T VT NonIsoCorrSpec Revised

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AParametricModelfortheDistributionoftheAngleofArrivalandthe

AssociatedCorrelationFunctionandPowerSpectrumattheMobileStation

AliAbdi1,JanetA.Barger2,andMostafaKaveh3

1Dept.ofElec.andComp.Eng.,NewJerseyInstituteofTechnology2Dept.ofElec.Eng.,UniversityofTexasatTyler

3Dept.ofElec.andComp.Eng.,UniversityofMinnesota

Abstract__OneofthemainassumptionsintheClarkesclassicchannelmodelisisotropicscattering,

i.e. uniform distribution for the angle of arrival of multipath components at the mobile station.

However,inmanymobileradiochannelsweencounternon-isotropicscattering,whichstronglyaffects

thecorrelationfunctionandpowerspectrumofthecomplexenvelopeatthemobilereceiver.Inthis

contribution,weproposetheuseoftheversatilevonMisesangulardistribution,whichincludesand/or

closelyapproximatesimportantdistributionslikeuniform,impulse,cardioid,Gaussian,andwrapped

Gaussian, for modeling thenon-uniform angle of arrivals at themobile.Basedon this distribution,

associatedcorrelationfunctionandpowerspectrumofthecomplexenvelopeatthemobilereceiverare

derived.Theutilityofthenewresultsisdemonstratedbycomparisonwiththecorrelationfunction

estimatesofmeasureddata.

Pointofcontact:AliAbdi

Dept.ofElectricalandComputerEngineeringNewJerseyInstituteofTechnology

323KingBlvd.Newark,NJ07102,USA

Phone:(973)5965621

Fax:(973)5965680

Email:[email protected]

Thework of thefirstand the third authors have beensupported inpart by theNationalScience Foundation, under the

WirelessInitiativeProgram,Grant#9979443.


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AParametricModelfortheDistributionoftheAngleofArrivalandtheAssociatedCorrelationFunctionandPowerSpectrumAliAbdietal.

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I.INTRODUCTION

Thereceivedsignalcorrelationfunctionsandpowerspectraatthemobilestation(MS)dependon

theprobabilitydensityfunction(PDF)oftheangleofarrival(AOA)ofascatteredwave.Clarkestwo-

dimensionalisotropicscatteringmodel(auniformAOAPDFover ),[ )givesrisetothezero-order

Besselfunction fortheautocorrelation function andtheU-shaped power spectrum for thecomplex

envelopeof multipathcomponentsattheMS[1,p.40-43].However,ithasbeenargued[2]-[6],and

experimentallydemonstrated[7]-[16] that thescatteringencountered in many environments is non-

isotropic, resulting ina non-uniform PDF for AOA at the MS. As has been discussed in[16], the

assumptionofauniformPDFfortheAOAintroducessmallerrorsonthefirstorderstatisticsofthe

received signal, but a significant error on thesecond order statistics, like correlation functions and

levelcrossingrates.

In[17]and[18],twogeometricallybasedangleofarrivalPDFsarederived;whilethreedifferent

models are assumedfor such a PDF byotherauthors:quadratic PDF [19], LaplacePDF [10],and

cosine PDF [20]. In addition to showing good fit to measurements, another important factor in

selectingacandidatefortheangleofarrivalPDFisthemathematicalconvenienceitprovidesto the

determinationofclosed-formsolutionsforthecorrelationfunctions,powerspectra,spectralmoments,

etc..ThepreviouslymentionedPDFsdonotsatisfythelatterrequirement.Achievingthisgoalisthe

mainpurposeofthispaper.

Theremainderofthepaperisorganizedasfollows.InSectionIIwepresentthetwoparametervon

Mises PDF as flexible model for the PDF of the angle of arrivals. The utility of this model is

demonstrated by the derivation of closed-form expressions for the correlation function and power

spectrum of the complex envelope at the MS in Section III. This is followed by a study of the

differences in the characteristics of the correlation function and power spectrum for cases which

deviatefromtheisotropicscatteringmodelintroducedbyClarke.Finally,measureddataisusedto

confirmthevalidityoftheproposedmodelinpredictingthescatteringcharacteristicsencounteredin

actualmobilecommunicationsituations.


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II.AFLEXIBLEPDFFORTHEAOA

A PDF was introduced by R. von Mises in 1918 to study the deviations of measured atomic

weightsfromintegralvalues[35].ThisPDFplaysaprominentroleinstatisticalmodelingandanalysis

ofangularvariables[21,pp.57-68].Lettherandomvariable representtheAOAofamultipath

component(scatterandspecular),receivedattheMS.ThevonMisesPDFforthescattercomponentof

, )(scatp ,isgivenby:

),[,)(2

)]cos(exp[)(

0

=

Ip

pscat , (1)

where (.)0I isthezero-ordermodifiedBesselfunction, ),[ p accountsforthemeandirectionof

AOAofscattercomponents,and 0 controlsthewidthofAOAofscattercomponents.Figs.1and

2show )(scatp inlinearandalsoinpolarcoordinates,respectively,fordifferentvaluesof and

0=p (non-zerovaluesof p simplyshiftFig.1circularlyorrotateFig.2).

AscanbeobservedinFig.1,for 0= weobtain = 21)(scat

p (isotropicscattering), while

= yields )()( pscat

p = (extremelynon-isotropicscattering),where (.) istheDiracdelta

function.Forsmall ,thisfunctionapproximatesthecardioidPDF[21,p.60],whichisrathersimilar

tothecosinePDF[20];whileforlarge itresemblesaGaussianPDFwithmean p andstandard

deviation 1 [21,p.60].Ingeneral,thevonMisesPDFcanapproximatethewrappedGaussian

PDFquitewell[21,p.66].ItisinterestingtonotethatthevonMisesPDFappearsinanumberofother

communication contexts. For example, this PDF is referred to as the Tikhonov PDF in partially

coherentcommunication[22,p.406],hasbeenusedinphase-lockedlooprelatedproblems[23],and

hasbeenshowntorepresentthePDFofthephaseofasinewaveinGaussiannoiseforlargesignalto

noiseratios[24].

Ausefulgeometricalinterpretationfor canbeobtainedbythefactthatitistheslopeofthe

tangenttothecurve )(scat

p ,inpolarcoordinates,atthepoint 2= .Thisresultcanbeobtained

using the relation )cossin()sincos( ++ [25, p. 176], where )(= represents a

curveinpolarcoordinatesandprimedenotesdifferentiationwithrespectto .AlookatFig.2verifies

thisresult.Forlarge ,say 3 , p scat ( ) takesaunidirectionalshape,ascanbeobservedinFig.2.


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InordertoobtainameasureforthespreadofAOA'sinthepolarcoordinatesforlarge ,weconsider

theinflexionpointsof p scat ( ) whichareapproximatelyequalto 1 ,assuming islarge[21,

p.60]. So, for a unidirectional scattering scenario (large ), the width of the AOA of scatter

componentsatMSisroughlyequalto 2 .For 10,3= ,weobtain o66 and o36 ,respectively.

III.CORRELATIONANDSPECTRUMOFTHECOMPLEXENVELOPEOFMULTIPATHCOMPONENTS

Let the received complex envelope at the MS due to scatter components be represented by

)()()( trjtrtr scatqscat

i

scat += ,with 1=j ,where )(trscati and )(trscat

q arethein-phaseandquadrature

scattercomponents. Then,the normalized autocorrelationfunctionof )(trscat , )(~

scatrr ,isgivenby

)(~

)(~

)(~

+= scatrrscat

rr

scat

rr qiiij , where

~( )

~( ) r r

scat

r r

scat

i i q q= and

~( )

~( ) r r

scat

r r

scat

i q q i= are normalized

autocorrelationandcrosscorrelationfunctionsof )(trscat

i and )(trscat

q .Notethatthetermnormalizedindicates 1)0(

~= scatrr .Itcanbeeasilyshownthat )]cos2[exp()(

~= m

scat

rr fjE [1,p.40],whereE

representsmathematicalexpectationand mf isthemaximumDopplerfrequency.Using )(scat

p in(1)

forcarryingouttheexpectationwithrespectto yields[26,p.357]:

)(

cos44)(

~

0

22220

+

=I

fjfImpm

scat

rr . (2)

For 0= ,(2)resultsinthecorrelationfunctionsfortheClarkestwo-dimensionalisotropicscattering

model, i.e. )2()2()(~

00 == mmscat

rrfJfjI

iiand 0)(

~= scatrr qi , where (.)0J isthezero-orderBessel

function. Note that for both analytical and numerical calculations, formula (2) and its real and

imaginarypartsaremuchsimplerthan,andthusmorepreferabletothecomplicatedresultsin[19].

The power spectrum of )(trscat can be obtained based on

222 )]()()()([)( ffpGpGfS mscatscatscat

rr += , mff [1, p.42], where and f are

relatedaccordingto )(cos 1 mff= , 2 isthetotalpowerfromallscattercomponentsattheinputof

thereceiverantenna,and (.)G isthereceiverantennagainpattern.Foraverticalmonopoleantenna,

23)( =G . So the normalized power spectrum of )(trscat can be defined as

23)(2)(~

= fSfS scatrrscat

rr , i.e.22)]()([)(

~ffppfS m

scatscatscat

rr += , mff . The term

normalizedcomesfrom 1)(~

=m

m

f

f

scat

rr dffS ,whichcanbeeasilyverified.Thisisinagreementwith


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1)0(~

= scatrr ,since )(~

scatrr and )(~

fSscat

rr areFouriertransformpairs.Byusingthe )(scat

p in(1)and

aftersomemanipulationsweobtain:

m

mpmp

m

scat

rr ffI

ffff

fffS

= ,

)(

))(1sincosh()cosexp(1)(

~

0

2

22, (3)

where cosh(.) isthehyperboliccosine.For 0= ,(3)reducestothepowerspectrumfortheClarkes

two-dimensionalisotropicscatteringmodel,i.e. 221)(~

fffS mscat

rr = , mff .

Forthespecularcomponent,thePDFofAOAcanbewrittenas:

),[),()( 0 =spec

p , (4)

where ),[0 is theAOAofthespecularcomponent.Letthereceivedcomplexenvelopeatthe

MSduetothespecularcomponentberepresentedby )(trspec . Then,thenormalized autocorrelation

function of )(trspec can be obtained based on )]cos2[exp()(~

= mspec

rr fjE andusingthegiven

)(spec

p :

)cos2exp()(~

0 = mspec

rr fj . (5)

Thenormalizedpowerspectrumof )(trspec , )(~

fSspec

rr , canbe easily obtainedby taking the Fourier

transformof(5):

mm

spec

rr fffffS = ),cos()(~

0 . (6)

Notethat 1)(~

)0(~

== m

m

f

f

spec

rr

spec

rr dffS .

Wheninadditiontothescattercomponents,aspecularcomponentisalsopresent,thePDFofthe

AOAofmultipathcomponents(scatterandspecular)attheMScanbeexpressedas[19]:

),[,1

)()()(

+

+=

K

Kppp

specscat

, (7)

whereKistheRicefactor,definedastheratioofthepowerinthespecularcomponenttothatinthe

scattercomponents. )(scatp and )(

specp arealsogivenin(1)and(4),respectively.For )(tr ,the

receivedcomplexenvelopeat MSdue tomultipathcomponents(scatterandspecular),we alsohave


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)(2

sin)(2cos)]()([)2(

~

0

21

2202

2

++=

I

IIIfb

pp

m

scat .

So,forthezerocrossingrateof )(trscati wearriveat:

m

ppscat

if

I

IIItrZCR 2

)(

sin)(2cos)]()([)}({

0

21

220

++= .

Theaboveclosed-formformulaisausefultoolforstudyingtheeffectofnonisotropicscatteringonthe

performanceof )}({ trZCR scati forestimatingthespeedofthemobile.For 0=p and 0= ,Clarkes

isotropic scattering, the above result reduces to mf2 , in agreement with [1, p. 60]; while for

2=p and 7= weobtain mf251.0 ,halfoftheisotropiccase.Therefore,wemayconclude

that )}({ trZCR scati is not a robust estimator against nonisotropic scattering and may change

significantly as p and vary. Higher order spectral moments likescatb3~ and

scatb4~ , needed for

developingmoreefficientandrobustestimatorsforthespeed[27],canbesimplyobtainedbytaking

higherorderderivativesof )(~

scatrr in(2).

IV.ACASESTUDY

Here we consider three different urban scenarios studied in [19] and derive the associated

autocorrelationfunctionsandpowerspectraofthecomplexenvelopeofmultipathcomponentsatthe

MS.LettheMSmovefromlefttoright.Forthefirstscenario,S1,scattercomponentsarriveattheMS

headon,i.e. p = 0 .Forthesecondscenario,S2,scattercomponentsarrivefromboththe p = 0 and

p = directionswithequalprobabilities.Forthethirdscenario,S3,scattercomponentsarriveatthe

MSfromadirectionperpendiculartoitsdirectionofmotion,e.g. p = 2 .ForS1andS3, )(scatp is

equal to )(2)cosexp( 0 I and )(2)sinexp( 0 I , respectively; while for S2 we have

)(2)coscosh()(5.0)(5.0 00 =+ == Ipp ppscatscat .ThesethreePDFsareplottedinFig.3for

3= ,togetherwithClarkesclassicisotropiccase 0= .

Thenormalizedautocorrelationfunctionsofthecomplexenvelopeduetothescattercomponents

fortheabovethreescenarioscanbederivedaccordingto(2):


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)(

44)(

~

0

22220

1,

+

=I

fjfImm

Sscat

rr , (10)

)(

44Re)(

~

0

22220

2,

+

=I

fjfI mmSscat

rr , (11)

)(

4)(

~

0

22220

3,

=I

fIm

Sscat

rr , (12)

where Re[.] gives the real part of its complexargument.Notice that for S1, the imaginarypartof

)(~ 1, Sscatrr ,i.e. )(

~ 1, Sscatrr qi ,isnonzero,contrarytoClarkesclassicisotropicscenario.Bysubstituting

(10),(11),and(12)into(8),wecanreadilyobtain~

( ) r rS1 ,

~( ) r r

S2 ,and~

( ) r rS3 .Figs.4and5show

~ ( ) r rS

i i

1 , ~ ( ) r rS

i i

2 ,and ~ ( ) r rS

i i

3 ,therealpartsof ~ ( ) r rS1 , ~ ( ) r r

S2 ,and ~ ( ) r rS3 ,for ( , ) ( , )K = 0 10 and

( , )1 3 , respectively (for the case of nonzero K, we considered p=0 ). Comparing with Clarkes

model ( , ) ( , )K = 0 0 shown in Figs. 4 and 5, we observe the noticeable effect of non-isotropic

scatteringindifferentscenariosonthecorrelationpropertiesattheMS.Fornonisotropicscattering,it

is possible to have large correlations when Clarkes isotropic scattering model predicts small

correlations.

Thenormalizedpowerspectraofthecomplexenvelopeduetothescattercomponentsforthethree

scenariosS1,S2,andS3canbeobtainedbasedon(3):

m

m

m

Sscat

rrff

I

ff

fffS

= ,

)(

)exp(1)(

~

022

1, , (13)

m

m

m

Sscat

rr ffI

ff

ff

fS

= ,)(

)cosh(1)(

~

022

2, , (14)

m

m

m

Sscat

rr ffI

ff

fffS

= ,

)(

))(1cosh(1)(

~

0

2

22

3, , (15)

NoticethatforS1,thepowerspectrum )(~ 1,

fSSscat

rr isnotsymmetricwithrespectto 0=f ,contraryto

Clarkesscenario.Bysubstituting(13),(14),and(15)into(9),wereadilyobtain )(~ 1

fSS

rr , )(~ 2

fSS

rr ,and


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)(~ 3

fSS

rr . Figs. 6 and 7 show )(~ 1,

fSfSscat

rrm , )(~ 2,

fSfSscat

rrm , and )(~ 3,

fSfSscat

rrm for 10= and 3 ,

respectively.ComparingwithClarkesclassicisotropiccase, 0= ,superimposedontheplotsinFigs.

6 and 7, we clearlysee thesignificant effectsof different non-isotropic scattering scenarios on the

powerspectrumattheMS.

V.EXPERIMENTALRESULTS

InthissectionweshowtheutilityoftheproposedAOAdistribution,andtheassociatedparametric

correlationfunctionindescribingthecorrelationpropertiesof arangeofavarietyofmeasureddata.

Firstwegiveabriefsummaryofthedataused(thedynamicstatisticalcharacteristicsofthisdataset

arediscussedin[28]).Inthesubsequentsubsections,wetalkaboutenvelopecorrelationandenvelope

extraction,twonon-isotropicscatteringcorrelationmodelforfittingtodata,estimationoftheenvelope

correlationofdataandparameterestimationforthetwomodels,andfinally,themeritsandlimitations

ofthetwomodelsinfittingtomeasureddata.

A.TestLocationsandDataCollection

Twositeswereusedforthecollectionofdata:asuburbanhousingdevelopmentinGreenville,

Texas,andanurbanareaamongthebuildingsofthecampusoftheUniversityofTexasatArlington,

Texas. These test locations were chosen to provide a random signal environment that would be

representative of actual operational situations. By choosing environments with irregular structures,

datareflectsthegeneralangularspreadofthesignalandnotyieldresultsfoundonlyin veryregular

environments.

Inthesuburbanarea,parkedvehicles,vegetation,andinconsistencyinthesetbackofthehousing

structures produced very irregular reflective conditions. Measurements for records #0011 through

#0015weretakeninthesuburbanneighborhoodcharacterizedbyhomesofawood/brickconstruction

withanominal7mofseparationbetweenthestructures.Thetransmitantennawasataheightof2m

aboveahillsidelocationtoproducealowgrazingangleoverthehomes,simulatingacellularradio

signalatagreaterdistance.Themeasurementsweremadewiththereceiveantennainatotalshadow


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environment. Records #0016 through #0018 represent data taken near a large intersection in a

suburbanareawiththetransmitantennaheightraisedtoabout5m.Fortheintersectionmeasurements

there was a significant direct ray component of the signal. During these sample periods there was

significanttrafficactivitynearthetestvehicle.

Theurbanenvironmentischaracterizedbybuildingsofonetosixstoriesinheight,heavytraffic,

largeopenareas,andhighlyinconsistentsetbackfromthestreet.Records#0019through#0022were

madeintheurbanlocationwiththetransmitantennaataheightofabout21m,producingasignalata

moderatelylowangleofradiationoverthebuildingssurroundingthepath.Amapoftheurbanlocation

isshowninFig.8.Buildingsshownareofmodernbrickconstruction.Buildingheightsareindicated

byanumberinsidetheoutlinesshowninFig.8.Theurbanlocationcontainedsignificantvehicular

traffic, and the measurements were made over paths that had no direct ray component, but

demonstratedthedeepshadowenvironment.

The UHF transmitter generated a 910.25 MHz carrier at a nominal power of 0.2 W (the

wavelengthwas0.33m).Signalsamplesweretakenatdistancesof0.16kmto0.25km.Thecollected

dataconsistoftwelvesetsofnarrowbandinphaseandquadraturecomponentstakenattwelvedifferent

locations.Eachdatasetrepresentsthesignaloveratraveleddistanceof47m,or7softime.These

signalswere digitized witha sampling rateof 35156.25 Hz, toproduce one sampleper 0.2 mmof

distancetraveled(28sintime)bythemovingreceiverplatform(atafixedspeedof6.7m/s).Hence,

thenumberofsamplesforeachrecordisapproximately250,000.Theantennaatthemobilereceiver

wasa1/4waveverticalstub,0.6cmindiameter,mountedontopofa1.9cmdiameteraluminumtube.

Thetubeservedasarathersmallgroundplane.Forthefixedtransmitter,thesameantennawasused

except a 1/4 wave diameter groundplane (flat) was attached tothe base ofthe antenna stub. Both

antennashadomnidirectionalpatterns.

B.EnvelopeCorrelationandEnvelopeExtraction

Sinceformanypractical applications, theenvelopeofmultipathcomponentsand itscorrelation

propertiesareofconcern,weconsidertheenvelopeofmultipathcomponent )()( trtz = .When )(tr is


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a complex Gaussian process with non-zero mean, the autocovariance function of )(2 tz ,

)]}]([)()]}{([)([{)( 222222 ++= tzEtztzEtzEzz

, can be expressed in terms of the

autocorrelation functionofthe complexenvelopeof multipath components )(tr ,havingbothscatter

and specular components [29]. The normalizedversion of )(22 zz , )(~

22 zz , assuming nospecular

component(whichmakes )(tr azero-meancomplexGaussianprocess),isgivenby[1,p.55]:

2

)(~

)(~ 22 =scat

rr

scat

zz, (16)

where )(~ 22 scat

zzisthenormalizedautocovariancefunctionof )(2 tz duetoscattercomponents.Again

normalizedindicates 1)0(~ 22 =scat

zz.

Inordertoextract )(tz ,themultipath(orfastfading)componentoftheenvelope,whichwe are

interested in, the shadow (or slow fading) component of the envelope must be removed. Let

)()()( tjRtRtR qi += denote the received complex envelope at the MS in the presence of both

multipath and shadow components, with )(tRi and )(tRq as the inphase and the quadrature

components. The shadow (or slow fading)component of theenvelope, )(tR , is represented by its

mean, ])([ tRE , which is a function of time because )(tR isnonstationary.Themultipath(orfast

fading) component of )(tR can be represented as ])([)()( tREtRtz = [1, p. 91]. A method for

estimating ])([ tRE from )(tR usingalocalslidingwindowisdescribedin[30].

C.TwoNon-IsotropicScatteringPropagationModel

Weconsider )(~ 22 scat

zzforfittingtothedata,accordingtotwopropagationmodels:thesimplenon-

isotropicmodelandthecomposite non-isotropic/isotropicmodel.In thesimplenon-isotropicmodel,

we assume that the MS receives the signals mainly from one direction. The AOA PDF and the

normalized autocorrelation function for the simple non-isotropic model are given in (1) and (2),

respectively.Forthecompositenon-isotropic/isotropicmodel,theMSreceivessignalsuniformlyfrom

alldirections,inadditiontoacomponentcenteredaroundaspecificdirection.TheAOAPDFandthe

normalizedautocorrelationfunctionforthecompositenon-isotropic/isotropicmodelcanbewrittenas:


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),[,2

1)1(

)(2

)]cos(exp[)(

0

+

=

Ip

pscat , (17)

)2()1()(

cos44)(

~0

0

22220

+

+

= mmpm

scat

rr fJI

fjfI

, (18)

where 10 indicatestheamountofdirectionalreception.Thecompositenon-isotropic/isotropic

modelreducestothesimplenon-isotropicmodelfor 1= ,andsimplifiestoClarkesmodelfor 0= .

Forthesimplenon-isotropicmodelandbasedon(2)and(16)wehave:

)(

cos44)(~

2

0

22222

0

22

+

=I

fjfImpm

scat

zz, (19)

whileforthecompositenon-isotropic/isotropicmodelandaccordingto(18)and(16)wederive:

)(

)()2()1(cos44)(~

20

2

002222

0

22

+

+

=I

IfJfjfI mmpmscat

zz. (20)

D.EstimationoftheEnvelopeCorrelationandtheParametersoftheTwoModels

Theestimateforthenormalizedautocovariancefunctionoftheenvelope-squaredduetomultipath

components, )(~ 22 zz

,wasobtainedusingthebiasedmethod[31,p.743],scaledsuchthat 1)0(~ 22 =zz

(the bias is very small because of the large number of samples 250000). Recall that (16) and

consequently(19)and(20)arevalidwhen )(tr isazero-meancomplexGaussianprocess(nospecular

componenthence 0)( =trspec and )()( trtr scat= ).ThismeansthatatleasttheunivariatePDFsofthe

envelope, )()( trtz = , and the phase, )(tr , have to be Rayleigh and uniform, respectively,

independent of each other (note that a univariate Rayleigh PDFfortheenvelope by itselfdoes not

implythat )(tr isazero-meancomplexGaussianprocess[32]).Accordingtothedataanalysisresults

partlyreportedin[33],records#0011,#0017,#0019,#0021,and#0022meettheaboverequirements.

So, for these records, equation (16), and subsequently (19) and (20) hold exactly, while for other

records,thethreeequationsmaybeconsideredasapproximations.However,ourcollecteddatathatis


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analyzedanddiscussedinthesequel,suggestthatirrespectiveoftheapparentunderlyingprobability

distributionforthecomplexenvelope )(tr , the parametricmodelsgiven in(19) and(20) givevery

goodfitstotheautocovariancefunctionoftheenvelope-squared.

In (19) and (20), the maximum Doppler frequency is 20=mf Hz. The unknown parameters

),( p in (19) for the simple non-isotropic model and ),,( p in (20) for the composite non-

isotropic/isotropic model were estimated by the nonlinear least squares method (which provides

consistent estimates [34, p. 207]): Simple),(

MSEminarg),(p

p

= where

= =

n scat

zzzzn

1

21Simple )](

~)(~[MSE 2222

with )(~ 22 scat

zz given in (19), while

Composite),,(

MSEminarg),,(

=p

p where = =

n scat

zzzzn

1

21Composite )](

~)(~[MSE 2222

with )(~ 22 scat

zz

given in (20) (note that MSE stands for mean-squared-error). In the estimation procedure we had

350=n and 1...,1,s,1014.0 31 ==

+ n .Thisindicatesthattheestimationwascarriedout

overthecoherencetime[1,p.72],whichrangesfrom 0= through mf1= .Theestimatedvalues

),( p and ),,( p forbothmodelsarelistedinTableI.Themean-squared-erroroftheClarkes

model, defined by = =

n

mzzfJn

1

220

1Clarke )]2()(

~[MSE 22

,isalsogiveninTableI,together

with the minimum values of SimpleMSE and CompositeMSE for all twelve data sets. In Fig. 9 andfor

severaldatarecords, )(~ 22 scat

zzsin(19)and(20)areplottedusingtheestimatedparameters ),( p and

),,( p ,togetherwith )(~

22 zz andtheassociatedClarkesresult )2(2

0 mfJ .Thecorrelationplots

ofrecords#0012,#0020,#0021,andalso#0015arenotshowninFig.9,asthefirstthreearesimilarto

thecorrelationplotof#0011,whereastheforthoneresemblesthatof#0014.

E.Discussion

Asexpected, )MSEmin()MSEmin(MSE CompositeSimpleClarke > forallthetwelvesetsofdata.Note

thesignificanterrorofClarkesisotropicscatteringmodelinpredictingtheamountofcorrelationfor

mostofthecases.Forexample,forrecords#0011and#0014,themaximumerrorinpredictingthe

correlations is about 0.5, which is not negligible at all. Regarding the characteristics of the two

proposed models, let us first focus on the simple non-isotropic model. In this case, only for few

situationsthemaximumdeviationofthemodelpredictionsfromtheempiricalcorrelationsoverthe


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range 10 ,theempirical

correlations approach zero slower than the predictions of Clarkes model. Notice the very slow

convergenceof the empirical correlationsto zero for record#0018, 3.3 = , followed well by the

simplenon-isotropicmodel.Onlyforonecase(record #0019), issmallandequalto0.6,yielding

aresultclosetothatgivenbyClarkesmodel.Onedisadvantageofthesimplenon-isotropicmodel,

whichgaveusthemotivationtodevelopthecompositenon-isotropic/isotropicmodel,isitsinabilityto

followtheoscillationsobservedintheempiricalcurvesofrecords#0013-#0017and#0022.Onthe

otherhand,thecompositenon-isotropic/isotropicmodelisabletofollowallkindsofvariationsofthe

empirical curves. Specifically, note the improvements obtained in data fitting for records #0013 -

#0017,#0019,and#0022.Ofcoursethoseimprovementsareachievedattheexpenseofaddingthe

parameter tothesimplenon-isotropicmodel.

Notethatsincethesimplenon-isotropicmodelshouldaccountfortheincomingwavesattheMS

fromalldirectionsjustbyasinglevonMisesPDF,estimatedvalues areallsmallforthismodel

( 3.3 ), which correspond to large AOA spreads ( o632 ). On the other hand, since the

composite non-isotropic/isotropic model is equipped with two PDF for modeling the directions of

incoming waves at the MS, von Mises PDF can take care of the directional reception muchbetter,

whichinturnyieldsmoderateandlargevaluesfor ( 8812 and 700 = ).Thiscorrespondsto

moderate and small values for the AOA spread of the directional incoming waves at the MS

( oo 33212 and o3.42 = ).

Regardingthephysicalinterpretationoftheestimatedvalues p forbothmodelsisTableI,letus

assumethata o20 toleranceinestimatingtheapproximateorientationstothefixedtransmitter,listed

inTableI,isacceptable(ofcoursethetolerancerange o20 shoulddependon ,butherewefixitto

makeasimpleconclusion).InTableI,aperpendiculardirectioncorrespondstoeither o90=p or

o270=p ,andaparalleldirectionisassociatedwitheithero0=p or

o180=p .Nowitiseasyto

verify that within the tolerance limit o20 , thesimple non-isotropic model correctly estimates p

onlyforrecords#0013,#0014,#0018,and#0021,whilethecompositenon-isotropic/isotropicmodel

successfully estimates p for #0011 - #0013, #0016, #0017, #0020 and #0022. This confirms the


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superiorityofthecompositenon-isotropic/isotropicmodelincharacterizingthepropagationofwaves

inwirelesschannels.

VI.CONCLUSION

In this contribution wehave proposed the application of the von Mises PDF for modeling the

distributionoftheanglesofarrivalofscattercomponentsreceivedatthemobilestation.Thisflexible

parametricfamilyofangularPDFsincludesavarietyofnon-isotropicscatteringscenarios,including

theisotropiconeasaspecialcase.ThisPDFforanglesofarrivalresultsinclosed-formandeasy-to-

use formulas for the correlation function and power spectrum of the complex envelope of scatter

componentsatthemobilestation.TheseformulasincludeClarkesBesselfunction-basedcorrelation

and U-shaped power spectrum as special cases. As a consequence, those correlation or spectral

parameterswhichareimportantforvarioussystemcalculationscanbeeasilyderived.Forexample,

spectralmomentsofthereceivedsignalatthemobilestation,usefulforestimatingthespeedofmobile,

canbeexpressedinsimpleclosedforms.Themathematicaltractabilityofthecorrelationfunctionand

power spectrum derived in this paper make them useful for other applications such as efficient

simulationof fadingchannels[36],Karhunen-Loeveexpansionofthemultipathfadingprocess[37],

derivationofclosed-formexpressionsfortheerrorprobabilitiesinchannelswithsignificantDoppler

spread,hencehavingfastvariations[38][39,pp.523-524][40,pp.365-367](inplaceof numerical-

onlycalculations[41]),etc..Theutilityoftheproposedmodelandnewresultshavebeenverifiedby

fitting the parametric model of the correlation function to the estimates of this function based on

measureddata.

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Woerner,andJ.H.Reed,Eds.,Boston,MA:Kluwer,1999,pp.1-9.


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TABLEI

ESTIMATEDVALUESFORTHEPARAMETERSOFTHESIMPLENON-ISOTROPICANDCOMPOSITENON-

ISOTROPIC/ISOTROPICMODELS,TOGETHERWITHTHEESTIMATIONERRORS

Clarkesmodel

Thesimplenon-isotropicmodel

Thecompositenon-isotropic/isotropmodel

Recordno.

Approximateorientationto

transmitter MSEClarke p min(MSESimple) p min(MSECompo

0011 Perpendicular 1.2E-1 2.4 19.8o 3.3E-4 38 280.8o 0.80 3.3E-40012 Perpendicular 9.3E-2 3.0 36.0o 1.1E-3 29 266.4o 0.78 2.5E-4

0013 Parallel 4.6E-2 1.2 0o 1.5E-2 700 0o 0.50 3.0E-3

0014 Parallel 1.6E-1 2.4 0o 7.0E-3 12 331.2o 0.80 2.9E-4

0015 Perpendicular 1.6E-1 2.4 0o 1.7E-2 36 331.2o 0.72 9.7E-4

0016 Perpendicular 2.3E-2 1.2 36.0o 9.8E-3 700 108.0o 0.42 3.1E-3

0017 Perpendicular 5.2E-2 1.5 19.8o 8.9E-3 700 79.2o 0.56 4.1E-40018 Parallel 2.5E-1 3.3 0o 3.4E-3 88 252.0o 0.86 7.8E-40019 Perpendicular 6.7E-3 0.6 0o 2.7E-3 700 237.6o 0.30 7.6E-4

0020 Perpendicular 1.1E-1 2.1 14.4o 1.5E-3 59 79.20o 0.74 1.5E-40021 Parallel 1.2E-1 2.1 10.8o 2.0E-4 39 259.2o 0.80 1.8E-4

0022 Perpendicular 6.9E-2 1.5 0o 1.1E-2 700 252.0o 0.60 1.3E-3


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Fig.1. VonMisesPDFfortheangleofarrivalofscattercomponentsatthemobilestation( 0=p ). 0=

5.0=

----- 3= ---- 10=

Fig.2. Von Mises PDF inpolarcoordinates for the angle of arrivalof scatter components at the

mobilestation( 0=p ).

0=

5.0=

----- 3= ---- 10=

3 2 1 0 1 2 3

0

0.2

0.40.6

0.8

1

1.2

p

tacs

0 0.2 0.4 0.6 0.8 1 1.2

0.2

0.1

0

0.1

0.2

p

tacs


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Fig.3. ThePDFofangleofarrivalofscattercomponentsatthemobilestationfordifferenturban

scenarios.

Clarkesscenario: 0=

S1scenario: 3=

-----S2scenario: 3= ----S3scenario: 3=

3 2 1 0 1 2 3

0

0.1

0.20.3

0.4

0.5

0.6

p

ta

cs


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Fig.4. Realpartofthenormalizedautocorrelationfunctionofthecomplexenvelopeatthemobile

stationduetomultipathcomponents(scatterandspecular),fordifferenturbanscenarios.

Clarkesscenario: 0=K , 0=

S1andS2scenarios: 0=K , 10=

S3scenario: 0=K , 10=

Fig.5. Realpartofthenormalizedautocorrelationfunctionofthecomplexenvelopeatthemobile

stationduetomultipathcomponents(scatterandspecular),fordifferenturbanscenarios.

Clarkesscenario: 0=K , 0=

S1andS2scenarios: 1=K , 3= , 00 =

S3scenario: 1=K , 3= , 20 =

0 0.2 0.4 0.6 0.8 1fm

1

0.5

0

0.5

1

eR

rr

0 0.2 0.4 0.6 0.8 1fm

0.75

0.5

0.25

0

0.25

0.50.75

1

eR

rr


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Fig.6. Normalized power spectrum of the complex envelope at the mobile station due to scatter

components,fordifferenturbanscenarios.Clarkesscenario: 0=

S1scenario: 10=

S2scenario: 10=

S3scenario: 10=

Fig.7. Normalized power spectrum of the complex envelope at the mobile station due to scatter

components,fordifferenturbanscenarios.

Clarkesscenario: 0=

S1scenario: 3=

S2scenario: 3=

S3scenario: 3=

0.75 0.5 0.25 0 0.25 0.5 0.75ffm

0

1

2

3

4

fmS

rr

tacs

f

0.75 0.5 0.25 0 0.25 0.5 0.75f f

m

0

0.5

1

1.5

2

2.5

3

3.5

fmSrr

tacs

f


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Fig.8. MapoftheArlingtonsiteforcollectionoftheurbandatacontainedinfiles#0019-#0022.

S.NEDDERM A N

WEST3R D ST.

WESTFIR STSTREET

WESTSECONDST

W.BORDERST.

W.SO UTHST.

W.NEDDERMANDR.

COOPERST.

YATESST.

COLLEGEST.

S.

WESTST.

S.

OAK

ST

.

S.P

EC

AN

ST

.

1/4MILE

FILE0019

15MPH

AREAOF

SIGNAL


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Fig.9. Normalizedautocovarianceoftheenvelope-squaredatthemobile,fordifferentrecords.

Empirical(obtainedbythetime-averagedestimatorofthecorrelationfunction[31,p.743])

Thesimplemodelin(19),with ),( p estimatedbynonlinearleastsquares

Thecompositemodelin(20),with ),,( p estimatedbynonlinearleastsquares

Clarkesmodel

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0011

Normalizedautocovarianceofen

velopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0013


velopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0014

Normalizedautocovarianceof

envelopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0016


envelopesquared

fm


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Fig.9.Continued.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0017


velopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0018


velopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0019


envelopesquared

fm

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Record #0022


envelopesquared

fm

T VT NonIsoCorrSpec Revised

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