RAPIDLY ADAPTIVE CFAR DETECTION BY MERGING INDIVIDUAL DECISIONS FROM TWO-STAGE ADAPTIVE DETECTORS

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1 RAPIDLY ADAPIVE CFAR DEECION BY MERGING INDIVIDUAL DECISIONS FROM WO-SAGE ADAPIVE DEECORS Analii A. Knnv, Sung-yun Chi and Jin-a Kim Research Cener, SX Engine Yngin-si, 694 Krea Absrac his paper addresses he prblem f arge deecin in adapive arrays in siuains where nly a small number f raining samples is available. Wihin he framewrk f w-sage adapive deecin paradigm, we prpse a new wsage (S) jin laded persymmeric-epliz adapive mached filer (JLP-AMF) deecr. his new deecr cmbines, using a jin deecin sraegy, individual scalar CFAR decisins frm w rapidly adapive deecrs: a S AMF deecr and a S LPAMF deecr. he frmer is based n a MI filer, which is an adapive array filer emplying a epliz cvariance marix esimar inrduced in []. he laer is based n an adapive LPMI filer ha uses diagnally laded persymmeric cvariance marix (CM) esimae inversin. he S JLP-AMF deecr ensures he cnsan false alarm rae (CFAR) prpery independenly f he anenna array dimensin M, he inerference CM, and he number f raining samples NCME be used fr esimaing his CM. his new deecr exhibis highly reliable deecin perfrmance, which is rbus he angular separain beween he surces, even when NCME is abu m/ ~ m, m is he number f inerference surces. he rbusness f he prpsed adapive deecr he angular separain is analyically prven and verified wih saisical simulain. Keywrds adapive array; CFAR deecin; persymmery; radar; epliz marix; w-sage deecin I. INRODUCION he adapive MI filers based n he epliz cvariance marix (CM) esimar prpsed in [] exhibi a superfas cnvergence rae in erms f he 3dB average signal--nise rai lss fr he lw-rank (LR) cvariance scenaris when he exac CM R f he inerference has a cliff-like eigenspecrum λ λ... λm >> λ m+ =... = λm = σ, () where λ k, k =,,..., M are he eigenvalues f R, M is he array dimensin; m is he number f exernal pwerful inerference surces (m M) and σ = p sands fr he addiive whie Gaussian nise pwer. MI filers require m/ raining samples achieve he 3dB average signal--nise rai (SNR) lss, while he diagnally laded persymmeric CM esimae inversin (LPMI) filers and he cnveninal LSMI filers require m and m samples, respecively. MI filers can achieve such excepinally rapid cnvergence rae if he angular separain beween surces des n clsely apprach he saisical resluin limi (SRL) defined in []. As demnsraed in [], he cnvergence perfrmance f he MUSIC-based MI filers may degrade due a pssible unaccepable drp in upu SNR when he angular separain beween surces is near he SRL. hugh using her direcin f arrival (DOA) esimars, such as R-MUSIC [3] r ESPRI [4], can make MI filers less sensiive he small angular separain beween surces, his apprach des n prvide a reliable remedy because perfrmance breakdwn resides in all knwn DOA esimars. Als, ne ha deecrs emplying he adapive MI filers inrduced in [] are n sricly CFAR deecrs; lss f he CFAR prpery is he price paid fr achieving he superfas cnvergence rae. Basic adapive deecrs (BADs), such as he generalized likelihd rai es (GRL), he adapive mached filer (AMF), and he adapive cherence esimar (ACE), pssess w invariance prperies relaive he unknwn CM f inerference. hese are he invariance f he deecr s upu saisics resuling in he CFAR prpery f he deecr, and he invariance f he deecin perfrmance lss cmpared wih he clairvyan deecr. BADs exhibi hese impran prperies nly if he number f raining samples significanly exceeds he array dimensin [5, 6]. Mrever, BADs pssess he sric CFAR prpery primarily because hey use a generic maximum-likelihd (ML) sample CM (SCM) esimar; his esimar ignres any a priri infrmain n he exac CM R. Ignring his infrmain resuls in he need fr a cnsiderable number f raining samples ensure he CFAR prpery and reliable deecin perfrmance. able I is based n he resuls repred in []. his able summarizes he rai qrad f he al number f real variables (all real and imaginary pars) cnained in he cmplex-valued enries f raining daa samples, he al number f free real variables cnained in he cmplex-valued enries f all he signal subspace (dminan) eigenvecrs f he exac CM R depending n a priri infrmain abu R incrpraed in he CM esimar. he free real variables are all he real and imaginary pars, wih n cnsrains impsed n hem, f all cmplex-valued enries in he dminan eigenvecrs. In able I, he quaniy QDS is represened as QDS = N3dB QS, where N3dB is he minimum number f raining samples required achieve he 3dB average SNR lss relaive he Wiener filer, and QS is he number f real and imaginary pars in ne raining sample (vecr). he quaniy

2 ABLE I. NUMBER OF REAL RAINING VARIABLES PER ONE FREE REAL VARIABLE DEPENDING ON SRUCURE OF RUE CM Srucure f rue CM al number f real variables in raining daa se, QDS al number f free real variables, QFRV General M M M M Lw-rank (LR) m M m M LR+Persymmery m M m M LR+epliz m/ M m M/ QFRV is represened as QFRV = NDEV QEV, where NDEV is he number f he dminan eigenvecrs f R and QEV is he number f free real variables in each dminan eigenvecr. Analysis f he qrad = QDS/QFRV rai in able I leads he cnclusin ha he fllwing cndiin f reliable adapive deecin is valid (a leas fr Gaussian inerference); qrad mus mee he cndiin qrad> ensure reliable adapive deecin; in her wrds, he rai f he al number f real variables in a raining daa se he al number f free real variables in he dminan eigenvecrs f he exac CM has exceed. his fundamenal cndiin has been verified fr general and fr lw-rank srucures f he rue CM; rigrus prfs, in erms f he prbabiliy densiy funcins fr he SNR a he upu f adapive filers, are given in [7] and [8, 9], respecively. he w remaining cases in able I have n ye been rigrusly prven; hwever, hese cases are cnfirmed by numerus Mne-Carl simulains. In many cases, BADs very aracive feaures discussed abve cann be used; hwever, hey can be achieved wihin he framewrk f a w-sage deecin apprach [5, 6]. his apprach appears be he nly feasible adapive deecin pin when he CFAR prpery cann be realized even hereically, e.g., in nn-hmgeneus raining cndiins ha ccur due he saisical discrepancy beween he raining daa and he primary daa, see fr deails [5, 6]. A similar wsage apprach may als be cnsidered fr hmgeneus raining cndiins. In he case f hmgeneus raining cndiins, a se f N independen idenically disribued (IID) raining samples allcaed fr any single primary range cell is divided in w subses f he NCME and he NCFAR samples. he frmer is used in esimaing he inerference CM design an adapive filer, while he laer is used in esimaing he scalar CFAR hreshld fr arge deecin. Fr nn-hmgeneus raining cndiins, hwever, NCFAR is he number f primary cells used fr adapive scalar CFAR deecin; NCME and NCFAR cann be raded-ff agains each her since bh crrespnd daa ses cnaining differen inerferences [5, 6]. his paper cninues ur previus wrk []. In his paper, we prpse a new adapive deecr in he framewrk f a wsage adapive deecin srucure [5, 6]. his srucure prvides a very simple sluin fr achieving he CFAR prpery in adapive deecrs emplying he MI filers suggesed in []. he prpsed deecr cmbines scalar CFAR decisins frm individual AMF deecrs emplying he LPMI and MI filers eliminae deecin perfrmance degradain resuling frm pssible SNR drp in MI filers. his new deecr is referred as he w-sage jin laded persymmeric-epliz adapive mached filer (S JLP-AMF) deecr. qrad II. PROBLEM FORMULAION In w-sage adapive deecrs, he required number f samples fr adapive CFAR cnrl NCFAR is independen f he anenna dimensin M. Fr a sufficienly large array dimensin M, herefre, any adapive array filer requiring significanly smaller raining sample suppr achieve mre efficien inerference suppressin han ha required fr he generic ML SCM esimar (N3dB M), will lead a mre efficien wsage adapive deecr han any f he basic CFAR adapive deecrs [5, 6]. hus, he superfas MI filer inrduced in [] is he filer f chice fr use in w-sage adapive deecrs. wever, he MI filers achieve a very high cnvergence rae (N 3dB m/) nly if he angular separain beween he inerference surces is n clse he SRL []. I shuld be ned ha w-sage adapive deecin guaranees nly sric CFAR cnrl; i des n eliminae deecin perfrmance degradain due a pssible upu SNR drp in MI filers in scenaris wih clsely spaced surces. hus, his paper s primary ask is discver such a wsage adapive deecr srucure based n MI filers ha can ensure rbus deecin perfrmance independenly n angular separains beween inerferers. analyze he S JLP-AMF deecr perfrmance, we cnsider a scenari wih a Swerling I arge embedded in he inerference wih CM R ha mees he cndiin (). ere, he primary sample y is mdeled as x ~ ( M, 0R, ) fr hyphesis y =, () x + a s, a ~ (0, p) fr hyphesis M where x is he bserved inerference plus receiver nise-nly daa vecr, s = s( θ ) is he nrmalized ( s s = ) array-signal seering vecr fr a arge frm a given direcin θ (θ denes DOA), and a represens he arge cmplex ampliude flucuains which pwer is p. III. ADAPIVE DEECORS UNDER SUDY Fllwing [5, 6], we specify he w-sage laded AMF (S LAMF) deecr as w y LSMI h (3) LAMF w Rˆ w LSMI CFAR LSMI where h is he scalar CFAR hreshld cnsan and w = Rˆ ( ) s, Rˆ ( ) = Rˆ + p I (4) β LSMI CME β βˆ CME CME NCME ˆ N = N, CME CME CFAR CFAR k k k = N = k= NCME + k k R ˆ x x R x x (5) wih xi ~ ( M, 0, R), i =,,, N being he IID samples having an M-variae zer-mean cmplex circular Gaussian disribuin wih cmmn cvariance R; he superscrip denes he ermiian ranspsiin. In his paper, we als inrduce he w-sage laded persymmeric AMF (S LPAMF) deecr as v y LPMI LPAMF v Rˆ v LPMI CFAR LPMI =, = ˆ ( β ) LPMI P LPMI LPMI RPCME h (6) where v U w w R s P, sp = U s P, wih U being he uniary marix given by P

3 = I J U (7) P ji jj where I and J respecively, are he M/-by-M/ ideniy and exchange marix (we cnsider nly he even M case), and he marix R ˆ () β is given by [] RPCME ˆ ( β) ˆ β ˆ R = R + p I, Rˆ = Re( U Rˆ U ) ; (8) RPCME RPCME RPCME P CME P mrever, he w-sage epliz AMF (S AMF) deecr as v y MI v Rˆ v MI CFAR MI AMF h (9) - where v = U w, w = Rˆ s MI MI MI RCME, s = U s, wih U being he uniary marix given by [ I jj] U = (0) where I and J respecively, represen he ideniy and exchange marix f dimensin M, and he marix ˆR is cmpued as RCME ˆ ˆ R = Re( U R U ), () RCME where Rˆ is he epliz CM esimae cmpued wih CME NCME raining samples using a nnieraive mdel-mached epliz CM esimar frm []. In frmulas (4) and (8) abve, β is he lading facr and ˆp is he nise pwer esimae. he new S JLP-AMF deecr is defined as an adapive deecr ha cmbines he scalar CFAR decisins assciaed wih he S LPAMF deecr (6) and wih he S AMF deecr (9) using he fllwing rule if η hi r η hi hen is rue, herwise is rue () LPAMF AMF where hi is he individual CFAR hreshld assciaed wih a given individual prbabiliy f false alarm PFAi. I fllws frm () ha he decisin sraegy adped in he S JLP-AMF deecr is declare ha he hyphesis is rue (he arge is presen) whenever a leas ne f he individual w-sage deecrs decides ha be rue. herem. he deecin prbabiliy f deecr () exceeds he maximum f he individual deecin prbabiliies. Prf. he prf is mied. his herem abve esablishes he self-adjusmen prpery f he S JLP-AMF deecr; fr he deecr (), he verall deecin prbabiliy PD is aumaically mainained a a level such ha PD value always exceeds he maximum f he individual deecin prbabiliies. Fr his reasn, if he angular separain beween surces is far enugh frm he SRL, he verall deecin perfrmance f deecr () is mainained a sme level exceeding he individual deecin prbabiliy f he S AMF deecr. If he S AMF deecr perfrmance shuld degrade due he presence f clsely spaced surces, he deecr () will aumaically mainain he PD value a sme level exceeding he individual deecin prbabiliy f he S LPAMF deecr ha is n sensiive he angular separain Δθ beween he surces because i uses he LPMI filer, which is n sensiive Δθ []. Deecr () rbusness he angular separain beween surces is ensured hereby. Ne ha deecr () als increases he verall prbabiliy f false alarm PFA, i.e., PFA > PFAi. he upper bund fr PFA CME fllws frm he inequaliy PFA PFAi ha can easily be prven assuming he hyphesis (n arge) is rue. IV. PERFORMANCE ANALYSIS his secin analyzes receiver peraing characerisics (ROCs) f he w-sage deecrs presened in Secin III. Fr his analysis, hmgenus raining cndiins are assumed when he full se f N = M secndary IID raining samples allcaed fr any single primary cell is divided in w subses f size NCME and NCFAR, NCME+NCFAR = N. In his analysis, we use he R MUSIC-based MI filer [] in bh f he S AMF deecr and he S JLP-AMF deecr. his MI filer is designed using he epliz CM esimar inrduced in [] wih all he parameer seings specified herein. In cmparaive perfrmance analysis, we als use he fllwing benchmark deecrs. Benchmark deecr (BD), BD is an pimal r clairvyan deecr achieving ulimae deecin perfrmance. his deecr cmprises he pimal Wiener filer - - w = R s /( s R s ) fllwed by he decisin rule p - y R s - BD s R s h. (3) he well-knwn expressin describes he ROC curve f he deecr (3) PD = exp[ ln P /( + q )], (4) FA where he upu SNR f he pimal Wiener filer is q = p sr s. (5) he ulimae deecin perfrmance (4) crrespnds NCME and NCFAR. Benchmark deecr (BD), BD, a w-sage deecr, cmprises pimal Wiener filer and scalar cell averaging (CA) CFAR deecr - y R s h, (6) - - BD sr ˆ RRs N where Rˆ = N x k = kxk is he sample cvariance marix. Cnsequenly, he deecr (6) uses all N raining samples fr adapive CFAR deecin (NCFAR = N). he ROC f he deecr (6) is given by N P = + h /( + q ) D, (7) / N where he mdified CFAR cnsan h = h/ N = P. FA Equain (7) crrespnds NCME and NCFAR = N. In all examples herein, a unifrm linear array f M = sensrs, a amming weighed seering vecr s uned θ = 0 (prviding a quiescen array paern wih 39 db sidelbes), and he lading facr β = fr bh he LSMI and he LPMI filers are used. Fig. cmpares he ROC f he S JLP-AMF deecr wih ha f he S AMF deecr, and wih ha f he S LPAMF and he S LAMF deecrs in Scenari B specified in [] as {m = 4, u =[ 0.8, 0.4, 0., 0.5], INR(dB) = [0, 0, 0, 0]}, u = sin(θ), θ = [θ, θ, θ3, θ4]. he curves labeled BD and BD represen he ROCs fr he crrespnding benchmark deecrs in he same scenari. One f he ROC

4 Fig.. ROCs fr adapive deecrs under sudy; Scenari B, m is n knwn Fig.. ROCs fr adapive deecrs under sudy; Scenari D, m is n knwn Fig. 3. ROCs fr adapive deecrs under sudy; Scenari D, m is n knwn Fig. 5. Esimaed prbabiliy f deecin PD versus Δθ/SRL; Scenari D, m is knwn Fig. 4. Esimaed prbabiliy f deecin PD versus Δθ/SRL; Scenari D, m is n knwn curves fr he BD is calculaed fr PFA = 0-4 frm (7), and anher is esimaed using he Imprance Sampling (IS) echnique ha emplys a s-called g-mehd [0]. We develped w mdified versins f he g-mehd: fr esimaing he prbabiliy f false alarm in case f arbirary exac inerference cvariance marix, and fr esimaing he deecin perfrmance fr he arge mdel specified in (). he perfec mach beween hese w ROC curves (beween he analyic and esimaed resuls) validaes he high accuracy f he mdified versins f he g-mehd: fr N is = 8,000 independen saisical rials, he average relaive sandard deviain errr is abu 5% in esimaing PFA and abu % in esimaing PD (PD 0.3). Similarly, he ROC curves fr he S LAMF, he S LPAMF, and he S AMF deecrs are cmpued using he mdified versins f he g-mehd, wih N is = 8,000. In Fig., w ROC pls, fr PFA=0-4 and PFAi= , are shwn fr each f he S LPAMF and S AMF

5 deecrs. Unfrunaely, he g-mehd cann be used fr he S JLP-AMF deecr; insead, we use he cnveninal Mne-Carl (MC) echnique wih N is = 0 7 verify PFA and wih N is = 50,000 calculae he ROC curves. In scenari B, he surces are sufficienly separaed in he angular crdinae. By he sufficienly separaed inerference surces we undersand he surces wih such he smalles angular separain beween hem ha is n clse he SRL. In his paper, we define he SRL as 0.5 FRL SRL = (8) /4 ( SNRarr ) where FRL = π/m is he sandard Furier resluin limi and SNR arr = 4 M SNR el is he array SNR wih SNR el being he elemen-level SNR. Fig. shws ha he S JLP-AMF deecr exhibis superir deecin perfrmance in a scenari wih sufficien angular separains beween surces: fr PD = 0.5, he SNR gain relaive he S LPAMF and he LAMF deecrs is.3 db and 5.4 db, respecively, while he SNR lss relaive he unrealizable BD is jus 0.5 db. Fig. pls he ROCs fr all he deecrs being sudied, and fr he BD and BD deecrs in Scenari D = {m = 3, u = [ 0.4, 0.0, sin(srl)], INR (db) = [0, 0, 0]} when he smalles angular separain beween surces is equal he saisical resluin limi SRL =.80 ( rad) calculaed frm (8) fr SNR el = 0 db. Fig. cnfirms he self-adjusmen prpery f he S JLP-AMF deecr. Indeed, he ROC curve fr his deecr ges abve ha fr he S AMF deecr; he laer represens he maximum f he w individual deecin prbabiliies assciaed, respecively, wih he S LPAMF and he S AMF deecrs a he specified individual prbabiliy f false alarm P FAi. Nex, we cnsider an unfavrable scenari in which he cnvergence perfrmance f he MI filer may degrade []. Fig. 3 pls he ROC curves and Fig. 4 pls he esimaed PD versus he relaive angular separain Δθ/SRL in Scenari D {m = 3, u = [ 0.4, 0.0, sin(srl)], INR (db) = [8, 0, 30]} when bh p and m are n knwn. In his scenari, he saisical resluin limi SRL =.034 ( rad) fr SNR el = max(inr, INR 3) = 30dB. Frm Fig. 3, hugh he S AMF deecr suffers essenial perfrmance degradain, fr he S JLP-AMF deecr, he ROC curve ges abve ha f he S LPAMF deecr a P FAi = and slighly abve ha a P FA = 0-6. Frm Fig. 4, he S AMF deecr cllapses fr 0.8SRL Δθ.7SRL, while fr he S JLP- AMF deecr, he PD curve is abve ha f he S LPAMF deecr independenly f he angular separain Δθ. Figs. 3 and 4 cnfirm ha he S JLP-AMF deecr is rbus. In he same Scenari D, Fig. 5 pls he esimaed PD as a funcin f he relaive angular separain Δθ/SRL under he assumpin ha m is knwn. Fig. 5 demnsraes ha when he number f inerference surces is knwn hen he S JLP- AMF deecr and he S AMF deecr (bh use he R MUSIC-based MI filer) are fundamenally rbus he angular separain. V. CONCLUSION In his paper, we have prpsed new rapidly adapive wsage JLP-AMF deecrs ha ake advanage f superfas adapive MI filers based n a nnieraive mdel-mached epliz cvariance marix esimar inrduced in []. hese deecrs pssess w remarkable feaures. Firs, hey exhibi reliable deecin perfrmances, which are rbus he angular separain beween he inerference surces, even when he raining sample size be used fr esimaing he inerference cvariance marix is abu m/ ~ m (m is he number f surces). his rbusness has been analyically prven and verified wih saisical simulains. Secnd, he w-sage JLP-AMF deecrs ensure he CFAR prpery independenly f he anenna array dimensin, he inerference cvariance marix, and he number f raining samples be used fr esimaing his marix. Fr he firs ime, his paper has demnsraed ha he necessary fundamenal cndiin f reliable adapive deecin is deermined by he rai f he al number f real variables in a raining daa se he al number f free real variables in he dminan eigenvecrs f he exac cvariance marix. his rai mus exceed ensure reliable adapive deecin. Finally, i is newrhy ha he denminar in (3), (6) and (9) can be rewrien as a sum f scalar samples, i.e., in he frm ha is specific fr he cnveninal CA CFAR es [, p. 5]. aving bained scalar represenains fr he primary vecrsamples assigned fr CFAR hreshlding, new scalar CFAR appraches [, 3] can be used in severely nnhmgeneus envirnmens prvide significan imprvemens in bh, false alarm regulain and deecin perfrmance. REFERENCES [] A.A. Knnv, C.. Chi and D.. Kim, Superfas cnvergence rae in adapive arrays, in Prc. f Inern. Cnf. n Radar 08, Brisbane, Ausralia, 7 30 Augus 08. [] S.. Smih, Saisical resluin limis and he cmplexified Cramér- Ra bund, IEEE rans. SP, vl. 53, n. 5, p. 60, May 005. [3] A.J. Barabell, Imprving he resluin perfrmance f eigensrucurebased direcin-finding algrihms, in Prc. ICASSP, Bsn, MA, pp , 983. [4] R. Ry and. Kailah, ESPRI-Esimain f signal parameers via rainal invariance echniques, IEEE rans. ASSP, vl. 37, n. 7, pp , July 989. [5] Y.I. Abramvich, B.A. Jhnsn, and N.K. 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