Automatic Censoring Best Linear Unbiased Constant False Censoring and Alarm Rates

Automatic Censoring Best Linear Unbiased Constant False Censoring and Alarm Rates

DetectorinLognormalBackground

Souad CHABBI(1),Toufik LAROUSSI(1)and Mourad BARKAT(2),IEEEFellow

(1)Département d’Electronique,Université MentourideConstantine, Laboratoire SISCOM,Constantine 25010,Algérie

(2)Department of ComputerEngineering,King Saud University,Riyadh,KSA.

E-mails: ,d

Abstract

We dealwiththe problem ofautomaticcensoringwith constantfalse censoringrate (CFCR)andautomatictarget detectionwithconstantfalsealarm rate(CFAR),i.e., CFCAR procedures,against non-stationary Lognormal clutter with unknown scale and shape parameters. We show that, in multiple target situations, automatic censoring detectors outperform those based on fixed point censoring.Forthat,westudy thecensoringanddetection performancesofthe AutomaticCensoringBestLinear Unbiased(ACBLU)CFCAR detector.Thecensoringand detectionalgorithmsareatwoinone built detector.They selectrepeatedlyasuitablesetoftherankedreference cellstoestimatetheunknownbackgroundlevelandset theadaptivethresholdaccordingly.To thisend,the Lognormalprobabilitydensityfunction(pdf)is reducedto aNormal one, viaalogarithmictransformation,andthe parametersareestimatedusingthe bestlinear unbiased estimators (BLUEs). Performances of this detector are carried outusing MonteCarlosimulations.

1.Introduction

The detection performance,inradarsystems,isrelatedto target models and background environments. The detectionprobability(Pd) is shownto besensitivetothe non-stationary clutter statistics andtothe number of spurioustargets that may appearinthereturnechos.The most important concernof suchsystems is tomaintainthe FAR at a desired constant value while developing adaptive thresholdprocedures thatguarantyarelative immunity against all kind ofbackground heterogeneities.

In multiple target situations, to circumvent the masking effects, fixed pointcensoring CFAR detectors based on order statistics [1-9] have been known to improvethedetectionperformancebyexcisingthe higher unwanted reference cells.However,these detectors give goodresults onlyifthenumberofinterferingtargets presentinthereferencewindowisknownapriori,which isnotalways availableinrealapplications.Forinstance, Figure1showsthePd versusthesignal-to-clutterratio (SCR), in presence of sixinterferingtargets havingan interference-to-clutterratioICR=SCR. Thatis,thisisthe case of a heterogeneous Lognormal environment for whichthereal numberofinterferencesexceedstheir a priori knownnumber.Consequently,aconsiderable degradation in the detection performance of the fixed point censoringBLU-CFARdetector [9] isobserved.

In ordertoimprovethePdinmultipletargetsituations, when no prior knowledgeof thenumber ofunknown

Figure1.DetectionprobabilitiesofaSwerlingItarget againstSCRfortheBLU-CFARdetectorinpresenceof sixinterfering targets in Lognormal clutter.

power outliers inthe referencecellsismadeavailable,the automaticcensoringtechniqueshave, fortheirpart,many contributedinthe improvementofthe detection performanceofthesedetectors.The well-known approaches proposed inthe literature are found in [10-12] fora Gaussian clutter, [13-14] for aLognormalclutterand [15-16] foraWeibullclutter. Therein,theauthors use orderedstatisticstodiscriminatebetween homogeneous andheterogeneousenvironments.Whereas,[13-15] use linearbiparametricadaptivethresholds basedonasimple approach forthefirstand ontheMaximum Likelihood Estimators(MLEs)forthe letter.[14-16] employthe Weber-Haykin (WH)adaptivethreshold toavoidthe distribution parameters estimation.

In thiswork,we studythecensoring anddetection performancesoftheACBLU-CFCAR detector in Lognormalclutterandmultipletargetsituations,without

anypriorknowledgeofneitherthenon-stationaryclutter

statisticsinwhichtheradaroperates nor thenumber of outliersthatmay be presentinthereferencewindow.The censoringand detectionalgorithmsareatwoinonebuilt detector,they usethesameadaptivethreshold. Whilethe first algorithm operates censoring under CFCR, the second performsdetection underCFAR. Based onthe censoringalgorithm,thedetectionalgorithm selects repeatedlyasuitablesetoftheranked referencecells surroundingthecell undertest toestimatetheunknown backgroundlevelandset theadaptivethreshold accordingly.Todothis,from alogarithmicamplifier,the Lognormal pdf is converted to a Normal one, i.e., location-scale (LS) type [9, 17], and then the BLUEs of location –scaleparameters aredeveloped to adjust

Target

Y Log. X

absent(H0).gk isthedetectionthresholdcoefficienttobe

Clutter

LED Amplifier X1

… XN/2

X0 XN/2+1 … XN

set to achieveadesiredPfagiven by

DesignPfc

X(1)≤···≤X(p)≤X(p+1)≤···≤X(N)

CensoringAlgorithm

DisignPfa

=Prob{X0

/H0

Tˆ

k

(2)

k TogetlinearestimatorsoftheNormallocation,µˆN−k,

DesignPfa

X(1)≤ X(2)≤···≤X(N-k)

andthescale,σˆN−k,parameters,let

X(i)s;i =1,K,N be

DetectionAlgorithm

orderedreferencesamplesinascendingorderofaNormal

H orH

distributionwithunknownparameters

µˆN−k

and

σˆN−k,

0 1

basedonkhighestpossiblycensoredunwantedsamples

Figure2. BlockdiagramoftheACBLU-CFCARdetector.

censoringand detection thresholds. CFCAR is thus achievedwhentheclutterislocallyhomogeneoussuch thatresilience againstlocalheterogeneitiesisguaranteed since BLUE lendsitself to censoring [9].

representing interfering targets [9]

N−k

µˆN−k =∑ai(N,k)X(i)

i=1

and

(3)

2.Computational Routines

N−k

σˆN−k =∑bi(N,k)X(i)

i=1

(4)

Many research works on modeling real applications

where

ais

and

bis

are weights that depend only on

proved that an excellent agreement with observed

intensityofthedatacanbeachievedusingaLognormal distribution.Itisadistributionofarandomvariabley

characterized by a scale parameter,µ, and a shape

system parameters,namelyNandk.Aftercensoringthek highestunwantedsamples, the(N-k) remaining onesare usedtoestimatetheenvironmentlevel.Inahomogeneous

environment,k=0 ,theentirereferencewindowisused.

parameter,σ,with

y =exp(x)

wherexisfromaNormal

These weights must be calculated once and for all

distributioncharacterizedbyalocationparameter,µ,and

accordingtoasuitableoptimizationcriterion.

µˆN−k

and

ascaleparameter,σ,[9].ReferringtoFigure2,alltarget

returnsareassumedtobeembeddedinLognormalclutter.

Thelinearenvelopedetector(LED)matchedfilteroutputs

σˆN−k

are equivarient if and only if[8,9]

N−k

Yis

are passed through a logarithmic amplifier to get

∑ai =1

i=1

(5)

Normaldistributedrandomvariables

Xis,beforestoring

and

them serially into a tapped delay line of length N+1;

corresponding to N reference cells surrounding the cell undertestX0. Thenaturallogarithm is assumed.The principaladvantageinherenttothe logarithmic transformationistheuseofLStypedistributiontoget

N−k

∑bi=0

i=1

Equivalently,theestimatorsµˆ(X)

(6)

andσˆ(X)oflocation-

equivariantlocation-scaleparameters.Inotherwords,we

scaleparametersbasedonsamplesXaresaidequivariant

would like to consider adaptive thresholds that ensure

if [8,9,17], for real constants r,

r∈(−∞,∞), and s,

constantprobabilityoffalsecensoring(Pfc)andconstant probability of false alarm (Pfa) for any values of the

s∈(0+,∞),

µˆ(X')=sµˆ(X)+r

and

σˆ(X')=sσˆ(X),

distributionparameters,i.e.,CFCAR.Indoingthis,we

whereX'= sX+r1.

assume that the

Yis

are independent and identically

Among all linear estimators, we focus on the best

linear unbiased estimators (BLUEs) to improve the

distributed(IID).Hence,thetransformedvariates

are also IID.

Xis,

detectionperformance[9].Theseestimatorsareunbiased, equivariant and minimum variance among all linear

estimators.Let

Z(i)s;i=1,K,N−k

beorderedvariates

2.1.Computational DetectionRoutine

TheACBLU-CFCARdetectionalgorithmisbasedonthe

fromastandardNormaldistribution(µ=0,σ=1),EZits known mean vector, B its known covariance matrix,

following hypothesis test[9],

D =[1,EZ]anauxiliary

[(N−k)x2]matrix.Recallthat,

eachvalueE(Z(i))of thevectorEZcanbecomputedas[1]

H1

> ∞

X0 Tˆg

=µˆN−k +gkσˆN−k

(1)

⎛N⎞

N−i

i−1

< k E(Z(i))=i⎜

⎟ ∫Z(i)[1−F(Z(i))]

[F(Z(i))]

f(Z(i),0,1)dZ(i)

(7)

H0

A target is declared present (H1) if

X0 exceeds the

where

⎝ i⎠−∞

F(Z(i))

is the cumulative distribution function

detectionthreshold

Tgk

.Otherwise,atargetisdeclared

(cdf) ofthestandard Normalpdff(Z

(i)

,0,1).

The BLUEsµˆN−k andσˆN−k

are given by [9]

⎡

⎡µˆN−k⎤ = DTB−1D−1DTB−1X=⎡a1 KaN−k⎤⎢

X(1)

⎤

⎥ (8)

⎢ ⎥ ( )

⎢ ⎥⎢ M ⎥

⎣σˆN−k⎦

⎣b1 KbN−k⎦⎢

⎣

⎥

(N−k)⎦

2.2.Computational CensoringRoutine

Thecensoringalgorithm usesthereferencecellsrankedin anascending order,according totheirmagnitudes.The initial population represented by the p lowest cells is assumedtobe homogeneous.Thus,pmust becarefully selected to yield a good performance in both homogeneous andheterogeneousenvironments.Itmustbe aslargeaspossibletoimprovedetectionandassmallas

Figure3.Probability of hypothesis test error of the

possibletodiscardanyclutter-plus-interferencesamples

[12].Thecensoringalgorithmconsistsofthefollowing tasks:

ACBLU-CFCAR detector, asa function of

ˆ

γc .

T is not available,g is foundresortingof Monte Carlo

k

Set c=0 (Auxiliary index)

Set d=1 (Homogeneoushypothesis)

simulation. Furthermore, the coefficient γc

of the

While c≤ N-p and d=1

censoringthresholdTˆ

c

isgivensothatalowprobability

Computeµˆp+c

If c N-p

andσˆp+c

(from(3) and (4))

ofhypothesistesterror,ec,isachievedinahomogeneous environment. Particularly, ecis defined as [13]

Selectγc (to satisfy design Pfc)

nhH

ec=Prob{X

(p+c+1)

/hHTˆ

k

}c=0,1,K,N−p−1

ˆ

(10)

X(p+c+1)

Tˆγ

< c

=µˆp+c+γcσˆp+c

(9) Here also, ananalyticalexpression of thepdf ofTγ

is not

hH

If nhH,

Set d = 0

available. Thus, ec is obtained through Monte Carlo

simulationsforeachvalueofcsuchthatthedesiredPfcis

maintained at each stepby setting

End

End

e =e

0 1

=L=e

N−p−1

=DesignP

fc

(11)

End

c=c+1Figure3showsec asafunctionof environment for (N, p)=(36,24).

γc inahomogeneous

c=c-1

k=N-p-c (Numberofinterferingtargets) 4. PerformanceAssessment

γc is the censoring threshold coefficient chosen to achievethedesiredPfc. Thenonhomogeneous hypothesis (nhH) refers to a heterogeneous environment, i.e., the

Here, we evaluate the performance of the ACBLU- CFCARdetectorwithabatteryofsimulationtests.We dealwiththesingle-pulsedetection,whichcorrespondsto

samples

X(p+c+1),...,X(N)

correspond to clutter-plus

bothSwerlingIandSwerlingIIfluctuatingmodels.We

interferencesampleswhilethehomogeneoushypothesis

assume the presence of

0≤m≤N−p

unknown

(hH)referstoahomogeneousenvironment,i.e.,X(p+c+1)

isaninterference-freecluttersample.Thesuccessivetests,

interferingtargets;m=0correspondstothehomogeneous case. The thermal noise isnegligible.

startingby

c=0,i.e.,the

(p+1)th

orderedsample,are

4.1.CensoringProbabilities

repeateduntilthereference cellunderinvestigationis declaredasaclutter-plus-interferencesampleorwhenall

theN−p highest samples are used up, i.e.,c=N−p.

3. ThresholdsSelection

Theimplementation oftheACBLU-CFCARdetector requiresthecomputationof thecensoring anddetection thresholds. Since an analyticalexpression of thepdf of

Theeffectivenessoftheautomaticcensoringalgorithm hasbeen first assessed,ina homogeneous environment. Table1 givesthecensoringprobabilitiesintheabsence of interferencesforPfc =10-2 and10-3.Itisclearthatthe censoring probabilities,i.e.,Prob{k=m},ismaximalfor k=0,whichcorrespondstothe event“anyreferencecellis censored”. Otherwise(k=0),wecan note thatwhenPfc decreases, thecensoringprobabilities tend to0.

Table1.Censoringprobabilities of the censoring algorithmin ahomogeneous environment for(N,p) =(36, 24).

fc

Figure4.Under-censoring probabilities against ICR

for the censoring algorithm with m and σ as

parameters.

Figure5.Under-censoring probabilities against ICR

for the censoring algorithm with σ and Pfc as

parameters.

In the presence of interfering targets, the censoring algorithm is characterized by the over-censoring

probability,P0 =Prob{k≥m}, and the under-

censoringprobability,Pu =Prob{km}. NotethatPu

may degrade the censoring and thus the detection

performances. Figures 4 and 5 show the Pu of the

Figure6.DetectionprobabilitiesagainstSCRforthe

Ideal,ACBLU-CFCARandBLU-CFAR detectorsina homogeneousenvironment withσas aparameter.

Figure7.DetectionprobabilitiesagainstSCRforthe

ACBLU-CFCAR andBLU-CFAR detectors inmultiple target situations with mandσas parameters.

4.2.Detection Probabilities

Inthissection, weshouldevaluatethedetection performance of the ACBLU-CFCAR detector by means of abattery ofsimulation tests.

censoringalgorithmversusICR,withm, σ

andPfc as

4.2.1. Homogeneous Environment

parameters.NotethatPu increaseswhenmincreases anddecreaseswhenPfc increases.Ontheotherhand, independentlyofm andPfc,anincreaseintheshape parameterσengendersanincrease inPu. Finally,the morepowerfultheinterferences,the closertozeroPuis. Itwillbeseeninthenextsubsectionthatadecreasein Puimprovesthedetectionperformance.From these Figures, wecanconcludethat theshape parameterσhas an important influenceonthecensoringperformance.

Inabsenceofinterferingtargets,Figure6showsthe

detection performanceofthe ACBLU-CFCARdetector and the BLU-CFAR [9] detector. The results are

compared withthose oftheidealdetector(N→∞).We

note that both ACBLU-CFCAR and BLU-CFAR

detectors give the same detection performance. Note thatanincreaseinσdegrades thedetectionperformance ofall kindof detectors.

4.2.2. Multiple Target Situations

In presence of interferingtargets,Figure7showsthePd ofthe ACBLU-CFCARandBLU-CFARdetectors againstSCR. ItisclearthattheproposedACBLU- CFCARdetectoralwaysoutperformstheBLU-CFAR

detectorforanynumberof interferences.TheCFAR- Loss getshigherwheneverthenumberofinterferences increases.Thisisprimarilyduetothe over-censoring characteristicoftheACBLU-CFCARdetector.Thishas adirectimpacton thehomogeneity of theresidual population.

5.Conclusion

Inthis paper,wehaveanalyzedandevaluatedthe censoringand detection performancesofthe ACBLU- CFCAR detectorinhomogeneousandheterogeneous Lognormalclutter. We havecompareditsdetection performance with that of the fixed point censoring BLU-CFAR detector.Simulationresultsshowthatthe detection performanceof bothdetectors isthesamein uniform clutter.However,inmultipletargetsituations, the ACBLU-CFCAR detector outperformsthe BLU- CFARdetector.

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