1.Schoolof Electronics and Information,Northwestern Polytechnical University,Xi'an 710072,China;

2.Science and Technology on Electromagnetic Scattering Laboratory,Beijing 100854,China

Application of DFT in bi-static RCS calculation of complex electrically large targets

Kuisong Zheng1,*,Tengjiang Ding1,HuiYu1,and Zhaoguo Hou2

1.Schoolof Electronics and Information,Northwestern Polytechnical University,Xi'an 710072,China;

2.Science and Technology on Electromagnetic Scattering Laboratory,Beijing 100854,China

To handle the electromagnetic problems of the bi-static radar cross section(RCS)calculation of scatterer in a wide frequency band,a fi nite-difference time-domain(FDTD)extrapolation method combining with discrete Fourier transform(DFT)is proposed.By comparing the formulas between the steady state fi eld extrapolation method and the transient fi eld extrapolation method, a novelextrapolation method combining with DFT used in FDTD is proposed when a transient fi eld incident wave is introduced.With the proposed method,the full-angle RCS distribution in a wide frequency band can be achieved through one-time FDTD calculation. Afterwards,the back-scattering RCS distributions ofa double olive body and a sphere-cone body are calculated.Numerical results verify the validity of the proposed method.

fi nite-difference time-domain(FDTD),radar cross section(RCS),discrete Fourier transform(DFT).

1.Introduction

The fi nite-difference time-domain(FDTD)method is an electromagnetic fi eld numerical method proposed by Yee [1]to solve the Maxwell's differential equations.This method has a wide application in many fi elds of the electromagnetic research,for instance,antenna radiation calculation,microwave devices calculation,electromagnetic scattering and radar cross section(RCS),lightpropagation in micro-opticalunits[2-7].The basic calculation unitfor FDTD method is the Yee cell(hexahedron)form.While, the rest of the main numericalmethods adopt the triangular surface,such as method of moment(MoM)[8,9]and multilevel fast multipole algorithm(MLFMA)[10,11].It is convenient that the triangle surface can be obtained by some commercialsoftware packages,such as 3DMax,AutoCAD and UG.Using the triangle surface data fi le of the objectobtained from the above software packages,we can getthe conventionalFDTDformthrough a certain transformation algorithm[12].By the process,we are convenient to handle the electromagnetic scattering ofcomplex targets with the FDTD method.

Calculating the RCS of the complex targetelectromagnetic scattering with FDTD method has two kinds of nearfi eld far-fi eld extrapolation methods[13,14].One is the steady state fi eld extrapolation method,the other is the transient fi eld extrapolation method.The formergets a full range of RCS distribution in entire space under a single speci fi c frequency,while the latter gets a single speci fi c azimuth of RCS distribution under a wide frequency band. However,these two kinds of extrapolation methods are not suitable when computing the full-angle RCS distribution ofelectrically large complex targets within a wide frequency band.In orderto achieve a full-angle RCS distribution within a wide frequency band,itruns the conventional FDTD method for many times to obtain all RCS data.Itis implied thatit spends a large number of run time to fi nish the whole calculation task.

In order to solve the above limitation for these two conventional extrapolation methods in a case of calculating RCS distributions of complex electrically large targets,we presented a novel extrapolation method,combining with the discrete Fourier transform(DFT)technique,which is suitable for the case under multiple azimuths and multiple frequenciesfora complex electrically large target.The proposed extrapolation method is illustrated to save a large of run time relative to the conventionalextrapolation method.

The remainderof this paper is organized as follows:we derived a near-fi eld far-fi eld extrapolation algorithm combining with DFT technique after comparing formulas of these two extrapolation methods in Section 2.Section 3 illustrates the FDTD discrete models from the triangle surface models of electrically large complex targets with a certain transformation algorithm.Section 4 presents nu-merical results of the bi-static RCS distribution of complex electrically large targets.Finally,concluding remarks are drawn in Section 5.

2.Formulas of RCS calculation

To calculate the scattering fi elds or radiation fi elds outside the computationaldomain with the FDTD method,we need to use the Huygens principle to extrapolate far fi elds from near fi elds obtained from the FDTD calculation.This kind of method is a so-called near-fi eld far-fi eld extrapolation method.According to the difference type of incident plane wave used in the FDTD method,the near-fi eld farfi eld extrapolation method includes two cases.One is the steady state fi eld extrapolation method.The other is the transient fi eld extrapolation method.The steady state fi eld extrapolation method is to calculate the bi-static RCS distribution undera single designated frequency and multiple azimuths.On the other hand,the transient fi eld extrapolation method deals with the back-scattering RCS distribution under a single designated azimuth within a wide frequency band.In the following paragraph,we list the nearfi eld far-fi eld extrapolation formulas for these two extrapolation methods mentioned above.

In a three-dimensionalcase,we getthe steady state fi eld extrapolation formula by the following process.According to the Green's function in free space,the vector magnetic potential A(r)and vectorelectric potential F(r)are given as

In(1),the symbol k stands for the wave number of plane wave in a scalar form,and its vector form is written as k. According to(1),the formulas forthe electric fi eld and the magnetic fi eld in the far-fi eld zone are

Transform the right-hand side of the fi rst expression in (2)into a spherical coordinate form.Then,combine the current moment f and magnetic current moment fm,and rewrite the electric fi eld parts in a spherical coordinate form as

where is the intrinsic impedance of the medium.According to properties of the plane wave,the magnetic fi eld H in the far-fi eld zone from the electric fi eld E is solved.As the near fi elds calculated by the FDTD method is usually expressed in a rectangular coordinate,we transform the current moment f and the magnetic currentmoment fmin a sphericalcoordinate to those expressed in a rectangular coordinate.After fi nishing the transformation,the electric fi elds in the far-fi eld zone is written as

where(θ,φ)is the azimuth of the observation pointatthe far-fi eld zone.From(4),it is noted that the steady state extrapolation method can getthe far fi elds in a single designated frequency and multiple azimuths through one-time FDTD calculation.For example,considering the far fi elds in two designated frequencies,we have to calculate the FDTD program for two times.

If the incident plane wave form in FDTD is a timedomain pulse,the extrapolation process from the timedomain near fi eld to the time-domain far fi eld is taken into consideration.In a case of using a time-domain pulse source,the expressions for the far zone electric fi elds are written as?

Note that the equations above are expressed in frequency domain with k=ω/c.The symbol‘?'stands for the fi elds placed in the source area,and the fi elds without the symbol‘?'locate in the observation area.By transforming the source in time domain with Fourieralgorithm,itis found that the incident pulse source possesses a wide frequency bandwidth.Therefore,it is notsuitable to directly calculate the far fi elds by(6).In order to solve this problem,the inverse Fourier transform is used to change(6) into a time-domain form.After transforming with the inverse Fourier transform,the expressions in time domainfor combining(5)and(6)are given by

where the expressions of w(t)and u(t)are expressed as

where(θ,φ)expresses the azimuth ofthe observation point in the far-fi eld zone.In(8),the symbol j stands forthe current density in time domain,and jmstands for the magnetic fl ux density in time domain.Using the Fourier transform,(7)and(8)are transformed into the forms in frequency domain.By the above process,the far zone fi elds are obtained within a certain frequency band.Therefore, according to(7)and(8),we getfar fi elds in a single designated azimuth within a wide frequency band through onetime FDTD calculation.

Comparing these two extrapolation methods,the steady state fi eld extrapolation method can gain far fi elds ata single speci fi c frequency and multiple azimuths through onetime FDTD calculation.While,the transient fi eld extrapolation method is suitable in a single designated azimuth within a wide frequency band.If the far fi elds for multiple frequencies and multiple azimuths are considered,these two extrapolation methods have to be run forseveraltimes. That is to say it is not adequate to run one-time FDTD program with the above extrapolation method to achieve RCS distribution for multiple frequencies and multiple azimuths.In orderto solve forthe limitation of extrapolation method,an originalalgorithm brie fl y mentioned in[12]is presented by using the transient fi eld incidentsource to calculate the bi-static RCS distribution of the targets.In this paper,we extend this algorithm to calculate the bi-static RCS distribution of complex electrically large targets.The idea is to apply the DFT algorithm to each tangentialelectromagnetic fi eld parton outputboundary when the waveform of incident source is a pulse in time domain.The transform formula is given as

whereΔt and n are,respectively,the time incrementand time step used in FDTD.E(nΔt,r)is the electric fi eld in time domain,and E(f,r)is expressed in frequency domain.By transforming with inverse DFT,we obtain the far fi eld of E(f,r)in frequency domain from those in time domain.

In(9),the symbol f stands for one or more operating frequency of interest.After fi nishing one-time FDTD calculation in this way,the electromagnetic fi elds in frequency domain can be gained at the same time.We,then, use the extrapolation formulas of(4)to getfar fi elds atthe operating frequency of interest.Ifelectrically smalltargets are considered,the advantages of the proposed extrapolation method are not so obvious in reducing the run time. However,if the complex electrically large targets are considered,this proposed method has an obvious advantage in saving calculation time.The next section illustrates the advantage of the proposed method.

3.Modeling ofelectrically large targets

In this section,the CADmodeling method is used to model the electrically large targetin[12].First,draw the geometric pro fi le of complex targetwith some commercialdrawing software packages,such as the AutoCAD,3dMax,UG. Second,mesh surfaces of target into triangular surfaces with the help of the drawing software mentioned above. Third,record every triangular surface's location information.Finally,get the FDTD discrete model of the entire complex targetby a certain transform algorithm,which is used to transform a triangular modelto an FDTD model.

The transform steps from the triangular surface to FDTD model are mainly presented as follows.First,settle the FDTD unit grid sizes ofΔx,Δy andΔz.Second, fi nd outthe maximum and minimum values in dimensions of the target along the x axis on the base of the triangle surface data fi le.Third,settle the maximum and minimum values in the y direction in a similar matter.Fourth,sweep the model with an FDTD cell along the x,y axis within the scope of the object.Simultaneously,judge whether the scan straight line parallel to the z axis intersects with the triangular surface on the target surface or not.If the scan straightline intersects with the surface,record the coordinate values of intersection points.We,thus,get the maximum and the minimum dimensions of the object in the z axis,and con fi rm the numerical dimensions of target.Finally,assign electromagnetic parameters to each of FDTD cells within the scope,and fi nish the FDTDgrid discretization of target.

Use the CAD modeling method above to fi nish the FDTDmodeling ofcomplex electrically large targetshown in Fig.1.The physical sizes of target are set as 2.32 m in length(x axis),2.32 m in width(y axis)and 0.39 m in height(z axis).Discretizing the target with FDTD unit is shown in Fig.2.

Fig.1 Geometric shape model

Fig.2 FDTD modeling for a complex target

4.Method validity

4.1 Metaldouble olive body

A double olive body is a rotating body formed by rotating along the symmetry axis.It includes two parts:the cone of the large tip and the cone of the small tip.Discretizing the double olive body with FDTD unitaccording to the geometry shape sizes provided in[15],as shown in Fig.3. In FDTD,parameters of calculation are setas discrete spatial intervalδ=0.001 m,time intervalΔt=δ/2c,and operating frequency f=1.57 GHz.The de fi nitions of a parallelpolarization wave and a verticalpolarization wave are the same as the[15].

Fig.3 FDTD modelof a metaldouble olive body

The simulated results in the two different polarization cases are plotted in Fig.4.For comparison,the measured results are also provided in Fig.4.In the fi gure,zero degree angle here means the incidentplane wave spreading along the small tip.As seen from Fig.4,the FDTD simulated results are in good agreement with the measured results. The average errors between the simulated results and the measured results keep within 2 dB,which is considered acceptable.The average error formula is de fi ned by

In(10),the symbol‘Com'stands for the results simulated by the proposed extrapolation method.The symbol‘Mea' stands for the measured results from the[15].If considering the measurementerror and the modeltoleranterror,it is reasonable to meeta betteragreementwith the measured results.

Fig.4 Back-scattering RCS of the metaldouble olive body

4.2 Metalsphere-cone body

The sphere-cone body consists of the hemisphere partand the cone part.The metalmodelafter the FDTD discretiza-tion is drawn in Fig.5 according to the geometry shape sizes provided in[15].In the FDTD calculation,discrete spatialintervalis setasδ=0.002 m,time intervalisΔt= δ/2c,and the operating frequency is f=0.869 GHz.The simulation results of sphere-cone body in these two different polarization cases are depicted in Fig.6.Zero degree angle here means an incident plane wave spreading along the cone tip.Meanwhile,the Cicero results are also given in Fig.6 for comparison.From Fig.6,the proposed extrapolation results in FDTD match well with the Cicero results.The average errors between these two methods keep in a range of 2 dB within proper acceptance.

Fig.5 FDTD modelof a sphere-cone body

Fig.6 Back-scattering RCS ofthe sphere-cone body

4.3 Complex electrically large target

Using the transient fi eld extrapolation formulas(5)-(8) and the DFT formula(9),we calculate the bi-static RCS distribution of the complex electrically large targetshown in Fig.1.To corroborate the effectiveness of the proposed extrapolation method,the results calculated by the commercial software Feko are also plotted in Figs.7-9 for comparison.In these fi gures,the solid line stands for the FDTD results,and the dotted line is for the Feko results. The proposed FDTD computing parameters are set with discrete spatial intervalδ=5 mm and time interval Δt=δ/2c.According to the coordinate system in Fig.1, the incident angle of incident wave are set asφi=180°andθi=90°,and the polarization angle isαi=0°.Calculate the bi-static RCS distribution in the xoy plane with receiving angleθr=90°and polarization angelαr=0°. In Figs.7-9,the operating frequencies of interest are set as f=2.6 GHz,f=3.0 GHz and f=3.6 GHz,respectively.An inspection of the curves in Figs.7-9 shows that numericalresults of these two methods are in an agreement with each otherwell.

Fig.7 Operating frequency f=2.6 GHz

Fig.8 Operating frequency f=3.0 GHz

Fig.9 Operating frequency f=3.6 GHz

Seen from the contrast curves in Figs.7-9,the proposed extrapolation method is effective for calculating the bi-static RCS of electrically large targets when the waveform of incidentwave is set to a Gaussian pulse.Note that the bi-static RCS distribution of the complex electrically large targetwas simulated on the workstation HP xw9400 CPU 2.4 GHz MEM 16.0 GB.To further show the advantage of the proposed extrapolation method,we recorded the run time and consumed memory for these two extrapolation methods.First,we used the steady state fi eld extrapolation method to calculate the bi-static RCS oftargetata single frequency point.The computermemory needs about 700 Mbytes.The run time costis 10 071.25 s for one operating frequency through one-time FDTDcalculation.Second,we use the transient fi eld extrapolation method forthe same case.The 880 Mbytes computer memory is required. However,we gain the bi-static RCS distribution of fi ve differentoperating frequencies through one-time FDTD calculation.It is noted that the entire run time cost is only 9818.45 s for fi ve operating frequencies through one-time FDTDcalculation.The run time costby the FDTDwith the proposed extrapolation method is 15%times less than that by the conventionalextrapolation method in FDTD.If the bi-static RCS distributions within a wide frequency band are considered,itis implied thatthe advantage ofproposed extrapolation method are especially more obvious than the conventionalextrapolation method.

Of course,the commercial software packages,such as CST,HFSS and Feko are very powerful to deal with various electromagnetic problems.As all know,we should make a concrete analysis of each speci fi c question.For example,HFSS and Feko are usually used to deal with electromagnetic problems in frequency domain.If a wide pulse source in time domain is considered,these two software packages will sweep all frequency points of interest included in the source.It will cost a large number of run time.Certainly,CST is a time-domain software package. However,if both severaloperating frequencies and whole azimuth are considered,CST willrun severaltimes to meet these two aspects,and costa lotofrun time.Therefore,for calculating bi-static RCS of electrically large target in a wide frequency band,the proposed extrapolation method in FDTD supplies a novelpowerfultoolto engineering researchers.

In summary,the fi rstand second cases verify the validity and accuracy of the proposed extrapolation method.The third case shows thatthe run time costby the proposed extrapolation method is,just costing a little more consumed memory,greatly less than thatspentby the originalextrapolation method.

5.Conclusions

This paper proposed a novel extrapolation method in FDTD to calculate the bi-static RCS distribution of the complex electrically large targets.The bi-static RCS distribution of the targetformultiple concerned frequencies and multiple azimuths were obtained by the proposed extrapolation method through one-time FDTD calculation.By comparing the calculation results of typical targets from the reference,the validity and accuracy ofthe proposed extrapolation method were proved.Finally,the bi-static RCS distribution of complex electrically large targets has been considered.The simulated results match well with those gained from the commercial software.The advantage of the proposed extrapolation method is to get the bi-static RCS distribution in a wide frequency band through onetime FDTD calculation.If several operating frequencies are considered,the run time cost is greatly less than that of the steady state fi eld extrapolation method.It has been pointed out that fi ve different operating frequencies are considered,85%of run time are saved while maintaining numericalerror within proper acceptance relative to these two conventionalextrapolation methods.

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Biographies

Kuisong Zhengwas born in 1980.He received his Ph.D.degree in radio science from Xidian University in 2006.Now he is an associate professor in Northwestern Polytechnical University.His main research interests focus on electromagnetic theory,electromagnetic radiation and scattering,and modern antenna design.

E-mail:kszheng@nwpu.edu.cn

Tengjiang Dingwas born in 1990.He received his B.S.degree in electronic and information engineering from Northwestern Polytechnical University in 2012.Now he is a graduate student in Northwestern Polytechnical University.His main research interests focus on electromagnetic computation.

E-mail:949206962@qq.com

Hui Yuwas born in 1991.He received his B.S.degree in communication engineering from EastChina Jiaotong University in 2012.Now he is a graduate student in Northwestern Polytechnical University. His main research interests focus on electromagnetic computation.

E-mail:450232763@qq.com

Zhaoguo Houwas born in 1983.He received his M.S.degree in condensed matter physics from Beijing Institute of Technology in 2007.In 2010,he received his Ph.D.degree in electromagnetic fi eld and microwave technology from Communication University of China.His main research interests focus on electromagnetic scattering and inverse scattering. E-mail:houzg@139.com

10.1109/JSEE.2015.00081

Manuscriptreceived September 02,2013.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61401361)and the Fundamental Research Funds for the Central Universities of China(31020150104).