Performance analysis of some CFAR detectors in homogeneous Pearson-distributed clutter

Transcription

1 SETIT 5 3 rd Intrnational Confrnc: Scincs of Elctronic, Tchnologis of Information and Tlcommunications arch 7-31, 5 TNISIA Prformanc analysis of som CFAR dtctors in homognous Parson-distributd cluttr iani Hilal Abdnour *, Soltani Faoui ** * nivrsité d Skikda, Déartmnt d ' élctroniqu, BP 6, Rout d EL Hadaik, Skikda, Algéri hamiani@yahoo.fr ** nivrsité d Constantin, Déartmnt d ' élctroniqu, Rout d Ain by, Constantin, Algéri f.soltani@caramail.com Abstract: In this ar, w analysis th rformanc of th -CFAR and -CFAR dtctors in th rsnc of homognous and non-gaussian cluttr. W assum that th cluttr lvl is modld as a Parson-distribution. Th targt in th tst cll is assumd Swrling II modl. W driv a closd form xrssion for th robability of fals alarm in homognous Parson distributd cluttr, and w show that th fals alarm robability dnd only on th thrshold multilir T and th numbr of rfrnc cll N. Ky words: CFAR, homognous cluttr, Parson distribution, Radar. 1. Introduction In radar systms, a constant fals alarm rat (CFAR) dtction is usd to dcid about th rsnc of a targt. Th uros of CFAR dsign is th maximisation of th robability of dtction whil maintaining a dsird fals alarm rat. To rlas ths constraints, CFAR dtction rocdurs involv th comarison of th rcivd signal with an adativ thrshold. Th rcivd signal is samld in rang by th rang rsolution clls. Th cluttr lvl in th tst cll is stimatd by avraging th oututs of th narby rsolutions clls. Th dtction thrshold is obtaind by scaling th nois lvl stimat by a constant T to achiv th dsign robability of fals alarm. This rocdur is known as th CA-CFAR (cll-avraging constant fals alarm rat) rocssor roosd by (Finn & al.,1968). This dtctor is otimal whn th cluttr owrs in th rfrnc window ar indndnt, and idntically distributd and Gaussian modlld. In som ractical alications, th cluttr rturns may not b uniformly distributd. In th rsnc of cluttr boundaris (cluttr dg) th fals alarm rgulation of th CA-CFAR dtctor dgrads drastically. To allviat this roblm th Gratstof constant fals alarm rat (-CFAR) dtctor and th Smallst-of constant fals alarm rat (- CFAR) dtctor wr roosd (Hansn 1979) (Trunk 1978). In th -CFAR dtctor th lading and lagging rfrnc samls ar saratly summd and th largr of th two is usd to st a thrshold, whil in th -CFAR th smallr is usd to st th thrshold. Rcnt studis (Pirc 1997) showd that th Gaussian distribution cannot aroriatly modl th cluttr rturns whos amlitud distribution is mor imulsiv and suggst that th cluttr can b modld as a Parson-distribution Figur 1. Block diagram of th CFAR rocssor. (Tsakalids & al., ) analy th rformanc of th CA-CFAR, OS-CFAR and PT-CFAR rocssors in homognous Parson distributd cluttr. In this ar w xtnd th rvious analysis

2 SETIT5 to th -CFAR and -CFAR rocssors and w driv xact xrssions for th fals alarm robability of this rocssors.. Problm formulation In a gnral CFAR rocssor, th squar-law dtctd signal is samld in rang by th rang rsolution clls. Th rang samls ar snt srially into a shift rgistr of lngth N+1=+1 as shown in Figur 1. Th lading samls and th lagging samls form th rfrnc window. Th data availabl in th rfrnc window ar rocssd to obtain th statistic, th stimat of th total nois owr. To maintain th robability of fals alarm () at a dsird constant valu whn th total background nois is homognous, th statistic is multilid by a scal factor T for a givn rfrnc window si N. th roduct T is th rsulting adativ thrshold. Th tst cll Q from th cntr of th window rfrnc is comard with th adativ thrshold to mak a dcision about th rsnc (H 1 ) or th absnc (H ) of a targt in th cll undr tst. H1 > q T < H (1) In our stu, w assum that th cluttr robability dnsity function dominats th background, and assums that in th homognous cas, th N rfrnc rang samls Q 1,..,Q N ar statistically indndnt and uniformly Parson distributd with a df givn by 1 qi Q ( qi ) = i = 1,..., N i 3 / q i - () Whr, is th disrsion or scal aramtr of th distribution. W assum that th targt in th tst clls is a slowly fluctuating targt of Swrling II modl. To valuat th rformanc of ths CFAR dtctors in such nvironmnts, w driv th robability of fals alarm () which is givn by = + + [ T Q ( q H ) dq ] ( ) d (3) Whr () is th robability dnsity function of th stimat. 3. Analysis of th -CFAR dtctor In th -CFAR rocssor, th cluttr lvl stimat is obtaind from th largr of two sarat sums comutd for th lading and lagging window = AX (, V ) (4) whr and = V = i= 1 q i q i i= + 1 (5) (6) whr =N/. Th df of th stimat dfind in (4) is givn by (Paoulis 1984) + ( ) = ( ) FV ( ) v ( ) F ( ) (7) Whr F (), F V () and (), V () ar th cumulativ dnsity function (cdf) and robability dnsity function (df) rsctivly of th lading window and th lagging window V. In th homognous background, th df of th statistic is rducd to ( ) = ( ) F ( ) (8) Sinc and V ar th avrag sum of Parson distributd random variabls, th disrsion aramtr of and V ar rsctivly. Hnc, th df of and th cdf of V ar givn by - 1 ( ) 3 / F = (9) ( ) = (1 Φ ( ) ) (1) Whr x dnots th cdf of th standard Gaussian distribution Φ( x) = x 1 y th df of th cluttr stimat is ( ) [1 Φ( )] 3 / (11) = (1) To comut th fals alarm robability, w assum that cll undr tst coms from a cluttr background. Hnc th df is givn by Q ( q 3 / - q 1 ) = (13) q

3 SETIT5 Substituting (1) and (13) into (3) yilds = ( Φ( (1 Φ( 3 / T )) using y, w obtain finally d (14) = ( Φ( ( Φ( 3 / T d (1) using y and aftr som mathmatical maniulations, th xact xrssion for th fals alarm robability for th homognous situation is = Erf [ y y ] Erfc[ T ] y (15) = Erf [ y y ] Erf [ T ] y () Whr Erf (x) is th rror function and Erfc (x) is th comlmntary rror function. Erf ( x) = x t dt (16a) Erfc( x) = 1 Erf ( x) (16b) Equation (15) shows that th fals alarm robability of th -CFAR dtctor dos not dnd on th aramtr of th Parson distribution but controlld only by th thrshold multilir T and th numbr of rfrnc cll N. 4. Analysis of th -CFAR dtctor In th -CFAR dtctor, th cluttr lvl stimat is th smallr of th lading and lagging windows = IN(, V ) (17) th df of th stimat dfind in (17) is givn by (Paoulis 1984) = ( )[1 F ( )] + ( )[1 F ( )] (18) V in th homognous background, th df of th stimat can b writtn as v ( ) = ( ) [1 F ( )] (19) substituting (9) and (1) into (19), th df of th stimat is ( ) [ Φ( ) 1] 3 / = () as in th -CFAR dtctor w obsrv that th fals alarm robability of th -CFAR dtctor dos not dnd on th aramtr of th Parson distribution but dnd only by th thrshold multilir T and th numbr of rfrnc cll N. 5. Rsults and Simulation In this sction, th thortical rsults obtaind in th rvious sction ar usd to dtrmin th thrshold multilir T. Th thrshold multilir T was dtrmind undr th hyothsis that th rfrnc window is in homognous Parson cluttr background. Th valu of T is numrically comutd by solving quation (15) for th -CFAR dtctor and () for th -CFAR dtctor for fixd rfrnc window si N and a dsign robability of fals alarm. Figurs and 3 show th thrshold multilir vrsus th numbr of cll rfrnc N for a dsign 1-4 and 1-6 rsctivly. To illustrat th fals alarm rat rformanc rgulation of th -CFAR and -CFAR dtctors in homognous Parson cluttr, w hav conductd ont Carlo simulations. Th cluttr samls background wr gnratd using th algorithm givn in (Chambrs 1976) for diffrnt disrsion aramtr. Th rsults shown in Figurs 4 and 5 wr carrid out using 1 runs for ach valu of th disrsion aramtr which confirm th thortical rsults. Th namic bhavior of th -CFAR and - CFAR dtctors in various dtction situations is illustrat in Figurs 6 and 7. In th simulation, th squar law dtctor oututs ar dictd as solid lin, th thrshold is surimosd on th inut. Isolatd targts ar introducd in clls, 3, 8, 9 with SNR= 8 db for modling clos and sacd targts and xtndd cluttr atch from clls 45 to 65 for modling non-homognous cluttr background. Th bst fals alarm rat rformanc at cluttr boundary is obtaind for th -CFAR dtctor, whil th - CFAR dtctor xhibits bttr rformanc for clos targts as xctd. assuming that th tst cll is in th cluttr background Substituting () and (13) into (3) yilds

4 SETIT CFAR dttor -CFAR dttor = CFAR dttor -CFAR dttor = Log 1 (T) Log 1 (T) Numbr of rfrnc clls Numbr of rfrnc clls Figur. Th thrshold multilir T of -CFAR and -CFAR dtctors as a function of cll numbr N, dsign=1-4 Figur 3. Th thrshold multilir T of -CFAR and -CFAR dtctors as a function of cll numbr N, dsign= Simulation thoritical 1-1 Simulation thoritical Probability of fals alarm Probability of fals alarm disrsion aramtr disrsion aramtr Figur 4. Simulation robability of fals alarm vrsus!" # $%&('*),+.-/ "698"%& :;=<?>: dsign =1 - Figur 5. Simulation robability of fals alarm vrsus 4@ A B6&A@ %CDBE&EF 68"6&HG IJKLON,P.QRSTV7WX"WYX"JK [=\?] dsign = CFAR thrshold CFAR thrshold 1 1 Outut (db) 8 6 Outut (db) Rang cll indx Rang cll indx Figur 6. Simulation of th -CFAR dtctor with N=8 and dsign =1-3 Figur 7. Simulation of th -CFAR dtctor with N=8 and dsign =1-3

5 SETIT5 6. Rfrncs Chambrs J.., allows C. L., Stuck B. A mthod for simulating stabl random variabls Journal of th Amrican statistical Association, Vol. 71, 354, , Jun Finn H.., Johnson R. S. Adativ dtction mod with thrshold control as a function of satially samld cluttr lvl stimats. RCA Rviw, 9, , St Hansn H.. Constant fals alarm rat rocssing in sarch radar. In Procdings of th IEE Intrnational Radar Confrnc, London, 35-33, Oct Paoulis A. Probability, Random Variabls and Stochastic Procsss. Scond dition, cgraw-hill, Nw York, Pirc R. D. Alication of th ositiv alha-stabl distribution. In IEEE signal rocssing worksho on highr- Ordr statistics, Banff, Albrta, Canada, 4-44, July Trunk G. V. Rang rsolution of targts using automatic dtctors. IEEE Transactions on Arosac and Elctronic Systms, Vol. AES-14, 5, , St Tsakalids P., Trinci F., Nikias C. L. Prformanc assssmnt of CFAR rocssors in Parson-distributd cluttr. IEEE Transactions on Arosac and Elctronic Systms, Vol. AES-36, 4, , Oct.. Wiss. Analysis of som modifid cll avraging CFAR rocssors in multil targt situations IEEE Transaction on Arosac and Elctronic systms, Vol. AES-18, 1, 1-114, Jan. 198.