Analysis of Resolution of Bistatic SAR

JOURNAL OF BLECTRONIC SCIENCBAND TECHINOLOGY OF CHINA VoL. 5, NO.3, SEPTEMBER 2007.255Analysis of Resolution of Bistatic SARTian-Ge Shao and Jian-Guo WangAbstract - - In this paper, the spatial resolutions at2. Bistatic SAR Geometrydifferent directions of bistatic synthetic aperture radarThe geometrical situation of BiSAR is sketched in Fig. 1.(BiSAR) bave been derived from the ambiguity function.Coapared wit monostaic sigal to noise rnto BiSAR's In retanglar cordinate %, VT and Vk represes theresolutions of a fixed point target are varying with slowvelocity vector of the transmitter and receiver respectively,time since BiSAR system is space-variant. Constraints forUr and URo are the position vectors of transmitter andthe assumption of space-invariant bistatic topology arereceiver, respectively, pointing to the center of the aperture. Pproposed in the paper. Moreover, under the assumption ofrepresents the position of the target point of interest.invariance, the chaoge of resolutions at different point inthe image scene is taken into account, and we havespecified two key parameters that affect resolutionsUnpVdirectly and analyzed the way how they infuence on theresolutions.Uan↑Index Terms- Ambiguity function, bistatic SAR,resolution.1. IntroductionThe rescarches of bistatic synthetic aperture radar(BiSAR) in last decade have attracted more and moreFig 1. Bistatic SAR geometry.attentions over the world, due to its predominant advantagesto monostatic one. Spatial resolution, as one of the crucialAccording to the stop-and-go approximation, the pointparameters of bitatic signal t如o noise ration sytem, is target response can be dsecried by slow time u and fast timediscussed in []-[4]. In [S], the resolution of BiSAR ist. Similarly, for the veiocities of transmitters and receiversdescribed with a simple geometry, which is only applicableare far smaller than the electronic propagation speed, Ur andfor the stationary or parallel mode. In [6], a method on theUr, which is the instantaneous position of transmitter anduse of gradient is brought out and can be applied to thereceiver with respect to slow time, can be represented asgeneral configurations. In [7] and [8], radar function andfollows:ambiguity function for BiSAR has been discussed. OpposedUr(u)=Uro(u)+uVx, Ux(u)=Uqo(u)+tuVx .to monostatic one, BiSAR configuration is spatially shiftedLet the transmitted signal be s(t)=H()exp(2x[2), fc beand its resolution is correspondingly space-shifted and time-carrier frequency, h([) be the complex envelope and theshifted. In this paper, spatial resolutions are deduced fromreceived signal from the point P can be expressed as:ambiguity function and a comprehensive insight into theproblem of resolutions is obtained through deep researchesf(t,u)= h(t-Fp)exp( -j2ttp).1)on the most influential factors of resolution. Firstly, thewhereresolutions at different directions are derived from ambiguityr,(1)==|p - U(2)+|P -U(a).function. Then the constraints for assumption of invariantsystem are put forward and region of required resolution inDoppler frequency without expansion is represented as:the image scene is demonstrated to tally depend on BiSARUro-PURo-P ]topology. Finally, the paper has specified the key parametersthat afect the resolution diretly and the way in which theyUro-P|VR Uo-」infuence on the resolution.中国煤化工Manuscript received July 2, 2006, revised October 10, 2006.The authorsare 、; School of Electronic Enginoering, University ofYHCNMHdyFunctionTheAccording to the definition of ambiguity function in [3],Electronic Science and Technology of China (UESTC), Chengdu, 610054,China. (-mails: swectpigeon@163 .com, jgwang@uestc .edu .cn).we can get expression of ambiguity function:JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA, VOL. s, NO.3, SEPTEMBER 2007g(a) is used to note the integral in (8), thenx(P,P)=-小]IGC.N dJJjCn dug(J)= Jexp[2nJju]du = sinc(rLf).(9Suppose (-Lew2|Vzl, Lex/2|VR|) to be the range of(3)integrating and Ler be the effective coherent integration time,determined by bistatic configuration, including the aperturelength and velocities of both transmitter and receiver.where P is an arbitrary point near the point P. fe(,u) andDifferent situations lead to different Ler which are cormplexfr(,u) are the received signals from P and P, respectively.and discussed as a separate problem in [9].Equation (3) shows that the ambiguity function is theAssume that transmitted signal is a narrow band signalcorrelation coefficient of the response of two point targets.and the frequency spectrum is narrow enough for P() toIn frequency domain, ambiguity function can be describedsuppose the first integral in (4) can be approximated as:asx(P,P)=2PC)xp[j2nf(r(u)-T(u){(国) is used to note the integral in (10), thenP()= jP(/)exp[j2nftJ]df.(11)xJexp[j2rf(Tp(2)- T(u)]du(4)Obviously, the first integral in (4) can be considered as thewhere PO) is the power spectrum of the transmitted signal.inverse Fourier transform of p(a) In other words, becauseThe first integral in (4) is caused by the complex envelope ofP(f) is the frequency spectrum of transmitted signals, p(a) istransmitted signal, while the latter integral is caused by thethe auto correlation of the complex envelope of thecarrier fequency. To simplify the ambiguity function, severaltransmitted signal H().From the above analysis, we can obtain the followassumptions can be made as follows on which all theequation as simplified bistatic ambiguity function:first-order Taylor expansion in the following text are based on|P|<|Uro|，|P|<|URo|， Lp<|Uxo| and L <|Uol.x(P,P)=. p()8UfJ)exp[j2nftJ].(12)LR and LT are the length of transmiting and receivingThe spatial resolution is mainly determined by the amplitudeantenna apertures, respectively.factor in (12).For convenience of expression, several definitions aremade.4. Bistatic SAR Resolution1) u is the time delay diference between two target pointsFrom (12), it is obviously that p(国) corresponds to timeat u=0.delay resolution while gfa) determines Doppler resolution.tg=-0Uro-P+|Uxo-P|-|Uxo-P1-IUxo-P). (5)Let 8elay and soppor be - 3dB widths of p(a) and gVa),respectively.2) fa is the Doppler frequency difference between twoUsing Taylor expansion of y with respect to P at p'=P,different target points at u=0.we can obtainU1o-P'Uko-P'Uro-PUxo-P|Uxp-P|i+Va|Uo-P1-P|URo-P||(P'-P).(13)URo-_P |In the following,中p is the unit vector in the direction of(6)Uro-P, ψp is the unit vector in the direction of Uko-P, βrepresents the bitatic angle, and A is unit vector along theDue to the above assumptions that length of synthetic aperturedirection of bisector of bistatic angle.is much smaller than the distance from target to airplanes,approximation can be made as follow by using first-order中p + Yp = 2cos(B/2)A(14)Taylor expansion:2nfe(Tp(u)- tp(u)) = 2πfTEJ + 2nfu.(7)τ==(@p+Yp)(*'- P)=2cos(β/2)A-(P'-P). (15)Fron中国煤化工that the optimum rangeSubstituting (7) into (4), the second integral in (4) can beresol!written asYHCNMHG。.(16)2cos(β12)SHAO et al: Analysis of Resolution of Bistatic SAR_257Similarly, using Taylor explanation of fs with respect to P' atslow time u will be studied first, then we will consider aboutP'= P, we can obtainthe problem that how does the resolution of different targetpoints change in the scene.I-(Uro-P)(Uro-PyY]Vr5.1 Space-Invariance Restriction|Uro-PBistatic angle is the determinant parameter of the+!- Um-PXUx PYYxr(P'-P). (1)resolution, so we consider the cosine of bistatic angle of point|Uko-PpP which can be expressed asLet 0rp and r be, respectively, the module and unit vectorcosβ=|JU,(u)-明+|Ur(u)- 1p' -|U-(u)-U(u)1- (26)of the first factor in the folder; 0ORp and日be, respectively, the2|U,(u)- Pl|U,(u)-P|module and the unit vector of the second factor in the folder.where β is the bistatic angle of point P varying with slow timeDefine 0w as the angle between r and 0. In particular, 4。canu. For convenience, suppose P= 0. The Taylor expansion ofrepresent the angle between transmitter's and receiver'scosp as a function of u at u=0 is listed below:velocity vectors when both the planes are boresight.And let E and Wsyv be respectively the module and unitcosβ=cosB,+k;u .(27)vector of wrp I+CORP 0, there arewhere Po is the initial βat u=0.apT+0xp0= ar=.(18)D.。P y'φ,(O) t V,(9)、Substituting (18) into (17), we havek=TU|URo|”2UplUxo| |Uxo|1Um) (28)fo=(@p「+aqxp):(P'-P)=-qx=.(P'-P). (19)where D=Uro-Uko, Vp=Vr-VR. Using (26)-(28), the changeUsing (12) and (19), the optimum azimuth resolution is alonginβwithachangeinuisgivenbythe direction of 3 and its complete expression can be described0β=.k。=Au.(29)as follows:√1-cos2 β。(20)Bistatic angle and thus resolution change with slow time u inawbistatic space-invariance system. Equations (17)-(20)Usually, the resolutions in ground plane are what we concemdemonstrate that azimuth resolution, depending on fa changeabout, so rmnge and 8Suximuth are projected into the plane groundvery little with slow time u because of Lp<<|URol andand range resolution and azimuth resolution in ground planeLr<<[Urol. On the other hand, range resolution iscan be obtained:comparatively more impressionable to change of u thanmnge. goud =8mm.C(I-Zz' )A(21)azimuth resolution, so we mainly use range resolution to2cos(B/2)determine the property of bistatic system:Ssznwth. groundOoplet-a-zz' )E.(22)<(30)In (21) and (22), z represents the unit vector of z inIf (30) can be satisfied, which means the change of rangerectangular coordinate xyz. The range resolution and azimuthresolution induced by the variance of bistatic system isresolution are not perpendicular to each other any longer, sosmaller than 1/8 of the origin valuel1), we suppose that thethey can not completely express the resolvability of two pointbistatic system can be considered as invariant during time Ou .targets. Define θ to be the angle between the directions of A(u)By using (16) and (27)-(30), we can fix the range of Au:and E(u), thenθ=cos "(A(u) .E(u))(23)Ou√1-cos2 A。| 2arccos[°cos(B,12)]-B, . (31)and the cross range resolution can be defined as follows:1simwmh yousHowever, equation (31) is not analytically solvable, so wes_mge =(24)sin(0)turm to numeric method. The relationship between the value ofThe area of the resolution cell in plane ground can berange resolution and slow time u at different angles betweenobtained:VT and VR is shown in Fig. 2 and the main parameters areSground = 8 mnge ground o8xinwh. ground .(25)listed in Table 1.中国煤化工5. Performance of ResolutionIYHCNMHGtion.B, (MHz)I VR (mn/s)|Vr|(m/s) Uno (km) URo(km)Because BiSAR system is space-variant, in this section,20035, 20,617,0,6the varying resolution of a fixed target point with respect to258JOURNALOF BLECTRONIC SCIENCE AND TECHNOLOGY OF CHNA, VoL.5, NO 3. SEPTEMBER 2007From those curves in Fig. 2, it is obviously that theFig. 3 is consistent with (32) and (33), which can well validatechanges of range resolution with the slow time u in onethe above analysis of BiSAR resolutions.coherent integration time under several different topologiesBiSAR Reconsructed Signalcan all perfectly tolerates the constraints in (30), including themnost quick-changing curve. The result demonstrates thevalidation of constraints in (30) and thus (31). In adition, thesimulations of point target image in the next segment 5.2 alsojustify (30).管01.3513Range (m)虽1.2(团)量1.15|BiSAR Reconstnucted Signal2 1.1020406080100120140160 180 2000Fig 2. Range Resolution as a function of slow time u.Another observation can be achieved from Fig. 2. The3varying rate of range resolution descends with the increase ofthe angle between Vr and VR. Take to the extremes, β changes-5-4-3-2-1012 3 4very lttle with slow time u at the condition that transmitter(band receiver have perpendicular trajectories while β drops theBiSAR Reconstructed Signalmost quickly with u wben the transmitter and receiver move inparallel trajectories. This result accords with the analysis in 5.2.5.2 Point Targets -2Consider a special topology. Suppose that:01) Both transmitter and receiver are boresight in stripmode.2) Transmitter's footprint is much more than that ofreceiver.3)| Vr1=| Vrl | UrlFl URol.-5-4-3-2-10123 4The expressions of resolutions (16) and (20) can besimplified as:Fig. 3. BiSAR Reconstruction image at dfferet Po: (a) 4w =0 (deg),(32)(b) 4w =10 (deg.), (c) y。=30 (deg).2Bcos(β12)A5.3 lmage Scenej.mu|=O%w_ r√1+2c0sφ +1(33)Assume that the system can satisfy the restrictions ofinvariance, in another word, the system can be considered as awhere Aew_R is the transmitter's antenna pointing angle. 4w isspace invariant system. Take a scene of image into account.identical to the angle between Vr and VR Obviously, bistaticAll the observations in this section are made at theangle β determines range resolution while ψw determinesobservation time instant u=0. Only the position of the targetazimuth resolution. Fig 3 shows the reconstruction image ofpoint P is considered as variable. Fig. 4 provides thepoint target at different Pw.information that how the resolution varies in different targetThe simulation parameters are listed in Table 1. In thispoint in the scene and point out the best area for imaging. Fig,experiment, the monostatic theoretical value of range and4(a) and Fig 4(b), respectively, show the twisting lines ofazimuth is 2m and 1m. When 4o =0, the transmittr parallelsconstant range resolution and azimuth resolution. Fig. 4(c)the receiver, directions of range and azimuth resolution aresuppl中国煤化工ution both in range andstill perpendicular, azimuth resolution is 2 m, double asazimu_ound that the two kindsmonostatic one. With the increase of wn the resolvability inof resYHCNM.HGinthesameareaintheazimuth decreases dramatically. The result of simulation inscene. So in order to obtain better image, the initial topologySHAO et al: Analysis of Resolution ofBistatic SAR259should be designed to satisfy the demand both in azimuth anddemand both in azimuth and range resolutions. By using therange resolutions.special case that transmitter and receiver are both boresight,with same altitude and module of velocity, it is demonstratedRange Resolution Cotoursthat resolutions mainly depends on two factors: bistatic angleβ and aogle between Vτ and VR (4w). Too large B and Pu cansignificantly deteriorate the resolvability of BiSAR system.1Some other factors which also play key roles in the0performance of resolution, like the ratio of transmitter's andreceiver's slant range, the ratio of transmitter's and receiver's;.-25module of velocity, and the squint angle of transmitter andreceiver respectively, will be the future research problem.Range (mx10)References(aAzimuth Rolution Contours[1] D. C. Lorti and J. J Bowman.“Tactical aircraft use bistaticradar," Microwave Systems News, vol.8, no. 9, pp. 49-52, Sep. .1978.2] I Walterscheid, A. Brenner, and J. H G Ender. "Geometryand system aspeets for a bistatic airbormne SAR experiment," inProc. of European Conference on Symthetic Aperlure Radar,Anosterdam, 2004, Pp.567-570.3] 0. Loffeld, H. Nies, V. Peters, and S. Knedlik,“Models and-2Iseful relations for bistatic SAR processing,” IEEETransactions on GRS, vol. 42, no. 10, pp. 2031-2038, Oct.2004.[4] I. Walterscheid, A. R. Brenner, and J. H. G Ender,“Results onRange (mx10")bistatic synthetic aperture radar," Electronics Letters, vol. 40,b)no. 19, pp. 1224-1225, Sep. 2004.Region of Required Resolutions] N. Willis, Bistatic Radar,Norwood, MA: Arech House, 1991.[6] G P. Gardillo,“On the use of gradicnt to determine bistaticSAR resolution", in Proc. of AP-S Ilntermational Symposium,Dallas, Texas, 1990, vol. 2, pp. 1031-1035.蚤-egion of essohrionless than3m7] M. Chemniakov,“Ambiguity function for bistatic SAR and its.0application in SS-BSAR performance analysis," Presented atthe Intemnational Conference RADAR 2003，Adelaide,Australia, 2003.8] T. Zeng, M. Cherniakov, and T. Long,“Generalized approachto resolution analysis in BSAR,” IEEE Transaclions on4-35325之131-036Aerospace and Electronic Systems, vol. 41, no.2, Pp.461-414,Range(mx10-Apr. 2005.[9] G. Nico and M. Tesauro.“Coverage and resolution propertiesFig. 4. Resolutions (a) Range resolution contours, (b) Azimuthof bistatic SAR configurations," in Proc. of the 6th Europeanresolution contours, (C) Region of less-than 3m resolution.Conference Symthetic Aperture Radar, Dresden, Germany,2006, pp. 567-570.[10} T-S. Yco, N.L. Tan, C.-B. Zhang, and Y.H, Lu,“A new6. Conclusionssubaperture approach to high squint SAR processing," IEEEIn this paper, we obtain the range and azimuth resolutionsTransaction on Geosciences and Remote Sensing, vol.39, no.5, pP.954-968, May 2001.of bistatic SAR through the ambiguity function. Differentfrom monostatic one, directions of range and azimuth are nolonger perpendicular to each other. Furthermore, thTian-Ge Shao was borm in Sichuan Province, China, in 1982.resolutions are changing continuously with the slow timeShe is DoW a postgraduate with School of Elctronic Engineering,University of Electronic Science and Technology of Chinabecause BiSAR system is space-variant. Restrictions orwhich BiSAR system can be considered as invariant are(UESTC). Her research interest includes bistatic synthesizedaperture radar.given in Section 5.1 and are justified by simulations. AnJian-Gnp Wang was hom in Sichuan Province, in 1954. He isanalysis of resolution varying with respect to the position inDOW中国煤化工nic Engineering, UESTC.the image scene substantiates that range and azimuthHis rLrocessing of synthesizedresolution cannot achieve optimum at the same area in theapertuEYHC N M H Gtarget rognitiono andscene and suitably designed topology is needed to satisfy theinformation collection, electronic reconnaissance and interference.