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REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To) 01-04-2009 Journal Article March 2007 - March 2008 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER DOA estimation and multi-user interference in a two-radar system N/A 5b. GRANT NUMBER N/A 5c. PROGRAM ELEMENT NUMBER 612311 6. AUTHOR(S) 5d. PROJECT NUMBER Maria Greco*, Fulvio Gini*, Alfonso Farina**, and Muralidhar Rangaswamy 2311 5e. TASK NUMBER HE 5f. WORK UNIT NUMBER 02 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Electromagnetic Scattering Branch *Univ. of Pisa, Italy N/A Air Force Research Laboratory, Sensors Directorate ** Selex-Sistemi Integrati 80 Scott Drive Rome, Italy Hanscom AFB, MA 01731-2909 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) Electromagnetics Technology Division SC: 437490 AFRL-RY-HS Sensors Directorate Air Force Research Laboratory 11. SPONSOR/MONITOR'S REPORT 80 Scott Drive Hanscom AFB, MA 01731-2909 NUMBER(S) AFRL-RY-HS-TP-2009-0006 12. DISTRIBUTION/AVAILABILITY STATEMENT Distribution A: Approved for UNLIMITED DISTRIBUTION 13. SUPPLEMENTARY NOTES Research support by Air Force Office of Scientific Research. Cleared for public release approved by 88 ABW/PA Public Affairs Office, WPAFB, OH: RY-08-0423 14. ABSTRACT In this paper,we investigate the impact of the presence of interfering radar on the target direction of arrival (DOA) estimation performed by the reference radar.The analyzed estimators are the pseudo-monopulse and the maximum likelihood techniques.The importance of the use of codes in multi-user radar system is highlighted in a simple scenario of two radars by calculating the root-mean square error of the estimators in different operational conditions and comparing them with the Cramer-Rao lower bounds. 15. SUBJECT TERMS Frequency-hop-coded signals, auto and cross-ambiguity function, monopulse radar, DOA estimation, maximum likelihood, Cramer-Rao lower bounds, radar networks 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OF Muralidhar Rangaswamy PAGES U U U UU 11 19b. TELEPHONE NUMBER (Include area code) 781-377-3446 Standard Form 298 (Rev. 8/98) Reset Prescribed by ANSI Std. Z39.18 ARTICLE IN PRESS SignalProcessing89(2009)355–364 ContentslistsavailableatScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro $ DOA estimation and multi-user interference in a two-radar system Maria Grecoa,(cid:2), Fulvio Ginia, Alfonso Farinab,1, Muralidhar Rangaswamyc,2 aDip.diIngegneriadell’Informazione,UniversityofPisa,viaG.Caruso,14-56122Pisa,Italy bSELEX-SistemiIntegrati,viaTiburtinaKm.12.400–00131Rome,Italy cAirForceResearchLaboratorySensorsDirectorate,80ScottDr.HanscomAirForceBase,MA01731-2909,USA a r t i c l e i n f o a b s t r a c t Articlehistory: Inthispaper,weinvestigatetheimpactofthepresenceofinterferingradaronthetarget Received3August2007 direction of arrival (DOA) estimation performed by the reference radar. The analyzed Receivedinrevisedform estimators are the pseudo-monopulse and the maximum likelihood techniques. The 22May2008 importance of the use of codes in multi-user radar system is highlighted in a simple Accepted29August2008 scenario of two radars bycalculating the root-mean square error of the estimators in Availableonline23September2008 differentoperationalconditionsandcomparingthemwiththeCrame´r-Raolowerbounds. Keywords: &2008ElsevierB.V.Allrightsreserved. Frequencyhop-codedsignals Autoandcross-ambiguityfunction Monopulseradar DOAestimation Maximumlikelihood Crame´r-Raolowerbounds Radarnetwork 1. Introduction transmit. In particular, it is possible to modify the waveform on a pulse-to-pulse basis, and electronically For many years, conventional radars transmitted, steeredphasedarrayscanquicklypointtheradarbeamin received, and processed the same waveform on every any feasible direction. Such flexibility calls for new pulse or burst within a coherent processing interval, methods of designing and scheduling the waveforms to independentlyoftheenvironment.Inthe1970s,adaptive optimize the radar performance. Then, an agile and processing began to be developed. For the first time, the diversewaveformradarsystemshouldbeabletochange processingof receivedsignalschanged dependingonthe on the fly the transmitted waveform based on the environment (noise, clutter, and interferences). Radars informationestimatedoraprioriknownontheenviron- begantobemoreflexibleonreceive. ment,onthetargetsandthejammers[1]. Now, modern radar systems have considerable flex- Moreover,ina radar networkeachsensorshouldalso ibility in their modes of operation, both on receive and beabletooperateandperformitstaskwithoutnegatively interfering with the other sensors and, possibly, to improve the performance of the whole network. Then, $ThisworkhasbeenpartiallyfundedbyAFOSRgrantFA8655-07-1- thewaveformsused bytheradarsin a complex network 3096. The views and conclusions contained herein are those of the shouldbedesignedandchangedonthefly,basedonthe authorsandshouldnotbeinterpretedasnecessarilyrepresentingthe clutter, target and interference echoes3; they should officialpoliciesorendorsements,eitherexpressedorimplied,oftheAir ForceOfficeofScientificResearchortheU.S.Government. guaranteegoodtargetdetectionandparameterestimation (cid:2)Correspondingauthor.Tel.:+390502217620;fax:+390502217522. E-mailaddresses:[email protected](M.Greco),[email protected] (F. Gini), [email protected] (A. Farina), Muralidhar.Rangaswamy@- hanscom.af.mil(M.Rangaswamy). 3Thechoiceofthetransmittedwaveformdependsnotonlyonthe 1Tel.:+390641502279;fax:+390641503755. desireddelayandDopplerresolution,butalsoonwheretheclutteror 2Tel.:+17813773446;fax:+17813778984. competingtargetsarelocatedinthedelay-Dopplerplane[5]. 0165-1684/$-seefrontmatter&2008ElsevierB.V.Allrightsreserved. doi:10.1016/j.sigpro.2008.08.013 ARTICLE IN PRESS 356 M.Grecoetal./SignalProcessing89(2009)355–364 indifferentscenariosandshouldallowanoptimalaccess the Crame`r-Rao lower bounds (CRLB). We focused our tothesametransmitchannel. attentiononlyontheinfluenceoftheinterferingradaron To perform good target detection and estimation, the DOA estimation, so in our scenario we did not frequency hop pulse train signals are often used in high simulatejammersorcorrelatedclutter. resolution radar systems. These signals arecharacterized The paper is organized as follows. In Section 2, the by an auto-ambiguity function (AAF) that exhibits a receivedsignalmodelisexplainedandthePMandtheML narrowthumbtackshapewithlowsidelobes.Incontrast, techniquesforDOAestimationaresummarized.InSection in application like multi-access communications, atten- 3, the results of the analysis are described. In Section 4, tionispaidindesigningasequenceoffrequency-hopped theCRLBfortheproblemathandisderivedandcompared coded waveforms with small cross-correlation functions. withtheroot-meansquareerror(RMSE)ofthePMandML In multi-user radar system scenarios both objectives are estimators. In Section 5, some conclusions on the use of desirable.Unfortunately,thereisatradeoffbetweenthese codesaredrawn. objectives. Frequency hop pulse trains based on Costas codes, for instance, are known to have almost ideal AAF 2. Problemstatement butnotverygoodcross-ambiguityproperties[2].Onthe contrary, frequency-coded signals based on linear con- The scenario is pictorially drawn in Fig.1. Tworadars gruences [3] have ideal cross- but unattractive auto- scan the same area transmitting frequency-coded bursts ambiguity properties. Some attempt to design multiple ofpulsesandlisteningtotheechoofapossibletarget.The access frequency hop codes with good cross-ambiguity complexenvelopeofthetransmittedunitarypowersignal function (CAF) has been done, for instance, in [4] where isgivenby the frequency hop patterns were constructed upon an extensionofthetheoryofquadraticcongruences. 1 XM uðtÞ¼p u ðt(cid:2)ðm(cid:2)1Þt Þ, (1) Thescenarioanalyzedinthispaperiscomposedoftwo ffiMffiffiffiffitffiffiffiffi m c cm¼1 radars transmitting in the same band, which can illumi- where natethesamearealookingforthesametarget;theycan uobseseervitehdersigthnealsaasmaefuonrctdioifnfeorfencrtoscso-daens.dWauetor-eawmrbitiegutihtye umðtÞ¼(e0xpðj2pfmtÞ 0elpsetwphtecr;e; (2) functionvaluesandtargetparameters,andweinvestigate theimpactofthepresenceofthetransmittedsignalofthe Misthenumberofsubpulsesforeachtransmittedpulse second radar (the interfering radar) on the first one (the of time duration T, t ¼T/M is the duration of each i c i referenceradar)intheestimationofthetargetdirectionof subpulse and {f } M is the sequence of frequencies m m¼1 arrival (DOA). We analyzed two DOA estimators, the relatedtothecodeusedbytheradar.ForCostasarrays,for pseudo-monopulse (PM) and the maximum likelihood instance, f ¼d /t, where d belongs to the sequence m m c m (ML)estimatorandwecomparedtheirperformanceswith d ¼{d ,d,y,d ,y,d },whichisapermutationofthe M 0 1 m M(cid:2)1 Fig.1. Radarscenario. ARTICLE IN PRESS M.Grecoetal./SignalProcessing89(2009)355–364 357 integer numbers J ¼{0,1,y,M(cid:2)1}. The choice of M {f } M , characterizing the frequency code is critical LPF A/D m m¼1 andofparamountimportanceindefiningthepropertiesof IF the auto- and cross-ambiguity functions of the trans- signal mitted signal. A detailed description of this topic can be LO foundin[5]. Correlator As known, the AAF represents the time response of a filter matched to a given finite energy signal when the (cid:2)/2 signal is received with a delay t and a Doppler shift n Trasmitted signal relativetothenominalzerosvaluesexpectedbythefilter LPF A/D [5].Then,theAAFdefinitionis jAðt;nÞj¼(cid:3)(cid:3)(cid:3)Z þ1uðtÞu(cid:3)ðtþtÞexpðj2pntÞdt(cid:3)(cid:3)(cid:3), (3) Fig.2. Receiverscheme. (cid:3) (cid:2)1 (cid:3) transmittedsignalu(n/f),wecanstatethatAisthevalue whereu(t)isthecomplexenvelopeofthesignal.TheCAF c of the complex AAF of the signal for some delay t and betweentwosignalsu (t)andu (t)issimilarlydefinedas4 1 2 Dopplertargetshiftf ;thenA¼A(t,f ).Thisisthecaseof D D jCðt;nÞj¼(cid:3)(cid:3)(cid:3)(cid:3)Z(cid:2)þ11u1ðtÞu(cid:3)2ðtþtÞexpðj2pntÞdt(cid:3)(cid:3)(cid:3)(cid:3). (4) trhefeerseingncealrabdaacrk.sIcf,atotnertehdebcyonatrataryrg,ethteilsluigmnianlarteecdeibveydthbye the reference radar is the signal transmitted by the interferingradar,AisthevalueofthecomplexCAF;that is A¼Cðt ;f Þ. If both signals are present, the received 2.1. Signalmodel R DR signalisgivenby Inthereferenceradar,thereceivedsignalisfirstdown- y¼aejjAðt;f ÞþbejyCðt ;f Þþd, (9) shiftedtoanintermediatefrequencyIFandamplified.The D R DR wherebejyisthecomplexamplitudeofthesignalrelating IF signal is then processed as in the scheme of Fig. 2, tothesecondradaranddistheunavoidablecontribution where LO is a local oscillator tuned on the IF frequency, ofthedisturbance(thermalnoiseplusclutter). LPF is a low pass filter and A/D is an analog-to-digital device. Before the digitalization, the inphase (I) and (Q) 2.2. Pseudo-monopulseestimator quadraturecomponentsofthetargetsignalare: (cid:4) dðtÞ (cid:5) In the analyzed scenario the reference radar is xIðtÞ¼acos 2p t tþ2pfDt(cid:2)j , (5) supposed to estimate the DOA of the target by using a c pseudo-monopulsetechniqueorthemaximumlikelihood (cid:4) dðtÞ (cid:5) estimator[6]. xQðtÞ¼asin 2p t tþ2pfDt(cid:2)j , (6) Inatypicalphasedarrayradar,asinglebeamisformed c on transmission and two or more beams are formed on where aexp(jj) is the complex amplitude of the target reception. We assume here that the system is a linear that depends on the radar-cross section (RCS) and on array radar that estimates the target DOA by using the the antenna gain, fD is the Doppler frequency, and sum channel P on transmission and two matched d(t)¼PmM¼(cid:2)01dm(cid:4)rect((t(cid:2)mtc/2)tc)isthefrequencycode. channels, the sum P and the difference D on reception. Afterthedigitalizationatasamplingfrequencyfc,the Thetwochannels,orantennapatterns,aredefinedasthe complexenvelopeofthereceivedtargetsignalisgivenby complex amplitude profiles versus target azimuth angle xðn=fcÞ¼aejj exp(cid:4)2pdðtnc=ffccÞnþ2pffDcn(cid:5). (7) dyTif.fTerheenscuempactthearnnnbeylfpDa(tyt)e;rtnheissedepnaottteerdnsbyarfePc(hyo)saenndhtehree as in [6], the (cid:2)3dB beamwidth is y ¼31. The PM B In the correlator of Fig. 2, the sequence of x(n/f) is c technique used here is very similar to the classical correlated with the sequence of samples of the trans- monopulse method, based on the ratio of the D and P mittedsignal.Thentheoutputsignalisgivenby5 channeloutputsyDandyP.ItusesNpulsestransmittedby x¼aejjNXS(cid:2)1uðn=fcÞx(cid:3)ðn=fcÞ¼aejjA. (8) tbheeamrasdaarreinetlhecetrtoimniec-aollny-toarrgemte(cThoaTn),icwalhlyilestteheereadntewnintha n¼0 constant angular velocity o rad/s. The number N of R ItiseasytocomparethetermAinEq.(8)withEqs.(3) pulses between the one-way (cid:2)3dB points is given by and(4).IfN islargewecanconsiderthesumasagood N¼y /(o T), where T¼1/PRF is the radar pulse repeti- S B R approximation of the integral. Moreover, if the received tion time and PRF is the corresponding pulse repetition signalx(n/f)isadelayedandfrequencyshiftedcopyofthe frequency.Theantennaintroducesanamplitudemodula- c tion on the target signal, in both the channels, that depends on the target azimuth position and on the 4ThecalculationofEqs.(4)and(5)isnotalwayseasy.Forsome instantaneousboresightofthearray. frequencycode it is useful to resort to the placement and difference BasedonEq.(9),itispossibletowritetheexpressionof matrices,asexplainedin[3,14]. 5WedidnotuseanywindowinthecalculationofEq.(8). the signal received on the sum and on the difference ARTICLE IN PRESS 358 M.Grecoetal./SignalProcessing89(2009)355–364 channels. Let us supposethat the Doppler frequenciesof isgivenby thetargetandoftheinterferenceradararethesamewith respect to the reference radar. In this case, the reference y^ ¼argmaxjyHM(cid:2)1gðyÞj2. (11) radar cannot distinguish between the two signals by ML y gHðyÞM(cid:2)1gðyÞ means of Doppler processing. Withoutlackof generality, Inthenextparagraphweinvestigatetheimpactofthe we assume that both Doppler frequencies are null; then presenceoftheinterferingradarontheDOARMSEwhen thereceivedsignalsaregivenby the estimation is performed by the PM and the ML ySðnÞ¼aAf2SðyT;nÞþbCfSðyI;nÞþdSðnÞ, (10a) techniques. yDðnÞ¼aAfSðyT;nÞfDðyT;nÞþbCfDðyI;nÞþdDðnÞ, (10b) 3. Simulationresults where n¼0,1,y,N(cid:2)1. With respect to Eq. (9), the Toevaluatetheimpactofthepresenceoftheinterfer- dependence of the complex amplitudes of target and ingradar,theRMSE,thevarianceandthebiasoftheDOA interfering signal on the antenna patterns has been estimator has been derived by running 104 Monte Carlo explicitly indicated. The first term in yP and yD is simulations. The disturbance is modeled as a complex due to the target signal; it depends on the target DOA yT through fS2(yT) in sum channel and through wwhhietreeGIatuhsesia2nN-pdriomceensss,ioinnalshidoertntnitoytamtioantrixd,(cid:6)tChNe(0ta,srgd2Ie)t, f (y )f (y ) in the difference channel. This is due to the S T D T andinterferencesignalamplitudesa anda aremodeled two-way antenna gain. The second term is due to the T I first as: (i) complex Gaussian independent random signaltransmittedbythesecondradarwhichhasaDOAy I variables, in short a (cid:6)CN(0,s2|A|2) and a(cid:6)CN(0,s2|C|2); and it depends on the one-way gain of the antenna T T I I and then as: (ii) deterministic unknown parameters. In pattern(fS(yI)andfD(yI)).dSanddDarethenoisesonthe the first case, the signal-to-noise (SNR) power ratio is twochannels. defined as SNR¼|A|2(cid:4)s2/s2 and the signal-to-interfer- The signal processor forms the monopulse ratio T d ence ratio as SIR¼s2/s2; in the second case, SNR¼ defined by r(n)¼Re{yD(n)/yS(n)} for each pulse where |a |2(cid:4)/s2andSIR¼|a /Ta|2I. Re{}denotestherealpart.Inabsenceofdisturbanceand T d T I The results of our analysis are shown in Figs. 3–7 for in the presence of only one target, the monopulse ratio thePMestimatorandinFigs.8–14fortheMLestimator. reduces to r(n)¼Re{fD(yT,n)/fS(yT,n)} from which the We set SNR¼20dB,7 SIR¼0dB and N¼16; the target angular location of the target for each pulse can be determined. Finally, we obtain y^ ¼PN(cid:2)1y^ðnÞ=N, where DOAhasbeensetequalto1.51(yT¼1.51).Weinvestigated y^ðnÞ isthetargetDOAestimateforTGeachpnu¼l0seT.G also other target DOA positions in the main beam of the TG antenna. The behavior of the estimators is similar, and We treat the situation in which the source can be thenweshowheretheresultsforonepositiononly. consideredstaticwithrespecttotheradarduringtheToT, Due tothe high value of SNR, the performance of the thatis,duringtherecordingoftheNpulses.Inthiscase, PMandMLestimatorsaremainlyaffectedbythepresence due to the scanning movement of the antenna only, the oftheinterferingradar;theyarethefunctionsoftheratio proposed estimator is biased and the bias can be calculated. It is easy to verify that b¼Efy^ (cid:2)y g¼ |C/A|. The value of C and Adepends on the code used by ðN(cid:2)1Þy =ð2NÞ then a DOA unbiased estimatoTGr is y^TG ¼ bothradarsandonthesynchronizationbetweentransmit PN(cid:2)1y^ðnBÞ=N(cid:2)b[6]. TG andreceiveofthereferenceradarandbetweenreference n¼0 TG andinterferingradar. Ifthereferenceradarissynchronizedinreceptionand transmissionandthereceiveristunedontheDopplerof 2.3. Maximumlikelihoodestimator the target, t¼0 and f ¼0, then A is the energy of the D transmitted pulse. In our analysis we considered A¼1 IfwerewriteEq.(10)invectornotation,thedatamodel (unit energy pulse) and 0p|C/A|p1. The worst case is is given by y¼aTg(yT)+aIf(yI)+d, where aT¼aA, and when|C/A|¼1;thisvaluecharacterizestwosynchronized aI¼bC, g(yT)¼[gSTgDT]T is the 2N(cid:5)1 steering vector of radars using the same code. The best case is when |C/ the target, [gP]n¼fS2(yT,n) and [gD]n¼fS(yT,n)fD(yT,n). A|¼0; this value characterizes the case of synchronized f(yT)¼[fSTfDT]T is the steering vector of the interference, radarsusingtwoorthogonalcodes. wtuirtbhan[fcSe]nv¼ecfSto(yrT,dn)iasndgi[vfeLn]nb¼yfDd(y¼T,n[)d.STTdhDeT]2T.NT(cid:5)h1e dMisL- a fuInncFtiigo.n3,owfethreepinotretrtfheereRnMceSED(OyTA),owftithheaPM(cid:6)eCsNti(m0,sat2o|Ar|a2s) estimator of yT,6 when the observed vector does not andaI(cid:6)CN(0,sI2|C|2).InFig.4,wereporttheTbiasofthTePM contain the interference and the vector d is modeled estimatorinthesamescenario.Thebiasoftheestimator asacomplexGaussianvectorwithacovariancematrixM, increaseswith|C/A|andtheRMSEgetsvaluesclosetothe 40% of the beamwidth. The behavior of both RMSE and 6Asshownin[6],Eq.(11)representstheMLestimatorofyTwhen biasarealmostsymmetricwithrespecttothepositionof theamplitudeaTismodeledasadeterministicunknownvalue.WhenaT thetargetyT.WhenyI¼yT,thebiasduetothepresenceof isarandomvariable,theMLestimatorhasnotaclosedform.Weadopt heretheMLestimatorfordeterministicaTevenifinourdatamodelthe complex amplitude is random; with an abuse of terminology we 7Withthis valueof SNRthe probabilityofdetectionof thefilter continuetoreferto(11)astheMLestimator. matchedtothetransmittedsignalisalmostunitary,evenforPFA¼10(cid:2)6. ARTICLE IN PRESS M.Grecoetal./SignalProcessing89(2009)355–364 359 10 1.2 |C/A|=0 |C/A|=0.35 |C/A|=0 |C/A|=0.02 |C/A|=0.02 |C/A|=1 0.8 |C/A|=0.17 |C/A|=0.17 |C/A|=0.35 |C/A|=1 es) 1 s) 0.4 e e egr gre d e 0 RMSE ( 0.1 Bias (d -0.4 -0.8 0.01 -1.2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 θI (degrees) θI (degrees) Fig. 3. RMSE vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, Fig. 6. Bias vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, N¼16,randomaandb,PMestimator. N¼16,deterministicaandb,PMestimator. 1.2 10 |C/A|=0 SIR=-10 dB 0.8 |C/A|=0.02 SIR=0 dB |C/A|=0.17 SIR=10 dB |C/A|=0.35 SIR=20 dB s (degrees) 0.04 |C/A|=1 E (degrees) 1 Bia -0.4 RMS 0.1 -0.8 -1.2 0.01 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 θI (degrees) |C/A| Fig. 4. Bias vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, Fig. 7. RMSE vs. |C/A|, yT¼1.51, SNR¼20dB, N¼16, yIAU((cid:2)71,71), N¼16,randomaandb,PMestimator. randomaandb,PMestimator. 10 10 |C/A|=0 |C/A|=0.35 SIR=-10 dB |C/A|=0.02 |C/A|=1 SIR=0 dB |C/A|=0.17 SIR=10 dB SIR=20 dB es) 1 s) 1 e e egr gre d e RMSE ( 0.1 RMSE (d 0.1 0.01 0.01 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 θI (degrees) |C/A| Fig. 5. RMSE vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, Fig. 8. RMSE vs. |C/A|, yT¼1.51, SNR¼20dB, N¼16, yIAU((cid:2)71,71), N¼16,deterministicaandb,PMestimator. deterministicaandb,PMestimator. ARTICLE IN PRESS 360 M.Grecoetal./SignalProcessing89(2009)355–364 10 1.2 |C/A|=0 |C/A|=0.02 0.8 |C/A|=0.17 |C/A|=0.35 |C/A|=1 egrees) 1 grees) 0.4 d e 0 RMSE ( 0.1 Bias (d -0.4 |C/A|=0 |C/A|=0.02 |C/A|=0.17 -0.8 |C/A|=0.35 |C/A|=1 0.01 -1.2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 θI (degrees) θI (degrees) Fig. 9. RMSE vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, Fig. 12. Bias vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, N¼16,randomaandb,MLestimator. N¼16,deterministicaandb,MLestimator. 1.2 10 |C/A|=0 |C/A|=0.02 0.8 |C/A|=0.17 |C/A|=0.35 |C/A|=1 0.4 grees) 0 grees) 1 e e d d s ( E ( Bia -0.4 MS R 0.1 -0.8 SIR= - 10 dB SIR=0 dB SIR=10 dB -1.2 SIR=20 dB 0.01 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 θI (degrees) |C/A| Fig. 10. Bias vs. interference DOA, yT¼1.51, SNR¼20dB, SIR¼0dB, Fig. 13. RMSE vs. |C/A|, yT¼1.51, SNR¼20dB, N¼16, yIAU((cid:2)71,71), N¼16,randomaandb,MLestimator. randomaandb,MLestimator. 10 10 |C/A|=0 |C/A|=0.35 SIR= - 10 dB |C/A|=0.02 |C/A|=1 SIR=0 dB |C/A|=0.17 SIR=10 dB SIR=20 dB MSE (degrees) 1 MSE (degrees) 1 R 0.1 R 0.1 0.01 0.01 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 θI (degrees) |C/A| Fig.11. RMSEvs.interferenceDOA,yT¼1.51,SNR¼20dB,SIR¼0dB, Fig. 14. RMSE vs. |C/A|, yT¼1.51, SNR¼20dB, N¼16, yIAU((cid:2)71,71), N¼16,deterministicaandb,MLestimator. deterministicaandb,MLestimator. ARTICLE IN PRESS M.Grecoetal./SignalProcessing89(2009)355–364 361 theinterferingradarisnullandtheRMSEðy^ Þfor|C/A|¼1 Thismeansthat,theoretically,iftheradarknowsthatan T islowerthanthatfor|C/A|¼0. interference is present, it is possible to find an efficient Similar remarks can be drawn from Figs. 5 and 6, unbiasedestimatorthatisnotsensitivetothecodeused where we report RMSEðy^ Þ and bias for deterministic by both reference and interfering radars. Unfortunately, T unknowna anda.Forsmall|C/A|,theRMSEðy^ ÞofthePM this never happens. In the existing systems, the radar T I T estimator is lower when the target and interference simply applies some techniques for the estimation of amplitudesaredeterministicthanwhentheyarecomplex target DOA evenwhen there is an interference, then the Gaussian random variables. Conversely, the RMSEðy^ Þ presence of an interfering radar biases the estimates T is more sensitive to high values of |C/A|¼0 when increasing the root mean square error more than one the amplitudes are deterministic than when they are order of magnitude with respect to the CRLB, as shown random. in Fig. 15. In Fig. 15, we report the square root of InFigs.7and8,RMSEðy^ Þisshownasafunctionofthe the Crame´r-Rao lower bound (RCRLB) of y in degrees, T T ratio |C/A| for different value of the SIR, in the case of RCRLBðy Þ¼ð180(cid:8)=pÞpffiCffiffiRffiffiffiLffiffiffiBffiffiffiðffiyffiffiffiffiffiÞffiffi, compared to the T T random and deterministic amplitudes, respectively. The RMSE(y ) of ML and PM estimators for |C/A|¼0 and |C/ T interference DOA y has been generated as a random A|¼1,asafunctionoftheinterferenceDOA.Thesolidline I variable uniformly distributed on the range [(cid:2)71,(cid:2)71].8 with white squares (labelled RCRLB) represents the The impact of the interference is apparent in both RCRLB(y )calculatedwhenallthetargetandinterference T figures. parameters are deterministic and unknown. The dotted Figs. 9–14 report the companion results for the ML estimator. The behavior is very similar. In the case of deterministic amplitudes the ML estimator outperforms thePMestimator;thecontraryintherandomamplitude 10 scenario.Itisworthremindingthatinthecaseofrandom RCRLBwi ML@|C/A|=1 amplitude,whatinthispaperistermed‘‘MLestimator’’is RCRLB PM@|C/A|=0 notthetrueMLestimator(fordetailssee[7]). ML@|C/A|=0 PM@|C/A|=1 s) 1 4. Crame´r-Raolowerboundsand ee gr performancecomparison e d E ( S TocompleteouranalysiswederivedalsotheCRLBfor M thetwocasesathand.Inthissectionwereportresultsand R 0.1 comments.Detailedderivationoftheboundsispresented inAppendicesAandB. When the observed vector y is complex Gaussian distributedwithmeanvaluelyandcovariancematrixCy, 0.01 in short y(cid:6)CN(l,C ), with unknown real parameters -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 y y v¼[w1,w2,ywn]T, the elements of the Fisher matrix are θI (degrees) givenby[8] ( ) Fig.15. RMSEofMLandPMestimatorsandRCRLBvs.interferenceDOA, ½JðvÞ(cid:7) ¼Tr C(cid:2)1ðwÞqCyðvÞC(cid:2)1ðvÞqCyðwÞ yT¼1.51,deterministicaandb. i;j y qw y qw i j þ2<e(qlHyðvÞC(cid:2)1ðvÞqlyðvÞ), (12) 10 qwi y qwj RCRLBwi ML@|C/A|=1 RCRLB@|C/A|=1 PM@|C/A|=0 whereTr{(cid:4)}isthetraceofamatrix,<e{(cid:4)}standsforthe ML@|C/A|=0 PM@|C/A|=1 realpartandi,j¼1,2,y,n. s) 1 e e 4.1. Deterministicamplitudes gr e d In this case Cy¼sd2I and ly¼aA(cid:4)g(yT)+bC(cid:4)h(yI). SE ( Decomposing the complex amplitudes a and b in their RM 0.1 modulusanphase,thatisa¼aejjandb¼bejc,thevector oftherealparameterstoestimateisv¼[y ajy bc]T. T I The 36 elements of the Fisher matrix are reported in AppendixA. As verified in Eq. (A.28), the CRLB of the target 0.01 parametersy,a,andjdonotdependontheratio|C/A|. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 T θ (degrees) I 8Thesearetheangularpositionsofthenullsoffirstrightandleft Fig.16. RMSEofMLandPMestimatorsandRCRLBvs.interferenceDOA, sidelobesofthesumchannel. yT¼1.51,randomaandb. ARTICLE IN PRESS 362 M.Grecoetal./SignalProcessing89(2009)355–364 line with white circles (labelled RCRLBwi) reports AppendixA. CRLBderivationfor the RCRLB(y ) calculated when the interference is not deterministicamplitudes T present (Eq. (A.29)). It should not seem to be an absurdthatforsomevaluesofyI,theRMSE(yT)issmaller First,letusdefinesomeusefulvectorsandmatrices than the RCRLB(y ), because both ML and PM estimators arebiased. T g_ðyÞ¼qgðyÞ; h_ðyÞ¼qhðyÞ (A.1) qy qy and 4.2. Randomamplitudes GðyÞ¼gðyÞgHðyÞ¼gðyÞgTðyÞ; HðyÞ¼hðyÞhHðyÞ¼hðyÞhTðyÞ, The comparisons in the case of complex Gaussian (A.2) distributed amplitudes are summarized in Fig.16. Again, the solid line with white squares (labelled RCRLB@|C/ qGðyÞ qHðyÞ A|¼1) represents the RCRLB(y ) calculated when both G_ðyÞ¼ ; H_ðyÞ¼ , (A.3) T qy qy targetandinterferenceDOAsareunknownand|C/A|¼1. where the n,mth element of the matrix is ½G_ðyÞ(cid:7) ¼ The dotted line with white circles (labelled RCRLBwi) n;m ½g_ðyÞ(cid:7) ½gðyÞ(cid:7) þ½gðyÞ(cid:7) ½g_ðyÞ(cid:7) , and ½H_ðyÞ(cid:7) ¼½h_ðyÞ(cid:7) ½hðyÞ(cid:7) reportstheRCRLB(y )calculatedwhentheinterferenceis n m n m n;m n m T þ½hðyÞ(cid:7) ½h_ðyÞ(cid:7) . notpresent(Eq.(B.3)).Thetwocurvesareclose,meaning n m In the case of deterministic amplitudes Eq. (12) that also in this case, knowing the presence of the reducesto interference,itcouldbepossibletofindanestimatorthat is not too sensitive to the values of |C/A|. Unfortunately, 2 (qlHðvÞql ðvÞ) alsointhecaseofrandomamplitude,theperformanceof ½JðvÞ(cid:7) ¼ <e y y ; i;j¼1;2;...6. (A.4) i;j s2 qw qw realistic estimators as the PM and ML are far from the d i j CRLBandtheimpactofthecodesisheavy. Knowing that l ¼aejjA(cid:4)g(y )+bejcC(cid:4)h(y), we can y T I calculate 5. Conclusions 2aejjAg_ðy Þ3 T 6 ejjAgðy Þ 7 orafdcTaohrdeensaeitimswooofrfpktahfroiasrmmpaoepudenrtbiyismttpowohoritgrahandlciagerhest.vteThnhaetintihmaevppearrcoytpsoeimfrutphsleee qlqyvðvÞ¼666666jbaeejjcjCAhg_ððyyTITÞÞ777777. (A.5) presenceofaradarontheotherhasbeenmeasuredinthe 664 ejcChðyIÞ 775 estimationofthetargetDOAperformedbythereference jbejcChðyÞ I radar. We considered only two multi-pulses algorithms, the ML and the PM, among all the DOA estimation Setting now g¼|C/A|, A¼jAjejWA and C¼jCjejWC ¼ techniques(seeforinstance[9–12],andreferencesthere- gjAjejWC,weobtainfortheFishermatrixelements in)notwiththepurposeoffindingthebestalgorithmin 2 the presence of an interference, but simply willing to J11¼s2a2jAj2jjg_ðyTÞjj2, (A.6) show that two very different techniques suffer the same d performanceimpoverishment.Weverifiedthattheinter- 2 ferencemainlyintroducesabiasintheDOAestimatethat J12¼J21¼s2ajAj2g_TðyTÞgðyTÞ, (A.7) depends on the position of the interferencewith respect d totheboresightoftheantennaandontheratio|C/A|,that J ¼J ¼0, (A.8) is, on the choice of the codes. For limiting this negative 13 31 effect,thecodeshouldbechosensuchthat|C/A|o1,thatis the codes used by the two radars should be almost J ¼J ¼ 2 abgjAj2 cosðW (cid:2)W þc(cid:2)jÞg_Tðy Þh_ðyÞ, 14 41 s2 C A T I orthogonal.Thisisnotalwaysaneasytask.Generallythe d code is designed for optimizing the frequencyand range (A.9) resolutionsoftheradar,thatis,foroptimallyshapingthe AAF. But often the best-shaped auto-ambiguity corre- J ¼J ¼ 2 agjAj2 cosðW (cid:2)W þc(cid:2)jÞg_Tðy ÞhðyÞ, sponds to a very poor CAF, that is, in our problem, high 15 51 s2 C A T I d valueof|C/A|.SomeasymptoticboundsonCAFofdifferent (A.10) classesoffrequencyhop-codedsignalshavebeenreported in[3,13–15].ForWelch–CostascodeoflengthNthebound 2 is N/N then |C/A| can easily get the unit value. Forcodes J16¼J61¼(cid:2)s2abgjAj2 sinðWC(cid:2)WAþc(cid:2)jÞg_TðyTÞhðyIÞ, baseduponlinearcongruencestheboundis2/N,thenthe d (A.11) maximum value of |C/A| is 0.5. Good results can be obtained with a set of the codes based upon extended 2 quadratic congruences for which the asymptotic bound J ¼ jAj2jjgðy Þjj2, (A.12) 22 s2 T seems to be 12/N. The adaptive use of different trans- d mitted waveforms to alleviate the impact of an inter- ferenceisnowunderanalysis[16]. J ¼J ¼0, (A.13) 23 32 ARTICLE IN PRESS M.Grecoetal./SignalProcessing89(2009)355–364 363 J ¼J ¼ 2 bgjAj2 cosðW (cid:2)W þc(cid:2)jÞgTðy Þh_ðyÞ, somealgebraweget 24 42 s2 C A T I d (A.14) CRLBðyTÞ¼½J(cid:2)1(cid:7)1;1 ¼ 1 jjgðyTÞjj2 . J ¼J ¼ 2 gjAj2 cosðW (cid:2)W þc(cid:2)jÞgTðy ÞhðyÞ, 2SNRjjg_ðyTÞjj2jjgðyTÞjj2(cid:2)g_TðyTÞgðyTÞgTðyTÞg_ðyTÞ 25 52 s2d C A T I (A.29) (A.15) 2 J ¼J ¼(cid:2) bgjAj2 sinðW (cid:2)W þc(cid:2)jÞgTðy ÞhðyÞ, 26 62 s2d C A T I AppendixB. CRLBderivationforcomplexGaussian (A.16) distributedamplitudes J33¼s22a2jAj2jjgðyTÞjj2, (A.17) theInotbhsiesrvcaesde,vseucptpoorsiinsgsatAillCNG(0a,ussas2i)anandwbitAhCNa(0m,seba2n), d value l ¼E{y}¼0 and a covariance matrix C ¼ y y J34¼J43¼s22dabgjAj2 sinðWC(cid:2)WAþc(cid:2)jÞgTðyTÞh_ðyIÞ, sveac2|tAo|r2Gis(ywT¼)+s½yb2T|C|y2IH(cid:7)T(yaI)n+dsd2(1I.2)Trheeduucneskntoown parameter (A.18) ( ) qC ðvÞ qC ðvÞ J35¼J53¼s22agjAj2 sinðWC(cid:2)WAþc(cid:2)jÞgTðyTÞhðyIÞ, ½JðvÞ(cid:7)i;j¼Tr C(cid:2)y1ðvÞ qywi C(cid:2)y1ðvÞ qywj ; i;j¼1;2. d (B.1) (A.19) Itiseasytoverifythat 2 J ¼J ¼ abgjAj2 cosðW (cid:2)W þc(cid:2)jÞgTðy ÞhðyÞ, 36 63 s2 C A T I qC qC d y¼ y¼s2jAj2G_ðy Þ (A.20) qw1 qyT a T and J44¼s22b2g2jAj2jjh_ðyIÞjj2, (A.21) qCy¼qCy¼s2jCj2H_ðyÞ. (B.2) d qw qy b I 2 I J ¼J ¼ 2 bg2jAj2h_TðyÞhðyÞ, (A.22) Replacing(B.2)in(B.1)andinvertingtheFishermatrix 45 54 s2 I I d wecanobtaintheCRLB. If the interference is not present, the Crame´r-Rao J ¼J ¼0, (A.23) 46 64 boundofy,canbeeasilycalculatedinverting[J] .After T 1,1 somealgebra 2 J ¼ g2jAj2jjhðyÞjj2, (A.24) 55 s2 I CRLBðy Þ d T 1 1 J56¼J65¼0, (A.25) ¼SNRTrf½ðI(cid:2)ðSNR(cid:4)GðyTÞ=1þSNR(cid:4)gTðyTÞgðyTÞÞÞ(cid:4)G_ðWTÞ(cid:7)2g. 2 (B.3) J ¼ b2g2jAj2jjhðWÞjj2. (A.26) 66 s2 I d Based on Eqs. (A.6)–(A.26), we can observe that the References Fishermatrixcanbewrittenas " A gB # [1] S. Suvorova, D. Musicki, B. Moran, S. Howard, B. La Scala, J¼ , (A.27) gBT g2C Multi step ahead beam and waveform scheduling for tracking of manoeuvring targets in clutter, ICASSP05, Philadelphia, USA, where A, B and C are three-dimensional sub-matrices. March2005. [2] S.W. Golomb, H. Taylor, Constructions and properties for Costas Then,theinverseofJcanbeexpressedas arrays,Proc.IEEE72(9)(September1984)1143–1163. "D E# 2 ðA(cid:2)BC(cid:2)1BTÞ(cid:2)1 (cid:2)1gðA(cid:2)BC(cid:2)1BTÞ(cid:2)1BC(cid:2)13 [3] Eu.pLo.nTitthleebtahuemor,yToifmlien–efarerqcuoenngcruyehnoceps,sIiEgEnEalTsraPnasrt.AI:ercoosdpi.nEglebctarsoend. J(cid:2)1¼ F G ¼64(cid:2)1ðC(cid:2)BTA(cid:2)1BÞ(cid:2)1BA(cid:2)1 1ðC(cid:2)BTA(cid:2)1BÞ(cid:2)1 75. Syst.17(4)(July1981)490–493. g g2 [4] J.R. Bellegarda, E.L. Titlebaum, Time–frequency hop codes based uponextendedquadraticcongruences,IEEETrans.Aerosp.Electron. (A.28) Syst.24(6)(November1988)726–742. It is worth noting that the sub-matrix D does not [5] N. Levanon, E. Mozeson, Radar Signals, IEEE Press, Wiley-Inter- science,NewJersey,USA,2004. dependong,thentheCRLBofyT,a,andjdonotdepend [6] M. Greco, F. Gini, A. Farina, Joint use of P and D channels for on g. Conversely, the bounds of the interference para- multipleradartargetDOAestimation,IEEETrans.Aerosp.Electron. metersareinverselyproportionaltog2. Syst.,inpress. [7] F. Gini, M. Greco,A. Farina, Multipleradartargetsestimation by Iftheinterferenceisnotpresent,theCRLBofy,a,and T exploiting induced amplitude modulation, IEEE Trans. Aerosp. j can be calculated simply inverting the matrix A. After Electron.Syst.39(4)(October2003)1316–1321.