1

UW

Bch

annel

chara

cterizatio

nand

modellin

g:

Som

eopen

issu

A.M

enouniHayar

EU

RE

CO

MInstitut,

Mobile

Com

munications

Departm

entSophia

Antipolis,France

Collaborators:

R.Saadane,R

.K

nopp,B.F

leury,T.Pederssen,M

.D

ebbah,N.M

ariyasagayam

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

2

outlin

e

•U

ltraW

ideB

andC

hannelSounding•

Em

piricalU

WB

Channelsubspace

Characterization

•Subspace

Analysis

Using

Information

Theoritic

Criteria

•U

WB

channel:Standardized

model

•P

hysicalmodel

analysis•

Am

aximum

entropyapproach

toU

WB

channelmodelling

•C

onclusions

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

3

motiv

atio

ns

•T

hirdgeneration

wireless

systems

andbeyond

(3Gand

4Gsystem

s).

•C

ompatibility

with

existingsystem

s.

•U

WB

capacityissues.

•U

WB

Applications:

-H

ome

based”L

ocationaw

areness”system

s.-

Cable

replacement...

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

4

Em

pirica

lEig

enanalysis

for

Indoor

UW

BChannels

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

5

Equip

ment

and

Measu

rem

ent

Setu

p

•M

easurements

carriedout

atE

urecomM

obilecom

munications

laboratory(R

Fequip.,com

puters,tables,chairs,m

etalliccupboard,glass

window

s,etc...)

•Frequency

domain

channelsounding•

Measurem

entE

nvironmentV

NA

remotly

controlledby

usingR

SIBprotocol

overE

thernet•

Measurem

entsbandw

idth3-9

GH

zw

ithdata

concatenationfrom

threem

easurements

groups:3-5

GH

z,5-7G

Hz

and7-9

GH

zw

ith2001

pointsfor

eachgroup

•calibration

forcables

andSM

Aconnectors

•O

ffline

signalprocessing

usingM

atlab

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

6

Equip

ment

and

Measu

rem

ent

Setu

p

•T

hefH

=3GHz

andthe

fL

=9GHz

Figure

1:C

hannelM

easurement

Setup

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

7

Measu

rem

ent

Enviro

nm

ent(1

)

•LO

Sscenario

•N

LO

Sscenario

usinglarge

obstaclebetw

eenTransm

itter(T

x)and

Receiver

(Rx)

•E

xperiment

areais

setby:

*T

xand

Rx

masts

at1m

abovethe

ground*

20cm

lineargrid

atT

xlocation

*50

cmlinear

gridat

Rx

location*

6m

separationbetw

eenT

xand

Rx

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

8

Measu

rem

ent

Enviro

nm

ent(2

)

The

environement

configurationw

erethe

scenariosare

conducted

Figure

2:C

hannelM

easurement

Setup

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

9

Measu

rem

ent

Enviro

nm

ent(3

)

The

measurem

entsconfiguration

isshow

nbelow

Figure

3:M

easurements

configurationw

ithseparation

distanceequal

to6m

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

10

Measu

rem

ent

Resu

lts

The

channelimpulse

responseversus

thedistance

typicalfor3

and6

metres

Figure

4:C

hannelM

easurement

inthe

time

domain

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

11

Eig

en-D

eco

mpositio

nO

fC

ovaria

nce

Matrix

(1)

The

radio-propagationchannelis

randomly

time-varying

dueto

variationsin

theenvironm

entand

mobility

oftransm

ittersand

receivers.Let

h(t)

=[hW,1 (t),h

W,2 (t),...,h

W,N

(t)]be

thechannelprocess

obtainedfrom

measurem

entsfor

Ndifferent

antennaconfigurations,

where

hW,i (t)

isexpressed

as

hi (t)

=gi (t)

+ni (t),i

=1..N

,(1)

where

ni (t)

iszero-m

eanadditive

white

Gaussian

noisew

ithpow

erspectraldensity

equaltoσ

2nat

allfrequencies

inthe

bandwidth

ofinterest.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

12

Eig

en-D

eco

mpositio

nO

fC

ovaria

nce

Matrix

(2)

The

covariancem

atrixis

Herm

itianand

positivedefinite.

Forthis

reason,a

unitarym

atrixU

existssuch

thatthe

Karhunen-L

oeve(K

L)

expansiongives

Kh

=U

ΛUH

=N∑i=

1

λi (h)ψ

i (h)ψHi

(h);UHU

=IN,

(2)

where

λ1 (h)≥

λ2 (h)≥

...≥λN

(h),ψi

isthe

i thcolum

nofU

andIN

isthe

N×N

identitym

atrixw

ithnum

berof

samples

equaltoN

.λi (h)

andψi (h)

arethe

i theigenvalue

andeigenvector

ofKh ,

respectively.

A.M

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obile

Com

munica

tion

Dep

t.

13

Eig

en-D

eco

mpositio

nO

fC

ovaria

nce

Matrix

(3)

The

channelprocessis

thendecom

posedon

two

sub-spaces:

Us,h

=[ψ

1 (h),ψ2 (h),...,ψ

L (h)],signalsub-space

with:

λ1 (h)≥

λ2 (h)≥

...≥λL (h),

LE

igenvalues(channelD

oF)

Un,h

=[ψL

+1 (h),ψ

L+

2 (h),...,ψN

(h)],noise

sub-space

with:

λL

+1 (h)≥

λL

+2 (h)≥

...≥λN

(h),noise

contribution

A.M

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obile

Com

munica

tion

Dep

t.

14

Eig

en-D

eco

mpositio

nO

fC

ovaria

nce

Matrix

(4)

Following

theeigen-decom

position,andlet

X=

‖hh ∗‖

2/‖hh ∗‖

2w

ecan

expressthe

channel-energym

oment

generatingfunction

as

GX

(s)=E

[esx]= ∫

+∞

−∞

esxfX

(x)dx

=L

(fX

(x))|s=−s

(3)

The

approximate

cumulative

distributionfunction

(cdf)can

beexpressed

interm

sof

theincom

pleteG

amm

afunction

as

FX

(x)=

xL

Γ(L

+1) ∏

i λi (h

)(4)

Then,

Fromthe

expressionof

cdfw

esee

thatthe

slopeof

log(cdf)gives

usan

ideaabout

thedegrees

offreedom

ofthe

channelorequivalently

itsinherent

diversity.A

.M

enouniH

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obile

Com

munica

tion

Dep

t.

15

Em

pirica

lR

esu

lts

The

empiricalresults

presentedin

thispaper

areobtained

fromtw

oscenarios

with

thefollow

ingspecifications:

•T

transmitter-to-receiver

distanceis

6m

eters.•

All

antennalocations

arein

thelaboratory

•E

achof

20T

xpositions

correspondsto

50R

xpositions,leading

to20x50=

1000com

plexfrequency

responses.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

16

020

4060

80100

120140

160180

−95

−90

−85

−80

−75

−70

−65

−60

−55

−50

−45

Tim

e in ns

Typicaly C

omplex R

esponse for UW

B in LO

S C

ase

10*log10(E|h(t)|²)

020

4060

80100

120140

160180

−100

−95

−90

−85

−80

−75

−70

−65

Typicaly C

omplex Im

pulse response for UW

B in N

LOS

Case

Tim

e in ns 10*log10(E(|h(t)|²)

LO

SC

ASE

NLO

SC

ASE

Figure

5:the

Average

Pow

erD

elayP

rofilein

LO

Sand

NLO

Ssitua-

tions.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

17

010

2030

4050

6070

0.7

0.75

0.8

0.85

0.9

0.95 1

Num

ber of Significant E

igenvalues

% of Energy

50MH

z100M

Hz

150MH

z200M

HZ

500MH

z1G

Hz

1.5GH

z2G

hz2.5G

Hz

010

2030

4050

60

0.7

0.75

0.8

0.85

0.9

0.95 1

Num

ber of Significant E

igenvalues

% of Energy

50MH

z100M

Hz

150MH

z200M

Hz

500MH

z1G

Hz

1.5GH

z2.5G

Hz

LO

SC

ASE

NLO

SC

ASE

Figure

6:Percentage

ofthe

capturedenergy

versusnum

berof

signifi-cant

eigenvaluesin

LO

Sand

NLO

Ssituations.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

18

−15

−10

−5

05

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2 0

Norm

alized Energy

CDF

Cum

ulative Distribution F

unction for 6GH

z

10 significant eigenvalues E

mpirical cdf from

scenario R

ayleighA

ll significant eigenvalues

0500

10001500

20002500

10 20 30 40 50 60 70 80

frequency band

Number of Eigenvalues

NLO

SLO

S

(a)(b)

Figure

7:(a)

Em

piricalC

umulative

Distribution

Functionand

(b)E

volutionD

oFfor

LO

Sand

NLO

Scases

UW

Bm

easurements

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

19

Subsp

ace

Analy

sisusin

gIn

form

atio

nT

heoritic

Crite

ria

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

20

Sta

tisticalC

lassifi

catio

nC

riteria

Information

Theoritic

Criteria

(1)W

axand

Kailath

(1985)presented

anew

approachfor

estimating

thenum

berof

signalsin

multichanneltim

e-seriesand

frequency-series,based

onstatistical

classificationcriteria

AIC

(Akaike

Information

Criterion)

andM

DL

(Minim

umD

escriptionLength).

The

covariancem

atrixis

Herm

itianand

positivedefinite.

Forthis

reason,an

unitarym

atrixU

hexists

suchthat

theK

arhunen-Loeve

(KL)

expansiongives

R=

Uh Λ

h UHh

=N∑i=

1

λi (h

)ψi (h

)ψHi

(h);

UHhU

h=

IN,

A.M

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obile

Com

munica

tion

Dep

t.

21

Sta

tisticalC

lassifi

catio

n

Information

Theoritic

Criteria

(2)A

kaikeproposed

thefollow

ingcriterion,

definedby:

AIC

=−

2.log(f(θ|θ))+

2.k

MDL

=−log(f(θ|θ))

+(1/2)k

.log(N)

where

theθ

isthe

maxim

umlikelihood

estimate

ofthe

parameter

vectorθ

andk

isthe

number

offreely

adjustableparam

etersinθ.

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obile

Com

munica

tion

Dep

t.

22

Info

rmatio

nT

heoritic

Crite

ria(3

)

•T

heA

ICis

AIC

(k)=

−2log (∏

pi=k+

1λi (h

)1

(p−

k)

1p−k ∑

pi=k+

1λi (h

) )N

(p−k)+

2k(2p−k)

(5)

•T

heM

DL

functionis

givenas

follows:

MDL

(k)=

−log (∏

pi=k+

1λi (h

)1

(p−

k)

1p−k ∑

pi=k+

1λi (h

) )N

(p−k)+log(N

)k(2p−

k+

1)4(6)

A.M

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obile

Com

munica

tion

Dep

t.

23

Estim

atio

nofth

edegre

es

offre

edom

The

number

ofdegrees

offreedom

,m

ainlythe

number

ofsignificant

eigenvalues,isdeterm

inedas

thevalue

ofk∈

0,1,...,p−1

which

minim

izesthe

valueof

(5)or

(6).In

thisw

ork,the

number

ofD

oFrepresents

thenum

berof

unitarydim

ensionindependent

channelsthat

constitutean

UW

Bchannel.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

24

Resu

lts

DoF

estimation

resultsN

umerical

Results

inLO

SC

ase

010

2030

4050

600

0.5 1

1.5 2

2.5 3x 10

4

index k

Criterion(k)

the AIC

and MD

L for 200MH

z of Bandw

idth LOS

case

AIC

Criterion

MD

L Criterion

the min of M

DL k=24

the min of A

IC: k=25

020

4060

80100

120140

1 2 3 4 5 6 7 8 9x 10

4

index k

Criterion(k)

the AIC

and MD

L for 6GH

z of Bandw

idth LOS

case AIC

Criterion

MD

L Criterion

the min of A

IC: k=

50

the min of A

IC: k=

47

200M

Hz

6G

Hz

Figure

8:T

henum

berof

UW

BchannelD

oFfor

LO

Ssetting.

A.M

enouniH

ayar,M

obile

Com

munica

tion

Dep

t.

25

Resu

lts

DoF

estimation

resultsN

umerical

Results

inN

LO

SC

ase

510

1520

2530

3540

450.5 1

1.5 2

2.5 3

x 104

index k

Criterion(k)

the AIC

and MD

L for 200MH

z of Bandw

idth NLO

S case

AIC

Criterion

MD

L Criterion

the min of A

IC: k=

29

the min of M

DL: k=

25

010

2030

4050

6070

0

0.5 1

1.5 2

2.5 3x 10

4

index k

Criterion(k)

the AIC

and MD

L for 6GH

z of Bandw

idth NLO

S case A

IC C

riterionM

DL C

riterion

the min of A

IC: k=

36

the min of M

DL: k=

36

200M

Hz

6G

Hz

Figure

9:T

henum

berof

UW

BchannelD

oFfor

NLO

Ssetting

.

A.M

enouniH

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obile

Com

munica

tion

Dep

t.

26

Rela

tionsh

ipbetw

een

num

ber

of(D

oF)

and

syste

mbandw

Following

[Sayeed],ifone

transmits

aband

limited

andtim

elim

itedsignal

overa

fadingchannelw

ithrm

sdelay

spreadTd

,the

channel(D

oF)

Nis

approximately

givenby

N=Td ·∆

W.

(7)

A.M

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obile

Com

munica

tion

Dep

t.

27

Rela

tionsh

ipbetw

een

num

ber

of(D

oF)

and

syste

mbandw

0500

1000

1500

2000

2500

3000

0

20

40

60

80

100

120

140

160

180

200

Fre

quency in

MH

z

Number of degrees of freedomN

LO

S e

mpiric

al v

alu

eLO

S v

alu

e b

ased o

n A

ICLO

S e

mpiric

al v

alu

eN

LO

S v

alu

e b

ased o

n A

IC

Figure

10:E

volutionofthe

number

ofdegreesoffreedom

forLO

Sand

NLO

Ssettings

A.M

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obile

Com

munica

tion

Dep

t.

28

UW

Bch

annel:

Sta

ndard

ized

model

ATapped

delayline

model

iscom

monly

adoptedfor

thechannel

impulse

responseof

UW

Bsystem

.For

thisreason

thissystem

functioncan

beexpressed

as

hh

=L−

1∑l=

0

K−

1∑k=

0

exp(jθ

kl )δ(t−

Tl −

τkl ),

(8)

where

Lis

thenum

berof

clustersand

Kis

thenum

berof

echoesin

eachcluster.

Sucha

model

representationhas

beenadopted

indefining

theIE

EE

802.15.3am

odel(J.

Foerster,C

hannelmodelling

Sub-Com

mittee;

Final

Report.

IEE

EP

802.15-02/490r1-SG3a,

Mar

2003)

A.M

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obile

Com

munica

tion

Dep

t.

29

UW

Bch

annel:

Sta

ndard

ized

model

Statisticalmodels

fordelays

τkland

Tl

•T

heN

eyman-Scott

model(Saleh-V

alenzuela).T

hisis

basedon

theprinciples

thatscattering

clustersare

describedby

poissondistribution

andthe

scatterersbelonging

thesam

ecluster

obeyto

thesam

edistribution.

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obile

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munica

tion

Dep

t.

30

UW

Bch

annel:

Sta

ndard

ized

model

•D

istributionsfor

tapam

plitudes

*Log-norm

al*

Nakagam

i*

PO

CA

-NaZ

U

•D

istributionsfor

tapphases

*U

niformover

[0,2π]*

Tw

opossible

(andequiprobable)

0,πphase

shiftsfor

eachm

ultipathcom

ponent

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obile

Com

munica

tion

Dep

t.

31

UW

Bch

annel:

Sta

ndard

ized

model

•T

heaverage

PD

P(also

dubbedsm

all-scaleaveraged

PD

P,SSA

-PD

P)

ofa

UW

Bchannelis

usuallyapproxim

atedas

one-sidedexponential,so

thatthe

averagepow

erassociated

with

thel−

thcluster

andthe

k−th

pathof

thiscluster

isgiven

by

Ea2kl =

P(τkl )

=Ω

0 exp (

Tl

Γ )exp (

τkl

γ ),

(9)

Where

Γand

γare

thedecay

constantsof

theclusters

andof

theechoes

insidethe

clusters,respectively

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obile

Com

munica

tion

Dep

t.

32

UW

Bch

annel:

Sta

ndard

ized

model

Figure

11:Saleh

Valenzuella

model

•IE

EE

802.15.3am

odel:Illustration

ofexponentialdecay

ofm

eancluster

power

within

clustersA

.M

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obile

Com

munica

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Dep

t.

33

Open

issues

•U

WB

channelmodelling

•capacity

analysis

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obile

Com

munica

tion

Dep

t.

34

UW

Bch

annelm

odellin

g:

Physica

lappro

ach

•M

ainpropagation

mechanism

s:R

eflection,diffraction,...

•U

ltraw

idebandw

idtheffects

analysis

A.M

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obile

Com

munica

tion

Dep

t.

35

Pro

pagatio

nm

ech

anism

s:U

ltraW

ide

band

case

•R

eflectioncoeffi

cientsfor

horizontaland

verticalpolarization

aregiven

by:

R(ψ,s)

=±√s

+2a−

κ √s

√s

+2a

+κ √

s(10)

with

τ=

σε ,β

=√εr −cos2ψ

εrsinψ

.and

a=τ/2

andκ

=β

forverticalpolarization

anda

=τ/2

andκ

=(εr β

)for

horizontalpolarization

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Com

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tion

Dep

t.

36

Pro

pagatio

nm

ech

anism

s:W

ide

band

case

The

resultingtransient

responsefor

thereflected

impulse

fieldr(t)

isthen

givenby:

r(t)= [

Kδ(t)

+4κ

1−κ

2

exp(−

at)

t

∑(−

1)n+

1nKnIn (a

t) ](11)

A.M

enouniH

ayar,M

obile

Com

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t.

37

Physica

lm

odelpara

mete

rseffect

at

larg

ebandw

idth

•Som

eresults

00.5

11.5

22.5

3

x 10−8

0 1 2 3 4 5 6x 10

19Im

pulse response, EpsilonR=10, Sigm

a=0.1, B=1GH

z

00.5

11.5

22.5

33.5

44.5

x 10−8

0 5 10 15x 10

9Im

pulse response, EpsilonR=10, Sigm

a=0.025, B=1GH

z

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38

Physica

lm

odel:

Ultra

larg

ebandw

idth

win

dow

ing

effect

−5

−4

−3

−2

−1

01

23

40

0.0

5

0.1

0.1

5

0.2

0.2

5

0.3

0.3

5

Tim

e in

ns

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39

Min

imum

entro

py

appro

ach

toU

ltraW

ide

Band

channelm

Idea:G

ivena

setof

measurem

ents,we

tryto

findthe

bestprocess

model

undersom

econstraints.

Entropy

rateof

aG

aussianprocess

The

entropyrate

ofa

stationaryG

aussianstochastic

processcan

beexpressed

as

h(χ)

=12log2π

e+

14π ∫π

−π

logS

(λ)dλ

(12)

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40

Min

imum

entro

py

appro

ach

toU

ltraW

ide

Band

channelm

Burg’s

Maxim

umE

ntropyT

heorem:

The

maxim

umentropy

ratestochastic

processXisatisfying

theconstraints

E[X

i X∗i+k ]=

αk ,

k=

0,1,.....,p,foralli,

(13)

isthe

pth

orderG

auss-Markov

processof

theform

Xi=

−p∑k=

1

ak X

i−k

+Zi

(14)

where

theZi

arei.i.d.∼

N(0,σ

2)and

a1 ,a

2 ,....,ap ,σ

2are

chosento

satisfyeqn.(13)

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41

Min

imum

entro

py

appro

ach

toU

ltraW

ide

Band

channelm

AR

processcoeffi

cientsestim

ation

R(0)

=−

p∑k=

1

ak R

−k

+σ

2(15)

and

R(l)

=−

p∑k=

1

ak R

l−k ,

l=1,2,....

(16)

These

equationsexactly

resemble

theY

ule-Walker

equations.T

hereare

p+

1equations

inthe

p+

1unknow

nsa1 ,a

2 ,....,ap ,σ

2.Therefore

we

cansolve

forthe

parameters

ofthe

processesfrom

thecovariances

usingfast

algorithms

suchas

theLevinson

andthe

Durbin

recursion.

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Min

imum

entro

py

appro

ach

toU

ltraW

ide

Band

channelm

Spectrumestim

ationA

fterthe

coefficients

a1 ,a

2 ,....,ap

havebeen

calculatedfrom

thecovariances,the

spectrumof

them

aximum

entropyprocess

isseen

tobe

S(l)

=σ

2

|1+ ∑

pk=

1ak e −

ikl| 2

(17)

This

isthe

maxim

umentropy

spectraldensity

subject

tothe

constraintsR

(0),R(1),R

(2),........,R(p).

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Resu

lts

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Conclu

sions

•C

lusteredbehavior

with

differentconstant

decay•

Saturationof

DoF

versusbandw

idthfor

LO

Sand

NLO

Sscenarios

•W

euse

(AIC

)and

(MD

L)

criteriato

estimate

thenum

berof

(DoF

)of

anU

WB

channelinan

in-doorenvironm

ent.•

We

pointedout

thatthe

number

ofD

oFbeyond

acertain

bandwidth

doesnot

scalelinearly.

•T

ime

domain

dispersionw

ithlarge

bandwidth

causescorrelation

between

samples

•M

aximum

entropyanalysis

shows

thatthe

channelinformation

doesn’tincrease

som

uchw

ithincreasing

thebandw

idth

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