Theoretical limits on localization in single input multiple output (SIMO) visible light systems

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Theoretical

limits

on

localization

in

single

input

multiple

output

(SIMO)

visible

light

systems

Furkan Kokdogan,

Osman Erdem,

Sinan Gezici

∗

DepartmentofElectricalandElectronicsEngineering,BilkentUniversity,Bilkent,Ankara06800,Turkey

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline31July2018

Keywords:

Cramér–Raolowerbound Estimation

Visiblelight Positioning

Singleinputmultipleoutput(SIMO) Receivedsignalstrength

Inthiswork,atheoreticalaccuracyanalysisisconductedforpositionestimationinvisiblelightsystems basedonreceivedsignalstrength(RSS)measurements.Consideringasinglelightemittingdiode(LED)at thetransmitterandmultiplephoto-detectors(PDs)atthereceiver,theCramér–Raolowerbound(CRLB) is derivedfor bothageneric three-dimensionalscenario and specificconfigurations ofthePDsatthe receiver. For the special case inwhichthe height ofthe receiveris known,a compact expression is derivedfortheCRLB,consideringauniformcircularlayoutandthesameelevationangleforallthePDs. Inaddition,theoptimalplacementofthePDsatthereceiverisinvestigatedbytakingtheeffectsofthe elevationangleparameterofthePDsintoconsideration.Theoptimal valuesare obtainedtheoretically and alsoverifiedbysimulations.Numerical examplesare providedtoillustratethe impactsofvarious systemparametersonlocalizationaccuracyandtocomparethetheoreticallimitsagainstthemaximum likelihoodestimator(MLE)forthereceiverposition.

©2018ElsevierInc.Allrightsreserved.

1. Introduction

Wirelessindoor localization hasbeen a popular research and development area in both academic and industrial communities [1–4]. Since the Global Positioning System (GPS) cannot provide highlocalization accuracy for indoor environments [5,6], various radio-frequency (RF) based solutions, such as WiFi and ultra-wideband (UWB) systems, are developed for indoor localization [1,2,7,8]. As an alternative to RF based localization, light emit-tingdiode(LED)based visiblelightpositioning (VLP) systemsare proposedandinvestigatedinrecentstudies [9].Accurateposition informationcan beobtainedviaVLPsystemssincea line-of-sight path is commonlypresent and significantly stronger than multi-path components in a visible light channel. In addition, visible lightsystemscansimultaneouslybeusedforilluminationanddata transmission,aswell[10,11].

PositionestimationinVLPsystemscanbe performedby utiliz-ingdifferenttypesofparameterssuch asreceivedsignal strength (RSS),time-of-arrival(TOA),time-difference-of-arrival (TDOA),and angle-of-arrival(AOA). RSS basedVLPsystems rely onestimating thepositionofavisiblelightcommunication(VLC)receiverbased onRSSmeasurements atthe receiverrelatedto anumberofLED

*

Correspondingauthor.

E-mailaddresses:kokdogan@ee.bilkent.edu.tr(F. Kokdogan), oerdem@ee.bilkent.edu.tr(O. Erdem),gezici@ee.bilkent.edu.tr(S. Gezici).

transmitters [12–17]. The study in [13] uses RSS measurements foraccurate positioningof deviceswith light sensing capabilities byutilizingexistingLEDlampsastransmittersandperforming tri-lateration. Particlefilteringbasedposition trackingisproposed in [12], where the position is estimated via RSS measurements. In [16],thebasic-framedslottedALOHA(BFSA)protocolisemployed inaVLCbasedindoorpositioningsystemwhichusesRSS informa-tion for estimating positions of VLC receivers. The work in [17] proposes a carrier allocation VLC system utilizing the intensity modulationanddirectdetection(IM/DD)methodandreports cen-timeterlevelpositioning accuraciesthroughexperimental studies. In additionto the RSSparameter, the TOAandTDOA parameters are also used forposition estimation in the literature [18,19]. In [19],anindoorVLPsystemthatemploysTDOAmeasurementsfrom three LEDtransmitters isproposed fortwo-dimensional position-ing.The useofthe AOAparameterisalsoconsidered forposition estimationinVLPsystems[20–22].Inaddition,[23] investigatesa hybridpositioningsystemthatutilizesbothAOAandRSS informa-tion, where the position estimationis performed via a weighted leastsquaresestimator.

The subject of this manuscript is on RSS based VLP systems in the presence of a single LED transmitter and multiple photo-detectors(PDs) atthereceiver, whichcanberegardedasasingle inputmultipleoutput(SIMO)system.Invisiblelightchannels,the line-of-sightcomponentissignificantlystrongerthanreflectedand scatteredcomponents.Forthisreason,theRSSparametercan

pro-https://doi.org/10.1016/j.dsp.2018.07.017

vide accurate positioninformationinVLPsystems anditis com-monlypreferredduetoitslow-cost andhighaccuracy [9].Inthe literature,there existsome studieson SIMOVLPsystemssuch as [14,21,24]. In[14],a VLPsystemconsistingofasingle LED trans-mitterandmultiple (three)optical receivers isdesigned, andthe positionof thereceiverunit is estimatedbasedon RSS measure-ments and the relative positions among the receivers. The study in[21] investigatesthe useofboth AOAandRSSinformation for three-dimensional positioning in an indoor localization scenario withasingleLEDtransmitterandmultipletiltedopticalreceivers. Thethree-dimensionalpositionisestimatedbasedonthegain dif-ferences among the tilted PDs, which are functions of both the AOAandthe RSS.In[24],an aperture-basedangulardiversity re-ceiverisproposedforaVLPsystemwhichemploysareceiverwith multiple elements, each consistingof an aperture anda PD. The RSSmeasurementsatthePDsareusedtoestimatethepositionof thereceiverandtheCramér–Raolowerbound(CRLB)iscalculated fortheconsideredmodel.Multipletransmitterand/or receiver el-ementsare alsoconsidered forvisible light communication (VLC)

systems and advantages in terms of data rate and capacity are investigated [25,26]. For example, [26] proposes novel angle di-versity receiver designs,calledpyramid receiver andhemispheric receiver, formultiple input multipleoutput (MIMO) VLC systems andanalyzes the effects of the PD elevation angle on the chan-nel capacity. The study in [27] investigates a three-dimensional multipleinputsingleoutput (MISO)VLPsystem,whichsolvesthe positioning problem via a modified particle swarm optimization (PSO)algorithmtoreducethecomputationalcomplexity.The pro-posed low-complexity estimator is shown to achieve positioning accuracyontheorderofcentimeters.

ToprovidebenchmarksforVLPalgorithmsintheliteratureand toillustratetheeffectsofsystemparametersonlocalization perfor-mance, theoretical accuracy limitsare obtained in various recent studies such as [18,23,24], [28–34]. In [28], the CRLB is derived for distance estimation based on RSS measurements in a visible light system, andits dependenceon system parameters, such as thesignalbandwidth,LEDconfiguration,andtransmitterheight,is investigated. The study in [18] presents the CRLBon TOAbased distance estimation in a synchronous VLC system, and analyzes theeffects ofvarious systemparameters, such asthearea ofthe photodetector,sourceopticalpower,andcenterfrequency,onthe estimation accuracy. In [29], the CRLBs and maximumlikelihood estimators(MLEs)areobtainedforsynchronousandasynchronous VLPsystems,andhybridTOA/RSSbaseddistanceestimationis pro-posed,whichutilizesboththetimedelayparameterandthe chan-nel attenuation factor. Also, comparisons are provided based on theanalyticalCRLBexpressionsforTOAbased,RSSbased,and hy-bridTOA/RSSbased distanceestimation.The workin [23],which focusesonAOAandRSSbasedthree-dimensionallocalization, de-rivesa CRLBexpression forRSSbasedthree-dimensional localiza-tion for a generic deployment scenario. In [33], direct and two-steppositioning approachesare investigatedforsynchronous and asynchronous VLP systems andCRLB expressions are derived for generic configurations. In addition,the effects ofcooperation are quantifiedforasynchronousVLPsystemsbasedonageneric CRLB expressionin[34].AsanalternativetotheCRLB,[31] and[32] fo-cusontheZiv–Zakaibound(ZZB)fordistanceestimationin asyn-chronousandsynchronousVLPsystems,respectively.Inparticular, [31] derivestheZZBinRSSbasedVLPsystemsandprovides com-parisonswiththemaximuma-posterioriprobability(MAP)andthe minimummean-squarederror(MMSE)estimators.

In this manuscript, theoretical accuracy limits, namely, CRLB expressions,areobtainedforSIMOVLPsystemsbasedonRSS mea-surements.First,ageneric three-dimensionallocalizationscenario isconsidered,andthenspecificconfigurationsofthePDsatthe re-ceiverare considered. Forthecasewitha known receiverheight

andauniformcircularlayoutforthePDs(whichisacommonand efficient configuration, as investigated in [26]), a compact CRLB expression is derived, which is used to obtain asymptotic CRLB expressions whenthe radiusofthe circularlayoutis significantly smaller(or,larger)thanthedistancebetweenthereceiverandthe projection ofthe LED transmittertothe floor.In addition, guide-lines areprovidedregarding theoptimalplacementofthePDsat thereceiverbasedontheCRLBexpressions.Furthermore,theCRLB expressions are compared against the MLE forthe receiver posi-tion. The main contributions and the novelty of the manuscript canbesummarizedasfollows:

•

TheCRLBisderivedforagenericthree-dimensionalSIMOVLP forthefirsttimeintheliterature.1

•

AnewcompactCRLBexpressionisobtainedwhenthereceiver is at a known height with identical PDs arranged in a uni-formcircularlayout(see Fig.2).Inaddition,asymptoticCRLB expressions are obtainedinthisscenariofordifferentsystem configurations(Lemma1andLemma2).

•

Under certain conditions, it is proved that it is optimal (in termsofCRLBminimization)toplacethePDstothemaximum possible radius in the uniform circular layout (Section 3.3, Proposition1).

•

The MLE forthe SIMO VLPsystemis derived and its perfor-manceiscomparedagainsttheCRLB.

In addition,various numericalexamples arepresented to investi-gatetheeffectsofsystemparameters,suchastheelevationangles ofPDs,layoutradius,andnumberofPDs,onlocalizationaccuracy. The rest of the manuscript is organized as follows: The sys-temmodelisdescribedinSection 2.TheCRLBexpressionsinthe generic three-dimensional scenario and the special cases with a known receiver height are obtained in Section 3, which also in-cludes asymptotic CRLB expressions and guidelines for optimal placement of PDs. Section 4 presents simulation resultsand dis-cussions, and Section 5 concludes the manuscript with remarks andfutureworkdirections.

2. Systemmodel

Consider aSIMOVLPsystemasillustrated inFig.1,wherethe transmitterconsistsofasingle LED andthe VLCreceiverconsists of N PDs. The location of the LED is represented by a three-dimensionalcolumnvectordenotedaslT,whichisknownbythe VLC receiver. Namely, lT

(

1

)

, lT

(

2

)

, andlT

(

3

)

are the elements of lT that specifythex, y,andzcomponents,respectively. (In prac-tice, the LED can periodicallybroadcast its location information.) The unknown location of the VLC receiver is represented by lR andthe locationofthe nth PD intheVLC receiverisdenoted by lR

+

an.In particular,lR

(

1

)

,lR

(

2

)

,andlR

(

3

)

specifythex, y,and z componentsof lR, respectively,andan

(

1

)

, an

(

2

)

,andan

(

3

)

are theelementsofan thatspecifythex,yandzcomponents, respec-tively.Thevectorsanareknownparameterswhichcanbeadjusted accordingto the layoutdesign inthe system. Based onthe Lam-bertianformula,theRSSobservation(measurement)atthe

nth

PD canbeexpressedas[25] PRn

=

m

+

1 2

π

PTcos m

_(φ

n

)

cos

(θ

n

)

Sn dn2

I

{θn≤θFOV,n

} +

η

n (1)

for n

∈ {

1

,

. . . ,

N

}

, where m is the Lambertian order of the LED,

PT isthetransmitpower, Sn isthearea ofthe

nth

PD,

φ

n isthe

1 _{Thestudyin[24] presentsaCRLBexpressionforasystememployingmultiple} PDs(eachwithaperture)atthereceiverbyconsideringatwo-dimensionalscenario (i.e.,knownreceiverheight)andperpendicularPDsatthereceiver.

Fig. 1. SIMO VLP system.

irradiation angle withrespect to the nth PD,

θ

n isthe incidence angleforthe

nth

PD,

θ

FOV,n isthefieldofview(FOV)ofthe

nth

PD,

I{·}

denotestheindicatorfunction,

d

n isthedistancebetweenthe

nth PD andtheLED,and

η

n isthemeasurementnoise atthe

nth

PD,which is modeled by a zeromeanGaussian randomvariable withvariance

σ

2

n [23]. Itisassumedthatthemeasurement noise

η

n isindependent acrossdifferentPDs.Also, distance

d

n between the

nth

PDandtheLEDcanbeexpressedas

dn

= 

lR

+

an

−

lT

.

(2)

Let nR,n denote the normal vector for the nth PD, as shown in Fig.1. Then, the cosine of theincidence angle forthe nth PD is givenby cos

(θ

n

)

=

nT R,n

(

lT

−

lR

−

an

)

dn

.

(3)

Assumingthat theLEDisabovetheVLCreceiverandfaces down-wards withthenormal vector nT

= [

0 0

−

1

]

T, thecosine ofthe irradiationangleisobtainedas

cos

(φ

n

)

=

lT

(

3

)

−

lR

(

3

)

−

an

(

3

)

dn

(4)

wherelT

(

3

)

,

l

R

(

3

)

, andan

(

3

)

representthe third elements of lT, lR,andan,respectively.Itisnotedthattheassumptionleadingto (4) is validin mostpractical scenarios since LEDsare commonly insertedattheceilingofaroomandfacedownwardstohave effi-cientillumination[18,22,28].

3. CRLBderivationsandreceiverdesign

In this section, a generic CRLB expression is derived for the SIMOVLPsystem,andspecificexpressionsareobtainedunder spe-cialcases. Then, optimum placement ofPDs ina VLC receiveris consideredforuniformcircularlayouts.

LetPR representavectorconsistingoftheRSSobservationsat thePDs;that is, PR

= [

PR1

· · ·

PRN

]

T_{, where P}

Rn isasin(1). The

aimistoestimatethelocationoftheVLCreceiver,lR,basedonPR. TocalculatetheCRLBforthisestimationproblem,the probability densityfunction (PDF) of PR conditioned on lR can be obtained from(1) asfollows: p

(

PR

|

lR

)

=



_N



n=1 1

√

2

π σ

n



exp



−

N



n=1

(

PRn

−

fn

(

lR

))

2 2

σ

2 n



(5) with fn

(

lR

)



(

m

+

1

)

PTcosm

(φ

n

)

cos

(θ

n

)

Sn 2

π

d2 n (6)

whereitisassumedthattheLEDisintheFOVofallthePDs.From (3) and(4), fn

(l

R

)

in(6) canbespecifiedas

fn

(

lR

)

=

PT

(

m

+

1

)

Sn

(

lT

(

3

)

−

lR

(

3

)

−

an

(

3

))

m 2

π



lT

−

lR

−

an



m+3

×

nT_R,n

(

lT

−

lR

−

an

) .

(7)

TheCRLBprovidesalowerlimitonmean-squarederrors(MSEs) ofunbiased estimators.Forthe consideredSIMOVLP system,the CRLBforestimatingthelocationoftheVLCreceivercanbestated asfollows:

E

 ˆ

lR

−

lR

2

≥

trace



J

(

lR

)

−1



C R L B (8)

where

ˆ

lR is an unbiased estimate of lR and J

(l

R

)

is the Fisher informationmatrix(FIM)givenby

J

(

lR

)

= E{(∇

lRlog p

(

PR

|

lR

))(

∇

lRlog p

(

PR

|

lR

))

T

_}

₍₉₎

with

∇

lR denotingthegradientoperatorwithrespecttolR [35].

From(5),theelementsoftheFIMin(9) canbecalculatedafter somemanipulationas

[

J

(

lR

)

]

i j

=

N



n=1 1

σ

2 n

∂

fn

(

lR

)

∂

lR

(

i

)

∂

fn

(

lR

)

∂

lR

(

j

)

(10)

for

i

,

j

∈ {

1

,

2

,

3

}

,where fn

(l

R

)

is asin(7) and

l

R

(

i

)

denotes the

ith elementoflR.Basedon(7), thepartialderivativesin(10) can beobtainedasfollows:

∂

fn

(

lR

)

∂

lR

(

1

)

=

cn

(

un

(

3

))

m

−

nR,n

(

1

)

dn2

+ (

m

+

3

)

un

(

1

)

nTR,nun dmn+5 (11)

∂

fn

(

lR

)

∂

lR

(

2

)

=

cn

(

un

(

3

))

m

−

nR,n

(

2

)

dn2

+ (

m

+

3

)

un

(

2

)

nTR,nun dmn+5 (12)

∂

fn

(

lR

)

∂

lR

(

3

)

=

cn

(

un

(

3

))

m

−

nR,n

(

3

)

dn2

+ (

m

+

3

)

un

(

3

)

nT_R,nun dmn+5

−

cnm

(

un

(

3

))

m−1nTR,nundn−m−3 (13) where cn



(

m

+

1

)

PTSn 2

π

and un



lT

−

lR

−

an (14)

withnR,n

(

i

)

andun

(

i

)

denotingthe ith elements ofnR,n andun, respectively.

The FIM in (9) can be evaluated via (10)–(14), and the CRLB fortheSIMOVLPsystemcanbecalculatedfrom(8).Theprovided CRLBexpressioncanbeemployedtoevaluateperformanceof prac-ticalSIMOVLPsystems.ItisnotedthattheCRLBexpression speci-fiedby(8)–(14) isgenericforanyconfigurationandorientationof thePDsattheVLCreceiverandisvalidforthree-dimensional sce-narios.SuchaCRLBexpressionhasnotbeenavailableforSIMOVLP systems in the literature. Three-dimensional localization can be requiredinsomepractical systems.Forexample,automatic store-house management systems inthe industry utilize navigation of locationawarerobots.Inordertoconstructahighly efficient sys-tem,three-dimensionalpositionsofroboticarmsshouldbeknown with highprecision. Such a system can be designedby utilizing VLPalgorithms,whichalsoprovideefficientilluminationand com-municationsamongrobots.

In the following sections,some special casesof the CRLB ex-pressionareinvestigated.

3.1. Known height and perpendicular PDs

Consider a scenario in which the height of the VLC receiver,

lR

(

3

)

, is known and the PDs point upwards such that nR,n

=

[

0 0 1

]

T _{and a}

n

(

3

)

=

0 for n

=

1

,

. . . ,

N.

In other words, all the PDsareatthesameknownheightandpointupvertically.Sucha scenariocan, forexample,be encountered inroboticapplications wheretheVLCreceiverismountedonthetopofarobotthat has aknownheight[9].

Inthisscenario, fn

(l

R

)

in(7) reducestothefollowing: fn

(

lR

)

=

(

m

+

1

)

PTSn

(

lT

(

3

)

−

lR

(

3

))

m+1 2

π



lT

−

lR

−

an



m+3

.

(15)

Then,the2

×

2 FIMcanbeobtainedbasedon

[

J

(l

R

)

]

i j in(10) for

i

,

j

∈ {

1

,

2

}

asfollows2: J

(

lR

)

=



J11 J12 J21 J22



(16) where J11

=

α

N



n=1 S2_n

σ

2 n

(

lRn

(

1

)

−

lT

(

1

))

2

||

lR n

−

lT

||

2m+10

,

(17) J22

=

α

N



n=1 S2_n

σ

n2

(

lRn

(

2

)

−

lT

(

2

))

2

||

lR n

−

lT

||

2m+10

,

(18) J12

=

J21

=

α

N



n=1 S2 n

σ

2 n

(

lRn

(

1

)

−

lT

(

1

))(

lRn

(

2

)

−

lT

(

2

))

||

lR n

−

lT

||

2m+10 (19) withlR n



lR

+

an,

l

Rn

(

i

)



lR

(

i

)

+

an

(

i

)

,and

α



(

m

+

1

)

2 4

π

2 P 2 T

(

m

+

3

)

2

(

lT

(

3

)

−

lR

(

3

))

2m+2

.

(20) From (16)–(19),the CRLBin(8) can be calculatedinclosed form as in (21). Based on (21), the CRLB can easily be evaluated for variousscenariosinwhichthePDsareatthesameknownheight andpointupvertically.

3.2. Known height, identical PDs, equal noise variances, and uniform circular layout

Acommonconfiguration forPDsata VLCreceiver isthe uni-formcircularlayout,asinvestigatedin[26].Inthissection,thePDs areassumedtobe identicalinthesense that theyhaveequal ar-eas; that is, Sn

=

S for n

=

1

,

. . . ,

N, andthe noise variancesare modeled to be thesame; i.e.,

σ

2

n

=

σ

2 for

n

=

1

,

. . . ,

N.

In addi-tion,itisassumedthatallthePDsareatthesameknownheight, tiltedwiththesameangle

β

,andareplacedinauniformcircular layout, as illustrated in Fig. 2-(a). (The motivations for this con-figuration can be found in [26].) It is notedthat if

β

∈ (

0

,

π

/

2

)

, the PDsface outwards with respectto the circular layout and if

β

∈ (−

π

/

2

,

0

)

,thePDsfaceinwards.

Inthisscenario,theradiusofthecircularlayoutisrepresented by R, andtheheight differencebetweentheLED andtheVLC re-ceiverisdenotedby H ; thatis,

C R L B

=

4π2 P2 T(m+1)2(m+3)2(lRn(3)−lT(3))2m+2

N n=1Sn 2 σ2 n (l_Rn(1)−lT(1))2+(lRn(2)−lT(2))2 ||lR n−lT||2m+10

(

N_n=1Sn2 σ2 n (lRn(1)−lT(1))2 ||lR n−lT||2m+10

)(

N n=1 Sn2 σ2 n (lRn(2)−lT(2))2 ||lR n−lT||2m+10

)

− (

N n=1 Sn2 σ2 n (lRn(1)−lT(1))(lRn(2)−lT(2)) ||lR n−lT||2m+10

)

2 (21)

2 _{Sincetheheightisknown,thereexistonlytwounknowns,l}

R(1)andlR(2),in

thisscenario.

Fig. 2. Uniform circular layout. (a) Three-dimensional view. (b) Top view.

lT

(

3

)

−

lR

(

3

)

=

H

.

(22)

Inaddition,theelementsofan canbewrittenas an

(

1

)

=

R cos

ψ

n

,

an

(

2

)

=

R sin

ψ

n

,

(23) an

(

3

)

=

0

,

for

n

=

1

,

. . . ,

N,

with

ψ

n



2

π

(

n

−

1

)

N

+ ¯ψ

(24)

where

¯ψ

is a random shift angle with 0

≤ ¯ψ <

2

π

/

N (see

Fig. 2-(b)). Then, the distance betweenthe LED and the nth PD canbeexpressed,from(22) and(23),as

dn

= 

lR

+

an

−

lT



=



(

dx

+

R cos

ψ

n

)

2

+ (

dy

+

R sin

ψ

n

)

2

+

H2 (25) where

d

x,

d

yand

D (see

Fig.1)aredefinedas

dx



lR

(

1

)

−

lT

(

1

),

(26) dy



lR

(

2

)

−

lT

(

2

),

(27) D





d2 x

+

d2y

.

(28)

Here, D is the magnitudeof the horizontal component(lying in the x-y plane) of the distance vector between the receiver unit andthe originoftheroom. Also,the normalvectorsforeachPD canbeexpressedas

nR,n

(

1

)

=

sin

β

cos

ψ

n

,

nR,n

(

2

)

=

sin

β

sin

ψ

n

,

(29)

nR,n

(

3

)

=

cos

β.

Based on the specifications/definitions in (22)–(29), the ele-mentsoftheFIMcanbeobtainedfrom(10)–(14) asfollows:

J11

= ˜

a N



n=1 A2_n (30) J22

= ˜

a N



n=1 B_n2 (31) J12

=

J21

= ˜

a N



n=1 AnBn (32) where

˜

a



(

m

+

1

)

2_P2 TS2H2m 4

π

2

_σ

2 (33) An



−

sin

β

cos

ψ

n dmn+3

+ (

m

+

3

)

un

(

1

)

(34)

×

sin

β

cos

ψ

nun

(

1

)

+

sin

β

sin

ψ

nun

(

2

)

+

cos

β

un

(

3

)

dmn+5

Bn



−

sin

β

sin

ψ

n dmn+3

+ (

m

+

3

)

un

(

2

)

(35)

×

sin

β

cos

ψ

nun

(

1

)

+

sin

β

sin

ψ

nun

(

2

)

+

cos

β

un

(

3

)

dmn+5

·

Then,theCRLBin(8) canbeobtainedas

C R L B

=

1 ˜ a

N n=1

(

A2n

+

Bn2

)

N n=1A2n

N n=1B2n

−



N n=1AnBn



2

.

(36)

In order to provide simple approximate expressions for the CRLBinthisscenario,thefollowingresultsarepresented.

Lemma1.

As D

/

R

−→

0,

the CRLB in

(36) can

be approximated as

C R L B

≈

2 J11 (37) where J11is given by J11

=

˜

c 2N



(

H2

+

R2

)

2sin2

β

−

2

(

H2

+

R2

)(

m

+

3

)

R

×

sin

β (

R sin

β

−

H cos

β)

+ (

m

+

3

)

2R2



R2sin2

β

+

H2cos2

β

−

H R sin 2

β )

] (38) with

˜

c



(

m

+

1

)

2_P2 TS2H2m 4

π

2

_σ

2

₍

_H2

₊

_R2

₎

m+5

.

(39)

Proof. As D

/

R

→

0, the distance from the LED to each PD be-comes approximately dn

≈

√

H2

₊

_R2

₌

_{d. Also, u} n

(

1

)

and un

(

2

)

canbeapproximatedas un

(

1

)

≈ −

R cos

ψ

n

,

un

(

2

)

≈ −

R sin

ψ

n

.

(40)

Then,substituting

d

n,

u

n

(

1

)

and

u

n

(

2

)

into(34), J11in(30) canbe statedaftersomemanipulationas

J11

≈ ˜

c N



n=1



d4sin2

β

cos2

ψ

n

−

2d2sin2

β(

m

+

3

)

R2cos4

ψ

n

+

2d2sin2

β(

m

+

3

)

R2cos2

ψ

nsin2

ψ

n

+

d2sin 2

β(

m

+

3

)

×

H R cos2

ψ

n

+ (

m

+

3

)

2sin2

β

R4cos6

ψ

n

+ (

m

+

3

)

2

Table 1

Listoftrigonometricsummationsforψnin(24).

Expression 1 Expression 2 Result

n cosψn n sinψn 0 n cos2_ψ n n sin2ψn N/2 n cosψnsinψn – 0 n cos3_ψ n n sin3_ψ n 0 n cos2_ψ nsinψn n cosψnsin2ψn 0 n cos4_ψ n n sin4ψn 3N/8 n cos3_ψ nsinψn n cosψnsin3ψn 0 n cos 2_ψ nsin2ψn – N/8 n cos5 ψn n sin5 ψn 0 n cos4 ψnsinψn n cosψnsin4ψn 0 n cos3 ψnsin2ψn n cos2 ψnsin3ψn 0 n cos6_ψ n n sin6_ψ n 5N/16 n cos5_ψ nsinψn n cosψnsin5ψn 0 n cos4_ψ nsin2ψn n cos2_ψ nsin4ψn N/16 n cos3_ψ nsin3ψn – 0

×

sin2

β

R4cos2

ψ

nsin4

ψ

n

+ (

m

+

3

)

2cos2

β

H2R2cos2

ψ

n

+

2

(

m

+

3

)

2sin2

β

R4cos4

ψ

nsin2

ψ

n

− (

m

+

3

)

2sin 2

β

×

H R3cos4

ψ

n

− (

m

+

3

)

2sin 2

β

H R3cos2

ψ

nsin2

ψ

n

)

(41) wherec is

˜

definedas

˜

c



a

˜

(

H2

₊

_R2

₎

m+5 (42)

witha being

˜

asin(33). Theexpressionin(41) canalsobestated as J11

≈ ˜

c



d4sin2

β

N



n=1 cos2

ψ

n

−

2d2sin2

β(

m

+

3

)

R2

×

N



n=1 cos4

ψ

n

+

2d2sin2

β(

m

+

3

)

R2 N



n=1 cos2

ψ

nsin2

ψ

n

+

d2sin 2

β(

m

+

3

)

H R N



n=1 cos2

ψ

n

+ (

m

+

3

)

2sin2

β

R4 N



n=1 cos6

ψ

n

+ (

m

+

3

)

2sin2

β

R4 N



n=1 cos2

ψ

nsin4

ψ

n

+ (

m

+

3

)

2cos2

β

H2R2 N



n=1 cos2

ψ

n

+

2

(

m

+

3

)

2sin2

β

R4 N



n=1

cos4

ψ

nsin2

ψ

n

− (

m

+

3

)

2sin 2

β

H R3 N



n=1 cos4

ψ

n

− (

m

+

3

)

2sin 2

β

H R3 N



n=1 cos2

ψ

nsin2

ψ

n



.

(43)

From (24), the summation terms in (43) can be calculated as shown in Table 1.Then, reorganizing the terms andsubstituting thevalueof

d (i.e.,

√

H2

₊

_R2_{)into(43), J}

11isobtainedasin(38). Basedonasimilarapproach,theotherelementsoftheFIMcan beobtainedas

J22

=

J11 (44)

From(44) and(45),theCRLBcanbeobtainedvia(8) as C R L B

=

J11

+

J22 J11J22

− (

J12

)

2

=

2 J11

.

(46)

Therefore, for D



R, the CRLB can be approximated asin (37), where J11isgivenby(38).

2

Lemma2.

As R

/

D

−→

0,

the CRLB in

(36) can

be approximated as

C R L B

≈

J11

+

J22

J11J22

−

J12J21

(47) where J11, J22, J12, and J21are given by (48)–(50).

Proof. As R

/

D

→

0,thedistancefromtheLEDtoeachPDcanbe approximatedas

d

n

≈

√

H2

₊

_D2

_

_{d. Substitutingtheexactvalues} of

u

n

(

1

)

and

u

n

(

2

)

(see(14))into(34),thefirstelementoftheFIM in(30) canbecalculatedaftersomemanipulationas

J11

≈ ¯

c N



n=1



d4sin2

β

cos2

ψ

n

−

2d2sin2

β(

m

+

3

)

cos2

ψ

n

× (

dx

−

R cos

ψ

n

)

−

2d2sin2

β(

m

+

3

)

cos

ψ

nsin

ψ

n

× (

dx

−

R cos

ψ

n

)(

dy

−

R sin

ψ

n

)

−

d2sin 2

β(

m

+

3

)

H

×

cos

ψ

n

(

dx

−

R cos

ψ

n

)

+ (

m

+

3

)

2sin2

β

cos2

ψ

n

× (

dx

−

R cos

ψ

n

)

4

+ (

m

+

3

)

2sin2

β

sin2

ψ

n

(

dx

−

R cos

ψ

n

)

2

× (

dy

−

R sin

ψ

n

)

2

+ (

m

+

3

)

2cos2

β

H2

(

dx

−

R cos

ψ

n

)

2

+

2

(

m

+

3

)

2sin2

β

cos

ψ

nsin

ψ

n

(

dx

−

R cos

ψ

n

)

3

× (

dy

−

R sin

ψ

n

)

+ (

m

+

3

)

2sin 2

β

H cos

ψ

n

(

dx

−

R cos

ψ

n

)

3

+ (

m

+

3

)

2sin 2

β

H sin

ψ

n

(

dx

−

R cos

ψ

n

)

2

(

dy

−

R sin

ψ

n

) )

(51) where

¯

c



a

˜

(

H2

₊

_D2

₎

m+5 (52)

witha being

˜

asin(33).From (24), the summationtermsin (51) can be calculated as specified in Table 1. Then, reorganizing the terms andsubstituting the value of d (i.e.,

√

H2

₊

_D2_{) into (51),}

J11canbesimplifiedtotheexpressionin(48).Followingasimilar approach,the remaining elements ofthe FIMcan be obtainedas in (49) and (50). Then, for D



R, the CRLB can be found as in (47).

2

For scenarios with known receiver heights, identical PDs and uniformcircularlayouts,Lemma1andLemma2provide approx-imate closed-form expressions forthe CRLB when the horizontal distance between the LED and the center of the receiver is sig-nificantlylargerorsignificantlysmallerthantheradiusofthe cir-cularlayout, respectively. Inother words, theCRLB expression in

J11

≈

¯

c 8N



4d4sin2

β

−

8d2sin2

β(

m

+

3

)(

dx2

+

R2

)

+

4d2sin

(

2

β)(

m

+

3

)

H R

+

sin2

β(

m

+

3

)

2



4



dx4

+

dx2dy2

+

R4



+

R2



27dx2

+

dy2



−

4 sin

(

2

β)(

m

+

3

)

2H R



4dx2

+

R2



+

4 cos2

β(

m

+

3

)

2H2



2dx2

+

R2



(48) J22

≈

¯

c 8N



4d4sin2

β

−

8d2sin2

β(

m

+

3

)(

dy2

+

R2

)

+

4d2sin

(

2

β)(

m

+

3

)

H R

+

sin2

β(

m

+

3

)

2



4



dy4

+

dy2dx2

+

R4



+

R2



27dy2

+

dx2



−

4 sin

(

2

β)(

m

+

3

)

2H R



4dy2

+

R2



+

4 cos2

β(

m

+

3

)

2H2



2dy2

+

R2



(49) J12

=

J21

≈

¯

c 4

(

m

+

3

)

Ndxdy



sin2

β



(

m

+

3

)



13R2

+

2D2



−

4d2



−

8 sin

(

2

β)

H

(

m

+

3

)

R

+

4 cos2

β(

m

+

3

)

H2



(50)

Lemma1isexpectedtobeaccurate whenthereceiverisdirectly undertheLEDwhereasthatinLemma2becomesaccurate asthe receiver islocated away fromtheprojection of theLED onto the

x

−

y plane (i.e.,thefloor).

3.3. Known height, perpendicular and identical PDs, equal noise variances, and uniform circular layout

In thissection, itis assumedthat allthe PDsare identical,at the same known height, point up vertically, andare placed in a uniformcircular layout(similarto [18,22,28]).Since thisscenario isa specialcaseofthescenario intheprevioussection for

β

=

0 (seeFig.2),theexactCRLBexpressioncanstill becalculatedfrom (36) bysimplifyingthe definitionsof An andBn in(34) and(35) as An



(

m

+

3

)

un

(

1

)

un

(

3

)

dmn+5

,

(53) Bn



(

m

+

3

)

un

(

2

)

un

(

3

)

dmn+5

.

(54)

Inaddition,theresultsinLemma1andLemma2canbeemployed toobtainsimpleapproximateexpressionsfortheCRLBinthiscase.

Corollary1.

When all the PDs point up vertically, the CRLB in

(36)

mul-tiplied by R2

_/(

_H2

₊

_R2

₎

m+5_{has the following limit as D}

_/

_R

_−→

_0:

lim D R→0 R2C R L B

(

H2

₊

_R2

₎

m+5

=

16

π

2

_σ

2 S2

₍

_m

₊

₁

₎

2

₍

_m

₊

₃

₎

2_P2 TH2m+2N

.

(55)

Proof. As D

/

R

−→

0,theCRLBin(36) convergesto theCRLB ex-pressionspecifiedby(37) and(38) inLemma1.WhenallthePDs pointupvertically,i.e.,when

β

=

0, J11in(38) becomes

J11

=

˜

c 2

(

m

+

3

)

2_H2_R2_N

_.

₍₅₆₎

Then,theCRLBin(37) isobtainedas

C R L B

=

4

˜

c

(

m

+

3

)

2_H2_R2_N

.

(57)

Inserting thedefinition ofc in

˜

(39) into(57),the followingCRLB expressionisobtainedfor

D

/

R

−→

0:

C R L B

=

16

π

2

_σ

2

₍

_H2

₊

_R2

₎

m+5 S2

₍

_m

₊

₁

₎

2

₍

_m

₊

₃

₎

2_P2

TH2m+2N R2

.

(58)

Therefore,as D

/

R

−→

0, thelimit oftheCRLBin(58) multiplied by

R

2

_/(

_H2

₊

_R2

₎

(m+5) _{canbeobtainedasspecifiedin(55).}

₂

Basedon(55) inCorollary1,theCRLBcanbeapproximatedfor

C R L B

≈

16

π

2

_σ

2

₍

_H2

₊

_R2

₎

m+5 S2

₍

_m

₊

₁

₎

2

₍

_m

₊

₃

₎

2_P2

TH2m+2N R2

.

(59)

ItisnotedthattheCRLBinmeters(i.e.,thesquare-rootof(59))is inverselyproportional tothetransmitpower,thearea ofthePDs, andthesquare-rootofthenumberofPDsinthisconfiguration.By calculatingthefirst-orderderivative,itcanbeshownthattheCRLB in(59) isamonotonedecreasingfunctionof

R if

(

H

/

R

)

2

>

m

+

4 (60)

andismonotone increasingotherwise.Forcommonroom dimen-sionsandVLCreceiverlayouts, H

/

R is expectedtobelargerthan 10 whentheLEDtransmitterisontheceiling.Therefore,the con-ditionin(60) issatisfiedinmostpracticalscenariosasthe Lamber-tianorder,

m,

cannotbeverylargeunlesstheLEDisverydirective (whichisnotverydesirableduetoilluminationpurposes).Hence, in the case of D



R and practical system parameters, the PDs should be placed at the boundary of the uniform circular layout to mini-mize the CRLB.

Corollary2.

When all the PDs point up vertically, the CRLB in

(36)

mul-tiplied by R2

_/(

_H2

₊

_D2

₎

m+5_{has the following limit as R}

_/

_D

_−→

_0:

lim R D→0 R2C R L B

(

H2

₊

_D2

₎

m+5

=

8

π

2

_σ

2 S2

₍

_m

₊

₁

₎

2

₍

_m

₊

₃

₎

2_P2 TH2m+2N

.

(61)

Proof. As R

/

D

−→

0,theCRLBin(36) converges totheCRLB ex-pressionspecifiedby(47) and(48)–(50) inLemma2.Whenallthe PDspoint upvertically, i.e.,when

β

=

0,the elementsoftheFIM in(48)–(50) reducetothefollowingexpressions:

J11

=

¯

c 2

(

m

+

3

)

2_H2_N

₍

_2d2 x

+

R2

)

(62) J22

=

¯

c 2

(

m

+

3

)

2_H2_N

₍

_2d2 y

+

R2

)

(63) J12

=

J21

= ¯

c

(

m

+

3

)

2H2Ndxdy

.

(64) Evaluating(47) forthisscenario,theCRLBbecomes

C R L B

=

4

(

D 2

₊

_R2

₎

¯

c

(

m

+

3

)

2_H2_R2_N

₍

_2D2

₊

_R2

₎

·

(65) Insertingthe definitionofc in

¯

(52) into (65), thefollowingCRLB expressionisobtainedforR

/

D

−→

0:

C R L B

=

4

(

D

2

₊

_R2

₎₍

_H2

₊

_D2

₎

m+5

˜

a

(

m

+

3

)

2_H2_R2_N

₍

_2D2

₊

_R2

₎

.

(66) As R

/

D

−→

0, the limit of the CRLB in (66) multiplied by

R2

_/(

_H2

₊

_D2

₎

m+5 _becomes lim R D→0 R2C R L B

(

H2

₊

_D2

₎

m+5

=

4

˜

a

(

m

+

3

)

2_H2_N Rlim D→0

(

D2

+

R2

)

(

2D2

₊

_R2

₎

=

2

˜

a

(

m

+

3

)

2_H2_N

.

(67) Then,based on thedefinition ofa in

˜

(33), the statement in(61) canbeobtainedfrom(67).

2

ForD



R, theCRLBcanbe approximatedfrom(61) in Corol-lary2as C R L B

≈

8

π

2

_σ

2

₍

_H2

₊

_D2

₎

m+5 S2

₍

_m

₊

₁

₎

2

₍

_m

₊

₃

₎

2_P2 TH2m+2N R2

.

(68)

It isagainnotedthat the CRLBinmeters (i.e.,thesquare-root of (68)) isinversely proportional tothe transmit power,the area of the PDs, andthe square-root of the number of PDsin this con-figuration. Anotherimportant property of theCRLB in (68) is its monotonedecreasingnaturewithrespecttotheradiusofthe uni-formcircularlayout.Hence,

the PDs should be placed at the boundary

of the uniform circular layout to achieve the minimum CRLB in this sce-nario.

Remark1. Considering practical receiver sizes and room dimen-sions,thecaseof D



R is expectedtobe quitecommonin real-life applications. Hence, the expressions in Lemma 2 and Corol-lary 2 hold approximately in most scenarios under the stated conditions. Onthe other hand,the case of R



D considered in Lemma1andCorollary1canbe observedwhentheVLCreceiver isdirectlyundertheLEDtransmitter;i.e.,when

D in

Fig.1isvery small.Theaccuracyoftheproposed expressionsisinvestigatedin thenextsection(seeFig.3).

As notedin Remark1,thecaseof D



R is quite commonin practicalapplications.Theexpressionin(68) canprovideasimple approximation forthe CRLBinthecaseofauniformcircular lay-out includingperpendicular andidenticalPDs thatare located at thesameknownheight.SincetheCRLBin(68) decreaseswith R,

itisoptimaltoplacethePDsattheboundaryoftheuniform cir-cularlayout(asstatedafterCorollary2).Inordertogeneralizethis resulttomoregenericconfigurations,thefollowingpropositionis presented.

Proposition1.Consider a configuration that is a superposition of multi-ple circular uniform arrays as in Fig.4-(a), and suppose that the PDs are identical, at the same known height, and point up vertically. Then, for D



R, the radii of all the circles should be set to the maximum possible value in order to minimize the CRLB.

Proof. Considera layoutthat isasuperpositionof K circles with differentradiidenotedby

R

1

,

R

2

,

. . . ,

R

K,andlet

N

k representthe numberofuniformlylocatedPDsatthe

kth

circle.Inthiscase,the totalnumberofPDs,denotedby

N,

becomes

N

=

K



k=1

Nk

.

(69)

Under the conditions in the proposition, J11 in (17) can be ex-pressedvia(25) as J11

=

α

N



n=1 S2

σ

2

(

lR

(

1

)

−

lT

(

1

)

+

an

(

1

))

2

(

dn

)

m+5 (70)

= ˜

α

K



k=1 Nk



i=1

(

lR

(

1

)

−

lT

(

1

)

+

ak,i

(

1

))

2

(

dk,i

)

m+5 (71)

where

d

k,i denotes thedistancebetweenthe LEDandthe ith PD atthe

kth

circle,

a

k,i

(

1

)

representsthe firstcoordinateofthe dif-ference betweenthereceiverlocation(lR) andthelocation ofthe

ith PDatthe

kth

circle,

α

isasin(20),and

α

˜

isdefinedas

˜

α



α

S2

σ

2

.

(72)

Similarto(23),

a

k,i

(

1

)

,

a

k,i

(

2

)

,and

a

k,i

(

3

)

canbeexpressedas ak,i

(

1

)

=

Rkcos

ψ

k,i

,

ak,i

(

2

)

=

Rksin

ψ

k,i

,

(73)

where

ψ

k,iistheangleofthe

ith

PDatthe

kth

circlewithrespect tothe

x-axis;

thatis(cf. (24)),

ψ

k,i

=

2

π

(

i

−

1

)

Nk

+ ¯ψ

k

(74)

with

¯ψ

k denoting the random shift angle for the kth circle (see Fig.2-(b)).

From ak,i

(

1

)

in (73), the definitions ofdx and D in (26) and (28),respectively,andbasedontheassumptionof

d

k,l

≈

√

H2

₊

_D2 (sinceD



R), J11in(71) canbewritteninthefollowingform:

J11

= ˜

α

K



k=1 Nk



i=1 d2

x

+

2dxRkcos

ψ

k,i

+

R2ncos2

ψ

k,i

(

D2

₊

_H2

₎

m+5 (75)

whichcanalsobeexpressedas

J11

= ˆ

c

⎛

⎝

Nd2_x

+

2dx K



k=1 Rk Nk



i=1 cos

ψ

k,i

+

K



k=1 R2_k Nk



i=1 cos2

ψ

k,i

⎞

⎠

(76) where

ˆ

c



α

˜

(

H2

₊

_D2

₎

m+5

.

(77)

FromTable1,theelementsin(76) canbecalculatedas Nk



i=1 cos

ψ

k,i

=

Nk



i=1 cos



2

π

(

i

−

1

)

Nk

+ ¯ψ

k



=

0 Nk



i=1 cos2

ψ

k,i

=

Nk



i=1 cos2



2

π

(

i

−

1

)

Nk

+ ¯ψ

k



=

Nk 2

.

Then, J11in(76) canbesimplifiedto

J11

= ˆ

c



Nd2_x

+

1 2 K



k=1 R_k2N_k2



.

(78)

Viasimilarcalculations,theother elementsoftheFIMin(18) and (19) areobtainedas J22

= ˆ

c



Nd2_y

+

1 2 K



k=1 R_k2N2_k



(79) J12

=

J21

= ˆ

cNdxdy (80) Defining G



N

(

d2_x

+

d2_y

)

(81) F



1 2 K



k=1 R_k2N2_k

,

(82)

theCRLBcanbewrittenas

C R L B

=

J11

+

J22 J11J22

−

J12J21

=

G

+

2F

ˆ

c

(

G F

+

F2

₎

.

(83) The partial derivative of the CRLB in (83) with respect to Ri is calculatedas

∂(

C R L B

)

∂(

Ri

)

= −

G2

∂

F

∂

Ri

+

2G F

∂

F

∂

Ri

+

2F2

∂

F

∂

Ri

ˆ

c

(

G F

+

F2

₎

2 (84) for i

∈ {

1

,

. . . ,

r

}

. Since

∂

F

∂

Ri

=

RiN2i

>

0, F

>

0, G

>

0 and c

ˆ

>

0, the partial derivative ofthe CRLBin(84) isnegative for all Ri

>

0 and i

∈ {

1

,

. . . ,

r

}

. Since all the partial derivatives are negative, the CRLB isa monotone decreasing function ofthe radius Ri for each i.Hence,itisoptimaltoplacethePDsattheboundaryofthe layout.

2

Proposition 1statesthat inauniformcirculararray configura-tionwithD



R, placingthePDstothemaximumpossibleradius leadstotheminimumCRLBinthecaseofidenticaland perpendic-ularPDsthatareatthesameknownheight,andthisresultholds foreachuniformcirculararrayinthepresenceofasuperposition of multiplecircularuniform arraysinthe givenVLC receiver.For example,consideringscenarios A,B,andCinFig.4,scenarioCis optimalundertheconditionsinProposition1intermsof minimiz-ingtheCRLBsinceitemploysthemaximumradiusforeach PD.

4. Numericalresults

Inthissection,numericalevaluationsoftheCRLBexpressionsin Section3areperformedinordertoinvestigatetheeffectsof differ-entparametersontheperformanceofaSIMOVLPsystemin vari-ousscenarios.Anemptyroomwithdimensions4 m.

×

4 m.

×

3 m. is consideredforthe simulations,where 3 m. correspondsto the height oftheroom.The Lambertianorderistakenas

m

=

1 [31], andtheLEDtransmitterislocatedatlT

= [

0

,

0

,

3

]

T (allinmeters) with a transmit power of PT

=

3 W, where theposition coordi-natesarewithrespecttothecenterofthefloor(i.e.,thecenterof thefloorisdefinedastheorigin).ToverifytheclosedformCRLB expressions in(37),(47),(55),and(61), whichare derived based on theknownheight assumption,theVLCreceiver isassumedto be placed on the top of an object (e.g., a robot) with a known height of0.5 m. (exceptforSection4.5inwhichtheheight is as-sumed tobeunknown.)Thus,theposition vectorforthereceiver isformedaslR

= [

lR

(

1

),

lR

(

2

),

0

.

5

]

T,where

−

2 m.

≤

lR

(

1

),

lR

(

2

)

≤

2 m. The areaofeach PD isconsidered as Sn

=

25 mm2

∀

n, and the FOVof eachPD is takenas

θ

FOV,n

=

75◦

∀

n [36]. Inaddition, [37, Eq. 6] and [38, Eq. 20] are employed to calculate the noise variances,

σ

2

n.Duringthosecalculations,theparametersarechosen asin [37] (seeTable I in[37]).While calculatingthe exactCRLB, the noise variancesare obtainedfor each lR separately since the noisevariancealsodependsonthereceivedRSS[37,38]. (Through-out the room and for all possible elevation angles, the extreme valuesof

σ

2 _{arecalculatedas}

_σ

2

max

=

1

.

8074

×

10−16 and

σ

min2

=

1

.

8012

×

10−16_{.)ThenumberofPDs(N),theelevationanglesfor} the PDs(

β

inFig. 2-(a)) andthe radiusof thecircular layout (R inFig. 2-(b)) arespecified inthefollowingsubsections.Then, the exact CRLB values are calculated via the expressions in (8)–(14), (17)–(21), or (33)–(36), and comparisons with the approximate CRLBexpressionsin(37),(47),(55),and(61) areperformed.

4.1. Accuracy of asymptotic results

Equation(37) inLemma1and(55) inCorollary 1present ap-proximate closed form CRLB expressions for the case of D



R

whereas(47) inLemma2and(61) inCorollary2provide approx-imate CRLBformulasforthecaseof D



R. Toinvestigatethe ac-curacyoftheseapproximations,numericalexamplesarepresented inFig.3-(a)and(b),wheretheradiusoftheuniformcircular lay-out istaken as R

=

0

.

15 m. andthe number ofPDs is specified as N

=

8. Since thenoise variances (

σ

2

n’s) atthe PDsdiffer only slightlyfromeach other atagivenlR,

σ

2 in(37), (47), (55),and (61) isreplacedbythemeanvalueof

σ

2

n’sataparticularposition vector lR while evaluating the approximate CRLB expressions. In Fig.3-(a),the elevationangleofthePDsistakenas

β

=

20◦ and

Fig. 3. TheapproximateCRLBexpressionscomparedwiththeexactCRLB.(a) Accu-racyofLemma1andLemma2.(b)AccuracyofCorollary1andCorollary2.

Fig. 4. Threedifferentscenarioswith12PDspointingupverticallyand3circles withdifferentradii.

the CRLBis evaluated based on the approximations in Lemma1

and Lemma 2 by locating the VLC receiver at various distances fromtheroomcenter(origin)inasingledirection(duetothe sym-metry).Inaddition,theexact2-dimensionalCRLBisevaluatedvia (33)–(36) andillustrated inthefigure.InFig.3-(b),thePDspoint upvertically(i.e.

β

=

0◦)andtheCRLBexpressions corresponding totheapproximationsinCorollary1andCorollary2arepresented, togetherwiththeexact2-dimensionalCRLB.Fromthefigures,itis observedthattheapproximationsinLemma2andCorollary2 pro-videacloseapproximation totheexactCRLBforawiderangeof distancessincethe condition of D



R is satisfiedinmany posi-tionsinapracticalsetting.Ontheotherhand,theapproximations in Lemma1 and Corollary 1 are quite accurate only around the pointwhichisdirectlyundertheLEDtransmitter(i.e.,

R



D). 4.2. Uniform circular layouts with various radii

Consideraconfigurationthatisasuperpositionofmultiple uni-formcircularlayoutswithvarious radii. Three differentscenarios (A,B,andC)areinvestigatedasshowninFig.4.Ineachscenario, 12 identicalPDsareplacedon3 uniformcircularlayoutswithradii

R1, R2,and R3,whereeach circlecontains 4 PDs. EachPD points upvertically withan elevationangleof

β

=

0.Inscenario A,the radii areset to R1

=

0

.

05 m., R2

=

0

.

1 m.,and R3

=

0

.

15 m., in scenario B, R1

=

R2

=

0

.

1 m. and R3

=

0

.

15 m., andin scenario C, R1

=

R2

=

R3

=

0

.

15 m., where 0.15 m. corresponds to the maximumpossiblevalue forthe radius inthe considered config-uration.Forthesethree scenarios, theexact CRLBsare calculated via(17)–(21) byplacingtheVLCreceiveratvariousdistancesfrom theroom center(origin). The results presentedin Fig.5 indicate that scenario C, in which all the PDs are located at the bound-aryofthelayout,yieldstheminimumCRLBs.Thisisinaccordance withProposition 1, which states that the radii of all the circles

Fig. 5. CRLB versus distance from the room center.

Fig. 6. CRLB contour (in cm) with respect toβand distance from the room center.

shouldbesettothemaximumpossiblevalueinordertominimize theCRLBforthecaseof

D



R (please alsoseethecomments af-terCorollary 1). Infact, in thisexample, thisresult holds forall possible values of D. Since it is optimal to place thePDs at the boundaryofthecircularlayoutundercertain conditions,scenario Cisemployedinthefollowingsubsections.

4.3. Elevated PDs at VLC receiver

Asinvestigatedin[26],tiltingthePDsplacedonauniform cir-cularlayoutcanprovidecertainbenefitsinsomecases.Inthispart, the effectsoftilting are investigatedforthe CRLBofa SIMOVLP system. Inorderto distinguishthe impactoftilting on theCRLB, eachPDiselevatedbythesameangle

β

,asillustratedinFig.2-(a). Forthe CRLBderivations inthisstudy,the LED transmitteris as-sumed to lie within the FOV of each PD, as stated in Section 3. The conditions underwhich thisassumption holdscan be speci-fiedbasedontheindicatorfunctionin(1) asfollows:

tan−1



D

−

R H



− θ

F O V

≤ β ≤ θ

F O V

−

tan−1



D

+

R H



(85)

InFig.6,theexactCRLBsarepresentedasa contourplotwith respect tothe elevationangle

β

andthedistance fromtheroom center. From the figure, the effects ofthe elevation angle of the