a bistatic point target reference spectrum for general bistatic sar processing

A Bistatic Point Target Reference Spectrum forGeneral Bistatic SAR Processing Robert Wang,Member,IEEE,Otmar Loffeld,Senior Member,IEEE,Qurat Ul-Ann,Holger Nies,Amaya Medrano Ortiz,and Ashraf SamarahAbstract—A bistatic point target reference spectrum(BPTRS) based on Loffeld’s bistatic formula(LBF)is derived in this letter. For LBF,the same contributions of the transmitter and receiver to the total azimuth modulation are assumed.This assumption results in the failure of LBF in the extreme configuration(i.e., spaceborne/airborne configuration).For general bistatic configu-rations,the azimuth modulations are unequal for the transmitter and receiver due to the different slant ranges and velocities. Therefore,the azimuth time-bandwidth products(TBPs)from the transmitter and receiver are different;in some cases(e.g., spaceborne/airborne case),one of them might be very small,which might even result in a serious error of the principle of stationary phase.This letter uses TBP to weight the azimuth phase modula-tion contributions of the transmitter and receiver to the common azimuth spectrum to approximately obtain the point of station-ary phase of the total azimuth phase history.Simulations show that the proposed BPTRS can work well for spaceborne/airborne configurations.Index Terms—Bistatic point target reference spectrum (BPTRS),time-bandwidth product(TBP).I.I NTRODUCTIONB ISTATIC synthetic aperture radar(BSAR)is characterizedby different locations for the transmitter and receiver and hence offers considerableflexibility in designing BSAR missions[1],[2].Time-domain methods can focus BSAR data well.However, these methods always have huge computational costs.The efficiency of processing can be highly improved by performing the focusing in the frequency domain.Therefore,frequency-domain methods are preferred.The basic idea of bistatic frequency-domain processing originally comes from monosta-tic SAR data processing,which is based on the monostatic point target reference spectrum(PTRS)[3].The bistatic PTRS (BPTRS)is an important basis for processing in the2-D fre-quency domain[4],[5].However,the slant range history of BSAR includes two range histories from the transmitter and receiver,which means that the range equation consists of a double square root term.This implies that the point target refer-ence spectra of the monostatic and bistatic cases are inherently different[3],[4].Therefore,BPTRS does not offer the analyti-cal simplicity of the single square root in monostatic SAR.Manuscript received September27,2007;revised November22,2007and April1,2008.This work was supported by the Germany Science Foundation under Grant Lo455/7-1BiFocus.The authors are with the Center for Sensorsystems,University of Siegen, 57076Siegen,Germany(e-mail:wang@zess.uni-siegen.de).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/LGRS.2008.923542Several bistatic point target reference spectra have been reported in[2],[4],and[8].In[2],BPTRS is approximately developed for the azimuth-variant configurations[6],[7].It uses the second-order Taylor expansion to calculate the points of stationary phase of the slant range histories from the transmitter and bining the two individual points yields an approximated bistatic point of stationary phase.The numer-ical method has also been used to derive the BPTRS[8]. It is applicable to high-resolution wide-swath SAR data in the azimuth-invariant bistatic configuration[10].In addition,a BPTRS has been derived based on the method of series rever-sion(MSR)[4].It has a good focusing performance for the gen-eral bistatic configuration,particularly for the high squint cases [11],[12].The method of[2]works well for the general(spaceborne/ spaceborne or airborne/airborne)configurations[9].However, for the spaceborne/airborne case,it will suffer from a significant degradation.For the numerical method,it facilitates the appli-cation of the standard SAR processors where only the conven-tional transfer functions need to be replaced[10];however,it is limited to processing the azimuth-invariant configuration[10]. This limitation restricts its application due to the fact that it places the stringent requirements on the orientation of theflight path of bistatic platforms.The aim of our research is to extend the result of[2]to improve its performance and to further focus the spaceborne/airborne configurations[13].In[2],Loffeld’s bistatic formula(LBF)consists of two com-ponents:quasi-monostatic phase term and bistatic deformation phase term.Due to the fact that a second-order approximation is used in the course of calculating the bistatic spectrum and equally weighted contributions of the transmitter and receiver to the azimuth modulation are assumed,LBF might fail in some extreme bistatic configurations(e.g.,spaceborne/airborne configuration:satellite used as a transmitter/receiver and air-craft used as receiver/transmitter).In the spaceborne/airborne configuration,azimuth signals from the spaceborne and air-borne platforms only cover a part of the total synthetic aperture time,particularly for the airborne case.Therefore,the airborne case has a very small time-bandwidth product(TBP),which might result in the failure of the principle of stationary phase. The different TBPs of slant range histories result in unequal contributions of the transmitter and receiver to the overall phase modulation of the azimuth signal.In this letter,we use TBP as a factor to the weight contributions of the transmitter and receiver to the common bistatic point of stationary phase.After that,using it to substitute the real point of stationary phase of BSAR,we can obtain the BPTRS in the general configuration.1545-598X/$25.00©2008IEEEFig.1.Geometry of the BSAR in the general configuration.The parameters corresponding to the receiver and transmitter are denoted with the subscripts “R”and“T,”respectively.The BPTRS will facilitate the development of the efficient frequency-domain algorithms for the general configuration. This letter is organized as follows.In Section II,we begin with the signal model of BSAR in the general configuration case and derive the PTRS.In order to verify our spectrum, some simulations are performed in Section III.Finally,some comments are given in Section IV.II.S IGNAL M ODEL OF BSARIn this section,we evaluate the signal model of BSAR based on a general bistatic geometry,shown in Fig.1.The mathematical symbols and their definitions used in this letter are given as follows.s Azimuth time variable.(s0R,r0R)Receiver-referenced coordinates defined asthe coordinates of image space.r0R,r0T Closest ranges of receiver and trans-mitter,respectively,to the point targetP(s0R,r0R).R R(s),R T(s)Instantaneous slant ranges from the receiverand transmitter,respectively,to the pointtarget P(s0R,r0R).v R,v T Platform velocities of the receiver andtransmitter,respectively.λ,f0Carrier wavelength and carrier frequency ofthe transmitted signal.c Speed of light.f,f a Range and azimuth frequency variables.s0R,s0T Zero Doppler times of the receiver andtransmitter,respectively.s P R,s P T Points of stationary phase of the slantrange histories of receiver and transmitter,respectively.The phase of the echoed signal after demodulation to base-band and range Fourier transform can be expressed as follows (a negative symbol and a second-order phase term from the transmitted signal are neglected):φb(s,f)=2πf+f0c[R R(s)+R T(s)]+f a s(1)whereR R(s)=r20R+(s−s0R)2v2R(2)R T(s)=r20T+(s−s0T)2v2T.(3)If s p is assumed to be the actual point of the stationary phase of(1),according to[5],our desired BPTRS can be written asφb(s p,f)=2πf+f0c[R R(s p)+R T(s p)]+f a s p.(4)Due to the fact that two square root terms are included in(1),the analytical point of stationary phase s p,regardless ofthe squint and aperture width,cannot be obtained easily,atleast analytically.In order to circumvent this limitation,twoequivalent terms are introduced into(1)[2]φb(s,f)=φRW(s)+φT W(s)(5)φRW(s)=2πf+f0cR R(s)+k r f a s(6)φT W(s)=2πf+f0cR T(s)+k t f a s(7)where k r and k t are weighting factors defined as follows:k r=TBP rTBP r+TBP t(8)k t=TBP tTBP r+TBP t.(9)TBP r and TBP t are the TBPs of the azimuth modulationsignals from the receiver and the transmitter,respectively.Forthe calculation of TBP,we use the composite synthetic aperturetime.Therefore,the weighting factor with TBP is also equal tothe one with an azimuth chirp rate.Obviously,k r+k t=1isalways satisfied.Via this weighting operation,the range historycontributions of the transmitter and receiver to the azimuthDoppler spectrum are restricted by their TBPs.In[2],theircontributions are equally weighted(k r=k t=1/2),i.e.,thetwo equivalent terms are given as follows:φR(s)=2πf+f0cR R(s)+12f a s(6a)φT(s)=2πf+f0cR T(s)+12f a s.(7a)To address the problem of the double square root term,we ex-pand(6)and(7)around s P R and s P T that satisfyφ RW(s P R)=0andφ T W(s P T)=0.The expanded series are truncated at thesecond-order term and expressed as follows:φRW(s)≈φRW(s P R)+12φ RW(s P R)(s−s P R)2(10)φT W(s)≈φT W(s P T)+12φ T W(s P T)(s−s P T)2(11)W ANG et al.:BISTATIC POINT TARGET REFERENCE SPECTRUM 519whereφ RW (s P R )=2πc v 2Rr 0R F 3R (f +f 0)2(12)φ T W (s P T )=2πc v 2Tr 0T F 3T (f +f 0)2(13)s P R =s 0R −k r cr 0R v 2R f aF R (14)s P T =s 0T −k tcr 0T v 2T f a F T(15)F R =(f +f 0)2−k r cf av R 2(16)F T =(f +f 0)2−k t cf a v T2.(17)Using (10)and (11),we getφb (s,f )≈φbW =φRW (s )+φT W (s ).(18)We apply the principle of stationary phase to (18)because(10)and (11)are the second-order polynomials.Based on the results in [2],the approximated point of stationary phase s W P can be formulated as follows:s W P=φ RW (s P R )×s P R +φT W (s P T )×s P Tφ RW (s P R )+φ T W(s P T ).(19)The substitution of (12)–(17)into (19)gives s W P=r 0T v 2RF 3R s 0R +r 0R v 2T F 3T s 0T r 0T v 2R F 3R+r 0R v 2T F 3T −cf a r 0R r 0T k r F 2R +k t F 2Rr 0T v 2R F 3R+r 0R v 2T F 3T .(20)In order to obtain the approximated BPTRS,substituting (20)for the actual point of stationary phase s p into (4)yields ΨB (f a ,f )=φb (s W P ,f )=2πf +f 0c[R R (s W P )+R T (s W P )]+f a s W P .(21)When the bistatic configuration reduces to a monostatic configuration,we have k r =k t =0.5.As a result,(20)and (21)can be simplified as follows:s W P =s 0−cr 02v 2(f +f 0)2−(cf a /2v )2f a(22)Ψ(f a ,f )=4πr 0c(f +f 0)2−cf a 2v2+2πf a s 0(23)where v is the platform velocity,s 0denotes the zero Doppler time,and r 0represents the slant range at the zero Doppler time.Equation (23)is the monostatic spectrum [5],which shows the general feature of ourBPTRS.TABLE ISAR S YSTEM P ARAMETERSc r 0R F R +2πcr 0T F T(24)ΨBDW=φ RW (s P R )(s W P −s P R )2+φT W (s W P −s P T )2=2πv 2Rv 2T F 3R F 3T c (f +f 0)2(r 0R v 2T F 3T +r 0T v 2R F 3R )×(s 0T −s 0R )−cf av 2R v 2T F R F T× k t r 0T v 2R F R −k r r 0R v 2T F T 2.(25)The proposed results in this letter can be regarded as anextended Loffeld’s Bistatic Formula (ELBF).III.S IMULATION R ESULTSIn order to verify the validity of our BPTRS,simulations with the parallel velocity vectors are carried out in this section.In the simulation,the following three configurations are designed.A)Azimuth invariant configuration (transmitter and receiverfollow the same velocity vector).B)Azimuth variant configuration (transmitter and receiverhave different velocity vectors).C)Spaceborne/airborne configuration (our TerraPAMIR ex-periment:TerraSAR-X is the transmitter,and Forschung-sgesellschaft für Angewandte Naturwissenschaften (FGAN)’s airborne PAMIR system will serve as the receiver).The system parameters are listed in Table I.In this simulation,the distances of the simulated point target P away from the scene center in the azimuth and range direc-tions are 200and 1000m,respectively,shown in Fig.2.As mentioned in the previous discussions,the analytical BPTRS cannot be obtained.We numerically assume the 2-D frequency phase of point target P to be an analytical BPTRS.It can be achieved by performing a 2-D Fourier transform for the simulated signal of P .For comparison,we use the BPTRS proposed in [2]and in this letter as the matched filter to focus520IEEE GEOSCIENCE AND REMOTE SENSING LETTERS,VOL.5,NO.3,JULY2008Fig.2.Simulated scene with one point target.the assumed analytical BPTRS.In addition,we also focus the assumed BPTRS using itself.It needs to be emphasized that no weighting function is used in this simulation.The azimuth impulse responses of the focusing results in the three configura-tions are shown in Fig.3.In Fig.3,analytical (ANA)represents the focusing output of the assumed BPTRS,ELBF is the focus-ing output of this letter,and LBF is the one proposed in [2].In Fig.3,for clearness,all azimuth impulse responses are enlarged by a time of 60using an interpolation operation.For A and B,the results in [2]and in this letter agree well with the analytical BPTRS.However,in configuration C (spaceborne/airborne configuration),using LBF suffers a significant degra-dation in the azimuth and ELBF still shows a nice focusing.To quantitatively compare the focusing performance,quality parameters corresponding to Fig.3are listed in Table II where impulse response width (IRW),peak sidelobe ratio (PSLR),and integration sidelobe ratio (ISLR)are used as criteria.From Table II,in configuration C,it can be seen that the IRW with ELBF focusing has an azimuth broadening of about 0.6%and the one with LBF has a broadening of about 50%.In addition,the PSLR and ISLR with LBF also have a significant degradation.Thus,the proposed BPTRS has a better perfor-mance with respect to the result of [2]and agrees well with the analytical one.IV .C ONCLUSIONIn this letter,the bistatic stationary point is obtained via considering the different contributions of the slant range histo-ries of the transmitter and receiver.For the proposed spectrum,the approximation comes from the difference between s p and s W P .The simulation shows that the spectrum works well in the general configuration.It can be considered as an extension or generalization of LBF.For future work,some processing algorithms for the general BSAR data will be developed based on the spectrum proposed in this letter.A CKNOWLEDGMENTThis work reported herein is part of the joint Deutsche Forschungsgemeinschaft research initiative Bistatic Explora-tion (PAK 59)of the Center for Sensorsystems (ZESS)and FGAN,which is gratefully acknowledged.The authorswouldFig.3.Azimuth impulse response output.(a)Comparison results in configu-ration A.(b)Comparison results in configuration B.(c)Comparison results in configuration C.TABLE IIQ UALITY P ARAMETERS OF THE P RECEDING T HREE CONFIGURATIONSW ANG et al.:BISTATIC POINT TARGET REFERENCE SPECTRUM521R EFERENCES[1]J.H.Ender,I.Walterscheid,and A.Brennner,“New aspects of bistaticSAR:Processing and experiments,”in Proc.IGARSS,Anchorage,AK, Sep.2004,pp.1758–1762.[2]O.Loffeld,H.Nies,V.Peters,and S.Knedlik,“Models and useful re-lations for bistatic SAR processing,”IEEE Trans.Geosci.Remote 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