jxu

Characteristics of the radio channelmodels;Envelope and spatial correlation;Level crossing rates and fade duration

08, Dec. 2001

jxu

Scattering environments in mobile radiosystem

The height and placement of the BSantennas

In the macro-cell:a fairly small angle ofarrival (AoA) spread

in the micro-cell:a larger AoA spread (thespecular component +…´) Riceandistributed envelope

Local scatterers usually surround MS.

No direct LoS component. Two-

dimensional isotropic scattering model.

Rayleigh distributed envelope

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Received signal correlation and spectrum1(4) Flat fading channel

tfgtftgtr ctqcI ππ 2sin2cos)()( )(−=

∑=

Φ=N

nnnI tCtg

1

)(cos)( ∑=

Φ=N

nnnQ tCtg

1

)(sin)(

The auto-correlation of r(t) is

τπτφτπτφττ cgQgIcgIgIrr fftrtrE 2sin)(2cos)())()(()( −=+=Φ

[ ] [ ] [ ]

[ ] [ ] ∑=

=+=Ω

Ω=

Ω=+=

N

nnQIp

nDp

mp

IIgIgI

CtgEtgE

fEfEtgtgE

1

222

,,

)()(

)2cos2

)cos2cos(2

)()()( τπθτπττφ θθθτ

[ ] [ ])cos2sin(2

)()()( , θτπττφ θθτ mp

QIgIgQ fEtgtgEΩ

=+=

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Received signal correlation and spectrum2(4) for 2-D isotropic scattering and an isotropic antenna

∫− =Ω

=π

πθθτπ

πτφ 0)cos2sin(

2

1

2)( dfm

pgIgQ

)2(2

)sin2cos(1

2)()()cos2cos(

2)( 00

τπθθτππ

θθθθτπτφππ

π mp

mp

mp

gIgI fJdfdGpfΩ

=Ω

=Ω

= ∫∫−

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Received signal correlation and spectrum3(4) Power density spectrum

• The power density spectrum of gI(t) and gQ(t) is the Fouriertransform of gIgI(t)or gQgQ(t). For the auto-correlation, thecorresponding psd is

• The autocorrelation of the received complex envelope g(t)=gI(t) + jgQ(t) is

• And its power spectral density (Doppler power spectrum)

)()()( fjSfSfS gIgQgIgIgg +=

)()()( τφτφτφ gIgQgIgIgg j+=

[ ]

≤

−

Ω

==)(0

)()/(1

12

)()(2

otherwise

fffff

FfS

m

mm

p

gIgIgIgI

πτφ

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Received signal correlation and spectrum3(4)

• With 2-D isotropic scattering and an isotropic antenna

gIgQ(t)=0 and Sgg(f)=SgIgI(f), so that

mc

m

cm

prr fff

f

ffffS ≤−

−−

Ω= ,

1

1

4)(

2π

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Received envelope and phase distribution 1(3)

Rayleigh fading

Ω−

Ω=

ppa

xxxp

2

exp2

)( π21

)()( =Φ xp t

•NLoS environments

•WSS complex Gaussian random process.

•2-D isotopic scattering,gI(t) and gQ(t) are IID zero-meanGaussian random variables at any time t1, with variancebo.The magnitude of the received complex envelope a(t)=|g(t)| has a Rayleigh distribution at any time t1.

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Received envelope and phase distribution 2(3)

Ricean fading•Specular component

•micro-cellular and mobile satellite applications

•gI(t) and gQ(t) are Gaussian randomprocesses with non-zero means mI(t) and mQ(t)

+−=

00

0

22

0 2exp)(

b

xsI

b

sx

b

xxpa

Where s2= mI2 (t)+ mQ 2 (t)

,)1(

2)1(

exp)1(2

)( 0

2

Ω+

Ω+−−

Ω+=

pppa

KKxI

xKK

Kxxp

Rice factor K = s2 /2bo

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Received envelope and phase distribution 3(3)

Nakagami fading•characterize rapid fading in long distance HF channels

•often used for the following reasons:

A) empirical justification.B) It can model fading conditions that are either more orless severe than Rayleigh fading. When m=1, theNakagami distribution becomes the Rayleigh distribution.C) Close relation between the Rice factor K and Nakagamishape factor m.

( ))12(

1 2

2

2

++=

−−−=

K

Km

mmm

mmK

Ω−

ΩΓ=Ρ

−

ppm

mm

a

mx

m

xmx

212

exp)(

2)( (m>=0.5)

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Level Crossing Rate (LCR)& Average Fade duration (AFD)

Two important second order statistics associated withenvelope fading

LCR: How often the envelope crosses a specified level AFD: How long the envelope remains below a specified level

t1 t2t3 t4

NR=4Signalstrength dB

Time,seconds

10

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LCR and AFD for a Rayleigh fading signal 1(2)

Useful for designing error control codes and diversityschemes

Level crossing rate

is the time derivative of r(t)

is the joint pdf

∫ −==2

2),( ρρπ efrdrRprN mR

r)rP(R,

rmsRR _/=ρ

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LCR and AFD for a Rayleigh fading signal 2(2)

Average fade duration

[ ]

[ ]

πρτ

τ

ρ

ρ

21

1)(Pr

1

2

2

0

m

R

R

R

f

e

edrrpRr

RrPN

−=

∫ −==≤

≤=

−

For the scattering environmentshown in left figure, the averageenvelope fade duration for it shown

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Spatial CORRELATIONS

τλ l

fv cm ==

)/2(16

)(

)/2(2

)(

2co

paa

cop

gIgI

lJl

lJl

λππ

µ

λπφ

Ω=

Ω=

Therefore, inpractice, sufficiently un-correlated diversity branches can beobtained at the MS by spacing the antenna elements a

distance apart.

For the case of isotropic scattering become, respectively

A fundamental question that arises is theantenna separation needed to provideuncorrelated antenna diversity branches.