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

Journal of Systems Engineering and Electronics Vol. 29, No. 1, February 2018, pp.48 57 Waveform design for radar and extended target in the environment of electronic warfare WANG Yuxi 1,*, HUANG Guoce 1, and LI Wei 1,2 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 Abstract: Transmit waveform optimization is critical to radar system performance. There have been a fruit of achievements about waveform design in recent years. However, most of the existing methods are based on the assumption that radar is smart and the target is dumb, which is not always reasonable in the modern electronic warfare. This paper focuses on the waveform design for radar and the extended target in the environment of electronic warfare. Three different countermeasure models between smart radar and dumb target, smart target and dumb radar, smart radar and smart target are proposed. Taking the signal-to-interferenceplus-noise ratio (SINR) as the metric, optimized waveforms for the first two scenarios are achieved by the general water-filling method in the presence of clutter. For the last case, the equilibrium between smart radar and smart target in the presence of clutter is given mathematically and the optimized solution is achieved through a novel two-step water-filling method on the basis of minmax theory. Simulation results under different power constraints show the power allocation strategies of radar and target and the output SINRs are analyzed. Keywords: waveform design, extended target, electronic warfare, clutter, water-filling method. DOI: 10.21629/JSEE.2018.01.05 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 environment 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 Manuscript received June 07, 2016. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61302153) and the Aeronautical Science Foundation of China (20160196001). 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 waveform 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 infinite wideband 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

WANG Yuxi et al.: Waveform design for radar and extended target in the environment of electronic warfare 49 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-jamming in 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 waveform 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). Letr(t) be the complex-valued receive filter impulse response and n(t) be a complexvalued, zero-mean channel noise process with PSD S nn (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 S cc (f), which means that S cc (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 y(t) =r(t) (x(t) h(t) +x(t) c(t) +j(t) +n(t)) (1) where the operator denotes convolution. Let y cnj (t) = r(t) (x(t) c(t)+j(t)+n(t)) and y s (t) =r(t) (x(t) h(t)) denote the interference and desired signal. The output SINR at time t 0 is denoted [7] as (SINR) t0 = y s (t 0 ) 2 E( y cnj (t 0 ) 2 ) = + + 2 R(f)H(f)X(f)e j2πft0 df R(f) 2 (S cc (f) X(f) 2 + J(f)+S nn (f))df

50 Journal of Systems Engineering and Electronics Vol. 29, No. 1, February 2018 + R(f) 2 (S cc (f) X(f) 2 + J(f)+S nn (f))df + + R(f) 2 (S cc (f) X(f) 2 + J(f)+S nn (f))df H(f)X(f) 2 S cc (f) X(f) 2 + J(f)+S nn (f) df (2) if and only if [H(f)X(f)e j2πft0 ] R(f) = S cc (f) X(f) 2 + J(f)+S nn (f), (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 (SINR) t0 W H(f)X(f) 2 S cc (f) X(f) 2 + J(f)+S nn (f) df H(f k )X(f k ) 2 X(f k ) 2 (3) + J(f k )+S nn (f k ) where W is the bandwidth of radar and jamming signal, K is the number of samples in frequency and 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), whereg(t) is a wide-sense stationary and a(t) is a rectangular window of duration T h. Thus h(t) is a finite-duration random process having support only in [0,T h ] and locally stationary within T h. The energy spectral variance (ESV) of h(t) can be defined [7] as σ 2 h (f) =E( H(f) μ h(f) 2 ) (4) where H(f) is the Fourier transform of h(t) and μ h (f) is the mean value of H(f), which is supposed to be 0. The output SINR for the stochastic target model [7] is (SINR) t0 σh 2(f) X(f) 2 W S cc (f) X(f) 2 df. (5) + J(f)+S nn (f) Compared with (3), the difference in (5) is that H(f) 2 is replaced by the ESV σh 2 (f). 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. max K X(f k ) 2 X(f k ) 2 + J(f k )+S nn (f k ) s.t. X(f k ) 2 P s J(f k )= P J, k =1, 2,...,K (6) K where P s and P J are the power constraints of radar and target. Note that for the coherence of mathematical expression, we still use to denote the uniform PSD of the clutter at frequency point f k in this and the following sections. Because the jamming spectrum within the bandwidth is white, the objective function (6) only depends on X(f k ) 2. With the known target s jamming power strategy, the function f( X(f k ) 2 )= X(f k ) 2 (7) + J(f k )+S nn (f k ) is concave about X(f k ) 2 and the power constraint is linear, so its optimal solution can be obtained via Lagrange multipliers L( X(f k ) 2,λ)= X(f k ) 2 + J(f k )+S nn (f k ) +

WANG Yuxi et al.: Waveform design for radar and extended target in the environment of electronic warfare 51 λ(p s X(f k ) 2 ). (8) Taking the derivate of L( X(f k ) 2,λ) with respect to X(f k ) 2 and setting it to zero yields the optimized X(f k ) 2 that maximizes the SINR. X(f k ) 2 is given by ( X(f k ) 2 H(f k ) = Scc 2 (f k)λ ) + J(f k )+S nn (f k ) (9) where (x) + max{0,x}, λ>0is chosen implicitly via X(f k ) 2 = P s. Obviously, with the assumption that the clutter PSD is constant within the bandwidth, the optimal waveform design distributes more power at the frequency point with a higher H(f k ) and lower J(f k )+ S nn (f k ) value. 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 min K J(f k ) X(f k ) 2 + J(f k )+S nn (f k ) P S s.t. X(f k ) 2 = K, k =1, 2,...,K J(f k ) P J. (10) Obviously, with the known X(f k ) 2, the kernel of (10) is convex and the power constraint of J(f k ) is linear, the optimized jamming waveform can also be achieved by Lagrange multipliers, i.e., L(J(f k ),γ)= X(f k ) 2 + J(f k )+S nn (f k ) + ( ) γ P J J(f k ). (11) Taking the derivate of L(J(f k ),γ) with respect to J(f k ) and setting it to 0, the optimized jamming waveform is J(f k )=( /γ X(f k ) 2 S nn (f k )) + (12) where γ is determined by the power constraint J(f k )=P J. 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 max min K X(f k ) 2 J(f k ) X(f k ) 2 + J(f k )+S nn (f k ) s.t. X(f k ) 2 P s, J(f k ) P J. (13) According to Sion s minimax theorem, the optimization problem (13) can be reformulated as min max K J(f k ) X(f k ) 2 X(f k ) 2 + J(f k )+S nn (f k ) s.t. X(f k ) 2 P S, J(f k ) P J. (14) Because the smart radar can optimally react to its opponent s jamming signal, the optimized X(f k ) 2 of (6) can be applied as the first step. Based on (9), (14) is reduced to min K J(f k ) X(f k ) 2 + J(f k )+S nn (f k )

52 Journal of Systems Engineering and Electronics Vol. 29, No. 1, February 2018 ( s.t. X(f k ) 2 H(f k ) = Scc(f 2 k )λ 1 ) + J(f k )+S nn (f k ) J(f k ) P J, X(f k ) 2 = P s. (15) 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(f k ) 2 > 0 and we have X(f k ) 2 +J(f k )+S nn (f k ) = H(f k ) 2 H(f k ) ( Scc(f 2 J(f k)+s nn (f k ) ) k )λ 1 H(f k ) λ 1 H(f k ) 2 H(f k ) ( Scc 2 (f J(f k)+s nn (f k ) ) k)λ 1 = H(f k ) ( Scc 2 (f J(f k)+s nn (f k ) )+J(f k )+S nn (f k ) k)λ = H(f k) 2 λ1 H(f k ). (16) On this basis, (15) can be simplified to min K ( H(fk ) 2 J(f k ) λ1 H(f k ) s.t. J(f k ) P J ) + ( H(f k ) Scc 2 (f k)λ 1 ) + J(f k )+S nn (f k ) = P s. (17) 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(f k ) with total jamming power constraint. (i) If X(f k ) 2 =0,wehaveJ(f k )=0;ifJ(f k ) 0, we have X(f k ) 2 0. (ii) For any two frequency sub-bands f m and f n,if J(f m ) > 0 and J(f n ) > 0, H(f m ) 2 i) if Scc(f 2 > H(f n) 2 m ) Scc(f 2 n ), we have J(f m)+ S nn (f n ) >J(f m )+S nn (f n ); H(f m ) 2 ii) if Scc(f 2 = H(f n) 2 m ) Scc(f 2 n ), we have J(f m)+ S nn (f m )=J(f n )+S nn (f n ); iii) if H(f m) 2 Scc(f 2 < H(f n) 2 m ) Scc(f 2 n ), we have J(f m)+ S nn (f m ) <J(f n )+S nn (f n ). (iii) For any two frequency sub-bands f m and f n,if J(f m ) > 0 and J(f n ) = 0,thenwehaveJ(f m )+ S nn (f m ) <S nn (f n ). (iv) For any two frequency sub-bands f m and f n, H(f m ) 2 Scc(f 2 H(f n) 2 m ) Scc(f 2 and S nn (f m ) < S nn (f n ); if n ) J(f n ) > 0,thenJ(f m ) > 0. 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 f k, 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(f k )>0, according to (i) we have X(f k ) 2 >0, X(f k ) 2 + J(f k )+S nn (f k ) = H(f k ) 2 λ H(fk ). (18)

WANG Yuxi et al.: Waveform design for radar and extended target in the environment of electronic warfare 53 Let x k = J(f k )+S nn (f k ), then the contribution of x k to the objective function (17) at sub-band f k is g k (x k )= H(f k) 2 λ H(fk ) 2 x k. (19) When x k > 0, λ > 0, we have g k(x k ) = x k λ H(fk ) 2 2 < 0 and g2 k (x k) x k x 2 > 0. The function k g k (x k ) is convex. For any two sub-bands f m and f n,there is H(f m) 2 Scc(f 2 > H(f n) 2 m ) Scc(f 2, without losing generality we n ) suppose the optimal solutions could be 0 < x m < x n. We define a positive number Δ which satisfies 0 < Δ x n x m.thenwehave 2 g m (x m )+g n (x n ) g m (x m + Δ) g n (x n Δ) = xn x n Δ g n (x) x dx xm+δ x m g m (x) dx. (20) x Since g(x) x is monotonic increasing, and x m <x m + Δ <x n Δ <x n, therefore we have g m (x m )+g n (x n ) g m (x m + Δ) g n (x n Δ) > xn xn g n (x) x n Δ x dx g m (x) x n Δ x dx = λ H(fn ) 2 ( x n Δ x n )+ S cc (f n ) λ H(fm ) 2 ( x n x n Δ) = S cc (f m ) ( x n λ H(fm ) x n Δ)( 2 λ H(fn ) 2 ) > 0. S cc (f m ) S cc (f n ) (21) From (21) we can see that there is always possible to find a positive Δ,where0 < Δ x n x m, satisfying 2 g m (x m )+g n (x n ) >g m (x m + Δ)+g n (x n Δ). (22) Obviously g m (x m ) and g n (x n ) cannot be the optimal solutions 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 waterfilling solution of J(f k ) with a total jamming power constraint. With the above proof, (17) can be solved by the second step water-filling algorithm. The optimized J(f k ) is given by H(f k ) J(f k )= (γ 2 ) + 1 Scc(f 2 k ) S nn(f k ). (23) Substitute (23) into (9), and the value of X(f k ) 2 can be also obtained. Finally, the results of the original minmax optimization problem (13) are as follows: H(f k ) J(f k )= (γ 2 ) + 1 Scc(f 2 k ) S nn(f k ) (24) ( X(f k ) 2 H(f k ) = Scc 2 (f k)λ 1 ) + J(f k )+S nn (f k ) (25) where constants γ 1 and λ 1 are determined by two power constraints J(f k )=P J and X(f k ) 2 = P s, respectively. From the above derivation process, we can see that the minmax optimization problem can be solved by the twostep 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(f k ) and 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 20 MHz. Denote { H(f k ) 2 } = {6.5, 5.3, 4, 4, 1.6} and {S nn (f k )} = {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 =1for each subband. 4.1 Jamming power fixed In this example, P J is fixedto20dbandp S can be changed from 10 db to 30 db. Fig. 2 shows the optimized jamming strategy against the radar s signal power P S when jammer is smart and radar is dumb. Because radar is dumb and the signal spectrum within bandwidth is uniform, when P S is small and the clutter is not strong, the target will allocate its jamming power for each sub-band. However, with the increasement of P S, the clutter becomes

54 Journal of Systems Engineering and Electronics Vol. 29, No. 1, February 2018 strong and the target will focus its limited jamming power on the sub-bands with bigger H(f k ) 2 to 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(f k ) 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-band 4, 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 P S.We can see that radar s power allocation strategy is contrast to the strategy of smart target in Fig. 2. When P S is small, radar will focus its limited signal power on the sub-bands with bigger H(f k ) 2. With the increasement of P S, 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. 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(f k ) 2 and S nn (f k ). For a sub-band with good performance which means that the sub-band has big H(f k ) 2 and small S nn (f k ), 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 pour its 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 output SINRs of three different cases against P S. Obviously, the output SINR that corresponds to smart radar and smart target lies between other two cases. Fig. 2 Smart target s jamming optimization strategy against dumb radar s P S Fig. 4 Smart target s jamming power allocation with smart radar Fig. 3 Smart radar s waveform optimization strategy against P S with dumb target Fig. 5 Smart radar s waveform optimization strategy against P S with smart target

WANG Yuxi et al.: Waveform design for radar and extended target in the environment of electronic warfare 55 Fig. 6 Output SINRs of three different countermeasure models against P S Fig. 7 Smart target s jamming optimization strategy against P J with dumb radar 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 fixedto20db 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(f k ) 2.WhenP J is 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 P J as shown in Fig. 8. Note that when P J 10 db, sub-band 1 which has the biggest H(f k ) 2 does 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(f k ) 2. Fig. 8 Smart radar s waveform optimization strategy against P J with dumb target When radar and target are all smart, the optimal jamming power 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(f k ) 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 P J.

56 Journal of Systems Engineering and Electronics Vol. 29, No. 1, February 2018 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 Fig. 9 Smart target s jamming optimization strategy against P J with smart radar Fig. 10 Smart radar s waveform optimization strategy against P J with smart target Fig. 11 The output SINRs of three different countermeasure models against P J 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. References [1] BELL M R. Information theory and radar waveform design. IEEE Trans. on Information Theory, 1993, 39(5): 1578 1597. [2] GUO D, SHAMAI S, VERDU S. Mutual information and minimum mean-square error estimation in Gaussian channels. IEEE Trans. on Information Theory, 2005, 51(4): 1261 1282. [3] YANG Y, BLUM R S. MIMO radar waveform design based on mutual information and minimum mean-square error estimation. IEEE Trans. on Aerospace and Electronic Systems, 2007, 43(1): 330 343. [4] ZHANG W, YANG L. Communications inspired sensing: a case study on waveform design. IEEE Trans. on Signal Processing, 2010, 58(2): 792 803. [5] TANG B, TANG J, PENG Y. MIMO radar waveform design in colored noise based on information theory. IEEE Trans. on Signal Processing, 2010, 58(9): 4684 4697. [6] CHEN C, VAIDYANATHAN P. MIMO radar waveform optimization with prior information of the extended target and clutter. IEEE Trans. on Signal Processing, 2009, 57(9): 3533 3544. [7] ROMERO R, BAE J, GOODMAN N. Theory and application of SNR and mutual information matched illumination waveforms. IEEE Trans. on Aerospace and Electronic Systems, 2011, 47(2): 912 927. [8] AUBRY A, MAIO A D, FARINA A, et al. Knowledge-aided (potentially cognitive) transmit signal and receive filter design in signal-dependent clutter. IEEE Trans. on Aerospace and Electronic Systems, 2013, 49(1): 93 117. [9] KARBASI S, AUBRY A, CAROTENUTO V, et al. Knowledge-based design of space-time transmit code and receive filterforamultiple-input-multiple-outputradar insignal-

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