LIU Tianpeng,WEI Xizhang,PENG Bo,LIU Zhen,SUN Bin,and GUAN Zhiqiang

1.College of Electronic Science and Technology,National University of Defense Technology,Changsha 410073,China;2.School of Electronic and Communication Engineering,Sun Yat-sen University,Shenzhen 518107,China;3.Beijing Institute of Tracking and Telecommunication Technology,Beijing 100094,China;4.Beijing Huahang Radio Measurement Institute,Beijing 100013,China

Abstract: Tolerance sensitivity limits the practical application of the cross-eye jammer. Previous literature has demonstrated that retrodirective cross-eye jamming with multiple antenna elements possesses the advantage of loose tolerance requirements compared to traditional cross-eye jamming. However, the previous analysis was limited, because there are still some factors affecting the parameter tolerance of the multiple-element retrodirective cross-eye jamming(MRCJ)system and they have not been investigated completely, such as the loop difference, the baseline ratio and the jammer-to-signal ratio.This paper performs a comprehensive tolerance analysis of the MRCJ system with a nonuniformspacing linear array. Simulation results demonstrate the tolerance effects of the above influence factors and give reasonable advice for easing tolerance sensitivity.

Keywords: electronic warfare (EW), electronic countermeasure(ECM),cross-eye jamming,radar active jamming,tolerance.

1.Introduction

Cross-eye jamming as an electronic warfare technique recreates the worst-case glint angular error into monopulse radar [1–4]. Although it was proposed more than sixty years ago,cross-eye jamming still attracts attention of the researchers in the electronic warfare community in the past twenty years. In 2000, the successful experimental testing on cross-eye jamming was first publicly announced.In 2010,du Plessis gave a comprehensive theoretical analysis of cross-eye jamming with a retrodirective antenna array[5–8]. The retrodirective antenna array possesses the advantage that the rediated field has a maximum back in the direction of arrival of the primary plane wave.

As a figure of merit, parameter tolerances of cross-eye jamming are defined as the tolerated errors of the system parameters to ensure a specified angular error to be achieved.It is well-known that cross-eye jamming can induce the largest angular error into the monopulse radar when the amplitude ratio approaches 1 and the phase difference is 180°between the signals through the two directions. However, these conditions are too strict to be satisfied in practical application[8–10].Although du Plessis indicated that the retrodirective implementation can reduce the tolerance requirements in[8],the limitation of the extreme tolerance sensitivity cannot be removed completely from a cross-eye jamming system with two antenna elements.

Multiple-element retrodirective cross-eye jamming(MRCJ) was proposed to ease the tolerance sensitivity by providing additional amplitude ratios and phase differences[11–18].The antenna array of the MRCJ system can be linear, circular or orthogonal.Basic tolerance analyses of the MRCJ system were performed by previous literature[13–15].It was proved that the MRCJ system can significantly reduce tolerance requirements compared to the traditional two-element cross-eye jamming system.

This paper will further study the parameter tolerances of the MRCJ system with the linear antenna array(L-MRCJ),considering the fact that the analyses in [13]and[14]are insufficient to account for the tolerance performance of the L-MRCJ system. Firstly, the analysis in [13] ignored the fact that the jamming signals from different jammer loops have amplitude and phase differences.Especially the large phase difference severely affects the jamming performance. Secondly, only the median case of the cross-eye gain was considered in [14] where the platform skin return was taken into account.Thirdly,both analyses in[13]and [14] were performed by assuming that the antenna elements of the linear retrodirective array were aligned with uniform spacing,which was not the best antenna configuration for the MRCJ system.

In this paper,the effects of the influence factors on parameter tolerances of the MRCJ system will be investigated.A general nonuniform-spacing linear retrodirective array is employed by the L-MRCJ system. The cross-eye gain of the L-MRCJ system is derived after giving the jamming geometry.Then,the process for tolerance solving is proposed according to the results in[8].The factors affecting the value of the cross-eye gain do affect the parameter tolerances of the L-MRCJ system. The influence factors,including the loop differences, the baseline ratio and the platform skin return, are discussed respectively. Finally,simulation results illustrate the effects of the influence factors,and valuable advice for building a practical L-MRCJ system is given.

2.Summary of previous results

2.1 Jamming geometry

The jamming geometry for tolerance analysis is given in Fig.1.The L-MRCJ system has a nonuniform-spacing linear retrodirective array,which comprisesNantenna elements(denoted by crosses).Multiple jammer loops of the L-MRCJ system have different baseline lengths but own the same phase center. Meanwhile, the phase-comparison monopulse radar (denoted by circles) is employed in the jamming geometry.

Fig.1 Jamming geometry for tolerance analysis of L-MRCJ

In the jamming geometry,the jamming range is denoted byr,the radar rotation angle is denoted byθr,the jammer rotation angle is denoted byθc,the half angular separation of thenth jammer loop(i.e.,the jammer loop consisted of antennanand antennaN?n+1)is denoted byθn,the angle of the apparent target is denoted byθs, the spacing of radar antenna elements is denoted bydp, and the baseline of thenth jammer loop is denoted bydn.

According to the relationshipand the approximation tanthe half angular separationθncan be calculated from the jamming geometry:

2.2 Cross-eye gain of L-MRCJ

A cross-eye jammer is an onboard jamming system where the aircraft or ship platform return must be taken into account.When cross-eye jamming combines range gate pulloff jamming,the platform skin return is isolated and cannot be considered.Under the isolated condition,the cross-eye gain of the L-MRCJ system (called as isolated cross-eye gain)is given from[13]as

with

whereanandφnare the amplitude ratio and the phase difference between the signals that pass through the two opposite directions in thenth jammer loop,Cnis defined as the loop parameter of thenth jammer loop that is used to characterize the differences between multiple jammer loops,anddenotes the real part of the complex number.

Without the isolated method, the platform skin return needs to be considered and added to the jammer signals.In presence of the platform skin return,the total cross-eye gain of the L-MRCJ system(called as statistical cross-eye gain)can be given from[14]as

whereis the platform scatter factor relative to the radar cross section of the ship or aircraft platform[14],asis the amplitude factor,andφsis the phase factor.

The angular error induced into the monopulse radar can be quantified by the monopulse indicated angle.When the radar rotation angle is zero,the monopulse indicated angle is the angular error.The monopulse indicated angleθican be obtained from the following relationship[13]:

with

whereβis the free-space phase constant.

Observing the expressions of the cross-eye gain in (3)and (5), we find that there are three factors involved in the cross-eye gain, which are the loop parameterCn, the baseline ratioFnand the platform scatter factorAs.These three influence factors will further affect the tolerance performance of the L-MRCJ system.

3.Process for tolerance solving

The process to solve parameter tolerances for an L-MRCJ system is deriving the closed-form solutions for the system parameters to achieve a specified angular error,which is similar to the process for a two-element retrodirective cross-eye jamming (TRCJ) system outlined in [8]. More complex than the TRCJ system,the factors between multiple jammer loops, e.g., the loop differences and the baseline ratio, need to be considered for the L-MRCJ system.In addition, it is impossible to derive the closed-form solutions for the specified system parameters(ai,φi)of the L-MRCJ system unless other parametersare endowed with constants, due to the large numbers of degrees of freedom.

The angular error induced into the monopulse radar can be characterized by the “settling angle”θswhich is the radar rotation angle where the apparent target exists [8],and is calculated from (6) when the monopulse indicated angle is specified to zero.Another factor characterizing the angular error is the “angle factor”Gθwhich is defined as the degree of the angular deflection of the apparent target and is given by

The process to determine the closed-form solutions of(ai,φi) to achieve a specified settling angle or an angular factor is translated into deriving the specified cross-eye gain when the radar rotation angle is the specified settling angle. The specified cross-eye gain magnitudeGSis derived from(6)by

where the approximation in (10) occurs whenθ1is assumed to be a small value.Actually,this assumption is reasonable because the jammer rangeris much larger than the baseline of the antenna arraydnin practical applications.

After obtaining the cross-eye gain magnitudeGS, the closed-form solutions of(ai,φi)can be calculated by substituting(10)into(3)and(5).

As stated in[5,6,13],there are the cases that the settling angle does not exist. This will occur under the condition below:

Given the relationships as follows:

the condition in(11)can be simplified as

whereGIis the marginal value of the cross-eye gain.WhenGS > GI, we cannot obtain the closed-form solutions of(ai,φi). It means that the specified angular error is large enough to break the monopulse radar’s lock.

In conclusion, the process for tolerance solving is divided into the following steps.

Step 1Specify the angular error and calculate the angle factorGθfrom(9).

Step 2Calculate the cross-eye gain magnitudeGSfrom(10).

Step 3Calculate the closed-form solutions of(ai,φi)by substituting(10)into(3)and(5).

The ranges of the values of(ai,φi)are the required tolerances for specified angular error.

4.Influence factors

4.1 Loop differences

Loop differences are specific to the MRCJ system.Multiple jammer loops have different jamming paths and circuit elements, resulting in the differences between the signals through different jammer loops. We define the signal differences between multiple jammer loops as the loop differences.The phase shift involved in the loop differences can be 180°which will affect the performance of the MRCJ system [13]. Although du Plessis has investigated the effect of loop differences between multiple loops in[17],the effect of the loop differences on tolerance performance is still unclear.

We use the factorto denote the loop parameters of thenth jammer loop,wherecnis the attenuation of the jamming signal through thenth jammer loop and?nis the phase shift of the jamming signal through thenth jammer loop. Using the loop parameters of the frist jammer loop as reference to quantify thenth jammer loops,we give the amplitude ratio and phase difference of the loop differences between the first and thenth jammer loop by

and

Utilizing the largest amplitude of|Cn|to normalize the loop differences,we have the relationship that|Cn|1.

4.2 Baseline ratio

The main difference between the antenna configuration in Fig.2 and the previous antenna configuration in the analyses[13]is that the baselinednis different.

When the antenna elements are aligned with uniform spacing,the baseline of thenth jammer loop is given from[13]as

wheredcis the baseline of the antenna array(i.e.,dc=d1),and the term

is the baseline ratio between the first and thenth jammer loop,and was also called as the “attenuated factor”in[13].

For the general case that the jammer employs a nonuniform-spacing linear retrodirective array in this paper,the baseline ratioFnis given by

The difference between the baseline ratios in (16) and those in (18) is that the latter can be any value except 0 and 1, while the former cannot.For example,for the case where the number of the antenna elementsN= 4, the constant baseline ratio in(16)is 1/3,while the ratio in(18)varies with the baselinedn.

Considering thatFnweakens the contribution of the inner jammer loop to the total difference-channel return, a largerFnis better for the MRCJ system from the perspective of obtaining a larger cross-eye gain or a larger angular error.

4.3 Platform skin return

Due to the variable platform phase, the statistical crosseye gainGCsis a distribution. We use the special statistical values of the cross-eye gain to account for the effect of platform skin return on the tolerance performance of the L-MRCJ system.

After assuming that the platform phaseφsfollows the uniform distribution,the median statistical cross-eye gain was derived in[14]and given by

where

The extreme statistical cross-eye gains were derived in[18]and given by

The±symbol means that the extreme values in (24)could be either the maximum or the minimum cross-eye gain.Whenthe plus and minus signs in(24)correspond to the maximum and the minimum cross-eye gains respectively. On the contrary, whenthe plus and minus signs in (24) correspond to the minimum and the maximum cross-eye gains respectively.

To quantify the effect of the platform skin return, the jammer-to-signal ratio(JSR)was given from[14]as

Notably,the JSR value is not constant for the case thatcnan >1.

Substituting (25) into (19) and (24) gives the relationships between the JSR and the special statistical cross-eye gains as

and

which will make it convenient to analyze the effect of the JSR value on the tolerance requirements on the L-MRCJ system.

5.Simulation results and discussion

This section mainly focuses on the effects of the above influence factors on the tolerance requirements on the L-MRCJ system.An L-MRCJ system with a four-element linear antenna array is employed in this paper without loss of generality.The simulation parameters for a typical jamming geometry used in [8,13] are also employed in this paper for a fair comparison as shown in Table 1.

Table 1 Simulation parameters

5.1 Effect of loop differences

Considering that the loop differences are independent of the baseline ratio and the JSR,we assume that the baseline ratio is 0.8 and the platform skin return is isolated. Other values of the baseline ratio and the JSR do not affect the conclusions.We investigate the effect of the phase difference Δ?2and the amplitude ratio Δc2,respectively.

The effect of the phase difference in the loop difference Δ?nis analyzed by plotting contours of the specified angle factor,when the amplitude ratio Δc2=?0.5 dB,and six cases of the phase difference Δ?2are considered, as shown in Fig. 2. The other system parameters are given as follows:a2=?0.5 dB,φ2= 180°,?1= 0°, and Δc2=?0.5 dB.The contours with specified angle factors illustrate the tolerated errors of the system parameters.To obtain the specified angular error, the combination ofanandφnneeds to be valued inside the contour.The infinite angle factor in Fig. 2 means that the victim radar cannot lock its target when the corresponding cross-eye gain magnitude is larger thanGIwhich approximates to 14.5 for the jamming parameters considered.The results in Fig. 2 show that the contour of the specified angle factor becomes smaller and smaller when the phase difference Δ?2in the loop differences increases from 0°to 180°. It means that the larger phase difference will lead to stricter tolerance requirements for the specified angle factor.Hence,an important conclusion from Fig.2 is that the phase difference severely affects the tolerance performance of the L-MRCJ system.

Fig.2 Contours of specified angle factor of an L-MRCJ system for(a1,φ1)when the phase difference varies

The phase difference in loop differences takes the cancelation between the channel returns and weakens the jamming performance of the jammer.When Δ?2=180°,the cancelation becomes the largest and the tolerance requirements for the specified angle factor are the strictest.

Furthermore, the existence of the phase difference in loop differences makes the contours of the specified angle factor skew as shown in Fig.2.This phenomenon will cause huge difficulties to design the optimum tolerance point for the specified angle factor.It can be found that the optimum tolerance point that endures the largest tolerated errors of the system parameters varies when the phase difference varies.Although the contours for the 180°case are not skew,the contours for the positive and negative crosseye gains reverse their positions,which is the worst case.

Furthermore, the comparison between Fig. 2(b) and Fig.2(f)demonstrates that the sign of the phase difference determines the skew direction of the contours.

The effect of the amplitude difference in loop differences is illustrated by Fig.3.The other system parameters are given as follows:a2=?0.5 dB,φ2= 180°, Δ?2=0°, andc1= 0 dB. The phase difference Δ?2= 0°and three cases of the amplitude ratio Δc2are considered.An observation from Fig.3 is that the jammer system with the largest amplitude difference has the smallest contours of the specified angle factor.It suggests that the existence of a large amplitude difference also makes the tolerance requirements on the MRCJ system strict. However, the effect of the amplitude difference is limited compared to the effect of the phase difference.Even a large amplitude difference of 10 dB can still obtain considerable tolerances of system parameters.For example,the tolerances ofa1andφ1for an angle factor of 7 are at least 1.55 dB and 10.2°as shown in Fig.3(c),respectively.

Fig.3 Contours of specified angle factor of an L-MRCJ system for(a1,φ1)when the amplitude difference varies

Another conclusion arising from Fig. 3 is that the amplitude difference in loop differences determines the place of the center of the contours.This will also make it difficult to design the optimum tolerance point.Taking the case Δc2=?2 dB as an example,the center is at the place thata1=0.38 dB as shown in Fig.3(a).Actually,the center of the contours can be computed from(3)under the values ofφ1=φ2=180°,and is given by

In conclusion, the loop differences severely affect the tolerance sensitivity of the L-MRCJ system, making the tolerance requirements stricter and the optimum tolerance point hard to be designed.

5.2 Effect of the baseline ratio

According to the definition of the baseline ratio, its value varies from 0 to 1.Without loss of generality,we consider two values of the baseline ratio which are 0.1 and 0.8,respectively. The contours of the specified angle factor of the MRCJ system for the two baseline ratios are plotted in Fig. 4 where the loop differences are assumed to be perfectly compensated for,i.e.,Cn=1,and the platform skin return is isolated.

An important observation from the comparison between Fig. 4(a) and Fig. 4(c) is that the contours are much smaller in Fig. 4(a) for the same specified angle factor.The small baseline ratio makes the tolerance requirements of system parameters stricter. This conclusion can also be obtained from the comparison between Fig. 4(b) and Fig. 4(d). Hence, the baseline length of the inner jammer loop should be designed as large as possible to obtain loose tolerance performance. However, the baseline ratio approaching 1 is not the best value in the practical application because it needs a high isolation between the antennas of the jammer loops.

Fig.4 Contours of specified angle factor of an L-MRCJ system for different baseline ratios

On the contrary,whenan >0.47 dB,the contours of the specified angle factor in Fig.4(a)are larger than those relevant contours in Fig.4(b).The reason for this observation is that the baseline ratio in the numerator of (3) weakens the contribution of the inner jammer loop, which needs a greater contribution of the outer jammer loop to achieve the specified angle factor.For the casean <0.47 dB where the angle factor in Fig.4 corresponds to the positive cross-eye gain,the large difference-channel return of the outer jammer loop needs a small amplitude and phase mismatching,i.e.,a1→1,φ1→180°,which results in strict tolerance requirements of (a1,φ1) compared to (a2,φ2). However,for the casean >0.47 dB where the angle factor corre-sponds to the negative cross-eye gain,the large differencechannel return of the outer jammer loop needs a large amplitude and phase mismatching,i.e.,resulting in loose tolerance requirements of(a1,φ1).

Furthermore,the baseline ratio of 0.8 is so large that the equivalent contours in Fig. 4(c) and Fig. 4(d) have little difference. It means that the decrease of the contribution of the inner jammer loop can be ignored when the baseline ratio is 0.8.

Hence,the tolerance requirements of(a1,φ1)are quite similar to those of(a2,φ2)when the baseline ratio is large enough. This is the reason why we only investigate the tolerance requirements of (a1,φ1) in Subsection 5.1 and Subsection 5.3 next. Hence, the moderate baseline ratio is advised. A large baseline ratio will result in the isolation problem, and a small baseline ratio will lead to poor jamming performance.The baseline ratio of 0.8 may be an applicable value for a practical jammer.

5.3 Effect of platform skin return

We consider two values of the JSR to investigate the effect of the platform skin return, which are 10 dB and 30 dB, respectively. The contours of the specified angle factor are plotted in Fig. 5 and Fig. 6 for the 10 dB and 30 dB JSR cases,respectively.

Fig.5 Contours of specified angle factor of an L-MRCJ system for(a1,φ1)when the JSR is 10 dB and a2 = ?0.5 dB,φ2 =180°

The relationships between the JSR and the statistical cross-eye gains in (26) and (27) are used to compute the specified angle factor.We assume that the loop differences are perfectly compensated for(Cn= 1)and the baseline ratio is 0.8.Hence,according to the conclusions from Subsection 5.2, we only need to investigate the tolerance requirements of(a1,φ1).The other parameters are given asa2=?0.5 dB,φ2=180°.

Fig.6 Contours of specified angle factor of an L-MRCJ system for(a1,φ1)when the JSR is 30 dB and a2 = ?0.5 dB,φ2 =180°

Considering that the JSR has two expressions in (25),the borderline between the two different JSR expressions is denoted asAon the left axis in Fig. 5 and Fig. 6. The value of the borderlineAis calculated byA= 1/c1, and is 0 dB whenCn= 1.Hence, the expressions of the JSR can be given by

It should be noted that,under the case that the JSR varies with system parameters ofanwhena1> A, the jammer can still achieve the specified angle factor with a constant value of the JSR.The reason is that,for a constant value of JSR,the parameterasincorporating the gain of the crosseye jammer and its antennas varies when the system parameters vary.

Different JSR definitions will affect the trend of the contours especially for the extreme case as shown in Fig. 5.When enhancing the value of the JSR,the effect of different JSR definitions will be weakened, which can be observed from Fig.6.

For the 10 dB JSR case, there are obvious differences between the curves of the two extreme cases and the median case as shown in Fig. 5. The first observation from the comparison between the extreme cases and the median case is that both the maximum case and the minimum case can obtain larger angle factors than the median case,and even obtain an infinite angle factor. However, the region with contours of infinite angle factors is very unstable,where an angle factor of 9 is included in the infinite region.Actually,the 10 dB JSR makes the extreme cross-eye gain suffer from drastic variation when the system parameters bounded by the region makea2+b2approacha2s.Hence,the tolerance performance of the MRCJ system for the 10 dB JSR case is dissatisfied due to the strict tolerance performance for the median case and the unstable tolerance performance for the extreme cases.

For the 30 dB JSR case, an infinite angle factor can be obtained both by the two extreme cases and by the median case as shown in Fig.6.Furthermore,the contours for the median case in Fig.6(b)are much larger than those in Fig. 5(b). Hence, the larger JSR will make the L-MRCJ system obtain a larger specified angle factor and looser tolerances. Another observation from the comparison between the contours of the extreme and median cases is that the three series of contours show good agreement with each other when the JSR is 30 dB. It suggests that a high JSR value can make the extreme cross-eye gain inclined to the median gain. We can further assert that the agreement between the median case and the extreme cases will be much better when the value of the JSR increases.This is because a high JSR makes the amplitude scalingasof the platform skin return become a negligible value compared to the jammer return, and makes the statistical cross-eye gains approximately equal to the constant isolated crosseye gain. It hence comes as no surprise that a high JSR value is advised for the cross-eye jammer.

6.Conclusions

A comprehensive investigation of parameter tolerances of an L-MRCJ system is presented in this paper. Especially,the effects of the influence factors on tolerance performance of the L-MRCJ system are analyzed.Valuable advice for building a practical L-MRCJ system is proposed.

The loop differences severely affect the tolerance performance of the L-MRCJ system.A large phase difference of loop differences makes the contours of the specified angle factor small and skew. Meanwhile, the amplitude difference of loop differences mainly determines the position of the center of contours.

A small baseline ratio makes the tolerance requirements on system parameters strict.Hence,the baseline length of the inner jammer loop should be designed as large as possible to achieve loose tolerance requirements in practice.

The tolerance performance of the L-MRCJ system for a low JSR is dissatisfied due to strict tolerance requirements for the median case and unstable tolerance requirements for the extreme cases. When the value of the JSR increases, the tolerance requirements for the median case become loose and the agreement between the median case and the extreme cases are much better.Hence,a higher JSR value is required for the L-MRCJ system to obtain better tolerance performance.