmitted waveform design method guarantees the parameter estimation performance effectively and provides useful guidance for waveform energy allocation.


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## Body

## 1. Introduction

Cognitive radar (CR) is a new radar system concept proposed in recent years. This system is inspired by the bat echolocation, which improves the system performance of the radar through using the feedback structure from the receiver to the transmitter to optimize the transmitted waveform based on the recognition of the target and the scene [1]. The transmitted waveform of the traditional radar is independent of the environment and each transmission repeats the same waveform. Therefore the research of the traditional radar is devoted to optimizing the receiver design through radar signal processing [2]. Different from the traditional radar, CR transmitter can adjust the transmitted waveform adaptively to achieve optimal matching with the environment according to the acquired information [3]. During the past decades, many adaptive waveform design methods for radar target parameter estimation have been developed by a lot of experts and scholars. MI is a useful information metric in information theory [4], which has been widely adopted in cognitive radar and other engineering fields [5, 6]. One important method for radar waveform design is to use information theory. Researchers such as Vaidyanathan applied information theory to the radar of the MIMO system [7]. Many radar experts are devoted to improving the parameter estimation performance by boosting MI [8–10]. The innovative study in [9] optimizes the transmitted waveform through maximizing the conditional MI between the radar target echo and the random target spectrum response. Kwon et al. studied multitarget detection at low SNR, using the maximum eigenvalue of the sample covariance matrix and the correlation coefficient between the transmitted signal and the echo signal to obtain a modified full correlation detector from the perspective of average mutual information [11]. Under certain assumptions, such as fully known noise PSD and white noise, better estimation performance can be obtained by maximizing MI, as shown in [12].However, the designed optimal waveforms under the environment of complex target model are not well known and do not consider the complex battlefield game environment, while in practice precise estimation of the real target spectrum is impossible and the game between radar and jammer is real existence in many cases. The mismatch of prior information of the real target and the ignorance of battlefield game environment might reduce the waveform performance transmitted by the radar transmitter.In this paper, a novel transmitted waveform design technique based on MI under the environment of complex battlefield game and complicated target model is presented. Minimax robust jamming and maximin robust waveform design methods are proposed successively to reduce the impact of insufficient prior information on the performance of the designed waveform. Our main contribution is that the imprecise estimation of target spectrum [12] is considered in the optimal jamming and the optimal transmitted waveform design strategies. In addition, we also establish a hierarchical game model of radar and jammer, which regards radar as the leader and jammer as the follower. The minimax robust jamming and maximin robust transmitted waveform techniques under the established hierarchical game model above based on the MI are developed successively. In summary, first of all, given that the real target spectrum is known, the optimal jamming and optimal transmitted waveform design methods for random target based on MI are proposed successively. Secondly, by considering the uncertainty of the target spectrum, the MI-based minimax robust jamming and maximin robust transmitted waveform techniques are proposed successively. In this paper, we consider the single target model and multitarget model, respectively; then the minimax robust jamming and maximin robust transmitted waveform techniques under the two different target models above based on MI are proposed, respectively. The minimax robust jamming and maximin robust transmitted waveform design methods optimize the performance under the most unfavorable condition of the jammer and the radar transmitter, respectively. Their behaviour with regard to the uncertainty of target spectrum is also analyzed. The MI-based robust jamming and robust waveform provide useful guidance for waveform energy allocation strategy. These two waveform design techniques are easy to implement in an intelligent jammer and a cognitive radar and will have important applications in electronic warfare. In the actual situation, the minimax and maximin robust waveform design methods proposed in this paper can provide the most favorable strategies for intelligent jammer and radar, respectively, in complex target environments. And the waveform design method proposed in this paper can be well applied to the battlefield game environment which regards radar as the leader and the intelligent jammer as the follower. In electronic warfare, radar and jammer can design their own waveform, respectively, according to the strategy of their opponent, which can improve their own performance and weaken the performance of their opponent. As the method proposed in this paper assumes that the radar is more powerful than the jammer, the radar finally won this electronic warfare, and the transmitted waveform designed by radar maximizes the estimation performance of radar.


## 2. Signal Model and Problem Formulation

For the jammer, to impair the estimation performance, the minimization of MI means that the radar target echo contains little information on the target, which will result in poor performance of the radar transmitter. But for the radar transmitter, the maximization of the MI will improve the estimation performance.


### 2.1. Model of Random Target and Optimal Jamming Design Based on MI

In this subsection, to minimize the estimation performance of a general radar system, the optimal jamming design method based on MI is proposed. The model of the random target is given in Figure1 [9, 13], where Figure 1(a) illustrates that the duration of the random target is finite. In this model, at denotes a window function with duration Th and gt represents a generalized stationary random process. Thus, the product ht=atgt is a generalized stationary random process within [0, Th]. The random target model is shown in radar signal processing system in Figure 1(b), where xt denotes the transmitted waveform signal and ht represents the signal model of random target. The spectrum response of xt can be denoted by Xf, and similarly Hf is the spectrum response of ht. rt denotes the signal model of receiver filter and nt is a noise process with the power spectrum density (PSD) Snnf. Likewise, ct represents a jamming component which is a Gaussian random process and the PSD of ct can be denoted by Jf.Figure 1
Random target model for waveform design based on MI. (a) Signal model for random target with finite durationTh. (b) Signal model for waveform design based on MI.
(a) (b)The energy spectrum variance (ESV) ofht is represented as follows [9, 13].(1)σH2f=ΕHf-μHf2In the expression of (1), E[·] represents the expectation of an input entity, and μHf denotes the mean value of Hf which is assumed to be 0.The signal model depicted in Figure1(b) can be applied for MI-based jamming and transmitted waveform design. Therefore, the expression of MI for the model of random target can be denoted by [13](2)MI=Ty∫BWln⁡1+σH2fXf2TyJfXf2+Snnfdfwhere Ty represents the duration of the radar target echo yt. The energy constraint of the jamming is set to be EX. Therefore, for the jammer to impair the performance of the radar transmitter, the optimization problem can be expressed as follows.(3)minJfMIJf(4)s.t.∫BWJfdf≤EXIn the expression of (4), BW is the bandwidth that the spectrum response of transmitted waveform and jamming are virtually limited to. The minimization of MI means that the radar target echo contains little information on the real target, which will result in poor performance for the radar transmitter. The equation of MI is expressed by the transmitted waveform of previous moment, the noise PSD, the target ESV, and the jamming PSD.The optimal jamming solution obtained by Lagrange multiplier method that minimizes the MI (2) under the energy constraint (4) can be denoted by [14](5)Jf=max⁡0,BfA+Dfwhere(6)Bf=-σH2fXf22Ty·Snnf+σH2fXf2,(7)Df=TySnn2f+σH2fXf2SnnfσH2fXf4,and A is a constant that can be derived from the energy constraint of the jamming.(8)∫BWmax⁡0,BfA+Dfdf≤EXThe results show that the optimal jamming solution based on MI can be obtained by water-filling operation which assumes that the target spectrum Hf and transmitted waveform of previous moment Xf are greater than zero at each sampling frequency.


### 2.2. MI-Based Transmitted Waveform Design

According to the jamming solved above, for radar transmitter to improve the parameter estimation performance, MI is also adopted as the criterion. The energy constraint of the transmitted waveform is also set to beEX. In (9), the MI is expressed by the jamming designed by jammer, the noise PSD, the target ESV, and the transmitted waveform spectrum. The designed optimal transmitted waveform should satisfy the following.(9)maxXf2MIXf2(10)s.t.∫BWXf2df≤EXThe maximization of MI means that the radar target echo contains much target information, which will result in rich parameter estimation performance for the radar.The optimal transmitted waveform obtained by Lagrange multiplier method which maximizes the MI (9) under the energy constraint (10) can be written as(11)X-f2=max⁡0,B-fA--D-fwhere(12)B-f=σH2f2Ty·Jf+σH2f,(13)D-f=TySnnfσH2f,and A- is a constant that can be derived from the energy constraint of the transmitted waveform.(14)∫BWmax⁡0,B-fA--D-fdf≤EXThe optimal transmitted waveform based on MI is obtained by performing a water-filling operation when the jamming spectrum, the target ESV, and the noise PSD are known. By using the designed transmitted waveform, the available energy can be used more efficiently in battlefield game environment, which will achieve better performance of the transmitted waveform. Simulation results show that the optimal jamming and optimal transmitted waveform actually lead to opposite energy allocation strategies.Note that in the designed jamming and transmitted waveform above, the real target spectrum response is assumed to be fully known, while in practice the real target spectrum is difficult to capture. When the target model is blurred, the designed jamming and transmitted waveform based on target prior information will not guarantee the performance of the radar or the jammer effectively, so it is critical to minimize the loss of the performance. Therefore, the robust jamming technique and robust transmitted waveform technique are considered next.


## 2.1. Model of Random Target and Optimal Jamming Design Based on MI

In this subsection, to minimize the estimation performance of a general radar system, the optimal jamming design method based on MI is proposed. The model of the random target is given in Figure1 [9, 13], where Figure 1(a) illustrates that the duration of the random target is finite. In this model, at denotes a window function with duration Th and gt represents a generalized stationary random process. Thus, the product ht=atgt is a generalized stationary random process within [0, Th]. The random target model is shown in radar signal processing system in Figure 1(b), where xt denotes the transmitted waveform signal and ht represents the signal model of random target. The spectrum response of xt can be denoted by Xf, and similarly Hf is the spectrum response of ht. rt denotes the signal model of receiver filter and nt is a noise process with the power spectrum density (PSD) Snnf. Likewise, ct represents a jamming component which is a Gaussian random process and the PSD of ct can be denoted by Jf.Figure 1
Random target model for waveform design based on MI. (a) Signal model for random target with finite durationTh. (b) Signal model for waveform design based on MI.
(a) (b)The energy spectrum variance (ESV) ofht is represented as follows [9, 13].(1)σH2f=ΕHf-μHf2In the expression of (1), E[·] represents the expectation of an input entity, and μHf denotes the mean value of Hf which is assumed to be 0.The signal model depicted in Figure1(b) can be applied for MI-based jamming and transmitted waveform design. Therefore, the expression of MI for the model of random target can be denoted by [13](2)MI=Ty∫BWln⁡1+σH2fXf2TyJfXf2+Snnfdfwhere Ty represents the duration of the radar target echo yt. The energy constraint of the jamming is set to be EX. Therefore, for the jammer to impair the performance of the radar transmitter, the optimization problem can be expressed as follows.(3)minJfMIJf(4)s.t.∫BWJfdf≤EXIn the expression of (4), BW is the bandwidth that the spectrum response of transmitted waveform and jamming are virtually limited to. The minimization of MI means that the radar target echo contains little information on the real target, which will result in poor performance for the radar transmitter. The equation of MI is expressed by the transmitted waveform of previous moment, the noise PSD, the target ESV, and the jamming PSD.The optimal jamming solution obtained by Lagrange multiplier method that minimizes the MI (2) under the energy constraint (4) can be denoted by [14](5)Jf=max⁡0,BfA+Dfwhere(6)Bf=-σH2fXf22Ty·Snnf+σH2fXf2,(7)Df=TySnn2f+σH2fXf2SnnfσH2fXf4,and A is a constant that can be derived from the energy constraint of the jamming.(8)∫BWmax⁡0,BfA+Dfdf≤EXThe results show that the optimal jamming solution based on MI can be obtained by water-filling operation which assumes that the target spectrum Hf and transmitted waveform of previous moment Xf are greater than zero at each sampling frequency.


## 2.2. MI-Based Transmitted Waveform Design

According to the jamming solved above, for radar transmitter to improve the parameter estimation performance, MI is also adopted as the criterion. The energy constraint of the transmitted waveform is also set to beEX. In (9), the MI is expressed by the jamming designed by jammer, the noise PSD, the target ESV, and the transmitted waveform spectrum. The designed optimal transmitted waveform should satisfy the following.(9)maxXf2MIXf2(10)s.t.∫BWXf2df≤EXThe maximization of MI means that the radar target echo contains much target information, which will result in rich parameter estimation performance for the radar.The optimal transmitted waveform obtained by Lagrange multiplier method which maximizes the MI (9) under the energy constraint (10) can be written as(11)X-f2=max⁡0,B-fA--D-fwhere(12)B-f=σH2f2Ty·Jf+σH2f,(13)D-f=TySnnfσH2f,and A- is a constant that can be derived from the energy constraint of the transmitted waveform.(14)∫BWmax⁡0,B-fA--D-fdf≤EXThe optimal transmitted waveform based on MI is obtained by performing a water-filling operation when the jamming spectrum, the target ESV, and the noise PSD are known. By using the designed transmitted waveform, the available energy can be used more efficiently in battlefield game environment, which will achieve better performance of the transmitted waveform. Simulation results show that the optimal jamming and optimal transmitted waveform actually lead to opposite energy allocation strategies.Note that in the designed jamming and transmitted waveform above, the real target spectrum response is assumed to be fully known, while in practice the real target spectrum is difficult to capture. When the target model is blurred, the designed jamming and transmitted waveform based on target prior information will not guarantee the performance of the radar or the jammer effectively, so it is critical to minimize the loss of the performance. Therefore, the robust jamming technique and robust transmitted waveform technique are considered next.


## 3. Maximin Robust Waveform Design

Taking the target spectrum uncertainty into account, the band model in [15] is adopted. For single target, assume that the real target spectrum exists in an uncertainty range ε where both the upper and the lower bound are known, that is,(15)Hf∈ε=lk≤Hfk≤uk,k=1,2…,Kwhere fk denotes the sampling frequency. The model of uncertainty single target is shown as Figure 2.Figure 2
Model of the uncertainty range of single target spectrum.For multiple targets, assume that each real target spectrum of the multiple targets exists within an uncertainty rangeεi which is confined also by known upper and lower bounds, that is,(16)Hif∈εi=lik≤Hifk≤uik,k=1,2…,Kwhere i=1,2,3,4…, which is used to distinguish different targets. The uncertainty class εi for each target is different. The model of uncertainty multiple targets is shown as Figure 3.Figure 3
Model of the uncertainty range of multitarget spectrum.In practice, uncertainty target model is widely adopted in robust waveform design because the uncertainty range can be captured through spectrum estimation [15]. The larger the difference between the upper and the lower bound, the greater the uncertainty of the target spectrum. Moreover, pay attention to the fact that the difference in amplitude between the upper and the lower bound of the blurred target spectrum could be different at each sampling frequency.Now the game model of radar and jammer is built: assume that the radar is in a leading position and the jammer is in a following position. The information of the leader and follower is not equal. Firstly, the strategy of the jammer is optimized according to the radar transmitted waveform of the previous moment. Then the radar selects its own strategy according to the jammer’s strategy.For each particular target spectrum, there exist an optimal jamming and an optimal transmitted waveform, respectively, for jammer and radar. However, the real target spectrum may vary in this uncertainty range, so the minimax robust technique for jammer and the maximin robust technique for radar are good approaches which guarantee the performance under the most unfavorable case. In this section, the minimax robust jamming technique and the maximin robust transmitted waveform technique for MI are proposed successively.


### 3.1. Minimax Robust Jamming Technique Based on MI

LetξJf,σH2f denote the optimization criterion MI. The expression of MI can be expressed by the jamming spectrum Jf and the target ESV σH2f or target spectrum Hf. Note that the expressions of σH2f for single target and multiple targets are different, which will be given in Sections 3.1.1 and 3.1.2, respectively. The minimax robust jamming design method should satisfy the following [12, 15, 16].(17)minJf⁡maxHf∈ε⁡ξJf,σH2f∫BWJfdf≤EXBased on the theory of robust signal processing in [16], the solution of this minimax optimization problem can be denoted as follows.(18)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX≤ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXFrom the right side of the inequality above, the minimax optimal jamming is optimal for the jammer when σH2f=σHworst2f. It minimizes the performance of radar transmitter. If other jamming spectrum is adopted, the performance of the jammer will be degraded. Meanwhile, the left side of the inequality indicates that σHworst2f is the most unfavorable target ESV for the minimax optimal jamming. If the minimax optimal jamming spectrum Jminmaxf is adopted, for all target ESV in the uncertainty range ε or εi, the MI performance will be better than the unfavorable case, at least as good as the case of σH2f=σHworst2f. Therefore the minimax optimal jamming for the most unfavorable target ESV within the uncertainty range is optimal. By ensuring the performance under the most unfavorable condition, the performance for all target spectra within the uncertainty range will not be worse than this case.


#### 3.1.1. Minimax Robust Jamming Technique for Single Target Based on MI

For the jammer, the upper bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the minimax robust jamming technique for single target based on MI should satisfy the following.(19)minJfmaxHf∈ε⁡MIJf,σH2f∫BWJfdf≤EXTheorem 1.
The solution to the minimax optimum problem described in (19) is (20)J~minmaxf=max⁡0,B~fA~+D~fwhere(21)B~f=-σU2fXf22Ty·Snnf+σU2fXf2and(22)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4.σU2f=Uf2 denotes the unfavorable target ESV for jammer, where Uf=uk,k=1,2,...,K represents the upper bound of the target uncertainty range, and A~ is a constant which can be derived by the following.(23)∫BWmax⁡0,B~fA~+D~fdf≤EX


#### 3.1.2. Minimax Robust Jamming Technique for Multiple Targets Based on MI

The minimax robust jamming technique for multiple targets based on MI should satisfy the following.(24)minJfmaxHif∈εi⁡MIJf,σH2f∫BWJfdf≤EXTheorem 2.
The solution to the minimax optimum problem described in (24) is(25)J~~minmaxf=max⁡0,B~~fA~~+D~~fwhere(26)B~~f=-σU2fXf22Ty·Snnf+σU2fXf2and(27)D~~f=TySnn2f+σU2fXf2SnnfσU2fXf4.Uif=uik,k=1,2,...,K represents the upper bound of i-th target uncertainty range, where σU2f=∑i=1MPiUif2-∑i=1MPiUif2 [13] in (26) and (27), M denotes the number of targets, Pi denotes the occurrence probability of i-th target, and A~~ is a constant which can be derived by the following.(28)∫BWmax⁡0,B~~fA~~+D~~fdf≤EXNote that the optimization problem in (24) which minimizes the MI for the multitarget model is similar to the problem described in (19). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems1 and 2.
To prove the conclusion above, the optimal problem should satisfy the following.(29)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX≤ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXFirstly, we prove the right side of inequality (29). The expression of MI can be denoted as follows.(30)MIJf=Ty∫BWln⁡1+σH2fXf2TyJfXf2+SnnfdfThe expression of σH2f in (30) is varied from single target to multiple targets, assuming that the most unfavorable target spectrum can be captured, which is Hworstf=Uf. Therefore the most unfavorable target ESV is σU2f. Similarly, σU2f is the upper bound of σH2f. The optimal problem is equivalent to designing the optimal jamming that minimizes the MI when the real target spectrum is Uf.
We determine an objective function by using the Lagrangian multiplier technique.(31)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf+λEX-∫BWJfdfThis is equivalent to minimizing LJf with respect to Jf; the expression of (31) can be converted into(32)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf-λ∫BWJfdfwhere LJf can be denoted as follows.(33)LJf=Ty·ln⁡1+Xf2σU2fTySnnf+Xf2Jf-λJfThe second order derivation of LJf with regard to Jf is greater than zero. Therefore, deriving LJf to Jf and setting it to zero yield the optimal jamming Jminmaxf, that is,(34)Jminmaxf=max⁡0,-R~f+R~2f-S~fA~+D~fwhere A~ is a constant which can be derived by(35)∫BWmax⁡0,-R~f+R~2f-S~fA~+D~fdf≤EXwhere(36)R~f=SnnfXf2+σU2f2Ty(37)S~f=σU2fTy(38)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4respectively.
We define the following(39)Q~f=-R~f+R~2f-S~fA~+D~fand use the first order Taylor approximation to (39) to yield(40)Qf=B~fA~+D~fwhere(41)B~f=-σU2fXf22Ty·Snnf+σU2fXf2.Thus the designed jamming can be denoted as follows.(42)Jminmaxf=max⁡0,B~fA~+D~fTherefore we obtain the following.(43)ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXThen we continue to prove that Hworstf=Uf is the most unfavorable target spectrum which means that σHworst2f=σU2f is the most unfavorable target ESV. Substituting the designed jamming into the expression of MI in (30) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is(44)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2·max⁡0,Qfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk≤Ty·∑k=1KΔf·ln1+Xfk2σU2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk=ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EXwhere Δf denotes the sampling frequency interval. Therefore, the most unfavorable target spectrum which maximizes the MI is Hworstf=Uf, and similarly the most unfavorable target ESV is σHworst2f=σU2f; the proof is complete.


### 3.2. Robust Transmitted Waveform Design Based on MI

According to the minimax robust jamming solved above, which is the strategy of the jammer, in order to improve the estimation performance, it is time for the radar to select its own strategy.LetξXf2,σH2f,Jminmaxf represent the optimization criterion MI. The MI criteria can be expressed by the transmitted waveform spectrum Xf, minimax robust jamming Jminmaxf, and the target ESV σH2f or target spectrum Hf. Note that the expressions of σH2f for single target and multiple targets are different, which will be given in Sections 3.2.1 and 3.2.2, respectively. The maximin robust transmitted waveform design method should satisfy the following [15, 16].(45)maxXf2minHf∈εξXf2,σH2f,Jminmaxf∫BWXf2df≤EXBased on the theory of robust signal processing in [16], the solution of this maximin optimization problem can be denoted as follows.(46)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXFrom the right side of the inequality above, the maximin optimal transmitted waveform is optimal for the radar transmitter when σH2f=σHworst2f. It maximizes the performance of radar transmitter. If another waveform spectrum is adopted, the performance of the radar will be degraded. Meanwhile, the left side of the inequality indicates that σHworst2f is the most unfavorable target ESV for the maximin optimal transmitted waveform. If the maximin optimal transmitted waveform spectrum Xmaxminf2 is adopted, for all target ESV in the uncertainty range ε or εi, the MI performance will be better than the unfavorable case, at least as good as the case of σH2f=σHworst2f. Therefore the maximin optimal transmitted waveform for the most unfavorable target ESV within the uncertainty range is optimal. By ensuring the performance under the most unfavorable condition, the performance for all target spectra within the uncertainty range will not be worse than this case.


#### 3.2.1. Robust Transmitted Waveform Design for Single Target Based on MI

For the radar, the lower bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the maximin robust transmitted waveform technique for single target based on MI should satisfy the following.(47)maxXf2minHf∈εξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 3.
The solution to the maximin optimum problem described in (47) is(48)X^maxminf2=max⁡0,B^fA^-D^fwhere(49)B^f=σL2f2Ty·Jminmaxf+σL2fand(50)D^f=TySnnfσL2f.σL2f=Lf2 denotes the unfavorable target ESV for radar, where Lf=lk,k=1,2,...,K represents the lower bound of the single target spectrum uncertainty range, and A^ is a constant which can be derived by the following.(51)∫BWmax⁡0,B^fA^-D^fdf≤EX


#### 3.2.2. Robust Transmitted Waveform Design for Multiple Targets Based on MI

The maximin robust transmitted waveform technique for multiple targets based on MI should satisfy the following.(52)maxXf2minHif∈εiξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 4.
The solution to the maximin optimum problem described in (52) is(53)X^^maxminf2=max⁡0,B^^fA^^-D^^fwhere(54)B^^f=σL2f2Ty·Jminmaxf+σL2fand(55)D^^f=TySnnfσL2f.Lif=lik,k=1,2,...,K denotes the lower bound of i-th target spectrum uncertainty range, where σL2f=∑i=1MPiLif2-∑i=1MPiLif2 in (54) and (55), and A^^ is a constant which can be derived by the following.(56)∫BWmax⁡0,B-fA-D-fdf≤EXNote that the optimization problem in (52) which maximizes the MI for the multitarget model is similar to the problem described in (47). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems3 and 4.
To prove the conclusion above, the optimal problem should satisfy the following.(57)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXFirstly, we prove the right side of inequality (57). The expression of MI can be denoted as follows.(58)MIXf2=Ty∫BWln⁡1+σH2fXf2TyJminmaxfXf2+SnnfdfThe expression of σH2f in (58) is still different from single target to multiple targets, supposing that the most unfavorable target spectrum can be captured, which is Hworstf=Lf. Therefore the most unfavorable target ESV is σL2f. Similarly, σL2f is the lower bound of σH2f. The optimal problem is equivalent to designing the optimal transmitted waveform that minimizes the MI when the real target spectrum is Lf.
We determine an objective function by using the Lagrangian multiplier technique.(59)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf+λEX-∫BWXf2dfThis is equivalent to maximizing LXf2 with respect to Xf2; the expression of (59) can be converted into(60)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf-λ∫BWXf2dfwhere LXf2 can be denoted as follows.(61)LXf2=Ty·ln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxf-λXf2The second order derivation of LXf2 with regard to Xf2 is greater than zero. Therefore, deriving LXf2 to Xf2 and setting it to zero yield the optimal transmitted waveform Xmaxminf2, that is,(62)Xmaxminf2=max⁡0,-R^f+R^2f+S^fA^-D^f.A^ is a constant which can be derived by(63)∫BWmax⁡0,-R^f+R^2f+S^fA^-D^fdf≤EXwhere(64)R^f=Snnf2Ty·Jminmaxf+σL2f2JminmaxfTy·Jminmaxf+σL2f(65)S^f=SnnfσL2fJminmaxfTy·Jminmaxf+σL2f(66)D^f=SnnfσL2fTyrespectively.
We define the following(67)N^f=-R^f+R^2f+S^fA^-D^fand use the first order Taylor approximation to (67) to yield(68)Nf=B^fA^-D^fwhere(69)B^f=σL2f2Ty·Jminmaxf+σL2f.Thus the designed transmitted waveform can be denoted as follows.(70)Xmaxminf2=max⁡0,B^fA^-D^fTherefore we obtain the following.(71)ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXThen we continue to prove that Hworstf=Lf is the most unfavorable target spectrum which means that σHworst2f=σL2f is the most unfavorable target ESV. Substituting the designed transmitted waveform into the expression of MI in (58) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is as follows.(72)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX=Ty·∑k=1KΔf·ln⁡1+Xmaxminfk2σH2fkTySnnfk+Xmaxminfk2Jminmaxfk∫BWXmaxminfk2df≤EX=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTySnnfk+max0,Nfk·Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk≥Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σL2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk=ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EXTherefore, the most unfavorable target spectrum which minimizes the MI is Hworstf=Lf, and similarly the most unfavorable target ESV is σHworst2f=σL2f; this guarantees that (73)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX,which is the left side of (57). Thus, the proof of Theorems 3 and 4 is complete.For the minimax robust jamming, the most unfavorable case is the upper bound of the target uncertainty range, and the lower bound of the uncertainty range is the most unfavorable case for the maximin robust transmitted waveform. The MI performance of the jammer and the radar for other target ESV within this uncertainty range will be better than the performance of these two unfavorable cases. Therefore, considering the uncertainty range of the target spectrum can optimize the system performance of the jammer and the radar. Through considering the hierarchical game model of radar and jammer, the maximin robust transmitted waveform is designed based on the minimax robust jamming. Although the jamming designed by the jammer can greatly impair the performance of the radar system, the radar is in a leading position and the jamming is intercepted by the radar system. Therefore, the transmitted waveform designed by the radar transmitter can finally guarantee the performance of the radar system. Furthermore, the robust jamming and robust transmitted waveform techniques based on MI provide useful guidance for waveform energy allocation.


## 3.1. Minimax Robust Jamming Technique Based on MI

LetξJf,σH2f denote the optimization criterion MI. The expression of MI can be expressed by the jamming spectrum Jf and the target ESV σH2f or target spectrum Hf. Note that the expressions of σH2f for single target and multiple targets are different, which will be given in Sections 3.1.1 and 3.1.2, respectively. The minimax robust jamming design method should satisfy the following [12, 15, 16].(17)minJf⁡maxHf∈ε⁡ξJf,σH2f∫BWJfdf≤EXBased on the theory of robust signal processing in [16], the solution of this minimax optimization problem can be denoted as follows.(18)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX≤ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXFrom the right side of the inequality above, the minimax optimal jamming is optimal for the jammer when σH2f=σHworst2f. It minimizes the performance of radar transmitter. If other jamming spectrum is adopted, the performance of the jammer will be degraded. Meanwhile, the left side of the inequality indicates that σHworst2f is the most unfavorable target ESV for the minimax optimal jamming. If the minimax optimal jamming spectrum Jminmaxf is adopted, for all target ESV in the uncertainty range ε or εi, the MI performance will be better than the unfavorable case, at least as good as the case of σH2f=σHworst2f. Therefore the minimax optimal jamming for the most unfavorable target ESV within the uncertainty range is optimal. By ensuring the performance under the most unfavorable condition, the performance for all target spectra within the uncertainty range will not be worse than this case.


### 3.1.1. Minimax Robust Jamming Technique for Single Target Based on MI

For the jammer, the upper bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the minimax robust jamming technique for single target based on MI should satisfy the following.(19)minJfmaxHf∈ε⁡MIJf,σH2f∫BWJfdf≤EXTheorem 1.
The solution to the minimax optimum problem described in (19) is (20)J~minmaxf=max⁡0,B~fA~+D~fwhere(21)B~f=-σU2fXf22Ty·Snnf+σU2fXf2and(22)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4.σU2f=Uf2 denotes the unfavorable target ESV for jammer, where Uf=uk,k=1,2,...,K represents the upper bound of the target uncertainty range, and A~ is a constant which can be derived by the following.(23)∫BWmax⁡0,B~fA~+D~fdf≤EX


### 3.1.2. Minimax Robust Jamming Technique for Multiple Targets Based on MI

The minimax robust jamming technique for multiple targets based on MI should satisfy the following.(24)minJfmaxHif∈εi⁡MIJf,σH2f∫BWJfdf≤EXTheorem 2.
The solution to the minimax optimum problem described in (24) is(25)J~~minmaxf=max⁡0,B~~fA~~+D~~fwhere(26)B~~f=-σU2fXf22Ty·Snnf+σU2fXf2and(27)D~~f=TySnn2f+σU2fXf2SnnfσU2fXf4.Uif=uik,k=1,2,...,K represents the upper bound of i-th target uncertainty range, where σU2f=∑i=1MPiUif2-∑i=1MPiUif2 [13] in (26) and (27), M denotes the number of targets, Pi denotes the occurrence probability of i-th target, and A~~ is a constant which can be derived by the following.(28)∫BWmax⁡0,B~~fA~~+D~~fdf≤EXNote that the optimization problem in (24) which minimizes the MI for the multitarget model is similar to the problem described in (19). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems1 and 2.
To prove the conclusion above, the optimal problem should satisfy the following.(29)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX≤ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXFirstly, we prove the right side of inequality (29). The expression of MI can be denoted as follows.(30)MIJf=Ty∫BWln⁡1+σH2fXf2TyJfXf2+SnnfdfThe expression of σH2f in (30) is varied from single target to multiple targets, assuming that the most unfavorable target spectrum can be captured, which is Hworstf=Uf. Therefore the most unfavorable target ESV is σU2f. Similarly, σU2f is the upper bound of σH2f. The optimal problem is equivalent to designing the optimal jamming that minimizes the MI when the real target spectrum is Uf.
We determine an objective function by using the Lagrangian multiplier technique.(31)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf+λEX-∫BWJfdfThis is equivalent to minimizing LJf with respect to Jf; the expression of (31) can be converted into(32)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf-λ∫BWJfdfwhere LJf can be denoted as follows.(33)LJf=Ty·ln⁡1+Xf2σU2fTySnnf+Xf2Jf-λJfThe second order derivation of LJf with regard to Jf is greater than zero. Therefore, deriving LJf to Jf and setting it to zero yield the optimal jamming Jminmaxf, that is,(34)Jminmaxf=max⁡0,-R~f+R~2f-S~fA~+D~fwhere A~ is a constant which can be derived by(35)∫BWmax⁡0,-R~f+R~2f-S~fA~+D~fdf≤EXwhere(36)R~f=SnnfXf2+σU2f2Ty(37)S~f=σU2fTy(38)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4respectively.
We define the following(39)Q~f=-R~f+R~2f-S~fA~+D~fand use the first order Taylor approximation to (39) to yield(40)Qf=B~fA~+D~fwhere(41)B~f=-σU2fXf22Ty·Snnf+σU2fXf2.Thus the designed jamming can be denoted as follows.(42)Jminmaxf=max⁡0,B~fA~+D~fTherefore we obtain the following.(43)ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXThen we continue to prove that Hworstf=Uf is the most unfavorable target spectrum which means that σHworst2f=σU2f is the most unfavorable target ESV. Substituting the designed jamming into the expression of MI in (30) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is(44)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2·max⁡0,Qfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk≤Ty·∑k=1KΔf·ln1+Xfk2σU2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk=ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EXwhere Δf denotes the sampling frequency interval. Therefore, the most unfavorable target spectrum which maximizes the MI is Hworstf=Uf, and similarly the most unfavorable target ESV is σHworst2f=σU2f; the proof is complete.


## 3.1.1. Minimax Robust Jamming Technique for Single Target Based on MI

For the jammer, the upper bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the minimax robust jamming technique for single target based on MI should satisfy the following.(19)minJfmaxHf∈ε⁡MIJf,σH2f∫BWJfdf≤EXTheorem 1.
The solution to the minimax optimum problem described in (19) is (20)J~minmaxf=max⁡0,B~fA~+D~fwhere(21)B~f=-σU2fXf22Ty·Snnf+σU2fXf2and(22)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4.σU2f=Uf2 denotes the unfavorable target ESV for jammer, where Uf=uk,k=1,2,...,K represents the upper bound of the target uncertainty range, and A~ is a constant which can be derived by the following.(23)∫BWmax⁡0,B~fA~+D~fdf≤EX


## 3.1.2. Minimax Robust Jamming Technique for Multiple Targets Based on MI

The minimax robust jamming technique for multiple targets based on MI should satisfy the following.(24)minJfmaxHif∈εi⁡MIJf,σH2f∫BWJfdf≤EXTheorem 2.
The solution to the minimax optimum problem described in (24) is(25)J~~minmaxf=max⁡0,B~~fA~~+D~~fwhere(26)B~~f=-σU2fXf22Ty·Snnf+σU2fXf2and(27)D~~f=TySnn2f+σU2fXf2SnnfσU2fXf4.Uif=uik,k=1,2,...,K represents the upper bound of i-th target uncertainty range, where σU2f=∑i=1MPiUif2-∑i=1MPiUif2 [13] in (26) and (27), M denotes the number of targets, Pi denotes the occurrence probability of i-th target, and A~~ is a constant which can be derived by the following.(28)∫BWmax⁡0,B~~fA~~+D~~fdf≤EXNote that the optimization problem in (24) which minimizes the MI for the multitarget model is similar to the problem described in (19). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems1 and 2.
To prove the conclusion above, the optimal problem should satisfy the following.(29)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX≤ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXFirstly, we prove the right side of inequality (29). The expression of MI can be denoted as follows.(30)MIJf=Ty∫BWln⁡1+σH2fXf2TyJfXf2+SnnfdfThe expression of σH2f in (30) is varied from single target to multiple targets, assuming that the most unfavorable target spectrum can be captured, which is Hworstf=Uf. Therefore the most unfavorable target ESV is σU2f. Similarly, σU2f is the upper bound of σH2f. The optimal problem is equivalent to designing the optimal jamming that minimizes the MI when the real target spectrum is Uf.
We determine an objective function by using the Lagrangian multiplier technique.(31)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf+λEX-∫BWJfdfThis is equivalent to minimizing LJf with respect to Jf; the expression of (31) can be converted into(32)LJf,λ=Ty∫BWln⁡1+Xf2σU2fTySnnf+Xf2Jfdf-λ∫BWJfdfwhere LJf can be denoted as follows.(33)LJf=Ty·ln⁡1+Xf2σU2fTySnnf+Xf2Jf-λJfThe second order derivation of LJf with regard to Jf is greater than zero. Therefore, deriving LJf to Jf and setting it to zero yield the optimal jamming Jminmaxf, that is,(34)Jminmaxf=max⁡0,-R~f+R~2f-S~fA~+D~fwhere A~ is a constant which can be derived by(35)∫BWmax⁡0,-R~f+R~2f-S~fA~+D~fdf≤EXwhere(36)R~f=SnnfXf2+σU2f2Ty(37)S~f=σU2fTy(38)D~f=TySnn2f+σU2fXf2SnnfσU2fXf4respectively.
We define the following(39)Q~f=-R~f+R~2f-S~fA~+D~fand use the first order Taylor approximation to (39) to yield(40)Qf=B~fA~+D~fwhere(41)B~f=-σU2fXf22Ty·Snnf+σU2fXf2.Thus the designed jamming can be denoted as follows.(42)Jminmaxf=max⁡0,B~fA~+D~fTherefore we obtain the following.(43)ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EX≤ξJf,σHworst2f∫BWJfdf≤EXThen we continue to prove that Hworstf=Uf is the most unfavorable target spectrum which means that σHworst2f=σU2f is the most unfavorable target ESV. Substituting the designed jamming into the expression of MI in (30) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is(44)ξJminmaxf,σH2f∫BWJminmaxfdf≤EX=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTySnnfk+Xfk2·max⁡0,Qfk=Ty·∑k=1KΔf·ln⁡1+Xfk2σH2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk≤Ty·∑k=1KΔf·ln1+Xfk2σU2fkTy·max⁡Snnfk,Xfk2·Qfk+Snnfk=ξJminmaxf,σHworst2f∫BWJminmaxfdf≤EXwhere Δf denotes the sampling frequency interval. Therefore, the most unfavorable target spectrum which maximizes the MI is Hworstf=Uf, and similarly the most unfavorable target ESV is σHworst2f=σU2f; the proof is complete.


## 3.2. Robust Transmitted Waveform Design Based on MI

According to the minimax robust jamming solved above, which is the strategy of the jammer, in order to improve the estimation performance, it is time for the radar to select its own strategy.LetξXf2,σH2f,Jminmaxf represent the optimization criterion MI. The MI criteria can be expressed by the transmitted waveform spectrum Xf, minimax robust jamming Jminmaxf, and the target ESV σH2f or target spectrum Hf. Note that the expressions of σH2f for single target and multiple targets are different, which will be given in Sections 3.2.1 and 3.2.2, respectively. The maximin robust transmitted waveform design method should satisfy the following [15, 16].(45)maxXf2minHf∈εξXf2,σH2f,Jminmaxf∫BWXf2df≤EXBased on the theory of robust signal processing in [16], the solution of this maximin optimization problem can be denoted as follows.(46)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXFrom the right side of the inequality above, the maximin optimal transmitted waveform is optimal for the radar transmitter when σH2f=σHworst2f. It maximizes the performance of radar transmitter. If another waveform spectrum is adopted, the performance of the radar will be degraded. Meanwhile, the left side of the inequality indicates that σHworst2f is the most unfavorable target ESV for the maximin optimal transmitted waveform. If the maximin optimal transmitted waveform spectrum Xmaxminf2 is adopted, for all target ESV in the uncertainty range ε or εi, the MI performance will be better than the unfavorable case, at least as good as the case of σH2f=σHworst2f. Therefore the maximin optimal transmitted waveform for the most unfavorable target ESV within the uncertainty range is optimal. By ensuring the performance under the most unfavorable condition, the performance for all target spectra within the uncertainty range will not be worse than this case.


### 3.2.1. Robust Transmitted Waveform Design for Single Target Based on MI

For the radar, the lower bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the maximin robust transmitted waveform technique for single target based on MI should satisfy the following.(47)maxXf2minHf∈εξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 3.
The solution to the maximin optimum problem described in (47) is(48)X^maxminf2=max⁡0,B^fA^-D^fwhere(49)B^f=σL2f2Ty·Jminmaxf+σL2fand(50)D^f=TySnnfσL2f.σL2f=Lf2 denotes the unfavorable target ESV for radar, where Lf=lk,k=1,2,...,K represents the lower bound of the single target spectrum uncertainty range, and A^ is a constant which can be derived by the following.(51)∫BWmax⁡0,B^fA^-D^fdf≤EX


### 3.2.2. Robust Transmitted Waveform Design for Multiple Targets Based on MI

The maximin robust transmitted waveform technique for multiple targets based on MI should satisfy the following.(52)maxXf2minHif∈εiξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 4.
The solution to the maximin optimum problem described in (52) is(53)X^^maxminf2=max⁡0,B^^fA^^-D^^fwhere(54)B^^f=σL2f2Ty·Jminmaxf+σL2fand(55)D^^f=TySnnfσL2f.Lif=lik,k=1,2,...,K denotes the lower bound of i-th target spectrum uncertainty range, where σL2f=∑i=1MPiLif2-∑i=1MPiLif2 in (54) and (55), and A^^ is a constant which can be derived by the following.(56)∫BWmax⁡0,B-fA-D-fdf≤EXNote that the optimization problem in (52) which maximizes the MI for the multitarget model is similar to the problem described in (47). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems3 and 4.
To prove the conclusion above, the optimal problem should satisfy the following.(57)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXFirstly, we prove the right side of inequality (57). The expression of MI can be denoted as follows.(58)MIXf2=Ty∫BWln⁡1+σH2fXf2TyJminmaxfXf2+SnnfdfThe expression of σH2f in (58) is still different from single target to multiple targets, supposing that the most unfavorable target spectrum can be captured, which is Hworstf=Lf. Therefore the most unfavorable target ESV is σL2f. Similarly, σL2f is the lower bound of σH2f. The optimal problem is equivalent to designing the optimal transmitted waveform that minimizes the MI when the real target spectrum is Lf.
We determine an objective function by using the Lagrangian multiplier technique.(59)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf+λEX-∫BWXf2dfThis is equivalent to maximizing LXf2 with respect to Xf2; the expression of (59) can be converted into(60)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf-λ∫BWXf2dfwhere LXf2 can be denoted as follows.(61)LXf2=Ty·ln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxf-λXf2The second order derivation of LXf2 with regard to Xf2 is greater than zero. Therefore, deriving LXf2 to Xf2 and setting it to zero yield the optimal transmitted waveform Xmaxminf2, that is,(62)Xmaxminf2=max⁡0,-R^f+R^2f+S^fA^-D^f.A^ is a constant which can be derived by(63)∫BWmax⁡0,-R^f+R^2f+S^fA^-D^fdf≤EXwhere(64)R^f=Snnf2Ty·Jminmaxf+σL2f2JminmaxfTy·Jminmaxf+σL2f(65)S^f=SnnfσL2fJminmaxfTy·Jminmaxf+σL2f(66)D^f=SnnfσL2fTyrespectively.
We define the following(67)N^f=-R^f+R^2f+S^fA^-D^fand use the first order Taylor approximation to (67) to yield(68)Nf=B^fA^-D^fwhere(69)B^f=σL2f2Ty·Jminmaxf+σL2f.Thus the designed transmitted waveform can be denoted as follows.(70)Xmaxminf2=max⁡0,B^fA^-D^fTherefore we obtain the following.(71)ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXThen we continue to prove that Hworstf=Lf is the most unfavorable target spectrum which means that σHworst2f=σL2f is the most unfavorable target ESV. Substituting the designed transmitted waveform into the expression of MI in (58) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is as follows.(72)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX=Ty·∑k=1KΔf·ln⁡1+Xmaxminfk2σH2fkTySnnfk+Xmaxminfk2Jminmaxfk∫BWXmaxminfk2df≤EX=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTySnnfk+max0,Nfk·Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk≥Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σL2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk=ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EXTherefore, the most unfavorable target spectrum which minimizes the MI is Hworstf=Lf, and similarly the most unfavorable target ESV is σHworst2f=σL2f; this guarantees that (73)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX,which is the left side of (57). Thus, the proof of Theorems 3 and 4 is complete.For the minimax robust jamming, the most unfavorable case is the upper bound of the target uncertainty range, and the lower bound of the uncertainty range is the most unfavorable case for the maximin robust transmitted waveform. The MI performance of the jammer and the radar for other target ESV within this uncertainty range will be better than the performance of these two unfavorable cases. Therefore, considering the uncertainty range of the target spectrum can optimize the system performance of the jammer and the radar. Through considering the hierarchical game model of radar and jammer, the maximin robust transmitted waveform is designed based on the minimax robust jamming. Although the jamming designed by the jammer can greatly impair the performance of the radar system, the radar is in a leading position and the jamming is intercepted by the radar system. Therefore, the transmitted waveform designed by the radar transmitter can finally guarantee the performance of the radar system. Furthermore, the robust jamming and robust transmitted waveform techniques based on MI provide useful guidance for waveform energy allocation.


## 3.2.1. Robust Transmitted Waveform Design for Single Target Based on MI

For the radar, the lower bound of the uncertainty range is taken as the most unfavorable target spectrum. Therefore the maximin robust transmitted waveform technique for single target based on MI should satisfy the following.(47)maxXf2minHf∈εξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 3.
The solution to the maximin optimum problem described in (47) is(48)X^maxminf2=max⁡0,B^fA^-D^fwhere(49)B^f=σL2f2Ty·Jminmaxf+σL2fand(50)D^f=TySnnfσL2f.σL2f=Lf2 denotes the unfavorable target ESV for radar, where Lf=lk,k=1,2,...,K represents the lower bound of the single target spectrum uncertainty range, and A^ is a constant which can be derived by the following.(51)∫BWmax⁡0,B^fA^-D^fdf≤EX


## 3.2.2. Robust Transmitted Waveform Design for Multiple Targets Based on MI

The maximin robust transmitted waveform technique for multiple targets based on MI should satisfy the following.(52)maxXf2minHif∈εiξXf2,σH2f,Jminmaxf∫BWXf2df≤EXTheorem 4.
The solution to the maximin optimum problem described in (52) is(53)X^^maxminf2=max⁡0,B^^fA^^-D^^fwhere(54)B^^f=σL2f2Ty·Jminmaxf+σL2fand(55)D^^f=TySnnfσL2f.Lif=lik,k=1,2,...,K denotes the lower bound of i-th target spectrum uncertainty range, where σL2f=∑i=1MPiLif2-∑i=1MPiLif2 in (54) and (55), and A^^ is a constant which can be derived by the following.(56)∫BWmax⁡0,B-fA-D-fdf≤EXNote that the optimization problem in (52) which maximizes the MI for the multitarget model is similar to the problem described in (47). The difference is that the expression of σH2f is varied from σH2f=Hf2 to σH2f=∑i=1MPiHif2-∑i=1MPiHif2.Proof of Theorems3 and 4.
To prove the conclusion above, the optimal problem should satisfy the following.(57)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXFirstly, we prove the right side of inequality (57). The expression of MI can be denoted as follows.(58)MIXf2=Ty∫BWln⁡1+σH2fXf2TyJminmaxfXf2+SnnfdfThe expression of σH2f in (58) is still different from single target to multiple targets, supposing that the most unfavorable target spectrum can be captured, which is Hworstf=Lf. Therefore the most unfavorable target ESV is σL2f. Similarly, σL2f is the lower bound of σH2f. The optimal problem is equivalent to designing the optimal transmitted waveform that minimizes the MI when the real target spectrum is Lf.
We determine an objective function by using the Lagrangian multiplier technique.(59)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf+λEX-∫BWXf2dfThis is equivalent to maximizing LXf2 with respect to Xf2; the expression of (59) can be converted into(60)LXf2,λ=Ty∫BWln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxfdf-λ∫BWXf2dfwhere LXf2 can be denoted as follows.(61)LXf2=Ty·ln⁡1+Xf2σL2fTySnnf+Xf2Jminmaxf-λXf2The second order derivation of LXf2 with regard to Xf2 is greater than zero. Therefore, deriving LXf2 to Xf2 and setting it to zero yield the optimal transmitted waveform Xmaxminf2, that is,(62)Xmaxminf2=max⁡0,-R^f+R^2f+S^fA^-D^f.A^ is a constant which can be derived by(63)∫BWmax⁡0,-R^f+R^2f+S^fA^-D^fdf≤EXwhere(64)R^f=Snnf2Ty·Jminmaxf+σL2f2JminmaxfTy·Jminmaxf+σL2f(65)S^f=SnnfσL2fJminmaxfTy·Jminmaxf+σL2f(66)D^f=SnnfσL2fTyrespectively.
We define the following(67)N^f=-R^f+R^2f+S^fA^-D^fand use the first order Taylor approximation to (67) to yield(68)Nf=B^fA^-D^fwhere(69)B^f=σL2f2Ty·Jminmaxf+σL2f.Thus the designed transmitted waveform can be denoted as follows.(70)Xmaxminf2=max⁡0,B^fA^-D^fTherefore we obtain the following.(71)ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXf2,σHworst2f,Jminmaxf∫BWXf2df≤EXThen we continue to prove that Hworstf=Lf is the most unfavorable target spectrum which means that σHworst2f=σL2f is the most unfavorable target ESV. Substituting the designed transmitted waveform into the expression of MI in (58) for any Hf∈ε or Hif∈εi, the integral is approximated by summation, which is as follows.(72)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX=Ty·∑k=1KΔf·ln⁡1+Xmaxminfk2σH2fkTySnnfk+Xmaxminfk2Jminmaxfk∫BWXmaxminfk2df≤EX=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTySnnfk+max0,Nfk·Jminmaxfk=Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σH2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk≥Ty·∑k=1KΔf·ln⁡1+max0,Nfk·σL2fkTy·maxSnnfk,Nfk·Jminmaxfk+Snnfk=ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EXTherefore, the most unfavorable target spectrum which minimizes the MI is Hworstf=Lf, and similarly the most unfavorable target ESV is σHworst2f=σL2f; this guarantees that (73)ξXmaxminf2,σH2f,Jminmaxf∫BWXmaxminf2df≤EX≥ξXmaxminf2,σHworst2f,Jminmaxf∫BWXmaxminf2df≤EX,which is the left side of (57). Thus, the proof of Theorems 3 and 4 is complete.For the minimax robust jamming, the most unfavorable case is the upper bound of the target uncertainty range, and the lower bound of the uncertainty range is the most unfavorable case for the maximin robust transmitted waveform. The MI performance of the jammer and the radar for other target ESV within this uncertainty range will be better than the performance of these two unfavorable cases. Therefore, considering the uncertainty range of the target spectrum can optimize the system performance of the jammer and the radar. Through considering the hierarchical game model of radar and jammer, the maximin robust transmitted waveform is designed based on the minimax robust jamming. Although the jamming designed by the jammer can greatly impair the performance of the radar system, the radar is in a leading position and the jamming is intercepted by the radar system. Therefore, the transmitted waveform designed by the radar transmitter can finally guarantee the performance of the radar system. Furthermore, the robust jamming and robust transmitted waveform techniques based on MI provide useful guidance for waveform energy allocation.


## 4. Simulation and Results

To demonstrate the validity of the MI-based robust jamming and robust transmitted waveform techniques proposed in this paper, a lot of simulation analyses are performed. The uncertainty ranges of the single target and multitarget spectrum are presented in Figures4 and 5, respectively, where the real single target and multitarget spectrum are denoted by the solid line. The main energy of the real single target is allocated near the normalized frequency -0.2, 0, and 0.4. For each target of the nominal multiple targets, the main energy is allocated near the normalized frequency 0.2, 0.4, 0.6, and 0.8, respectively, with the occurrence probability 0.1, 0.2, 0.3, and 0.4. The upper and the lower bound at each sampling frequency are represented by the deviation bounds. The amplitude of the upper bound is the real amplitude that added a random value, and similarly the lower bound is the real amplitude that subtracted a random value.Figure 4
Bounded single target spectrum samples.Figure 5
Bounded multitarget spectrum samples.In Figures6 and 7, the total energy of the transmitted waveform of previous moment is 1W, and the main energy is allocated near the normalized frequency of 0.3; the target spectrum response Hf in Figure 6 is the same as the solid line in Figure 4. As σH2f and Hf have similar shapes, σH2f is not presented in Figure 6. The real ESV of multiple targets is illustrated in Figure 8. The duration of the target echo yt is supposed to be Ty=1.5s. The energy constraint of the noise signal is 1W. The optimal jamming under the real target spectrum and the robust jamming under the most unfavorable target spectrum are also illustrated in Figures 6 and 7. Both the two jamming energy constraints are assumed to be 1W. In Figure 6, both the two jamming techniques distribute the finite energy in frequency bands where both the real target spectrum response and the transmitted waveform of previous moment are relatively strong. But in Figure 7, both the two jamming techniques distribute the finite energy only in frequency bands where the transmitted waveform of previous moment is relatively strong, because the value of the nominal ESV of multiple targets is too small compared to the value of the waveform.Figure 6
Jamming results for single target.Figure 7
Jamming results for multiple targets.Figure 8
Nominal ESV for multiple targets.Assume that the total energy of the jamming varies from 1 to 7 W, the MI units (MIs) corresponding to the optimal jamming for real target spectrum, the optimal jamming for the most unfavorable target spectrum, the robust jamming for the most unfavorable target spectrum, the wide-band jamming for the most unfavorable target spectrum, and the LFM (linear frequency modulation) jamming for the most unfavorable target spectrum are compared in Figures9 and 10 for single target and multiple targets, respectively. Simulation results show that the MI of the optimal jamming for real target spectrum is the smallest, which reaches its best performance for the jammer; the reason is that the real target spectrum is adopted and the optimal jamming is designed based on the real target spectrum. When using the most unfavorable target spectrum for the jammer, that is, the upper bound of the uncertainty range, we can get the MI corresponding to the optimal jamming for the most unfavorable target spectrum. As expected, the MI corresponding to the robust jamming for the most unfavorable target spectrum is between the above two. That is because the prior knowledge of the real target for the designed robust jamming is less. However, it is better than the MI corresponding to the optimal jamming for the most unfavorable target spectrum because the minimax robust technique improves the most unfavorable performance of the jammer. The wide-band jamming indicates that the jamming spectrum is a straight line over the entire frequency band, and the LFM jamming means that the instantaneous frequency of the jamming signal changes linearly with time. Both of these two jamming techniques do not contain the information about the target, noise, and transmitted waveform of previous moment. Therefore the MIs corresponding to the wide-band jamming and the LFM jamming for the most unfavorable target spectrum are larger than the above-mentioned three cases.Figure 9
MI performance of the jamming results for single target.Figure 10
MI performance of the jamming results for multiple targets.According to the prior information of the robust jamming shown in Figures6 and 7, which denote the strategies of the jammer, the optimal waveform for real target spectrum and the robust waveform for the worst case are shown in Figure 11 for single target and Figure 12 for multiple targets. Both the waveform energy constraints are assumed to be 1W and distribute the finite energy in frequency bands where the target spectrum response is strong and the jamming spectrum response is weak.Figure 11
Waveform results for single target.Figure 12
Waveform results for multiple targets.Assume that the total energy of the transmitted waveform varies from 1 to 7 W, the MIs corresponding to the optimal transmitted waveform for real target spectrum, the optimal transmitted waveform for the most unfavorable target spectrum, the robust transmitted waveform for the most unfavorable target spectrum, the wide-band transmitted waveform for the most unfavorable target spectrum, and the LFM transmitted waveform for the most unfavorable target spectrum are compared in Figure13 for single target and Figure 14 for multiple targets. Simulation results show that the MI of the optimal transmitted waveform for real target spectrum is the largest, which reaches its best performance for the radar; the reason is that the real target spectrum is adopted and the optimal transmitted waveform is designed based on the real target spectrum. When using the most unfavorable target spectrum for the radar, that is, the lower bound of the uncertainty range, we can get the MI corresponding to the optimal transmitted waveform for the most unfavorable target spectrum. As expected, the MI corresponding to the robust transmitted waveform for the most unfavorable target spectrum is between the above two. That is because the prior knowledge of the real target for the designed robust transmitted waveform is less. However, it is better than the MI corresponding to the optimal transmitted waveform for the most unfavorable target spectrum because the maximin robust technique improves the most unfavorable performance of the radar. The wide-band transmitted waveform indicates that the transmitted waveform spectrum is a straight line over the entire frequency band, and the LFM transmitted waveform means that the instantaneous frequency of the transmitted waveform signal changes linearly with time. Both of these two transmitted waveforms do not contain the information about the target, noise, and jamming, which is similar to the wide-band jamming and the LFM jamming. Therefore the MIs corresponding to the wide-band transmitted waveform and the LFM transmitted waveform for the most unfavorable target spectrum are smaller than the above-mentioned three cases.Figure 13
MI performance of the transmitted waveform results for single target.Figure 14
MI performance of the transmitted waveform results for multiple targets.


## 5. Conclusion

In this paper, through assuming that the real target spectrum is known, the MI-based jamming and radar transmitted waveform techniques are proposed firstly. The designed jamming and transmitted waveform under the hierarchical game model of radar and jammer are proper for restricted energy condition. Then, the uncertainty range of the target spectrum is taken into account. The target model has been adopted, which assumes that the real target spectrum exists in an uncertainty range defined by the known upper and lower bounds. According to the uncertainty range above, the minimax robust jamming and maximin robust transmitted waveform have been designed successively. Although the jamming designed by the jammer can greatly impair the performance of the radar system, the radar is in a leading position and the jamming is intercepted by the radar system. Simulation results show that the transmitted waveform designed by the radar transmitter can finally guarantee the estimation performance of the radar system and provide useful guidance for waveform energy allocation. Although the data used in this paper does not come from the real world experimentation, the data has a certain representativeness, and it can provide reference for future hardware or actual device implementation.



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*Source: 4057849-2019-03-26.xml*