Shixian Gong,Xizhang Wei,Xiang Li,and Yongshun Ling

1.School of Electronic and Information,National University of Defense Technology,Changsha 410073,China;

2.Electronic Engineering Institute,Hefei 410073,China

Mathematic principle of active jamming against wideband LFM radar

Shixian Gong1,*,Xizhang Wei1,Xiang Li1,and Yongshun Ling2

1.School of Electronic and Information,National University of Defense Technology,Changsha 410073,China;

2.Electronic Engineering Institute,Hefei 410073,China

The inherent mathematic principle of active jamming against the wideband linear frequency modulated(LFM)radar is investigated.According to different generation strategies,the active jamming methods are reclassifed into three groups,i.e., non-coherent jamming(NCJ),convolution jamming(CJ)and multiplying jamming(MJ).Based on the classifcation,the mathematic principles of different active jamming groups are put forward,which describe the relationships between the modulated signals and the jamming results.The advantages and disadvantages of different groups are further analyzed,which provides a new perspective for the study of jamming/anti-jamming methods and a potential for engineers to integrate similar jamming methods into one jammer platform.The analyses and simulation results of some typical active jamming methods prove the validity of the proposed mathematics principle.

linear frequency modulated(LFM),mathematic principle,active jamming.

1.Introduction

The wideband linear frequency modulated(LFM)signal is widely used in modern radar,such as synthetic aperture radar(SAR)and inverse synthetic aperture radar(ISAR) systems,which are playing important roles in electronic reconnaissance,space surveillance and missile defense systems for their capability of all-weather,all-day and getting high-resolution one dimensional(1D)and two dimensional(2D)images of targets.Recently,active jamming for countering SAR/ISAR is widely researched and a lot of active jamming methods for the wideband LFM radar have been proposed.These jamming methods are usually classifed as noise jamming and deception jamming in electronic counter measures(ECM)[1,2].Usually among the noise jamming methods,there is non-coherence between the transmitted jamming signal and the radar signal[3,4].To improve the coherence between the noise and the radar signal,some smart noise jamming methods or coherent noise jamming are also presented[5–7],which generate jamming signals by modulating the radar signals with noise. For the deception jamming methods to the wideband LFM radar,there also have been various new techniques.For example,frequency-shift jamming[8–10]and interrupted sampling repeater jamming[11]can make a series of false targets in the 1D range imaging results of the wideband LFM radar.Digital false-target image synthesizer jamming [12–14]and 2D false target jamming[15,16]can provide 2D imaging jamming against the wideband LFM radar.

Although so many active jamming methods are presented,little attention has been paid to the mathematical relations of those methods.Actually,with the rapid development of active jamming,traditional classifcation cannot well refect the inherent principle of various techniques.It seems more reasonable to classify the methods according to the ways of jamming signal generation especially for the following two reasons.

(i)It can be noted that for some methods,such as frequency-shift jamming and interrupted sampling repeater jamming,are similar in means of jamming signal generation and jamming results,however,the presented principle of these two jamming methods are different. Therefore,it is diffcult to make a comparison and evaluation of their performances.

(ii)From the engineer’s point of view,the two main issues in jammer designing are how to effectively generate the jamming signals and how to integrate several jamming methods into one jammer.However,so far there has not been any convenient principle to provide valuable suggestions for the jammer engineers.

In this paper,according to the mathematic way of jamming generation,the active jamming methods are reclassifed into non-coherent jamming(NCJ),convolution jam-ming(CJ)and multiplying jamming(MJ).Based on the classifcation,we present the mathematic principle of each group,which can simplify the study of the active jamming methods and provide a new way for the research of new active jamming/anti-jamming methods.

The rest of the paper mainly consists of fve sections. Section 2 presents the new classifcation of the active jamming methods.In Section 3,according to the signal processing fows of the wideband LFM radar,we put forward the mathematic principles of different active jamming groups,which can well describe the mathematic relationships between the modulated signal and the imaging results.Section 4 makes some further discussions on the mathematic principles and characteristics of different active jamming groups.Section 5 carries out some simulations for several representative active jamming methods to verify the proposed mathematic principle.Section 6 concludes the whole paper.

2.Reclassifcation of active jamming methods

Traditionally,active jamming methods are classifed as noise jamming and deception jamming based on their jamming effects,however,it is not helpful for the engineering application.Herein,in this section,according to the means of the jamming signal generation(as shown in Fig.1),the active jamming methods for the wideband LFM imaging radar are reclassifed into three different kinds,i.e.,NCJ, CJ and MJ.

Fig.1 Block diagram of the jammer

The signal transmitted by the wideband LFM radar is assumed as

where fcis the carrier frequency of the radar,s0(ˆt)= rect(ˆt/Tp)exp(jπ γˆt2), Tpis the pulse width, rect(ˆt/Tp)=1 for|t|≤ Tp/2 and zero otherwise,γ is the chirp rate,B=γTpis the bandwidth of the signal, ˆt=t−tmand tm=mT denote the fast time and the slow time respectively,T is the pulse repeat interval(PRI),and m is the pulse number.

In the NCJ group,it is not necessary to receive and store the radar signal.Then,its baseband signal can be expressed as

where m1(ˆt,tm)is the modulated signal of the NCJ group.

For the CJ group and the MJ group,the jamming signals are generated by modulating the radar baseband signal,which can be respectively described as

where⊗represents the convolution operation,c is the light velocity,Rjmis the distance between the jammer and the radar in the mth radar pulse,m2(ˆt,tm)and m3(ˆt,tm)are the corresponding modulated signals.

In ideal situations,the transmitted jamming signals for the three jamming groups can be summarized as follows:

where fjis the center frequency of the NCJ signal.

According to(4),some existing active jamming methods are reclassifed in Table 1.Then,the active jamming methods,which have the same ways of generating jamming signals,can be easily altered by just changing the modulated signals.Therefore,the above reclassifcation gives a potential for engineers to integrate similar jamming methods into one jammer platform.Furthermore,from the presented classifcation,we can attain a great advantage for researching the mathematic principle among the jamming signal,1D and 2D imaging results for active jamming methods,which will be presented in the next section.

Table 1 Classifcation of active jamming methods for wideband LFM radar

3.Mathematic principles of different active jamming groups

In this section,by analyzing the pulse compression progress of the jamming signal,we illuminate the mathematic relationship between the pulse compression results (e.g.,1D and 2D imaging results)and the modulated signals for each active jamming group.Usually,the stretch processing is applied to process extremely high bandwidth LFM waveforms[19].According to the signal processing progress introduced in[20],we suppose that the low pass flter(LPF)of the radar h(t)with the frequency spectrum is H(f)(H(f)=1 for|f|≤fstopand zero otherwise,fstopis the cut-off frequency).Commonly,fstopis the half of the radar sample frequency and determines the radar imaging scope along the range.The reference signal of the radar can be expressed by

where sref(t) = rectref(t/Tref)exp(jπ γt2), rectref(t/Tref)=1 for|t|≤ Tref/2 and zero otherwise,and Rrefis the reference range.

3.1NCJ

For the NCJ group,the jamming signals received by the radar can be described by

(i)After mixing with the reference signal,performing low pass fltering and the Fourier transform along the fast time,we can attain the 1D imaging result of the NCJ group as

where∗represents the complex conjugate operation, Sref(f)is the spectrum of sref(t),and M1(ˆf,tm)is the 1D spectrum of m1(ˆt,tm).From(7),it can be seen that the bandwidth of the mixer’s output signal is much bigger than fstopbecause m1(ˆt,tm)is incoherent with the radar signal.Therefore,a majority of the NCJ energy is eliminated by the LPF.

(ii)By performing the Fourier transform on the 1D imaging result along the slow time,the 2D imaging result of the NCJ group can be expressed as

Then,the magnitudes of the 1D and 2D imaging results of the NCJ group can be approximated by

where|ˆf|≤fstop.

Since modulated signals are incoherent with the radar signal,the presented imaging results cannot directly demonstrate the relationship between the results and the modulated signals.However,from(7)and(8),we can observe that during the progress of pulse compression,a majority of the jamming energy is removed by the LPF and no signal processing gain is gotten,therefore,the NCJ group performs poorly and is rarely applied in practice nowadays.

3.2CJ

In the CJ group,the jamming signals received by the radar can be described as

(ii)By executing the Fourier transform along the slow time,we can attain the 2D imaging result of the CJ group as

where a(tm)=1 for 0≤tm≤Tmand zero otherwise, Tmis the total coherent time of 2D imaging.The second part of(14)can be described as

where fjcis the cross Doppler frequency of the jammer. Substituting(15)into(14)and collecting terms yield

where⊗2represents the 2D convolution operation,and M2(t,fm)is the 1D spectrum of m2(ˆt,tm)along the slow time.By approximating the sinc function by the impulse function δ,the magnitudes of the 1D and 2D imaging results for the CJ group can be respectively expressed as

This deduced mathematic result expressly indicates the relationship between the jamming results and the modulated signals for the CJ group.Then,for any active jamming method which uses the convolution operation to generate the jamming signal,the 1D and 2D imaging results of them can be approximated by the modulated signals and their 1 D spectra along the slow time respectively.However, to ensure all of the jamming energy can be involved into the radar,the time width of the modulated signals along the fast time must be smaller than fstop/γ.

3.3MJ

Many active jamming methods,like frequency-shifting jamming[10]and interrupted sampling repeater jamming [11],use the multiplying operation to build the jamming signal.For the MJ group,the jamming signals received by the radar can be written as

(i)By mixing with the reference signal,performing the low pass fltering and Fourier transform along the fast time, and multiplying with exp(−jπˆf2/γ),1D imaging result of the MJ group can be attained as

(ii)Also,by performing the Fourier transform along the slow time,the 2D imaging result of the MJ group can be attained aswhere M3(ˆf,fm)is the 2D spectrum of m3(ˆt,tm).

Also,in(19)and(20),we approximate the sinc function by the impulse function(δ).Then,the magnitudes of the 1D and 2D imaging results of the MJ against the wideband LFM radar can be respectively rewritten as

The above equations denote the relationship between the imaging results and the modulated signals for the MJ group.Thus,for all of the active jamming methods to the wideband LFM radar which use the multiplying operation to build the jamming signal,the 1D and 2D imaging results of them can be approximated by 1D spectrum and 2D spectrum of the modulated signals respectively.Also,because of the LPF,the frequency of the modulated signal along the fast time must be smaller than fstopto ensure all of the jamming energy can be involved into the radar.

4.Further discussions on the classifcation

The above researches deduce the relationship between the imaging results and the modulated signals.Based on the relationship,we can further build the universal mathematic principle and analyze their practical characteristics.

4.1Summary of the mathematic principle

According to(9),(17)and(21),we get the summarization as follows and present the more detailed analysis of the relationship between the imaging results and the modulated signals.

(i)The mathematic principle for the relationship between the 1D imaging results and the modulated signals can be represented as

(ii)The mathematic principle for the relationship between the 2D imaging results and the modulated signals can be represented as

Equation(23)illustrates that as the jamming signal is incoherent with the radar signal,the jamming result of the NCJ group covers the whole azimuth scope;the jamming results of CJ and MJ groups along the azimuth are determined by the frequency components of the modulated signals along the slow time.

Based on the above summarization,we can achieve that for the given active jamming methods to the wideband LFM radar,the approximate 1D and 2D imaging results of jamming can be easily attained after using the above mathematic principle to fnd the modulation methods and the modulated signals of them.Moreover,by inversely applying the proposed mathematic principle,new jamming methods can be pursued for desired jamming results.Then,the proposed universal mathematic principle not only brings a great convenience to research the active jamming methods but also provides a new perspective for seeking the anti-jamming methods.

4.2Characteristics analysis

From the presented classifcation and deduced results,we can get further discussions on the characteristics of different active jamming groups,such as computational cost, hardware requirement and jamming to signal ratio(JSR)in imaging results.

(i)Computational cost

Here,we detail the computational complexity of NCJ, CJ and MJ groups when generating the jamming signals. In order to make the comparison more fair,we assume that the length of the radar signal is Ns.From(2)and(3),it canbe seen that the computational cost of the NCJ group iszero,the computational cost of the CJ group is determined by Nsand the length of the modulated signals along the fast time Nc,but the computational cost of the MJ group is in direct proportion to Ns.Computational cost for the generation of the jamming signal is shown in Table 2.

Table 2 Computational cost for the generation of jamming signal

Since the computational cost of the CJ group is in direct proportion to the length of modulated signals along the fast time,the range size of the jamming result of the CJ group is limited by the computational capabilities of the jammer. However,the MJ group can cover a large range scope in the jamming result but just cost little computation because the range characteristics are determined by the spectrum of the modulated signal which has no effect on the computational cost of generating the jamming signal.

(ii)Hardware requirement

Different jamming groups have different hardware requirement to generate the modulated signal.In the NCJ group,the structure of the jammer is the simplest for the radar signal is not needed to be received,stored and modulated.On certain conditions,frequency measuring equipment is required to ensure the center frequency of the NCJ signal as same as the radar carrier frequency.However,in CJ and MJ groups,the jammers need some equipment to receive,store and process the radar signal.Moreover,for the MJ group,an extra hardware is needed to measure the chirp rate of the radar signal.

(iii)Jamming performance

From(22)and(23),we can infer that the JSR in pulse compression results is in direct proportion to EJ/NJ, where EJis the jamming energy entering in the radar and NJis the covering cells of the jamming result.Therefore, as a majority of the jamming energy is removed by the LPF and the jamming result covers whole imaging cells, the NCJ group performs very poorly when counter to the wideband LFM radar.Whereas,for the CJ and MJ groups, all the jamming energy can be involved into the radar,so it can increase the JSR in pulse compression results by reducing the covering cells of the jamming results,which is very useful for promoting the research of the new jamming method with high JSR performance.

In conclusion,each group of the active jamming against the wideband LFM radar has its advantages and disadvantages.Thus,it should choose the jamming group according to the hardware performance and the mission requirements of the jammer.

5.Simulation results

In this section,the simulation results of noise jamming methods for three groups are frstly put forward to prove the validity of the proposed mathematic principle.Moreover,the CJ and MJ methods are then used to generate the false target jamming for further comparison.

If it is not specially stated,the wideband LFM ISAR system is used,and the true target consisting of an ideal scatter point is chosen to provide a maximum out power. The parameters are illustrated in Table 3.

Table 3 Simulation parameters

5.1Noise jamming methods for each group

This subsection devotes to the simulation results of the noise jamming method for NCJ,CJ and MJ when the input JSR is 20 dB.The parameters of noises are illustrated in Table 4.

Table 4 Parameters of noises

According to the traditional noise jamming theory, herein the carry frequency of the jammer and the bandwidth of the noise in the NCJ method are set to be the same as the radar signal,which can provide the best jamming effect.The time width of the modulated noise for CJ is set as 0.1 μ s,and the bandwidth of the modulated noise along the fast time for MJ is set as 5 MHz.Then,the jamming results of CJ and MJ have the equivalent size along the range domain.

In Fig.2,the 1D and 2D imaging results with NCJ are presented.Just as the analysis introduced in Subsection 4.2,due to the energy loss of jamming as well as the different gain between jamming and the true target,the powerof the jamming signal is lower than that of the target signal in the 1D imaging results and is far lower for 2D imaging although the input JSR is 20 dB.

Fig.2 Spot noise jamming method

Fig.3 and Fig.4 show the imaging results with the noise jamming method for CJ and MJ respectively.It can be seen from Fig.3(a)–(b)that the 1D image of CJ is determined by the characteristic of modulated signals along the fast time,and Fig.4(a)–(b)prove that for MJ it is determined by the 1D spectrum of modulated signals.For 2D imaging,because the spectra of CJ and MJ signals along the slow time are not limited,the jamming results will cover the whole azimuth range,as shown in Fig.3(c)–(d)and Fig.4(c)–(d).However,we can restrict the bandwidth of the spectrum to decrease the covering cells,which can improve the JSR performance.From the comparison of Fig.2, Fig.3 and Fig.4,we can get that the NCJ has poor jamming performance,but CJ and MJ have the same jamming performance when counter to the wideband LFM ISAR.

It should be stated that because the sinc function is approximated by the impulse function δ,in Fig.3 and Fig.4, there are small differences between the modulated signals and tube imaging results.

Fig.3 Convolution noise jamming method

Fig.4 Multiply noise jamming method

5.2False target jamming methods for CJ and MJ

For further comparison of CJ and MJ,the simulation results of the false target jamming are given in Fig.5 and Fig.6,respectively,where the input JSR is 10 dB.Assume that the false target has three scatter points with the location and refection coeffcient(–3 m,10 000 m,1),(3 m, 10 000 m,1)and(0 m,10 003 m,1).The corresponding relationships between the imaging results and the modulated signals shown in Fig.5 and Fig.6 will give a further and clear demonstration of the presented mathematic principle.

Fig.5 False target jamming for CJ

From Fig.5(a)and(b),it can be seen that the relationship between the range value of the scatter points and the data of the modulated signal along the fast time is R=cˆt/2+RΔjmfor the CJ method.Therefore,when generating the modulated signal of CJ,only the false target’s characteristics are required.However,as shown in Fig.6(a)and(b),in the MJ method,the range value of the scatter points is calculated from the frequency components of the modulated signal along the fast time by R=−cˆf/2γ+RΔjm.Therefore,we should attain both the chirp rate of the radar signal and the false target’s characteristics to build the modulated signal of MJ.As the modulated signal’s frequency characteristics of CJ and MJ just cover several cells(Fig.5(c)and Fig.6(c)),the 2D imaging results of the false target jamming method for CJ and MJ both obtain a high processing gain(Fig.5(d)and Fig.6(d)),which corresponds to the conclusions in Subsection 4.2.

To be concluded,no matter which jamming method it is,the corresponding relationships between the modulated signals and the jamming results can be attained according to the presented universal mathematic principle.What is more,it should be explained that although the JSR performance and the jamming results are the same for CJ and MJ groups as shown from Fig.3,Fig.5 and Fig.6,the modulated signals,the modulated methods and the corresponding relationships between jamming imaging results and modulated signals are totally different.

Fig.6 False target jamming for MJ

6.Conclusion and discussion

In this paper,based on the means of the jamming signal’s generation,we reclassify the active jamming methods to the wideband LFM radar into three different kinds, i.e.,NCJ,CJ and MJ.By analyzing the pulse compressing progress of the jamming signal,the mathematic results which denote the relationship between the modulated signal and imaging results for each active jamming group are given.Then,the universal mathematic principle of active jamming against the wideband LFM radar is proposed by synthesizing those mathematic results.Moreover,further discussions on computational cost,hardware requirement, and jamming performance for different active jamming groups are introduced.The proposed universal mathematic principle not only gives a great convenience to research the active jamming methods,but also provides a sound guidance to the jammer engineers in jammer designing.Simulation results demonstrate the effectiveness of the proposed mathematic principle.

[1]N.J.Li,Y.T.Zhang.Asurvey of radar ECM and ECCM.IEEE Trans.on Aerospace and Electronic,1995,31(3):1110–1120.

[2]F.Neri.Introduction to electronic defense systems.Raleigh, NC:SciTech Publishing Inc.,2006.

[3]K.Dumper,P.S.Cooper,A.F.Wonsl,et al.Spaceborne synthetic aperture radar and noise jamming.Proc.of the Radar Edinburgh International Conference,1997:411–414.

[4]Z.A.Han,Y.M.Pi,J.Y.Yang.Analysis of jamming on inverse synthetic aperture radar(ISAR).Proc.of the Society of Photo-Optical Instrumentation,2005:462–469.

[5]B.Lv.Simulation study of noise convolution jamming countering to SAR.Proc.of the International Conference on Computer Design And Appliations,2010:4130–4133.

[6]Y.Wei,R.Hang,S.X.Zhang,et al.Study of noise jamming based on convolution modulation to SAR.Proc.of the International Conference on Computer,Mechatronics,Control and Electronic Engineering,2010:169–172.

[7]X.Z.Feng,X.J.Xu.A 2-D correlated noise depressive jamming technique to synthetic aperture radar.Proc.of the IET International Radar Conference,2009:1–4.

[8]W.Q.Wang,J.Y.Ca.A technique for jamming bi-and multistatic SAR systems.IEEE Geoscience and Remote Sensing Letters,2007,4(1):80–82.

[9]Y.Zhang,S.Q.Yang,W.X.Zheng.Pulse weighted frequency modulation jamming technique countering ISAR.Systems Engineering and Electronics,2008,30(2):249–252(in Chinese).

[10]B.Lv,Q.Feng,N.C.Yuan.A study on frequency-shifting blanket jamming to LFM pulse-compression radar.Modern Radar,2009,31(1):9–17.(in Chinese)

[11]D.J.Feng,H.M.Tao,Y.Yang,et al.Jamming de-chirping radar using interrupted sampling repeater.Science China:Information Sciences,2011,54(10):2138–2146.

[12]R.T.Ekestorm,C.Karow.All-digital image synthesizer for countering high-resolution imaging radars.Monterey:Naval Postgraduate School,2000.

[13]P.E.Pace,D.J.Fouts,S.Ekestorm,et al.Digital false-target image synthesizer for countering ISAR.IEE Radar,Sonar and Navigation,2002,149(5):248–257.

[14]D.J.Fouts,P.E.Pace,C.Karow,et al.A single-chip false target radar image generator for countering wideband imaging radars.IEEE Journal of Solid-State Circuits,2002,37(6): 751–759.

[15]S.S.Zhang,X.M.Jin.A fast algorithm for SAR coherent jamming signal generation.Acta Electronica Sinica,2009,37(1): 108–111.(in Chinese)

[16]S.K.Xu.Active deception jamming technique based on midcourse radar target characteristics.Changsha:National University of Defense Technology,2012.(in Chinese)

[17]D.Adamy.Radar jamming techniques.The Journal of Electronic Defense,2010,33(1):44–46.

[18]X.S.Wang,J.C.Liu,W.M.Zhang,et al.Mathematic principles of interrupted-sampling repeater jamming(ISRJ).Science in China-Series F:Information Sciences,2007,50(1):113–123.

[19]B.R.Mahafza,A.Z.Elsherbeni.Matlab simulations for radar systems design.Washington:Chapman&Hall/CRC Press LLC,2004.

[20]Z.Bao,M.D Xing,T.Wang.Radar image technology.Beijing:Publishing House of Electronics Industry,2004.(in Chinese)

Biographies

Shixian Gong was born in 1984.He received his B.S.and M.S.degrees from National University of Defense Technology,Changsha,China,in 2007 and 2010.Currently,he is pursuing his Ph.D. in the School of Electronic and Information,National University of Defense Technology,Changsha,China.His research interest is electromagnetic characteristic of target and radar jamming.

E-mail:gsx263642571@nudt.edu.cn

Xizhang Wei was born in 1976.He received his Ph.D.degree from National University of Defense Technology,Changsha,China,in 2002.Presently, he is a professor in School of Electronic and Information,National University of Defense Technology. His major research interests are radar target recognize and the electromagnetic characteristic of target. E-mail:liweier@nudt.edu.cn

Xiang Li was born in 1967.He received his B.E.degree from Xidian University,Xi’an,China,in 1989, and the M.Sc.and Ph.D.degrees in electrical engineering from National University of Defense Technology,Changsha,China,in 1995 and 1998,respectively.He is currently a professor with the School of Electronic and Information,National University of Defense Technology,Changsha,China.His research interests include radar signal processing,radar imaging and automatic target recognition.

E-mail:lixiang01@vip.sina.com

Yongshun Ling was born in 1937.He was elected academician of Chinese Academy of Engineering in 1997.He is currently a professor in Electrical Engineering Institute,Hefei,China.His research interests include electronic countermeasures,radar stealth,infrared jamming,and infrared stealth.

10.1109/JSEE.2015.00008

Manuscript received December 18,2013.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61271442).