Cramér-Rao bound and statistical resolution limit investigation for near-field source localization

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Cramér-Rao bound and statistical resolution limit investigation for near-field source localization

Tao Bao, Mohammed Nabil El Korso, Habiba Hafdallah Ouslimani

To cite this version:

Tao Bao, Mohammed Nabil El Korso, Habiba Hafdallah Ouslimani. Cramér-Rao bound and statistical

resolution limit investigation for near-field source localization. Digital Signal Processing, Elsevier,

2016, 48, pp.137-147. �10.1016/j.dsp.2015.09.019�. �hal-01421517�

Cramér–Rao bound and statistical resolution limit investigation for near-field source localization

Tao Bao

^a

, Mohammed Nabil El Korso

^b

^,∗ , Habiba Hafdallah Ouslimani

^b

a DepartmentofElectronicsEngineering,NorthwesternPolytechnicalUniversity,Xi’an710129,China bParisOuestUniversity,LEMEEA4416,50ruedeSèvres,92410Villed’Avray,France

1. Introduction

Therehasbeentremendous interestintheinvestigation ofar- ray signal processing during the last three decades due to its importance in radar, sonar, seismology and communication sys- tems[2–9].One canclassify sourcelocalization, inthecontextof array processing, intothe so-called far-field andnear-field cases.

Thecontext offar-fieldsourcesassumesplanar wavefrontsofthe propagating waves as they reach the array. Nevertheless, when thesources arelocated intheso-callednear-fieldregion, thisas- sumptionisnolongervalid.Moreprecisely,thesourcewavefronts mustbeassumedspherical,whichrequiresboththerangeandthe direction-of-arrival (DOA) parameter estimation to carry out the localizationprocedure[10–12].Onecanfindseveralestimational- gorithmsadaptedforthenear-fieldsourcelocalization[13–17]and somerelatedperformanceanalysisinordertoquantifytheestima- torperformanceintermsofthemeansquareerror(MSE)[18–22].

However, to fully characterize estimator performance, one has, additionally, to perform the resolvability analysis of two closely spacedsources.Recently,someworksconsideredbothfarfieldand near field scenario. In [23–25] the authors studied, respectively, theasymptoticresolvabilityoffar-field,deterministic,discreteand stochasticsourcesignals, whereas,in[26] theauthorsfocused on

✩ ThisworkwaspartiallypresentedduringtheInternationalConferenceSAM-12 [1].

*

Correspondingauthor.

E-mailaddress:m.elkorso@u-paris10.fr(M.N. El Korso).

polarized far-field sources and in [27], the authors studied the angular resolution limit for far-field sources in the presence of modeling errors. Moreover, in [28], the authors provide a novel methodologytoderiveanapproximationoftheresolutionlimit.

Nevertheless,tothebestofourknowledge,noworkshave been done toquantifytheresolvabilityoftwo closelyspacednear-field sources in the non-uniform linear array (NULA) context for the general case¹ (general case, means that the time varying source signals, the noisevariance, the rangeandthebearing areall un- knownparameters).Consequently, thegoalofthispaperis tofill thislackbyaddressingthefollowingquestion:“Whatis,inanear- field source scenario, the minimum source separation required, underwhichtwonear-fieldsourcescanstillbecorrectlyresolved?”

Tothis end, one commontool is thedistance resolutionlimit (DRL)oftwocloselyspacedsources,definedastheminimumdis- tance withrespect to the sources for which allows a correct re- solvability.TheDRLcanbederived basedononeofthefollowing three fundamental approaches: i) the analysis of the mean null spectrum[30],ii)thedetectiontheory[19,23](usingabinaryhy- pothesis test anda proper generalizedlikelihood ratio test [27]), oriii)theestimationaccuracy(namely, basedontheCramér–Rao

1 In[29],theauthorsstudiedtheresolvabilityinthenearfieldcontextforthe specialoptimisticcaseinwhichthesourcesignalsandthenoisevarianceareas- sumedknown,andtheonlyunknownparametersaretherangeandthebearing.

Whereas,in[28],theauthorsderivedtheresolutionlimitwithrespecttothebear- ingparameteronly,withoutconsideringtherangeasaparameterofinterestaffect- ingtheresolution.

bound(CRB)tool).Moreprecisely,in[31],Smithproposedthefol- lowingDRLcriterion:“Twosourcesareresolvableiftheseparation betweenthem,denotedbyd_NF,islessthanthestandarddeviation oftheseparationestimation”.Sincethestandarddeviationcanbe approximatedusingtheCRBby

√

CRB

(

d_NF

)

[31],thus,Smith’scri- terion statesthat the DRL is givenasthe particular value ofthe distanced_NF betweentwo sources which isequal to

√

CRB

(

d_NF

)

. ThismeansthattheDRListhesolutionofthefollowingequation DRL

= √

CRB

(

DRL

)

.

ItisworthnotingthatthisDRL,whichisbasedontheCRB,is widely andcommonly usedfor thefollowing reasons: i) it takes thecouplingbetweentheparametersintoaccount,ii)itavoidsthe well-knowndrawbackofthemeannullspectrumapproachwhich is valid fora specific given high-resolution algorithm and hence lacks generality[24],andfinally,iii) itis relatedtothe detection theorybasedapproachthankstoatranslatorfactorwhichisgiven in[11].

In order to derive the DRL, we first investigate and develop explicitclosed-form expressions ofthe deterministic CRBfortwo closely spaced near-field spaced sources (which, to the best of our knowledge, has not been derived²). Then, we resort Smith’s criterion to derive the DRL. Thus, we deduce and analyze the relationship between the minimum distance of resolvability and the minimumsignal-to-noise ratio (SNR) requiredto resolve two closelyspacednear-fieldsources.Finally,weproposeafast,nearly optimal,arraydesignproceduretoenhancethecapacityofresolv- ability fora given constraint(number of sensors, array aperture, etc.).Numericalsimulationsaregiventoassessthevalidity ofthe proposedexpressions(forboththedeterministicCRBandtheDRL) andtheenhancementduetotheproposedarraygeometrydesign.

Forthe restpart ofthis paper,the following notation will be used.A lowercasebold letterdenotes avector, andan uppercase onea matrix.Vectors areby defaultincolumnorientationunless specified.Upper scripts

·

^{T and}

·

^{H denote,}respectively, the trans- pose and the trans-conjugate of a matrix. The operators diag

( · )

, adiag

(·)

^,

·

^and

{·}

^denote,respectively,thediagonaloperator, theanti-diagonal operator, theEuclidean normandthe realpart.

tr

{·}

anddet

{·}

denotethetraceandthedeterminantofamatrix;

whereas,

^and

⊗

^{denotetheHadamardandtheKroneckerprod-} ucts,respectively. IL denotestheidentitymatrixofsize L

×

^L.

[·]

i and

[·]

i,j represent, respectively, the i-thvector element and the

(

i

,

j

)

-thmatrixelement.Finally,

ε ˆ

denotesanyasymptoticallyun- biasedestimatorof

ε

.

This paper is structured as follows. Section 2 introduces the datamodelandtheunderlyingassumptions.InSection 3,wede- rivetheCRBofaNULAfororthogonalandnon-orthogonalsources.

We deduced the DRL in Section 4. Discussionof the results and conclusionarepresentedinSections5and6,respectively.

2. Systemmodels

Assume a linear (possibly non-uniform) array composed of N sensorswithlocationsd₀

·

^d,d1

·

^d,

· · ·

^,dN−1

·

^{dthatreceivesasignal} emittedbytwonear-field,narrow-bandsourcess1

(

t

)

ands2

(

t

)

,as showninFig. 1.Letd_n

·

^{ddenotethelocationofthe}

(

n

+

¹

)

-thsen- sor,inwhichdistheunitandthereferencesensoristhefirstone (i.e.,d₀

=

^0).Furthermore,forsake ofsimplicity,the sensorloca- tionsareexpressedasmultiplicationofacommonunitbase-lined.

Thismeansthatd0,d1,

· · ·

^,dN−¹areintegers.Theobservedsignal, xn

(

t

)

,atthet-thsnapshotofthe

(

n

+

¹

)

-thsensorisgivenby

x_n

(

t

) =

^s1

(

t

)

e^jτn1

+

^s2

(

t

)

e^jτn2

+

^vn

(

t

),

n

= ⁰ , 1 ,· · · ,

N

− ¹ ,

t

= ¹ , 2 ,· · · ,

T

(1)

2 Moreprecisely,in[18–22],onlytheonesinglesourcecasewasunderconsider- ation.

Fig. 1.Localizationoftwosourcesinthenear-fieldregionusinganon-uniformlin- eararray.Symbols•^and◦^{representthepositionofasensorandthepositionofa} missingsensor,respectively.

wherevn

(

t

)

denotesacomplexcircularwhiteGaussiannoisewith zero-mean and unknown variance

σ

², which is assumed to be uncorrelated both temporally and spatially. The i-th source sig- nal, with a carrier frequency equal to f0 is given by si

(

t

) = α

i

(

t

)

e^j(2πf0t+ψi(t)),i

=

¹

,

2,where

α

i

(

t

)

and

ψ

i

(

t

)

aretherealam- plitude andtheshiftphaseofthe i-thsource,respectively,and T is thenumber ofsnapshots.Moreover, the time delay

τ

ni associ- atedwiththesignalpropagationtime fromthei-thsourcetothe

(

n

+

¹

)

-thsensorisrepresentedas[23]

τ

ni

= ² π

ri

λ

1 +

^d2nd2

r²_i

− ^2d

^nd

^sin θ

i

r_i

− 1

,

n

= 0 , 1 ,· · · ,

N

− 1 (2)

where

λ

isthesignalwavelength,r_iand

θ

i

∈ [

⁰

, π /

2

]

^aretherange andthe bearingof thei-th source,respectively. Itis well known thatiftherangeisinsidetheso-calledFresnelregion[17],i.e.,if

0 . 62

d³_N−1d³

λ

1/2

<

r_i

< ^2d

2 N−¹d²

λ ⁽³⁾

then,thetimedelay

τ

ni canbeapproximatedby

τ

ni

≈ ω

idn

+ φ

id²_n

+

O

d²

r²_i

(4)

in which

ω

i and

φ

i are the so-calledelectricangles for the i-th sourcewhichareconnectedtothephysicalparametersbythefol- lowingrelationships

ω

i

= − 2 π

^d

λ ^sin (θ

i

) (5)

and

φ

i

= π

^d

2

λ

r_i

cos

²

(θ

i

) (6)

Then,basedonapproximationoftimedelayin(4),theobservation atthet-thsnapshotofthe

(

n

+

¹

)

-thsensorbecomes

xn

(

t

) =

^s1

(

t

)

e^j

ω1d_n+φ1d²_n

+

^s2

(

t

)

e^j

ω2d_n+φ2d²_n

+

^vn

(

t

) (7)

Consequently,thet-thobservationvectorcanbeexpressedas x

(

t

) = [

x0

(

t

) · · ·

xN−¹

(

t

)]

^T

=

A

(

p₁

,

p₂

)

s

(

t

) +

v

(

t

) (8)

where s

(

t

) = [

^s1

(

t

),

s₂

(

t

)]

^{T, v}

(

t

) = [

^v0

(

t

), · · · ,

v_N−1

(

t

)]

^{T, p}i

= [ ω

i

, φ

_i

]

^{T for i}

=

¹

,

2 and A

(

p₁

,

p₂

) = [

^a

( ω

1

, φ

1

),

a

( ω

2

, φ

2

) ]

^{. The}

(

n

+

¹

)

-thelementofthesteeringvectora

( ω

i

, φ

i

)

isgivenby

[

^a

( ω

i

, φ

i

)]

n+1

= ^e

^{j(ωidn+φidn2)}

. (9)

Weassumethattheunknownparametervectoris

ε ₌

p₁^T

,

p^T₂

, ψ

₁^T

,ψ

^T₂

, α

₁^T

, α

₂^T

, σ

²

_T

(10)

where

ψ

_i

= [ ψ

_i

(

1

), · · · , ψ

_i

(

T

) ]

^T,

α

_i

_{= [} α

i

(

1

), · · · , α

i

(

T

) ]

^{T. It is} worth noting that the unknown vector parameter

ε

is assumed tobe deterministic anditssize dependsonthe numberofsnap- shotsT.

Underthedeterministicmodel,thejointprobabilitydistribution function(pdf)ofthefullobservations

χ = [

^xT

(

1

) · · ·

^xT

(

T

) ]

^T,under i.i.d.assumption,iswrittenasfollows

p

( χ | ε ) = ¹ π

^NT

det (

R

)

^e

−(χ−μ)^HR⁻¹(χ−μ)

(11)

inwhichR

= σ

²I_NT and

μ _{= [}

s^T

( 1 )

A^T

(

p₁

,

p₂

), · · · ,

s^T

(

T

)

A^T

(

p₁

,

p₂

) ]

^T

. (12)

3. Cramér–Raobounddefinitionandderivation

LetE

(ˆ ε ₋ ε _)(ˆ ε ₋ ε )

^T

bethecovariancematrixofanestimate of

ε

.Letusassumethat

ε ˆ

isanasymptotically unbiasedestimate ofthetrueparametervector

ε

,andletus,inthefollowing,define theCRBforthe consideredmodel[32].The covarianceinequality principlestatesthatunder asymptoticconditions[33],wehave

MSE

[ˆ ε _]

_i

₌

E

[ˆ ε _]

_i

_{− [} ε _]

_i

²

_{≥ [}

CRB

( ε _)]

_i,i

(13)

wheretheCRBisgivenastheinverseoftheso-calledFisherinfor- mationmatrix(FIM)asfollows

CRB

( ε ) =

^FIM−1

( ε ) (14)

Sinceweare workingwithacomplexcircularGaussian obser- vation model, the i-th row,k-th columnelement ofthe FIM for theunknownrealparameter vector,givenin(10),canbe written as[34]:

[

^CRB−1

( ε ) ]

i,k

= [

^FIM

( ε )]

i,k

=

^tr

R⁻¹

∂

R

∂ [ ε ]

i

R⁻¹

∂

R

∂ [ ε ]

k

+ ²

∂ μ

^H

∂ [ ε ]

i

R⁻¹

∂ μ

∂ [ ε ]

k

=

^NT

σ

⁴

∂ σ

²

∂ [ ε _]

_i

∂ σ

²

∂ [ ε _]

_k

⁺

2

σ

²

∂ μ

^H

∂ [ ε _]

_i

∂ μ

∂ [ ε _]

_k

(15)

3.1.FIMderivation

From(15)we caneasily seethat thenoise variance

σ

² isde- coupledfromtheothersparameters.Thus,forsimplicityandwith- out loss of generality, we compute the FIM without taking into accountthe noise variance.Consequently, thevariance parameter isomitted fromthe vector parameter

ε

. So that the structureof theFIMisasfollows(forproof,pleaserefertoAppendix A):

FIM

( ε ) = ² σ

²

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

Fp₁p₁ Fp₁p₂

F_p2p1 F_p2p2

Fp₁ψ₁ Fp₁ψ₂

0

Fp₁α2

F_p2ψ1 F_p2ψ2 F_p2α1

0

F_ψ1p1 F_ψ1p2

F_ψ2p1 F_ψ2p2

0

F_α1p2 F_α2p₁

0

FL L

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(16)

where³ L

= [ ψ

^T₁

, ψ

^T₂

, α

₁^T

, α

^T₂

_]

^T.

3 Forsakeofsimplicityandtoavoidanymisunderstanding,weusedthefollowing notation,Fuv,wherethelowerscriptu v denotestheconsideredpartoftheFIM whichcorrespondstothederivationaccordingtothevectorparametersuandvas shownin(7),respectively.

3.2. Analyticalinversion

First,wenotethatthe4T

×

^{4T matrixF}L L in(16)canbepar- titionedintodiagonalsub-blocksas

FL L

=

⎡

⎢ ⎢

⎣

Fψ₁ψ₁ Fψ₁ψ₂ F_ψ2ψ1 F_ψ2ψ2

0

Fψ₁α2

F_ψ2α1

0 0

F_α1ψ2

F_α2ψ1

0

F_α1α1 F_α1α2 F_α2α1 F_α2α2

⎤

⎥ ⎥

⎦ ⁽¹⁷⁾

inwhich,theexpressionofeachsub-blockisgiveninAppendix A.

Second,the FL L matrixis fullyinverted byapplyingtheSchur complement[35]formulatwiceasfollows:

1) Using theSchur complementwithrespect tothe lower right block matrixin(17)representedby thedottedlinepartition, oneobtains

F_α1α1 F_α1α2 F_α2α1 F_α2α2

₋1

=

S⁻¹

−

^S−1Fα1α2F⁻_α1¹_α1

−

^F−α1¹α1F_α2α1S⁻¹ F⁻_α1¹_α1

+

^F−α1¹α1F_α2α1S⁻¹F_α1α2F⁻_α1¹_α1

(18)

inwhich

S

=

F_α2α2

−

F_α1α2F⁻_α1¹_α1F_α2α1

. (19)

2) Now,weusetheSchurcomplementwithrespecttothelower rightblockmatrixin(17)representedbythesolid lineparti- tionandweobtain

F⁻_{L L}¹

=

⎡

⎣

Fψ₁ψ₁ Fψ₁ψ₂ 0 Fψ₁α2

Fψ2ψ1 Fψ2ψ2 Fψ2α1 0 0 F_α1ψ₂ Fα1α1 Fα1α2

F_α2ψ1 0 F_α2α1 F_α2α2

⎤

⎦

−¹

=

A−^BD−1C₋1

−

A−^BD−1C₋1 BD⁻¹

−D⁻¹C

A−BD⁻¹C₋1

D⁻¹+D⁻¹C

A−BD⁻¹C₋1 BD⁻¹

(20)

where

A

=

Fψ₁ψ₁ Fψ₁ψ₂ F_ψ2ψ1 F_ψ2ψ2

,

B

=

0

Fψ₁α2

F_ψ2α1

0

,

C

=

0

F_α1ψ₂ F_α2ψ1

0

and

D

=

Fα1α1 Fα1α2

F_α2α1 F_α2α2

.

3) Finally,having theinverse ofFLL matrixone can apply again thismethodtocomputeanalyticallytheSchurcomplementof thelowerrightblockofthematrixFIM

( ε )

in(16).

Aftertediouscalculation,weareabletoexpressinclosedform the Schur complement of the matrix F_LL, which corresponds to CRB⁻_{P P}¹:

CRB⁻_{P P}¹

= ² σ

²

⎡

⎢ ⎢

⎣

( 1 , 1 , 2 ) ( 1 ) ( 1 ) ( 1 ) ( 1 , 2 , 4 ) ( 2 ) ( 1 ) ( 2 , 1 , 2 ) ( 2 )

( 2 ) ( 2 ) ( 2 , 2 , 4 )

⎤

⎥ ⎥

⎦

(21)

inwhich P

= [

^pT1

,

p₂^T

]

^{T andfunctions}

, ,

and

aregivenin Appendix B.

Consequently,insteadofinvertinga

(

4T

+

⁴

) × (

4T

+

⁴

)

matrix toobtainCRBp_ip_k,wecaninvertjusta2

×

^{2 matrixasfollows}

CRBp_ip_k

=

σ²

2

ϒ

⁻¹

(

i

) for

i

=

^k

, ∀

ⁱ

= ¹ , 2

−

^σ₂²

ϒ

⁻¹

( 1 )ℵ ( 2 ) for

i

=

^k

, ∀

ⁱ

,

k

= ¹ , 2 (22)

where

ℵ (

i

) = ¹

(

i

, 1 , 2 )(

i

, 2 , 4 ) −

²

(

i

)

(

i

, 2 , 4 ) − (

i

)

−(

ⁱ

) (

i

, 1 , 2 )

, =

( 1 ) ( 2 )

and ϒ(

i

) =

(

i

, 1 , 2 ) (

i

) (

i

) (

i

, 2 , 4 )

− ℵ ( 3 −

ⁱ

),

which,finally,leadstothefollowingresult CRB_pipi

= σ

²

2

1

h

(

i

, 1 )

h

(

i

, 2 ) −

^m

(

i

)

²

h

(

i

, 2 ) −

m

( 1 )

−

^m

( 1 )

h

(

i

, 1 )

,

∀

ⁱ

= ¹ , 2 (23)

where

h

(

i

,

j

) = (

i

,

j

, 2

j

)

+ (

j

)(( 3 −

ⁱ

, 3 −

^j

, 2 ( 3 −

^j

))(

j

) − ( 3 −

ⁱ

))

²

( 3 −

ⁱ

) − ( 3 −

ⁱ

, 1 , 2 )( 3 −

ⁱ

, 2 , 4 ) + (( 3 −

ⁱ

,

j

, 2

j

) − ( 3 −

ⁱ

)(

j

))

²

( 3 −

ⁱ

) − ( 3 −

ⁱ

, 1 , 2 )( 3 −

ⁱ

, 2 , 4 ) , (24)

and

m

(

i

) = (

i

) + ( 1 )(( 3 −

i

, 2 , 4 )) − ( 3 −

i

)( 2 ))

²

( 3 −

ⁱ

) − ( 3 −

ⁱ

, 1 , 2 )( 3 −

ⁱ

, 2 , 4 ) + (( 3 −

i

, 1 , 2 )( 2 ) − ( 3 −

i

))

²

( 3 −

i

) − ( 3 −

i

, 1 , 2 )( 3 −

i

, 2 , 4 ) . (25)

3.3. CRBforthephysicalparametervector

κ

Evenifthemodel(7)isusuallyusedinarraysignalprocessing, itsCRBrelatedto

ε

doesnotbringusphysicalinformation.Then, it is interesting to analyzethis CRB regardingthe bearing

θ

and ranger.

Thatis,thenewunknownphysicalparametervectorbecomes

κ ₌

g

( ε ) =

θ

1

,

r₁

, θ

2

,

r₂

,ψ

₁^T

, ψ

^T₂

, α

^T₁

, α

^T₂

^T

(26)

FromCRB(

ε

),wededuceCRB(

κ

)byusingthefollowingformula (see[36,p.45])

CRB

( κ ) = ∂

g

( ε )

∂ ε

^CRB

⁽ ε ) ∂

g

( ε )

∂ ε

T

(27)

wheretheJacobianmatrixisgivenby

∂

g

( ε )

∂ ε ⁼ λ π

d

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

−2 cos1(θ1)

0 0 0 0

r₁sin(θ1) cos²(θ1)

−^r2₁

dcos²(θ1)

0 0 0

0 0

_{−2 cos}¹_(θ

2)

0 0

0 0

^r2sin(θ2)

cos²(θ2)

−^r2₂ dcos²(θ2)

0

0 0 0 0

^π_λ^dI_4T

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

(28)

Using(27)andtheJacobianmatrixabove,weobtain

CRB (θ

_i

) = σ

²

2

λ

²h

(

i

, 2 ) 4 π

²d²

cos

²

(θ

i

)

h

(

i

, 1 )

h

(

i

, 2 ) −

^m

(

i

)

²

, (29)

and

CRB (

r_i

) = ^σ

²

2 λ²m(i)²

h(i,2)d²sin²(θ_i)+^2k(i)r_idsin(θ_i)+^h(i,1)r²_i

π

²d⁴cos⁴(θ_i)

h(i,1)h(i,2)−^m(i)² .

(30)

3.4. Thespecialcaseoftwoorthogonalsources

Inthissubsection,weconsidertheparticularcaseoftwonear- field orthogonal sources.Thiskindofsignal iscommonlyusedin radarandradiocommunicationtofacilitatethetaskofseparating signals [37]. From theabove results, we deducethe CRBfortwo near-field orthogonal sources. The assumption of two orthogonal sourcesmeansthat[38]

T t=1

s^H₁

(

t

)

s2

(

t

) = ^{0 (31)}

Substituting(31)in(18),weobtain

=

⁰2×2

(32)

Consequently, the CRB_pipk for near-field orthogonal sources is givenby

CRBp_ip_k

=

_σ2

2

ℵ (

i

) for

i

=

^k

, ∀

ⁱ

= ¹ , 2

0

for

i

=

k

, ∀

i

,

k

= 1 , 2 (33)

whichleadstothefollowingresults

CRB ( ω

i

) = σ

²

2

(

i

, 2 , 4 )

(

i

, 1 , 2 )(

i

, 2 , 4 ) −

²

(

i

) , (34) CRB (φ

i

) = σ

²

2

(

i

, 1 , 2 )

(

i

, 1 , 2 )(

i

, 2 , 4 ) −

²

(

i

) , (35) CRB (θ

i

) = σ

²

2

λ

²

(

i

, 2 , 4 ) 4 π

²d²

cos

²

(θ

_i

)

(

i

, 1 , 2 )(

i

, 2 , 4 ) −

²

(

i

) , (36)

and

CRB (

ri

) =

^σ2

2 λ²r²_i

(i,2,4)d²sin²(θi)+2(i)ridsin(θi)+(i,1,2)r²_i π²d⁴cos⁴(θi)

(i,1,2)(i,2,4)−²(i) .

(37)

4. TheminimumSNRrequiredtoresolvetwocloselyspaced near-fieldsources

The aimof this section is to derive a closed-form expression oftheminimumSNR,denotedbySNR_min,requiredtoresolvetwo closely spaced near-field sources. Byusing Smith’scriterion [39], the DRL is theparticular value of the distancedNF between two sources forwhich

√

CRB

(

dNF

)

isequaltodNF,wherethedistance betweentwonearfieldsourcesisdefinedas

dNF

=

r²₁

+

^r2₂

− ^2r

1r2

cos (θ

2

− θ

1

). (38)

Consequently,thenewunknownphysicalparametervectoris

j

( κ ) =

r²₁

+

^r2²

−

^2r1r2cos

(θ

2

− θ

1

),

r1

, θ

2

,

r2

, ψ

₁^T

, ψ

^T₂

, α

₁^T

, α

^T₂

T

(39)

Asin(27),weobtainCRB

(

d_NF

)

fromthefollowingformula

CRB (

dNF

) = ∂

j

( κ )

∂ κ

^CRB

⁽ κ ) ∂

j

( κ )

∂ κ

T

1,1

(40)

From (21), (22) and using the relation (27), one notices that CRB

( κ ) =

^CRB_σ⁽2^κ) does not depend on

σ

². Denoting CRB

(

d_NF

) =

CRB(dNF)

σ² equation(40)canbewrittenas

CRB (

dNF

) =

∂

j

( κ )

∂ κ

^CRB

⁽ κ ) ∂

j

( κ )

∂ κ

T

1,1

(41)

Fig. 2.DRLvs.theratioofSNRsinthecaseofaULAof6sensors,100snapshots andSNR1+SNR2=30 dB.

Consequently,oneobtains

CRB (

d_NF

) =

4 i,j=1

∂

j

( κ )

∂ κ

1,i

∂

j

( κ )

∂ κ

1,j

[

^CRB

( κ )]

i,j

(42)

where

∂

j

( κ )

∂ κ

1,1

= − ∂

j

( κ )

∂ κ

1,3

= −

^r1r2

^sin (θ

2

− θ

1

)

d_NF

, (43)

and

∂

j

( κ )

∂ κ

1,2

= − ∂

j

( κ )

∂ κ

1,4

=

^r1

−

^r2

cos (θ

2

− θ

1

)

dNF

. (44)

LetusdefinetheSNRas

SNR = α

₁

²

₊ α

₂

²

σ

²

^{. (45)}

Furthermore, from (40) and (42), one has CRB

(

dNF

) =

^CRB_σ^(d2^NF). Since,CRB

(

dNF

)

is

σ

² independentandtheSNRminisgivenbythe SNRforwhichthefollowingequationholds

σ

²

CRB ( DRL ) =

^r2₁

+

^r₂²

− ^2r

1r₂

cos (θ

2

− θ

1

) (46)

Consequently,from(45)and(46),oneobtains

SNR

_min

=

α

1

²

+ α

2

²

CRB ( DRL )

r²₁

+

^r2₂

− ^2r

1r₂

cos (θ

2

− θ

1

) ⁽⁴⁷⁾

Remark. Weshould precise that usingthe definitionofthe total SNR inthe deterministic case(i.e.,SNR

=

^SNR1

+

^SNR2, inwhich SNR1

=

^||α_σ¹2^||2 and SNR₂

=

^||α_σ²2^||2) to characterize the SNR_min we losethe informationrelated to theratio ofSNRs, i.e., ^SNR_SNR¹

2. Con- sequently,one has to take a precaution using the SNRmin, since, foragivenDRLonemayrequireahigherSNRminiftheratio ^SNR_SNR¹

2

is far from 1 as represented in Fig. 2. Indeed, Fig. 2 shows the variation ofthe DRL w.r.t. ^SNR_SNR¹

2 subjectto a fixed total SNR (i.e., SNR1

+

^SNR2

=

^constant).Furthermore,wecan notice,bysimula- tion,thatthesmallestSNRminpossibleforagivenDRLisobtained inthecaseof ^SNR_SNR¹

2

=

^1.

Fig. 3.Comparisonbetweennumericalandanalytical CRB(ω1)andCRB(φ1)with (θ1,r1)=(30^◦,10λ)and(θ2,r2)=(60^◦,20λ).Linesandmarkersrepresentthenu- mericalandanalyticalCRBs,respectively.

5. Numericalinvestigation:CRB,DRLandgeometrydesign NumericalresultsarepresentedinthissectionforaNULAwith sixsensors(withthefollowinglocations:0, 3

·

^d,4

·

^d,5

·

^d,6

·

^d, 9

·

^{d) thatreceives asignal emittedbytwo sourcesintheFresnel} region inwhich their amplitudes andphasesare generated from arealizationofan i.i.d.Gaussiandistributionwithzeromeanand unit variance unlessspecified. The number ofsnapshots is given by T

=

^100.

5.1. NumericalanalysisoftheCRB

In this section, we analyze the proposed closed-form expres- sions giveninSection 3by comparing theanalytical resultswith thenumerical resultstoillustrate thevalidity ofourderived CRB closedformexpressions.Moreprecisely,

•

^{In Fig. 3,we compare the analyticalCRBs givenin (23)with} thenumericalexactCRBs givenin(6).Thisfigure showsthat theproposedclosed-formexpressionsareinagoodagreement withthetruenumericalCRBs,whichvalidatesourderivedex- pressions.Furthermore,we have thesame resultsfor CRB

(θ )

andCRB

(

r

)

withrespectto(29)and(30).

•

^{From(24),wenoticethat}

∀

ⁱ

=

¹

,

2,h

(

i

,

1

)

is O

(

N³

)

,however h

(

i

,

2

)

is O

(

N⁸

)

,which meansthat theestimation oftheso- calledsecondelectricalangle

φ

iismoreaccuratethanestimat- ingtheso-calledfirstelectrical angle

ω

i.Thisiscorroborated byFig. 3.

•

^{From (22) and using the inversion lemma matrix, we can} quantify the effect of the estimation accuracy of one source ontheotheronebythefollowingrelationship

CRB_p1p1

= σ

²

2 ℵ ( 1 ) + ℵ ( 1 )

CRB_p2p2

ℵ ( 1 )

= σ

²

2 ℵ ( 1 )(

I^2×2

+ ²

σ

²

^CRBp2p2

^{ℵ ( 1 )) (48)}

andinthesamewayweobtain CRB_p2p2

= σ

²

2 ℵ ( 2 )(

I^2×2

+ ²

σ

²

^CRBp1p1

^{ℵ ( 2 )) (49)}

Thus,inthegeneralcase,theestimationerroroftheelectrical anglesofonesourceaffectstheelectricalanglesestimationof

Fig. 4.VariationoftheCRBw.r.t.r1withfixedr2=^{2 m andforequalsourcepower.}

theother source.Whereas, from(48) and(49),we note that thisisnotthecaseifthesourcesareorthogonal,meaningthat theestimationaccuracyofonesourceneitherdependsonnor affectstheestimation accuracyofthe other source.However, itis obviousthat CRBp_ip_i is affected by theposition ofboth sources, moreprecisely, by the distancebetweenthe electric anglesofbothsources(because

(

i

,

k

,

p

)

dependson

ω

and

φ

evenintheorthogonalcase,andthusCRBp_ip_i dependson thedistancebetweentheelectricalangles).

•

^{Unlikethecaseofonesource,theCRBfortwosourcesarenot} phase-invariant. However, we notice that for two orthogonal sources the CRBbecomes phase-invariant. Numericalsimula- tionshowstheimprovementduetoorthogonality.

•

^{For a sufficient number of sensors, CRB}θ_iθ_i and CRBr_ir_i are O

(

1

/

N³

)

.

•

^CRBriri isbearing-invariantandrange-invariant.

•

^For

λ,

ri

∝

^{d,thedependenceontherangeis O}

(

r²_i

)

,meaning that nearer thesource is, the better the estimation(keeping inmindtheFresnelconstraints).ThisiscorroboratedbyFig. 4.

Thedependenceoftherangeonthebearingis O

(

1

/

cos⁴

(θ

i

))

. For

θ

_i closeto

π /

2, we observe that CRBr_ir_i goes toinfinity butfasterthanCRB_θiθi (cf.Fig. 5forCRB(

θ

1)).

•

^{Fig. 6 showsthat, fora different value of bearing andfor a} fixedSNR,thehigherthefrequencyis,thelowertheCRBriri. 5.2. Correlationfactor

Inthefollowing,weinvestigatetheeffectofthecorrelationfac- torontheDRL.Wedefinethecorrelationfactor

ξ

betweenthetwo sourcesas[40]

ξ =

^sH1s2

^s1

^s2

⁽⁵⁰⁾

inwhichsi

= [

^si

(

1

), . . . ,

si

(

T

) ]

^{T denotesthei-thsignalvector.We} usethe same simulation parameters asin Fig. 3,expect the fact thesignalsourcessatisfy(50)foragiven

ξ

.From(34)to(37),we knowthattheorthogonalitybetweentwosourcesimpliestheun- coupling property. Then, since the sources are decoupled,

ξ =

^0, the estimation will be improved (at least for any efficient algo- rithm). In Fig. 7, we compare the correlation effect of the real amplitude of

ξ

on the DRL. Note that, fora given SNR, DRL be- comesgreaterasthevalueoftherealamplitudeof

ξ

increases.In otherwords,thesmallerthecorrelationbetweenbothsources, the

Fig. 5.CRB(θ1)vs.SNRforr1=10λ,(θ2,r2)=(60^◦,20λ)andvaluesofθ1asshown inthelegend.LinesandmarkersrepresenttheanalyticalandnumericalCRBs,re- spectively.

Fig. 6.CRB(r1)vs.SNRfor(θ1,r1)=(30^◦,10λ),(θ2,r2)=(60^◦,20λ)andvaluesof f0asshowninthelegend.Linesandmarkersrepresenttheanalyticalandnumeri- calCRBs,respectively.

better theresolutionlimit performance.Meanwhile, DRLachieves its lowestvalue when

ξ =

^{0,whichmeansthattwo signalsenjoy} aminimum(best)resolutionlimitbetweenthemiftheyhavethe orthogonalrelationship.

5.3. SNR_minbehaviorw.r.t.centralangle

Letusdefinethecentralangleas

θ

c

= θ

1

+ θ

2

2 (51)

InFig. 8,we haveplottedthe SNR_min requiredtoresolves two closelyspacedsourcesfordifferentvaluesof

θ

c.Itshouldbenoted that fora givenresolutionlimit,asthecentralangle

θ

c increases, theminimumsignaltonoiseratiorequiredincreasesalso.

5.4. Minimumresolutionlimitboundary

In practical applications, one mayneed the knowledge of the physicalboundary, whichdelimitstheresolvabilityregion (mean-

Fig. 7.The effect of the amplitude of the correlation factor on the DRL.

Fig. 8.TheSNRminversusthedistancecurvesfromanon-uniformarrayforvarious centralangles.

ing the region for which two near field sources are resolvable).

Althoughthisquestion is worth consideration,to thebest ofour knowledge, no work has been done on it in the literature. In thissubsection, the minimum resolution limit boundary for two sourcesisinvestigatedanddiscussedfordifferentSNRs.

Followingsimilarstepsasthoseemployedin Section4,weob- taintheminimumresolutionlimitboundaryoftwosourcesfordif- ferentSNRvalue.SuchboundaryisrepresentedinFig. 9,inwhich thestarsymbolrepresentsthecoordinatesofthefirstsourcew.r.t.

thereferencesensor

(

xo

,

yo

) = (

0

,

0

)

,inwhich

(θ

1

,

r1

) = (

45^◦

,

6

λ)

,

α

1

²

=

^{1 and}

α

2

²

=

^{1. For different SNR, the second source} must be outside this boundary to ensure a correct resolvability givenbySmith’scriterion.Asexpected,onenoticesthatforhigher SNR, one obtains a small boundary. Furthermore, we can notice thatthisboundaryismorebearingsensitivethanrangesensitive.

Assaidbefore,thisboundaryisaphysicallimitation.Nevertheless, thislattercanbecompensatedbyaproperarraygeometrydesign whichistheaimofthenextsection.

5.5.Effectofsensors’position

Theremainingpartofthissectionisdevotedtothenumerical analysisoftheeffectofsensorarraygeometryontheDRL.Assume

(θ

1

,

r1

) = (

30^◦

,

6

λ)

,

(θ

2

,

r2

) = (

60^◦

,

10

λ)

and

α

1

²

= α

2

²

=

^1.

Fig. 9.DRL boundary for 6-sensors.

Table 1

Differentarraygeometricconfigurations.

Array type Geometric configuration

Type 1 • • • • • ◦ ◦ ◦ ◦ •

Type 2 • ◦ • ◦ • ◦ • ◦ • •

Type 3 • ◦ ◦ ◦ ◦ • • • • •

Type 4 • ◦ ◦ • • • • ◦ ◦ •

Type 5 • • • • • • • • • •

KeepthearrayapertureA

=

^{9 (inunitofd}

= λ/

2)fixedandN

=

^6, thereare10positionsavailableforsixsensors.

InFig. 10,weplotthe DRLwhichcomparestheimpact ofthe different sensorarray geometry w.r.t. SNRmin.Five types ofarray configurationsareconsidered,asshowninTable 1.Type1totype 4arelistedinTable 1withN

=

^{6 sensors,Type5isanULAcon-} figurationwithN

=

10 sensors.

From comparisonofsimulation results,one canfurther notice that

•

^{Forthesame array aperture,sameSNR andsame numberof} sensors,theDRLisgreatlyaffectedbythearraygeometriccon- figuration.

•

^{By comparing the results given from Type 1 to Type 4, the} configurationthat putstwo sensors attheextremity andthe restinthemiddleseemstobethebestgeometryconfiguration (atleast,itisthecaseforA

=

^{9 and N}

=

6 sensors).

Fig. 10.TherequiredSNRmintoresolvetwounknowncloselysourcesfordifferent arraygeometries.

Fig. 11.TherequiredSNRmintoresolvetwounknowncloselysourcesforthebest&

worstperformanceofdifferentsensors.

Table 2

Thebestandworstarraygeometricconfigurationsfordifferentnumberofsensors.

Array type Geometric configuration SNRmin

Type 6 • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ • ^{lowest SNR}min

Type 7 • ◦ ◦ ◦ • • ◦ ◦ ◦ • lowest SNRmin

Type 8 • • ◦ ◦ ◦ ◦ ◦ ◦ • • highest SNRmin

Type 9 • ◦ ◦ ◦ • • • ◦ ◦ • ^{lowest SNR}min

Type 10 • • • ◦ ◦ ◦ ◦ ◦ • • highest SNRmin

Type 11 • ◦ ◦ • • • • ◦ ◦ • ^{lowest SNR}min

Type 12 • • • ◦ ◦ ◦ ◦ • • • highest SNRmin

Type 13 • • • • ◦ ◦ ◦ • • • highest SNRmin

•

^{Type4,withsixsensors, isslightlyworsethan theULAcon-} figurationwith10sensors.

Furthermore, to better comprehend the resolvability perfor- mance as a function of array geometric configuration, consider Fig. 11,wherewecomparetheminimumsignaltonoiseratiowith respecttothebestandworstperformance ofdifferentnumberof sensors. Theconsidered type ofsensor geometryconfigurationis listinTable 2.

From thisfigure, we can see that the Type 11represents the best achievable performance, and Type 8 has the worst perfor- mance. Furthermore, on one hand, we can notice that with the same array aperture, Type6 (with3 sensors)andType13 (with 7 sensors)have,approximately,the sameperformance (Type 6is only 0.2 dB worse than the Type 13); and, on the other hand, one can notice that Type6 (with3 sensors)has a better perfor- manceinregard toType12(whichhas6sensors)by1.5 dB.This meansthatdependingonthearraygeometry,onecanhavebetter performance using a lessnumber of sensors. The relatively poor performance ofthetype8,10,12and13isduetothesensorge- ometry configuration.Consequently, withthe same aperture,one canoptimallydesignthearrayusingtheproposedmethodbelow, inordertoobtainbetterperformanceevenwithlesssensors.

5.6. Sensorsarraygeometrydesign

Asdiscussedabove,theresolvabilityisnotonlyconstrainedby thenumberofsensors,butalsobythearray geometry,whichaf- fectstheoptimalresolutioncapability.

In the following, we give a solution to design an optimalar- rayinordertoenhancetheresolvabilityand/ortocompensatethe performancelossduetothereductionorfailureofcertainsensors.

The optimalmethod isgiven by minimizing SNRmin over sen- sors’ positionamongallpossibilities usingthederived expression (47).However, thisschemeisnot feasiblefora modelwithlarge apertureandsmallnumberofsensors.Moreprecisely,foraperture ofsize A and N sensors,onehas C^N_A+1 possibilities.Toovercome thisdrawbackweproposeasub-optimalfastprocedureasfollows.

Step 1:We placetwo sensors ateach extremitytoobtain the largestaperturethatleadstothebestresolution.

Step 2:We placeonly onesensor by minimizing SNRmin with respecttothe A

−

2 remainingpositions.

Step 3: We iterate the second step by placing the

(

n

+

¹

)

-th sensoratoncebyminimizingsequentiallytheSNR_minwithrespect tothe A

−

ⁿ

+

1 remainingpositions.

In thefollowingpart,weshow that theproposed sub-optimal method produces a nearly optimal result with low complexity.

More precisely, the complexity cost ratioof the optimal method overtheproposedsub-optimalmethodisgivenby

f

(

N

,

A

) =

^Nopt N_prop

=

²

⁽

^A

⁺

¹

^{) !}

N

!(

^A

−

^N

+

¹

)!(

^A

(

A

−

¹

) − (

A

−

^N

+

¹

)(

A

−

^N

+

²

)) (52)

where N_opt meansthenumberofrequirediterationsbyusingthe optimalmethodandN_prop meansthatbyusingtheproposed sub- optimalmethod.

In the examples studied in this section, we consider several cases: Case 1) N

=

^{4, A}

=

^{7; Case 2) N}

=

^{5, A}

=

^{8, so that we} maycomparetheresultsofthetwomethods,asshowninFig. 12.

One notes that our proposed method is nearly identical to the optimal method.Inaddition, from(52),we have f

(

4

,

7

) =

^{6 and} f

(

5

,

8

) =

^{7,whichmeansthattheproposedmethodissixtoseven} times fasterthan the optimalone. Ifwe consider a moreimpor- tantaperture,e.g.,forN

=

^{6, A}

=

^{25,wenoticethattheproposed} method decreasessignificantly the complexity cost by f

(

6

,

25

) =

2558 times compared with the optimal one with the same per- formance.Consequently,wenoticethattheproposedmethodisof lowcomplexity,especiallyforthedesignofsparsearrays.

Oneshouldnotethatfortheabovediscussion,weassumethat derived closed formexpression of SNR_min is given. If no expres- sion ofthe SNRmin is given, one has,first,to numericallyinverse