Adaptive detection of range-spread target in compound-Gaussian clutter without secondary data

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Adaptive detection of range-spread target in compound-Gaussian clutter without secondary data

Abdelmalek Mennad, Arezki Younsi, Mohammed Nabil El Korso, Abdelhak M. Zoubir

To cite this version:

Abdelmalek Mennad, Arezki Younsi, Mohammed Nabil El Korso, Abdelhak M. Zoubir. Adaptive

detection of range-spread target in compound-Gaussian clutter without secondary data. Digital Signal

Processing, Elsevier, 2016, pp.90-98. �10.1016/j.dsp.2016.09.002�. �hal-01421514�

Adaptive detection of range-spread target in compound-Gaussian clutter without secondary data

Abdelmalek Mennad^a,∗, Arezki Younsi^a, MohammedNabil El Korso^b, Abdelhak M. Zoubir^c

aLaboratoireMicro-OndeetRadar,EcoleMilitairePolytechnique,P.O.Box17,16111BordjElBahri,Algeria bParisOuestUniversity,LEMEEA4416,50ruedeSèvres,92410Villed’Avray,France

cSignalProcessingGroup,TechnischeUniversitätDarmstadt,Merckstr.25,64283Darmstadt,Germany

1. Introduction

Range-spreadtargetdetectionhasreceivedanincreasingatten- tionin thelast two decades [1], whichismainly dueto therise ofhighresolutionradar(HRR). Indeed,anHRRcan resolveasin- gle target asmultiple scatteringcenters,depending on the range extentofthe targetandthe rangeresolutionoftheradar system [2]. The improvement of the range resolution capabilitiesresults ina reductionofthe clutter powerineach rangecell, henceim- proving the detection performance. It is worth mentioning that range-spread target detectors are useful in many situations, e.g., forthedetectionoflargeshipswithcoastalradars ordetectionof acluster ofpoint liketarget, movingatthe samevelocity andin closespatialproximitytoeachother[3].

Nevertheless,performingrange-spreadtargetdetectionrequires to develop a proper strategy since classical adaptive detection schemes, originally proposed for point-like targets, may lead to poor performances[4,5]. This is mainly dueto the fact that the target signal may contaminate the secondary cells, while these aregenerallyassumedtobetarget-freeintheclassicalcontext of point-liketargetdetection.

* Correspondingauthor.

E-mailaddress:[email protected](A. Mennad).

Range-spread targetdetectionunderGaussiandistributedclut- terhasbeenextensivelyinvestigated.In[6],theauthorsproposed a detector for spatially distributed targets embedded in white Gaussiannoise,basedonaprioriknowledgeofthetargetscatterer density. Amodified generalized likelihoodratiotest (MGLRT)de- tectorwasproposedin[7]toaccountforasingularestimatedclut- tercovariancematrix.Furthermore,afastconvergingdetectorwith boundedconstantfalsealarmrate(CFAR)propertywas derivedin [8], under the assumption of dominant thermal noise at the re- ceiver. On the other hand,an adaptivedetection ofrange-spread target inhomogeneous andpartially homogeneous environments was addressedin[1]and[9].Specifically,[1]and[9]assume that thesecondarydataaresignal-freecomponentsandthattheyshare thesamecovariancematrixstructure,butwithpossiblyfluctuating power.

Nevertheless, forHRRoperating atlow grazingangles orin a maritime environment, the Gaussian model of the clutter is no longer valid. More precisely, in [10], the clutter has been suc- cessfully modeledasacompound-Gaussianprocess.Acompound- Gaussianprocess istheproductofarealpositiverandomprocess, namedtexture,andacomplexGaussianprocess,namedspeckle.In thelastfew years,manyresultshavebeenobtainedfortheadap- tivetargetdetectionincompound-Gaussianclutter.Asanexample, Gini andFarinaproposed the so-calledgeneralizedmatchedsub- spacedetector(GMSD)[11].TheGMSD,whichisaCFARdetector, aimsto detectanunknowndeterministic signal,butknowntolie

inagivensubspace,corruptedbyacompound-Gaussianclutter.In [12],theauthorshavederived anadaptivedetectorforrangeand Doppler distributed targets, embedded in a compound-Gaussian clutter,assumingtarget-freesecondarycells.

Inpractice,thecluttercovariancematrixisoftenunknownand commonlyestimatedfromsignalfree-targetsecondarydatainthe contextof compound-Gaussian process.In [13], theauthors used theapproximatedmaximumlikelihoodmethod(AML),toestimate thecluttercovariancematrix,fromasetofhomogeneousandsig- nal free-targetsecondary data. Nevertheless,to respect the non- singularity condition of the estimated covariance matrix,a large numberofsecondarydataisrequired,whichinpractice,cancon- tain interfering targets or other kind of non-homogeneities [14].

Tocopewiththeissueoflowsample,structuralinformationabout theclutter covariancematrixcanbeexploited. Specifically,inthis paper, we take advantage of modeling the clutter as an autore- gressive process. Such modeling does not rely on a covariance matrixestimationstepandconsequently,avoidstheneedforlarge homogeneous secondary data set. This methodology has already beenappliedinthecontextofradar.Moreprecisely, adaptivede- tection in unknown colored noise, modeled by an autoregressive process is addressed in [15], for known point-like target signals andin[16]foratargetsignalknownuptoascalingfactor.In[17]

and[18],theauthorsmodeledthespecklecomponentoftheclut- terby acomplex Gaussian autoregressive process oforder P, for range-spreadtarget detectioninhomogeneous andheterogeneous environment.

Nevertheless,inthispaper, foreach rangecell andforan au- toregressive process of order P and N samples, we propose to predict the P first samples in a backward manner, and samples from P +^{1 until N in a forward manner, using only cells un-} der test. The proposed scheme is compared with some existing ones,especially therecentalgorithms presentedin[17] and[18], whichuseonlytheforwardprediction,byconsideringjust N−^P samples, unlike the proposed algorithm which uses all available samples. A performance analysis in terms of false alarm proba- bility and detection probability is performed for different target models.Ournumericalresultsindicatethattheproposeddetector outperformstheautoregressive GLRT(ARGLRT) detectorfrom[17]

and[18] in homogeneous andheterogeneousenvironment, espe- cially when few samples are available. In addition, the proposed methodoutperformstheGLRTdetectorbasedontheAMLestima- torofthecluttercovariancematrix,whichrequiresalargenumber ofsecondarydata[19].

This paper is organized as follows. In section 2, we describe thedatamodel.Insection3,wepresenttheclassicalARmodeling andtheproposedforward–backwardARmodeling.Section4isde- votedtotheproposed detectionalgorithm.Simulation resultsare discussedinsection5andfinallyaconclusionisgiveninsection6.

2. Problemsetup

Letusconsidera radarthat collects N pulsesduring the time ontarget.Forthei-thrangecell,thesepulsesformthefollowing complexvaluedreceivedvector

zi=^[zi(1) , . . . ,zi(N)]^T,i=¹, . . . ,L (1) where [.]^{T denotes the transpose operator and L is the number} ofrange cells under test. Forsake ofsimplicity, we assume that the received data vectors are independent among range cells [1, 11]andthatthetargetiscompletelycontainedwithin Lcells.

Thedetectionproblemisformulatedasthefollowingbinaryhy- pothesistestingproblem

H0: ^zi=^ci, i=¹, . . . ,L

H1: ^zi=^ci+αip, i=¹, . . . ,L (2)

in which, c_i=^[ci(1) , . . . ,c_i(N)]^T denotes the clutter vector, p= [p(1) , . . . ,p(N)]^T =

1,e^j2πfd, . . . ,e^j2π(N−1)fdT

is the N-dimen- sionaltimesteering vector,where fd denotesthenormalizedtar- getDopplerfrequency.Inaddition,weassumethat f_disknown[9, 16,18–20],orefficientlyestimated,seee.g.,[12,21].Finally,theun- knowncomplexamplitudes αiareaccountingforboththechannel attenuationandthetargetcrosssection.

According to the compound-Gaussian model, the clutter ci for the i-th range cell, is modeled as the product of two inde- pendent random variables. Specifically, ci=√τix, where x is a N-dimensional zero-mean complex circular Gaussian vector, rep- resentingthe speckle component, andτi denotes the variance of the underlying conditional Gaussian vector, named texture and represents the local clutter power, for each range cell. In short- handnotation,wewritex∼CN(0,M),whereMisthenormalized covariancematrixofthespeckle,such thattrace(M)=^N.Weas- sume that τi andx aremutually independent,consequently, one hasE

cic^H_i

=τiM=^Mc,withMc denotingtheclutter covariance matrix and [.]^{H indicates the conjugate transpose operator. The} conditionalprobabilitydensityfunctions(pdf)ofzi underH0 and H1canbeexpressed,respectively,as

f(z_i|H0;τi,M)= ¹

(π τi)^N|^M|^exp

−^ziHM−1zi τi

(3)

f(zi|H1;αi,τi,M)

= ¹

(π τi)^N|^M|^exp

−(z_i−αip)^HM⁻¹(z_i−αip) τ_i

(4)

inwhich,|.|^denotesthedeterminant.

3. ARmodeling 3.1. Forwardmodeling

LetusassumethatthecluttermodelisgivenbyanARprocess of order P. Then, for n=^P +¹,. . . ,N and i=¹,. . . ,L, we can write

c_i(n)= − P k=1

a_i(k)c_i(n−^k)+^wi(n) (5)

whereai= [^ai(1),. . . ,ai(P)]^{T isthe P}-dimensionalvectorofcom- plex forwardAR coefficientsintherangecell i andw_i(n) iszero meancomplexGaussian whitenoisewithvariance τi [22].Under theARmodel[16],wecanrewrite(2)as:

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

H0: ^zi(n)= −_P

k=¹a_i(k)z_i(n−^k)+^wi(n) i=¹, . . . ,L;ⁿ=^P+¹, . . . ,N H1: ^zi(n)= −_P

k=1a_i(k) (z_i(n−^k)−αip(n−^k)) +αip(n)+^wi(n)

i=¹, . . . ,L;ⁿ=^P+¹, . . . ,N

(6)

Theconditionalpdfs(3)and(4)canbeapproximatedforN^P [18,23],respectively,by

f(zi|H0;τi,ai) ¹ (π τ_i)^{N−P exp}

−(z˜i+Ziai)^H(z˜i+Ziai) τi

(7) f(zi|H1;αi,τi,ai) ¹

(π τ_i)^N−P

×exp

−(˜zi+Ziai−αi(p˜+ ˜^{P a}i))^H(z˜i+Ziai−αi(p˜+ ˜^{P a}i)) τi

(8)

wherethe(N−^P)dimensionalvectors˜z_iandp˜ aregivenbyz˜_i= [^zi(P+¹),. . . ,zi(N)]^{T and}p˜= [^p(P+¹),. . . ,p(N)]^{T,whereas,the} (N−^P)×^P-dimensionalmatricesZ_i andP˜ areexpressedby

Zi=

⎡

⎢⎢

⎢⎢

⎣

zi(P) zi(P−1) . . . zi(1) zi(P+1) zi(P) . . . zi(2)

... ... . .. ...

zi(N−1) zi(N−2) . . . zi(N−P)

⎤

⎥⎥

⎥⎥

⎦

and

P˜ =

⎡

⎢⎢

⎢⎢

⎣

p(P) p(P−¹) . . . p(1) p(P+1) p(P) . . . p(2)

... ... . .. ...

p(N−1) p(N−2) . . . p(N−P)

⎤

⎥⎥

⎥⎥

⎦.

Inthe above equations,we used samplesfrom P+^{1 to N to} modelthereceivedobservationinaforwardmanner.Inthefollow- ing,weproposetoextendthismodelingbyaddingtheprediction ofsamples from 1 to P,in a backward manner inorder to gain more information, thus enhancing the performance of the pro- poseddetector.

3.2. Theproposedforward–backwardmodeling

Inthebackwardprediction,weproposetopredictthe(n−^P)- thsample fromthe P futuresamples.Thus, forn=^P+¹,. . . ,N andi=¹,. . . ,L,weobtain

ci(n−P)= − P k=1

bi(k)ci(n−k+1)+wi(n) (9)

whereb_i= [^bi(1),. . . ,b_i(P)]^{T isthe P}-dimensionalvectorofcom- plex backward AR coefficients in the i-th range cell and wi(n) denotesazeromeancomplexGaussianwhitenoise withvariance

τi [22].

In the backward context, we can rewrite (2), for n= ^P + 1,. . . ,N andi=¹,. . . ,L,

⎧⎪

⎨

⎪⎩

H0: ^zi(n−^P)= −_P

k=¹b_i(k)z_i(n−^k+¹)+^wi(n) H1: ^zi(n−^P)= −_P

k=¹b_i(k)(z_i(n−^k+¹)

−α_ip(n−^k+¹))+α_ip(n−^P)+^wi(n).

(10) Let

zi= [˘^zi^Tz˜^T_i]^{T (11)}

inwhich ˘zi= [^zi(1),...,zi(P)]^{T and}z˜i= [^zi(P+¹),...,zi(N)]^{T. In} thefollowing,samplesofthevector^z˘iarepredictedbyabackward mannerandsamples ofthe vector z˜i are predictedby a forward manner.

3.2.1. PredictionunderH0hypothesis

Under H0 hypothesis and from(10), thebackward prediction of˘^ziisgivenby

ˆ˘

zi= − ˘Zibi (12)

with

Z˘_i=

⎡

⎢⎢

⎢⎢

⎣

z_i(P+¹) z_i(P) . . . z_i(2) z_i(P+²) z_i(P+¹) . . . z_i(3)

... ... . .. ...

z_i(2P) z_i(2P−¹) . . . z_i(P+¹)

⎤

⎥⎥

⎥⎥

⎦. (13)

Using(6),theforwardpredictionof˜z_iisgivenby

ˆ˜

z_i= −Z_ia_i. (14)

Consequently, from (12) and (14), and under H0, the forward–

backwardpredictionofzi isgivenby

ˆ z_i=

⎡

⎣zˆ˘_i ˆ˜

z_i

⎤

⎦= −^Zid_i (15)

with Z_i=^bdiag(Z˘_i,Z_i) and d_i= [^bTi,a^T_i]^{T. The residual of the} forward–backwardpredictionofziisgivenby

w_i=^zi− ˆ^zi=^zi+^Zid_i (16) Then, theconditional pdf(3),under H0,canbe approximated forN^{P by[23]}

f(z_i|H0;τi,d_i) ¹ (π τi)^Nexp

−(z_i+^Zid_i)^H(z_i+^Zid_i) τi

(17)

3.2.2. PredictionunderH1hypothesis Letusdenote

si= [˘s^T_i˜s^T_i]^T (18) with

˘

s_i= [^si(1), ...,s_i(P)]^T, (19)

˜

s_i= [^si(P+¹), ...,s_i(N)]^{T (20)}

in which, si(n)=^zi(n)−αip(n),for n=¹,. . . ,N. Under H1 hy- pothesisandfrom(10),thebackwardpredictionof^s˘i conditioned to αi,isgivenby

ˆ˘

si= − ˘^Sibi (21)

with

˘Si=

⎡

⎢⎢

⎢⎢

⎣

s_i(P+¹) s_i(P) . . . s_i(2) si(P+2) si(P+1) . . . si(3)

... ... . .. ...

s_i(2P) s_i(2P−¹) . . . s_i(P+¹)

⎤

⎥⎥

⎥⎥

⎦. (22)

On the other hand,from(6), theforward predictionof s˜i condi- tionedto αi,canbeexpressedby

ˆ˜

s_i= − ˜^Sia_i (23)

with

˜S_i=

⎡

⎢⎢

⎢⎢

⎢⎣

s_i(P) s_i(P−¹) . . . s_i(1) s_i(P+¹) s_i(P) . . . s_i(2)

... ... . .. ...

s_i(N−¹) s_i(N−²) . . . s_i(N−^P)

⎤

⎥⎥

⎥⎥

⎥⎦

. (24)

Finally,from(21)and(23),theforward–backwardpredictionofs_i underH1 isgivenby

ˆ si=

⎡

⎣ˆ˘si

ˆ˜

s_i

⎤

⎦ (25)

anditserrorvectorisexpressedas

wi=^si− ˆ^si= w˘

˜ w

(26)

inwhich

˘

w= ˘^si+ ˘^Sib_i (27)

and

˜

w= ˜si+ ˜Siai. (28)

Inaddition,using(10)and(19),wehave

˘

s_i= ˘^zi−α_ip˘, (29)

S˘_i= ˘^Zi−α_iP˘ (30)

withp˘= [^p(1),. . . ,p(P)]^{T and} P˘=

⎡

⎢⎢

⎣

p(P+¹) . . . p(2) ... . .. ... p(2P) . . . p(P+¹)

⎤

⎥⎥

⎦. (31)

Then,from(27),(29),(30)andafterstraightforwardmanipulations, weobtain

˘

w= ˘^zi+ ˘^Zib_i−αi

p˘+ ˘^{P b}i

. (32)

Usingthesamemethodologyforw,˜ wehavefrom(6)and(20):

˜

s_i= ˜^zi−αip˜. (33)

Ontheotherhand,(6)and(24)leadto

S˜_i=Z_i−α_iP˜ (34)

Consequently,combining(28),(33)and(34),weobtain

˜

w= ˜^zi+Z_ia_i−α_ip˜+ ˜^{P a}i

(35) Finally,from(32)and(35),theforward–backward residualvector ofsiconditionedto αiisgivenby

w_i=^zi+^Zid_i−α_i(p+^{P d}i) (36) withP=^bdiag

˘ P,P˜

= P˘ 0

0 P˜

.

Insummary,usingthe samemethodologyasfor H0, we con- cludethattheconditional pdfgivenin(4),underH1,canbe ap- proximatedforN^{P by}

p(z_i|H1;α_i,τ_i,d_i) ¹ (π τi)^N

×^exp−(zi+^Zidi−αi(p+^{P d}i))^H(zi+^Zidi−αi(p+^{P d}i)) τi

(37) 4. Theproposedadaptivedetectionalgorithm

4.1.Homogeneousenvironment

Ina homogeneousenvironment, itis commonlyassumedthat the clutter parameters do not change from cell to cell, which meansthatd^d1=. . .=^dL,and ττ1=. . .=τL [17,18].

The optimum decision rule for target detection is the well known Neyman–Pearson criterion, which leads to the likelihood ratio test. Nevertheless, in our context, the parameters α ₌ [α1,. . . ,αL]^, τ and d are unknown. We therefore resort to the suboptimalGLRTdetectorgivenby

_GLRT(z₁, . . . ,z_L)=

maxα,τ,df(z₁, . . . ,z_L|H1;α,τ,d) maxτ,d f(z₁, . . . ,z_L|H0;τ,d)

H1

H≷0

δ (38)

whereδdenotesthedetectionthresholdwhichissetaccordingto thedesiredvalueofthefalsealarmprobability[24].

Assuming independent z_i for i = ¹,. . . ,L and using (17) and (37), the conditional joint pdfs of the total observationvec- tors z1,. . . ,zL under H0 andH1 are, respectively, approximated by

f(z1, . . . ,zL|H0;τ,d)

¹

(π τ)^{N Lexp}

− L i=¹

(z_i+^Zid)^H(z_i+^Zid) τ

(39)

and

f(z₁, . . . ,z_L|H1;α,τ,d) ¹ (π τ)^{N L}

×^exp

− L

i=¹

(z_i+^Zid−αi(p+^{P d}))^H(z_i+^Zid−αi(p+^{P d})) τ

(40) in which, the estimation of the unknown parameters are per- formedbythemaximumlikelihoodmethodinthefollowing.

4.1.1. Maximumlikelihoodestimateof α

UnderH1 hypothesis,wecanrewrite(40)as:

f(z1, . . . ,zL|H1;α,τ,d) ¹ (π τ)^{N Lexp}

− L i=1

|^gi−α_iψ|² τ

(41) inwhichg_i=^zi+^Zid andψ=^p+^{P d.Themaximumlikelihood} estimator(MLE)of αi isgivenby

ˆ

αi=^{arg min}_α

i |^gi−αiψ|²=ψ^Hg_i

ψ^Hψ. (42)

Plugging(42)into(41),weobtain f(z₁, . . . ,z_L|H1; ˆα,τ,d)

¹

(π τ)^{N Lexp}

− L i=1

(zi+Zid)^HP^⊥_ψ(zi+Zid) τ

(43)

wheretheorthogonalprojectorontothesubspacespannedbyψ is givenby P^⊥_ψ=^IN−^ψψ_ψHψ^H inwhichIN istheN×^{N identitymatrix} andαˆ_{= [ ˆ}α1,. . . ,αˆL]^T.

4.1.2. Maximumlikelihoodestimatesof τandd

Tofind themaximum likelihood estimate ofτ under H0 and H1, we maximize (39) and(43) with respect to τ, respectively.

Aftersomemanipulations,weobtain

ˆ τ_H0=

_L

i=¹

z_i+^Zid_H0H

z_i+^Zid_H0

N L (44)

ˆ τ_H1=

_L

i=¹

zi+Zid_H1H

P^⊥_ψ

zi+Zid_H1

N L (45)

Following the same methodologyas in [18], the MLEof the for- wardpart,i.e.,aunderH0 andH1,respectively,isgivenby

ˆ a_H0= −

_L

i=¹

Z_i^HZ_i −¹ _L

i=¹

Z^H_i z˜_i

(46)

ˆ a_H1= −

_L

i=¹

Z_i^HP^⊥_˜ φZ_i

−¹ _L

i=¹

Z^H_i P^⊥_˜ φz˜_i

(47)